NOVEL RECONFIGURABLE RF AMPLIFIERDESIGN TECHNIQUES

A THESIS SUBMITTED TO THE GRADUATE DIVISION OF THEUNIVERSITY OF HAWAI'I IN pARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

IN

ELECTRICAL ENGINEERING

AUGUST 2005

ByKendall Ching

Thesis Committee:

Wayne A. Shiroma, ChairpersonKazutoshi Najita

Eric Miller

25115xUJ \fl

This thesis is dedicated to my family.

To my brother, 1 dedicate this thesis because although his architecture PhD will make

him my academic superior, he will never be my financial superior (I hope). But all

kidding aside, even though he never washes the dishes and does not use enough soap, 1

know he will be there if1 ever need him.

To my parents 1 dedicate this thesis because they have dedicated their lives to me. They

selflessly raised me and instilled in me the values 1have today, and to this day 1 still find

myself learning to be a better person by their example. The fact that they live in a pig sty

and raised me to be a super fat kid will nonetheless be overshadowed by their hard work,

kindness, and respect for others. It is a testament to their character that 1never heard a

single bad word spoken about them in the 25 years that 1have been alive, and if1 end up

being half the person that my parents are today, I will consider myself a success.

iii

ACKNOWLEDGMENTS

The author would first like to thank Northrop Grumman Space and Technologies

for the funding ofthe project, and specifically Matt Nishimoto, Larry Lembo, and

Michael Tamamoto for their input and freedom in letting me choose the direction of the

research. Outside the EE realm, the author would like to thank Tomoe Sato, who

provided much needed sustenance during the author's research efforts, and Mary, the

roommate who did not kick me out ofthe house whenever I broke her dishes or ate her

food.

Inside the world ofHolmes Hall, the author would like to recognize his fellow

"Shiromites," who have made the journey a little less hectic. Chenyan Song, who always

surprised everyone with her microwave specials, Grant Shiroma, who inspired the author

with his crazy, "work 80 hours a week and get sick before the IMS deadline" lab routine,

and Justin Roque, whose love ofNaruto and incessant bold statements (within the first

hour ofmeeting him he told me, "I can run faster then all of you") always provided

humor in the author's life. The author would also like to mention his fellow graduate

students outside the office who have made these past couple of years a blast. Eric Young,

a fellow CB and confidant whose been there since the HKN days, Jaren Goya, an AZ

companion and an all around decent guy, Jason Akagi, a fellow hiking and traveling pal

whose been there through some good times, Byungkwon "BK" Kim, the soju gulping

Korean the author wishes he had known earlier, Luis Ortiz Hernandez, the funniest (and

only) Mexican the author knows who introduced him to international travel, and Alex

Vergara, the philosopher, comedian, and belligerent drunk who the author always

iv

enjoyed lifting and talking with. In addition, the author would like to thank Dr. Ryan

Miyamoto, the funniest Nihonjin, who acted like a 70 year old but provided much needed

humor, guidance, and insight into my research. Along with these people, the author

would also like to mention Shuhei, Nik, Chris, Dave, Steven, Blaine, Daniel, Ed, William,

Jodie, Rory, Tom, the old school EE fellas, the basketball intramural crew "Da Bus"

(cause we take everyone to school), the 3rd place A division softball intramural group,

and the C division intramural softball champs, as those people who have also created

lasting memories during the authors quest for his MS.

Lastly, the author would like to express his sincerest thanks to his committee

members (this may seem like brown-nosing...which it is, but I can honestly say that none

of the following are lies); Dr. Kazutoshi Najita, who was the author's first electrical

engineering professor in EE 101 and who inspires him to stay active mentally and

physically, Dr. Eric Miller, the author's EE 326 professor who is one of the most genuine

and nice people that he knows, and last but not least, Dr. Wayne Shiroma, who has

taught, guided, mentored, and befriended the author for the past 6 years. Without him,

the author would have never achieved half of what he has done, and would have never

experienced the wonders ofBoston, Zion, Kalalau, or Ryogoku sumo practices. The

author is a better person today for knowing Dr. Shiroma, and for everything he has done

the author will be forever grateful.

v

ABSTRACT

Reconfigurable RF amplifiers utilize tunable microelectromechancial matching

networks to optimize their performance over varying operating conditions such as

frequency, temperature, or function. Although this reconfigurability allows for greater

functionality, novel design techniques must be developed to fully utilize the benefits.

Three different reconfigurable amplifier design issues are investigated in this thesis.

The first technique involves a method for creating a fully autonomous, self­

reconfigurable maximum-gain amplifier. Using seven different output power

measurements at arbitrary yet distinct input/output impedances, the necessary S­

parameters can be extracted to design a maximum-gain amplifier. Simulations support

the technique, but measurements are not as conclusive due to the unavailability of a fully

functional tunable matching network.

The second technique examines the benefits ofvariable capacitors and resistors in

stabilization networks to improve the potential gain of an amplifier at different operating

frequencies. A combination of stabilization networks at the gate of the device provided

an increase in gain of up to 6.5 dB over traditional stabilization methods.

The last design technique is a way for an autonomous reconfigurable amplifier to

monitor the operation of its tunable matching network. Since there are reliability issues

with reconfigurable systems, design equations that characterize the matching network

were used to detect failures in the system.

VI

TABLE OF CONTENTS

Acknowledgements ivAbstract viList of Tables viiiList of Figures ixChapter 1: Introduction 1

Organization ofThesis 4Chapter 2: Method for Designing a Self-Reconfigurable Maximum-Gain Amplifier 6

Theory ofProposed Method 8Simulations and Calculations 14Fabrication and Measurements 17

Chapter 3: Adaptive Stabilization Networks .22Frequency Dependent Stabilization Networks .26Adaptive Stabilization Networks .30Simulations 32

Chapter 4: Characterization ofReconfigurable Matching Networks 46TITL Characterization Theory 51Simulation and Calculations 54

Chapter 5: Conclusion 56Suggestions for Future Work : 57

List ofPublications and Presentations 59Appendix 1: Matlab Code to Calculate Sl1, S22, and SI2S21 61Appendix 2: Calculating CSwitch with Gain Measurements 85References 88

VB

LIST OF TABLES

1-1 Design differences between conventional static amplifiersand reconfigurable amplifiers , , , ,.4

3-1 Characteristics of the different configurations of frequency-dependentstabilization networks for a common-source device configuration .. , .28

3-2 Schematics and characterization of the effective frequency-dependentstabilization networks for a common source device configuration .29

3-3 Component values and maximum gain for the best performing ASNin order to optimize for gain at various frequencies .45

V111

LIST OF FIGURES

Figure Page

1-1 Mobile military multi-frequency communications showing thebenefit of reconfigurable amplifiers in a headset overtraditional amplifiers 2

1-2 Schematic of a reconfigurable amplifier using tunable input andoutput matching networks 3

2-1 An application of a reconfigurable amplifier in a typical receiver 8

2-2 Matching conditions of a reconfigurable simultaneous conjugatematch amplifier 9

2-3 Simulated and calculated 811 of the stabilized FHX35LG from1 - 10 GHz in dB vs. angle 14

2-4 Simulated and calculated 822 of the stabilized FHX35LG from1 -10 GHz in dB vs. angle 15

2-5 Simulated and calculated 821812 of the stabilized FHX35LGfrom 1 - 10 GHz in dB vs. angle 15

2-6 Stability parameters K and B1 of the device, which showthat the device is unconditionally stable 16

2-7 Simulated maximum stable gain and unilateral conjugate matchedgain at 1, 5, and 10 GHz 17

2-8 "Transistor" setup showing the active device, stabilization network,bias tees, and SMA(male)-SMA(male) connectors 18

2-9 Two stub networks used to provide the input and output matchingnetworks. Port (a) is connected to the transistor, and port (b) isconnected to the 50-(2 port of the network analyzer.. .18

2-10 Extracted and measured (a) 811 magnitude and (b) 811 phase ofthe ATF-I0736 19

2-11 Extracted and measured (a) 822 magnitude and (b) 822 phase ofthe ATF-I0736 20

IX

2-12 Extracted and measured (a) SII magnitude and (b) SII phase ofthe ATF-26884-STR 20

2-13 Extracted and measured (a) S22 magnitude and (b) S22 phase ofthe ATF-26884-STR 21

3-1 Amplifier design flowchart for an unconditionally stable device.The highlighted boxes demonstrate the steps that are needed tostabilize the device 22

3-2 Block diagram of an RF amplifier to describe the stability ofan amplifier 24

3-3 Graphical relationship between Mu and /S21/, for a typical device 26

3-4 Different types ofresistive loading to improve stability .27

3-5 (a) Series resistor/inductor stabilization network connected inshunt with the gate of a FET. (b) Equivalent circuit of (a) atlow frequencies, when the inductor acts as a short.(c) Equivalent circuit of (a) at high frequencies, when theinductor acts as an open 27

3-6 (a) Adaptive stabilization network consisting of a variableresistor and inductor shunted to the gate of a FET. (b) Mu and(c) GTmax ofthe FET with the adaptive stabilization networkoptimized for gain at 10 GHz (solid line) and 50 GHz (dashed line) .31

3-7 (a) Stabilization characteristics of the simulated HEMT modelshowing that the device is not unconditionally stable from0- 50 GHz. (b) GMSG of the HEMT 32

3-8 (a) Resistive loading stabilization network consisting of a resistorin series with the gate of the HEMT. (b) Mu and (c) Maxgain 33

3-9 (a) Resistive loading stabilization network consisting of a resistorin series the drain of the HEMT. (b) Mu and (c) Maxgain of theHEMT for varying values ofresistance .34

3-10 (a) Resistive loading stabilization network consisting of a resistorin shunt with the gate of the HEMT. (b) Mu and (c) Maxgainof the HEMT for varying values ofresistance 35

3-11 (a) Resistive loading stabilization network consisting of a resistor

x

in shunt with the drain of the HEMT. (b) Mu and (c) Maxgainof the HEMT for varying values ofresistance 36

3-12 (a) Resistive loading stabilization network consisting of a resistorconnected to the gate and drain of the HEMT. (b) Mu and(c) Maxgain ofthe HEMT for varying values ofresistance .37

3-13 (a) Adaptive stabilization network consisting of a series resistorand capacitor shunted to the gate of the HEMT. (b) Mu andmaxgain of the HEMT for a static resistor of 100 Q and varyingvalues of capacitance. (c) Mu and maxgain of the HEMT for aconstant capacitance of 1 pF and varying values ofresistance 38

3-14 (a) Resistive loading stabilization network consisting of a resistorin series with the gate of the HEMT. (b) Mu and (c) Maxgain .39

3-15 Effect of varying capacitance on the stabilization of circuit 2-14a .40

3-16 (a) Adaptive stabilization network consisting of a series resistorand inductor shunting the gate of the HEMT. (b) Mu andmaxgain of the HEMT for a static resistor of 450 Q and varyingvalues of capacitance. (c) Mu and maxgain of the HEMT for aconstant capacitance of 1 pF and varying values ofresistance .42

3-17 (a) Adaptive stabilization network consisting of a series resistorand inductor connecting the gate and drain ofthe HEMT.(b) Mu and maxgain of the HEMT for a constant inductor of1 nH and varying values ofresistance .43

3-18 The four simulated combination adaptive stabilization networksthat utilize networks that affect both low and high frequencies 43

3-19 (a) Configuration ofthe most effective ASN in providingunconditional stability and gain optimization. (b) Resultinggain ofthe ASN when optimized for operation at variousfrequencies 44

4-1 Schematic of the NGST TITL using MEMS switches and acapacitively loaded 47

4-2 Smith chart coverage of a sample variable capacitive loadedline using 8 MEMS switches at (a) 20.2 GHz and (b) 60 GHz .48

Xl

4-3 Schematic of the NGST TITL model in the ON and OFF statethat will be used in the investigation 50

4-4 Signal flow graph of an input reconfigurable matching networkand a unilateral amplifier that was used to solve for theoverall gain 52

4-5 Advanced Design System simulation setup to verify the iterativeTITL characterization procedure 54

4-6 Simulated and calculated (a) capacitance and (b) Sus ofthesetup in Fig. 4-4 55

xu

CHAPTER!

INTRODUCTION

A radio-frequency (RF) amplifier is one of the most important components in any

wireless system since an amplified signal is necessary to overcome losses associated with

free space. Unfortunately, an amplifier is also one of the most complex components of

any system due to the many factors involved in the design, such as stability, noise, gain,

power, bandwidth, efficiency, and linearity, among other things. Because of the many

design factors involved in an amplifier, sacrifices must often be made when designing an

amplifier. A perfect example of this is the amplifier gain-bandwidth product [1], which

characterizes the increase in gain of a device at the cost of bandwidth, and vice versa.

Due to this inherent tradeoff in amplifier design, most amplifiers are optimized for a

single operating condition such as frequency, noise figure, power, or efficiency.

The proliferation of operation-specific amplifiers implies that an increasing

number of amplifiers are needed for multipurpose systems. A typical scenario where this

is apparent is in mobile military communications, in which several different

communication links (GPS, IRIDIUM, and cellular), all operating at different frequency

bands, must be maintained, as in Fig. 1-1. The US military currently carries several

radios to handle all of the different links, which works well for a stationary command

center, but is quite cumbersome for mobile units that must move around with this

equipment [2].

One solution to this dilemma is the use of reconfigurable amplifiers, which can

tune and adapt to changing operating conditions [3], [4]. In the example pictured in Fig

1-1, the use of reconfigurable amplifiers in a system would allow a soldier to

1

rr

ReconfigurableI Headset("I( ( ( (

Freq 1Traditional I ' ..,Headsets I

1~'

Freq2

......-r(((

Fig. 1-1: Mobile military multi-frequency communications showing the benefit ofreconfigurableamplifiers in a headset over traditional amplifiers.

communicate on any frequency band with a single headset rather than having to switch

between different radios. In addition to size reduction, these reconfigurable amplifiers

also enable versatility, multifunctionality, and robustness in a communication system,

thereby reducing the overall cost of a system.

Conventional and Reconfigurable RF Amplifiers

Conventional RF amplifiers differ from their low-frequency counterparts in that

they use matching networks at the input and output of a device to minimize the

reflections associated with high-frequency signals. These matching networks are an

important part of the amplifier that determines many factors such as the noise figure,

gain, and power, but their functionality is limited because they only work at a single

frequency and are fixed after manufacturing. The limited functionality of these fixed

matching networks is the reason why conventional, "static" RF amplifiers are operation

specific.

2

son

Source

TunableI.nput

MatchingNetwork

Device

TunableOutput

MatchingNetwork

50n

Fig. 1-2: Schematic of a reconfigurable amplifier using tunable input and output matching networks.

On the other hand, if these static matching networks were altered in real time, a

single amplifier would be able to optimize its performance at different frequencies or

changing environmental conditions. This is the premise of reconfigurable amplifiers,

where static matching networks are replaced by tunable matching networks, as shown in

Fig. 1-2. This is not an entirely new concept, but it is only with the invention of

microelectomechanical systems (MEMS) and its application in RF systems [5] that these

tunable matching networks have become feasible. Over the past five years, MEMS

tunable matching networks have utilized double stub [6], triple stub [7], and capacitively

loaded transmission line [8] techniques to produce working reconfigurable amplifier

prototypes.

However, as is the case with most new technologies, novel design techniques

must be developed to fully utilize their capabilities. In many respects, the characteristics

of reconfigurable RF amplifiers are sufficiently different from static RF amplifiers (see

Table 1-1) that many of the conventional amplifier design techniques do not take

advantage of the potential of reconfigurable amplifiers. The research in this thesis covers

some of these issues and attempts to remedy them by introducing novel design techniques.

3

Table 1-1: Design differences between conventional static amplifiers and reconfigurable amplifiers.

Differences Conventional Static Amplifiers Reconfigurable Amplifiers

Matching Networks Fixed Matching Networks Tunable matching networks

Operating Optimized for a single operating Must be optimized for multi-Characteristics condition operational conditions in real time

Gain Performance Gain optimized at one frequencyGain can be optimized for many

frequencies

StabilityMust be stable with adequate gain at a Must be stable with adequate gain over

single operating frequency a broad range of frequencies

MaintenanceMatching networks static-no Tunable matching networks must be

maintenance needed checked for signs of degradation

The thesis is organized into three different main chapters, each discussing a different

reconfigurable amplifier design issue.

Organization of Thesis

Chapter 2 investigates the design of self-reconfigurable amplifiers that can

optimize themselves to operate in any situation. Recent reconfigurable amplifier

technology has enabled the production of prototype reconfigurable amplifiers that can

only function with the help of operators that control the tunable matching networks.

When used in a remote system, however, an intelligent algorithm must be created so that

these tunable matching networks can self adjust according to a specified operating

condition. Chapter 2 investigates an algorithm that does exactly this for the case of a

self-reconfigurable maximum gain amplifier. The ultimate goal is the design of a

maximum gain amplifier without any previous knowledge ofthe active device.

For the self-reconfigurable amplifier algorithm described in Chapter 2, a stable

device is needed to ensure proper operation. Static amplifier stabilization techniques that

4

maximize gain over a single operational frequency work well for static amplifiers, but do

not work when an amplifier has to function over a broad frequency range like

reconfigurable amplifiers. Chapter 3 investigates the use of tunable resistors and

capacitors in a reconfigurable stabilization network to provide stability while optimizing

itself for gain over many different operating conditions. The effectiveness of this

technique is demonstrated with a proprietary high electron mobility transistor (HEMT).

Following the investigation of stability, the focus of the next chapter is the self

characterization of the tunable matching networks. As mentioned earlier, tradeoff is

inherent to amplifier design, and in this case, the benefit of self-reconfigurable amplifiers

is not without drawbacks. Reliability is one issue with MEMS architectures that make up

the tunable matching networks, and as such, these systems must be periodically checked

for degradation. In Chapter 4, a method for in-situ monitoring of the matching network is

devised. When these calculated models are compared to measured models, a quick

analysis can determine if the matching network is still fully functional.

Finally, Chapter 5 presents the major conclusions of this thesis and suggests some

areas of future investigation.

5

CHAPTER 2METHOD FOR DESIGNING A SELF-RECONFIGURABLE

MAXIMUM-GAIN AMPLIFIER

With the advent of MEMS, reconfigurable matching networks (RMN) have been

realized in various ways [6], [7], [8], but a common limitation is that they require

operator assistance and cannot be used autonomously in remote conditions. A more

attractive solution is autonomous RMNs that self-reconfigure themselves. To this end, an

algorithm that incorporates intelligence into the system to allow se1f-reconfigurability is

investigated in this chapter. The algorithm can be used in conjunction with the

aforementioned, previously published RMNs to realize a fully autonomous, self-

reconfigurable amplifier that adjusts itself for maximum gain under varying operating

conditions.

A specific application where a fully autonomous self-reconfigurable amplifier

would be useful is in mobile military command center communications, as explained in

Chapter 1. On the battlefield, several different communication links operating at

different frequency bands must be maintained. Militaries throughout the world currently

carry several radios to handle all of the different links, but with the method proposed

here, a single radio with reconfigurable matching networks (RMN) is all that is needed.

Rather than switching between different radios whenever a different link is established,

our method allows the RMN to self-optimize according to whatever frequency band is

being used.

The starting point for determining basic design criteria such as stability and

proper matching conditions requires knowing the S-parameters of the active device. The

6

simplest solution is to use catalogued S-parameters from a datasheet, but this can be

inaccurate because such data is not device-specific and is valid at only one particular

condition. In reality, a transistor's performance can change significantly based on its

placement on the wafer, surrounding temperature, DC bias point, or input power [9].

A more accurate and flexible approach would be to solve for a particular device's

S-parameters under real-time operating conditions and configure the matching networks

accordingly. This way, an accurate conjugate match can be achieved over varying

operational conditions, resulting in an amplifier that is no longer device-specific, but

instead is smart enough to self-reconfigure itself for maximum gain.

This chapter proposes an algorithm for the real-time determination of the active

device's S-parameters, and requires only the following:

1. A predictable and reliable tuning network that can tune to arbitrary source and

load impedances, e.g. [6], [7], [8].

2. A set of output power measurements at distinct yet arbitrary input/output

terminations that are provided by the aforementioned tuning networks.

The number of measurements required in this process is identical to that of a

standard two-port network analyzer calibration1, but the method proposed here only

requires power measurements at one port. Also, in our method, the device-under-test

need not be disconnected from the system, but rather in-situ measurements can be

sampled using a directional coupler as shown in Fig. 2-1. The algorithm presented here

allows all of the necessary design parameters for a maximum-gain amplifier to be

1 A full-two port network analyzer calibration requires 7 measurements: open/shortlload at each of the twoports, and a through between the two ports.

7

/1J )

. )

/2

Antenna

) ) Reconfigurable Amplifier

) )

Tunable Matching Networks

DirectionalCoupler

IF

Fig. 2-1: An application ofa reconfigurable amplifier in a typical receiver.

obtained, whether it be stability parameters, simultaneous conjugate matching conditions,

or the unilateral figure ofmerit.

The resulting algorithm, when used with tunable matching networks, makes it

possible to create an autonomous, self-adaptable, self-reconfigurable system.

Simulations presented here validate the proposed concept, demonstrating that a

maximum-gain amplifier can be designed in real time without any prior knowledge of the

device.

Theory of Proposed Method

In contrast to standard network analyzer gain measurements that require circuitry

to detect both input and output signal power levels, the method presented here does not

require that the gain and input power be known. This simplification decreases the cost

and complexity, since the only information required for our method is the output power

level.

8

1i:ntableOtItput

MakldalNetwork

Fig. 2-2: Matching conditions of a reconfigurable simultaneous conjugate match amplifier.

The nomenclature in the following analysis is taken from the general model of an

amplifier (Fig. 2-2). The transducer gain of an amplifier can be expressed in terms of its

device S-parameters as [10]

S S rwhere f. =S + 12 21 L

In II I-Sr22 L

(2.1)

Multiplying the input power available from the source to the transducer gain produces the

output power delivered to the load,

If f s = r L = 0, (2.2) simplifies to

Pout (0,0) =PAVS IS2112

.

Substituting (2.3) into (2.2) gives

9

(2.2)

(2.3)

(2.4)

Equation (2.4) is the basis for our proposed method, and will be used to calculate

Sl1, S22, and (S21S12), which are the only parameters needed to determine stability and

matching conditions for a maximum gain amplifier. This fact is significant, as it allows

for the design of a non-unilateral, conjugately matched, maximum-gain amplifier without

having to know the values of each ofthe four S-parameters.

SII can be calculated by first reconfiguring the output matching network so that

r L = O. This reduces (2.4) to

(2.5)

where the two unknowns Sl1r and Slli are the real and imaginary parts ofSl1, respectively.

Pout(O,O) is also unknown, but can be measured by reconfiguring the matching networks

so that rL = rs = O. Given output power measurements at two unique values of rs, we

can solve for SII.

The solution to this set of equations is quite complex, and was solved with the aid

of Mathematica, a computer-aided symbolic solver. The symbolic solution is rather

lengthy, and is included in Appendix 1.

Two constraints in the solution for SII are worth mentioning. The first addresses

the values of rs that can be presented to the transistor; rs can be of almost any arbitrary

value, but must conform to the following:

10

1. rs should not have a magnitude of Irl=O or Irl=1 at any frequency within the

band.

2. The two values of r s should differ from one another at each frequency within the

band.

If these rules for r s are not followed, mathematically the solution will not converge since

the rules prevent (2.5) from yielding redundant solutions.

The second constraint in the solution exists because of the squared term in (2.5).

Mathematically, this results in two solutions for Sll that differ from each other in that the

magnitude of one is larger and the other smaller than unity. However, since IS111 of an

unconditionally stable device is always less then unity, and (2.1) is only valid for a stable

amplifier, the correct solution is the one that is less then unity.

S22 can be solved for in the same way as Sl1, except that the input matching

network is reconfigured so that r s = O. This reduces (2.4) to:

(2.6)

Output power measurements for two different values of r L, with the same constraints as

r s, will then yield the solution for S22.

Once the values of Sl1 and S22 have been obtained the product of S21 and S12 can

be calculated by reconfiguring the input and output matching networks to two arbitrary

combinations of r sand r L. This leaves (2.4) with two unknown variables, (S21S12)r and

(S21S12)i, which are the real and imaginary parts of (S21S12), respectively.

11

The values ofrsand r L that can be used in the calculation of (821812) have similar

rules as described in the procedure for calculating 811 •

1. r s and r L should not have a magnitude of Irl=o or Irl=1 at any frequency within

the band.

2. The two combinations of r s and r L should differ from one another at each

frequency within the band. For instance, r s = r 1 and r L = r 1 would be a valid

combination and would be different than r s = r 1 and r L = r 2.

With the values of 811 , 822, and 821812 known, the stability parameters can be

calculated to determine whether the device is stable or not. If it is, the simultaneous

conjugate matching conditions [10] can be calculated and the RMNs reconfigured for

maximum gain. If it is not stable, negative feedback can be added (e.g. through a

MEMS-switch activated stabilization networki and the process can be repeated.

It should be noted that the entire procedure requires only two distinct, arbitrary

values of matching network terminations (e.g. r 1 and r 2) for the seven single port power

measurements required for this method. This means that the reconfigurable networks

only need to reconfigure to two distinct configurations. Compared to a standard two-port

network analyzer calibration, which requires four distinct calibration standards of fixed

values (open, short, load, through) and seven measurements using two ports, the resulting

process is much simpler.

The entire procedure, as it would work on a fully autonomous amplifier as it

switches from one frequency of operation to another, would work as follows:

2 Stabilization will be covered in depth in Chapter 3

12

1. Input and output RMNs reconfigure so that ls=lL=O. Pout(O,O) is measured.

2. Output RMN presents lL=O. Input RMN reconfigures to arbitrary lSI = 11.

Pout(lI,O) is measured.

3. Output RMN presents lL=O. Input RMN reconfigures to arbitrary ls2 = 12.

Pout(12,0) is measured.

4. Input RMN presents ls=O. Output RMN reconfigures to arbitrary lLl = 11.

Pout(O,ll) is measured.

5. Input RMN presents ls=O. Output RMN reconfigures to arbitrary lLZ = 12.

Pout(O,l2) is measured.

6. Input RMN presents 1 S3 = 11 and output RMN presents 1 L3 = 11. Pout(lI, 11) is

measured.

7. Input RMN presents 1 S4 = 12 and output RMN presents 1 L4 = 11. Pout(12, 1 I) is

measured.

8. 8 11 , 822, and 8218 12 are calculated from the seven output power measurements.

9. Stability parameters and simultaneous conjugate matching conditions are

calculated.

10. Tunable matching networks reconfigure to appropriate impedances for maximum

gam.

13

Simulations and Calculations

To validate the theory, a simulation was conducted from 1 - 10 GHz using the

Agilent RF simulator, Advanced Design System (ADS). A stabilized nonlinear model of

the Fujitsu FHX35LG GaAs FET, included in the software library, was biased at VDS =

3V and Vos = -O.2V and used as the active device.

The procedure was carried out as described earlier. Seven simulated power

measurements were recorded using arbitrary values for all r s and rL configurations. A

Matlab code was then used to calculate Sl1, S22 and S21S12 from the power measurements.

A copy of this code is including in Appendix 1, and contains all the necessary formulas

and equations to replicate this process. The calculated S-parameters were then compared

to the simulated S-parameters as shown in Figs. 2-3, 2-4, and 2-5, respectively. The

validity of the proposed method is confirmed in these graphs.

90

180 ~-+--J----+------j-~-+--+--+--...j....:,l------: 0

270

Fig.2-3: Simulated and calculated SII of the stabilized FHX35LG from 1 - 10 GHz in dB vs. angle.

14

90

15

180 f----+----+----;r_-_+_---t-----l

270

o

Fig.2-4: Simulated and calculated 822 of the stabilized FHX35LG from 1 - 10 GHz in dB vs. angle.

90

simulated-

180 f---_+_----'~-_____cr_-_+_-!1l-+---l 0

270

Fig.2-5: Simulated and calculated 821812 of the stabilized FHX35LG from 1-10 GHz in dB vs. angle.

15

After solving for these device parameters, the stability parameters, K and B1,

were calculated, as shown in Fig. 2-6. Since these values were calculated from the S-

parameters, these values are also quite accurate, as would be expected. The stability

parameters demonstrate the unconditional stability of the device, allowing for the design

of the matching networks using the simultaneous conjugate match formulas [10].

For application in a self-reconfigurable system, the proposed method was used to

design a maximum gain amplifier at 1, 5, and 10 GHz without any previous knowledge of

the device. The resulting gain at these frequencies, along with the maximum transducer

gain of the device, is shown in Fig. 2-7 for comparison. At all frequencies, the

conjugately matched gain of the device is equal to the maximum allowable gain of the

device.

Fig. 2-7 demonstrates the true value of this technique. Instead of designing

maximum-gain amplifiers using S-parameters taken from complex 2-port measurement

3~t---------------------,

2.5 i \ : .

1.5

-K-simulated

........ • K-calculated I

-81-simulated

• 81-calculated

2 3 4 5 6GHz

7 8 9 10

Fig. 2-6: Stability parameters K and B1 of the device, which show that the device is unconditionallystable.

16

Fig.2-7: Simulated maximum stable gain and unilateral conjugate matched gain at 1,5, and 10 GHz.

systems, a designer can do the same thing with seven simple I-port measurements. This

algorithm, when used in conjunction with an intelligent RMN that can reconfigure itself,

opens the door for fully autonomous self-reconfigurable amplifiers.

Fabrication and Measurements

To validate the theory through measurements, an experiment was conducted from

2 - 3 GHz using two packaged FETs mounted on Rogers RT/duroid 5880 substrate. A

stabilization network was added to the FET to provide unconditional stability. Bias tees

were connected external to the dielectric mount, and SMA male-male connectors were

placed outside the bias tees so that various loads could be attached to the device. This

setup, shown in Fig. 2-8, is the "transistor" whose parameters we solved for, and a

measured baseline of these "transistor" S-parameters were recorded with a vector

network analyzer.

17

Bias Tee

StabilizationNetwork

Sl\fA connector

Fig. 2-8: "Transistor" setup showing the active device, stabilization network, bias tees, andSMA(male)-SMA(male) connectors.

In the proposed approach, any type of impedance transformer can be used, but for

simplicity, the transistor terminations r s and r L were provided by two different stub

networks on Rogers RT/duroid 5880 substrate. As was the case with the lO-step

procedure described earlier, the transistor terminations were made so that rSl = rLl and

r S2 = r L2. This way, the same Matlab code, included in Appendix 1, could be used in the

calculation of the S-parameters. The stub network impedance transformers are shown in

Fig. 2-9, with port (a) always connecting to the transistor and port (b) connecting to the

(a) (b)

Fig. 2-9: Two stub networks used to provide the input and output matching networks. Port (a) isconnected to the transistor, and port (b) is connected to the 50-Q port of the network analyzer.

18

50-0 load of the network analyzer.

With the stub networks, the extraction procedure was carried out as described

earlier. For each case, the proper impedance transformer was connected to the

"transistor", and the source power was kept constant.

The first FET measured was an Agilent ATF-10736 GaAs FET, biased at VDS =

2V and IDS = 25mA. A 750-0 shunt resistor at the gate provided unconditional stability.

Measured and calculated S-parameters are compared in Fig. 2-10 for SII and Fig. 2-11 for

S22. Instead of the more traditional polar plots, the results are graphed on a line graph so

that the error between the two can be clearly observed.

The second device was an Agilent ATF-26884-STR GaAs FET, biased at VDS =

3V and IDS = 10mA with a 220-0 series resistor added to the drain for stability. The

same materials and procedures were used for both transistors so that the method would

not be compromised during these tests. Figs. 2-12 and 2-13 demonstrate the accuracy of

the approach in extracting the magnitude and phase ofSII, and S22, respectively.

~

, .• .... . ,. ..~

, -. - . ,, ., ,,, I.,

--Measured I• • • • Calculated

H-MeaSUred I ~ .. ,- - - - Calculated I • .............. t

• ~• ,

,-,.. • ,

"' I

~ •-'I32.82.4 2.6

GHz2.2

180

135

Cl 90Gl~ 45Gl

~II.

o-45....

U; ·90

-135

-180

232.82.4 2.6GHz

2.2

-5

·252

iii'~ -10

~=2 -15CIlG::E.... -20....I/)

(a) (b)

Fig. 2-10: Extracted and measured (a) Sll magnitude and (b) Sll phase of the ATF-I0736.

19

- ,I,

I

I,,

I I, II, I

I. I , I

.., I, " , , , ,I I

,I , , , ,I ,

I ,'-0. , I

,I'I

, #". • ,II,

", , .• ",

-- fv'eaSUred

l

! , )

- - - - Calculated

'")00

~,

~r

" ,~,

,

~... ,

~,

-- fv'easured I '-"- , ,

- - - - Calculated I ,~

32.82.4 2.6GHz

2.2

o-45

-90

-135

-180

2

180

135

'6l 90

! 45ellfIl

.!Q,NNVI

32.82.4 2.6GHz

2.2

-20

2

o

iii'~ -5

-8::J

~ -10

III:::iiN -15NVI

(a) (b)

Fig.2-11: Extracted and measured (a) 822 magnitude and (b) 822 phase of the ATF-I0736.

The graphs show that the calculated S-parameters from the measurements are not

as consistent as those from the simulations. This is due to the constant fluctuations of the

data from the measurement procedure. During the measurements, both the magnitude

and phase had small fluctuations, most likely due to the movement of the cables as each

impedance transformer was taken on and off. Although these small fluctuations do not

make much of a difference for a single measurement, the overall effect increases when

the error from several different measurements are used together to solve a set of

, . - .--.... ~ .. ~ .. ,- -. ,,

--fv'easured I- - - - Calculated

.......

"'"'"'- ---

" ~-- fv'easured I "-- - - • Calculated I "'-..: .

5

iii'0~

ell'tl::J

:I:l -5cClIII:::ii.... -10iii

-15

2 2.2 2.4 2.6GHz

2.8 3

180

135

'6l 90ell~ 45ellfIl 0III.c

-45Q,........ -90VI

-135

-180

2 2.2 2.4 2.6GHz

2.8 3

(a) (b)

Fig. 2-12: Extracted and measured (a) 8 11 magnitude and (b) 8 11 phase of the ATF-26884-8TR.

20

I' ".t I

I I I'..I I

, , I.."

I, , ., I',. • I

, I I

"" I • I

"II

"MeaSUred, I ,. . . • Calculated

.I

~ I

~ ,~

I

·-............ , I

~ .. ••

--Measured I ." I.• • •• Calculated I ~

5

iii' 0:E-.g -5:::I

:I:!C

:: -10::l!!

:::l -15III

-202 2.2 2.4 2.6

GHz

(a)

2.8 3

180

135

'6l 90

! 45ell1II 0.!0. -45C'II~ -90

-135

-180

2 2.2 2.4 2.6GHz

(b)

2.8 3

Fig. 2-13: Extracted and measured (a) S22 magnitude and (b) S22 phase of the ATF-26884-STR.

equations. In a simulated environment that has no fluctuations from outside influences,

the theory is sound, but when various fluctuations in the power measurements are

introduced the accuracy suffers. More accurate estimation would undoubtedly be

achieved with reliable integrated reconfigurable matching networks.

Due to the errors in the calculated S-parameters, the product of S21 and S12 was not

calculated since the resulting values would not be accurate. Although the calculated S-

parameters were not as good as that of the simulated results, the data shows a definite

correlation between the calculated and measured data, particularly the SII results in Fig.

2-12 of the ATF-26884-STR. Here, the magnitude differs by no more than 3 dB and the

phase exhibits a reasonable amount of accuracy. Overall, the calculated phase data from

all measurements show a good correlation with the measured data, proving that the

general theory works. Although the extracted magnitude does not compare as favorably

as the phase, it is still apparent that the two are linked.

21

CHAPTER 3ADAPTIVE STABILIZAnON NETWORKS

In Chapter 2, a method for designing a self-reconfigurable amplifier was created.

However, this method is only applicable for a stable amplifier, a concern that is

mentioned in the second footnote on page 12. Here, it states that the problem can be

solved, "through a MEMS-switch activated stabilization network," but the type of

stabilization network to include with a reconfigurable amplifier is the question that will

be investigated in this chapter.

Stability is one of the most important and fundamental principles in amplifier

design. Without stability, an amplifier has the potential to oscillate, whereby it will then

cease to function properly. The importance of stability in amplifier design is outlined in

the basic RF amplifier design process shown below, where the highlighted boxes are the

steps devoted to amplifier stabilization to ensure that the resulting amplifier will not

GivenTransistor

[ I

DeaigntabiJizationNetworks

CondjtiooaJlyStable

CaltulateStabilityetworkJ

Bilatel"lllDe ign

oilateralDesign

D ignMatching

etworks

Fig.3-1: Amplifier design flowchart for an unconditionally stable device. The highlighted boxesdemonstrate the steps that are needed to stabilize the device.

22

oscillate.

In the design of RF amplifiers, stabilization is usually determined by the S­

parameters of a device, the matching networks, and the terminations at the input and the

output. One characteristic of an unstable amplifier is the presence of negative resistance

at the input or output port ofthe transistor. In terms of reflection coefficient, Cn and rout.

as shown in Fig. 3-2, negative resistance is present whenever the magnitude of either is

greater than unity.

Since Cn and rout are dependent on r s and r L, which can be set at any arbitrary

value depending on the situation, there are two different classifications in which stability

is categorized. Using the labels in Fig. 3-2, the two types of stability are described as

follows [11]:

1. Unconditional stability: ICnl < 1 and Irout! < 1 for all passive load and source

impedances (i.e., 0 < Irsl < 1 and 0 < Ird < 1). In this case, the amplifier will

always be stable no matter what the matching networks, r s and r L, are.

2. Conditional stability: ICnl < 1 and Irout! < 1 for certain values of passive load and

source impedances (i.e. 0 < Irsl < 1 and 0 < Ird < 1). In this case, the amplifier

will be stable for certain values ofr s and r L, and unstable for other values.

Intuitively, it may seem that all amplifiers should be built with unconditionally

stable devices so that there is no chance of oscillation. However, it is not uncommon to

see static amplifiers (i.e., those without reconfigurable matching networks) designed with

a device that is conditionally stable. This is because the potential gain of a device is

usually reduced when a device is made unconditionally stable. Designers are able to

23

son

9-- InputTransistor

OutputMatching

[8]Matching

Network .. ~ I+- ... Network

son

f s fin f out f L

Fig. 3-2: Block diagram of an RF amplifier to describe the stability of an amplifier.

make stable amplifiers from conditionally stable devices because the matching network

of a static amplifier does not change. This means that r s and rL do not change once they

are designed, so if these values are within the stable regions of the conditionally stable

device, a stable amplifier with more gain can be realized.

Reconfigurable amplifiers, however, have matching networks (and therefore r s

and rL) that change depending on the frequency of operation, amplifier function (i.e. low

noise, high gain, etc.), and environmental constraints. Because of this, a device used in a

reconfigurable setup must be made unconditionally stable so that the amplifier will

remain stable no matter what value of r sand r L the tunable matching networks present.

Therefore, novel ways to achieve unconditional stability while preventing large

reductions in gain is especially valuable in the design ofreconfigurable amplifiers.

Stability parameters are often used to determine the unconditional stability of an

amplifier. The most common of these is the use of K, 11, and B1 [11], which indicate

whether a device is unconditionally stable, conditionally stable, or unstable. However, if

one is merely interested if a device is unconditionally stable, like those in reconfigurable

amplifiers, a single parameter, Mu, can be used [12], as shown below.

24

(3.1)

A value of Mu > 1 implies that a device is unconditionally stable. If Mu < 1, the device

is either unstable or conditionally stable. Additionally, larger values of Mu suggest

increased stability, but at the cost of gain. Ideally, a device will have a value ofMu that is

close to unity for maximum gain performance.

In conjunction with Mu, the gain of a device is used to determine the

effectiveness of a particular stabilization network in the following way: if two different

stabilization networks both provide unconditional stability to a particular device, the most

effective one will be judged by the amount of available gain left after stabilization. For

this investigation, "maxgain3" is used to calculate the available gain of a device, and is

defined as:

Maxgain=GT,max , Mu ~ 1

CiMsG,Mu< 1

When Mu ~ 1 (unconditionally stable device), the maximum transducer power gain of a

transistor, GTmax [10], is calculated as:

(3.2)

G Tmax is the available gain of a device when it is unconditionally stable. On the other

hand, ifMu < 1 (device conditionally stable/unstable), the maximum stable gain,

3 Maxgain is a parameter defmed in Agilent's RF simulator Advanced Design System.

25

12 -r----------------.

10

8

N 6t/)

4

2

O..L--.,....---...,....---.....,-------,----!

1.51 1.54 1.58Mu

1.61 1.65

Fig.3-3: Graphical relationship between Mu and IS2d, for a typical device.

821

GMSG =S'12

(3.3)

is calculated instead. GMSG represents the maximum amount of gain available from a

conditionally stable or unstable device assuming that it will be stabilized so that Mu = 1.

Because of this, values of GMSG will always be greater or equal to values of G Tmax, since

the gain of a device decreases as Mu increases (see Fig. 3-3). Therefore, values of Mu

that are close to unity result in higher gain.

Frequency-Dependent Stabilization Networks

The most common way to unconditionally stabilize a transistor is through

resistive loading, as shown in Fig. 3-4 [10]. Here, the use of series or shunt resistive

stabilization networks placed either at the input or output of the device brings about

stability by reducing the amount of gain of the device. However, since gain is the main

purpose of the amplifier, a designer is often in the predicament of deciding whether to

decrease stability at the cost of gain or vice versa. Sometimes, attempts to achieve

26

Fig.3-4: Different types of resistive loading to improve stability.

unconditional stability over a broad bandwidth may result in larger reductions in gain at

frequencies where the device may already be stable, a particularly harmful situation if

these reductions occur at the operating frequency.

A better way to achieve unconditional stability is to use frequency-dependent

elements such as capacitors or inductors in conjunction with resistive loading techniques

[13]. This is because a normal device is usually more unstable at certain frequencies and

(a)

(b)=

(c)

Fig.3-5: (a) Series resistor/inductor stabilization network connected in shunt with the gate ofa FET.(b) Equivalent circuit of (a) at low frequencies, when the inductor acts as a short. (c) Equivalentcircuit of (a) at high frequencies, when the inductor acts as an open.

27

Table 3-1: Characteristics of the different configurations of frequency-dependent stabilizationnetworks for a common-source device configuration.

Device Connection Lumped Elements Characteristics Filter Type

1 Shunt at gate Series RC Stabilization at high f Highpass

2 Shunt at gate Parallel RC Does not stabilize N/A3 Shunt at gate Series RL Stabilization at low f Lowpass4 Shunt at gate Parallel RL Does not stabilize N/A5 Series at gate Series RC Does not stabilize N/A6 Series at gate Parallel RC Stabilization at low f Highpass

7 Series at gate Series RL Does not stabilize N/A8 Series at gate Parallel RL Stabilization at high f Lowpass

9 Shunt at drain Series RC Stabilization at high f Highpass

10 Shunt at drain Parallel RC Does not stabilize N/A11 Shunt at drain Series RL Stabilization at low f Lowpass

12 Shunt at drain Parallel RL Does not stabilize N/A13 Series at drain Series RC Does not stabilize N/A14 Series at drain Parallel RC Stabilization at low f Highpass

15 Series at drain Series RL Does not stabilize N/A16 Series at drain Parallel RL Stabilization at high f Lowpass

17 Gate to drain Series RC Stabilization at high f Highpass

18 Gate to drain Parallel RC Does not stabilize N/A19 Gate to drain Series RL Stabilization at low f Lowpass20 Gate to drain Parallel RL Does not stabilize N/A

more stable at others. If resistive loading can be applied in the unstable frequency

regions and removed from the stable frequency regions, the device will stabilize in the

unstable region at the same time gain is preserved in the stable region.

An example of a frequency-dependent stabilization network is shown in Fig. 3-5a,

which is comprised of a series resistor/inductor circuit in shunt with the gate of a FET.

At low frequencies, the inductor acts as a short and the resistor stabilizes the device,

leading to the equivalent circuit in Fig. 3-5b. At high frequencies the inductor acts as an

open and the resistor does not have any effect, as shown in the equivalent circuit of Fig.

3-5c. Similarly, a capacitor and resistor can also be used together to form a frequency-

dependent stabilization network.

28

These frequency-dependent stabilization networks are basically low-pass or high

pass filters. Depending on the stabilization characteristics of the transistor, the

appropriate type of filter response can be used to stabilize the device and preserve gain.

The characteristics of different frequency-dependent stabilization networks for a

common-source device are outlined in Table 3-1.

In the table, half of the stabilization network combinations can be eliminated

because they are not compatible with an amplifier. Networks that have parallel lumped

elements and are shunted to the device are not compatible because they short the gate or

drain to ground at certain frequencies. Also, networks that have series lumped elements

Table 3-2: Schematics and characterization of the effective frequency-dependent stabilizationnetworks for a common source device configuration.

Low-fre uenc stabilizationInductive

Low ass filter

stabilization

Hi h ass filterLow ass filter

Capacitive Inductive

Hi h ass filter

c:o~(J)c:c:o()

c:.~

o

c:on(J)c:c:o()

"*(!)

c:o~(J)c:c:8c:.~

i(!) Lowpass filter Highpass filter

29

in series with the device are not compatible because at certain frequencies an open circuit

is created at certain frequencies that does not let signals pass. These defective

stabilization networks are labeled, "Does not stabilize," in Table 3-1. The functional

stabilization networks and a schematic of each are listed in Table 3-2. There are a total of

10 different frequency-dependent stabilization networks that are categorized by device

connection, lumped elements, and stabilization frequency.

Adaptive Stabilization Networks (ASN)

The previous section discussed ways to stabilize a device over a wide band while

maintaining high gain at a single frequency. This section discusses how to maintain high

gain over the wide band.

Since static amplifiers only operate at one frequency, most stabilization networks

are designed to stabilize a transistor while preserving gain at one frequency. However,

the entire motivation for designing reconfigurable amplifiers is that they can work at

multiple frequencies, and as such maximum gain needs to be preserved over a wider

frequency band for best performance.

A solution to this problem is the use of adaptive stabilization networks (ASN),

which differ from conventional stabilization networks in that they are able to adapt to

varying operational conditions. Using variable resistors, capacitors, and inductors in

frequency-dependent stabilization networks, adaptive stabilization networks can optimize

for gain while simultaneously providing unconditional stability over a wide band.

30

- Configuration 1: RI, Ll.Gain optimized for 10 GHz

Configuration 2: R2, L2.Gain optimized for 50 GHz

50

=(a)

20

Iii' 15~ ....c "'.

'iij 10 "

Cl><lU 5::E

~-~ 0

10 20 30 40 50 0 10 20 30 40

Freq,GHz Freq,GHz

(b) (c)

o

1.1

1.4 -r--------------,

:i 1.2

1.3

Fig.3-6: (a) Adaptive stabilization network consisting of a variable resistor and inductor shunted to thegate of a FET. (b) Mu and (c) GTmax of the FET with the adaptive stabilization network optimized forgain at 10 GHz (solid line) and 50 GHz (dashed line).

An example of the way an ASN works is demonstrated in Fig. 3-6. The

schematic of Fig. 3-6a is a frequency-dependent stabilization network with a variable

resistor and inductor connected to the gate of a FET. Figs. 3-6b and 3-6c show the

resulting values ofMu and maxgain of the device with the ASN in place. For operation

at 10 GHz, the lumped elements change to Rl and Ll, thereby providing unconditional

stability (Mu > lover frequency band) and a gain of 14 dB at 10 GHz. When the

amplifier changes its operating frequency to 50 GHz, the lumped elements switch to R2

and L2, which provide unconditional stability and a gain of 8 dB at 50 GHz. For both

cases, the increase in potential gain that is generated by the adaptive stabilization network

is in excess of 3 dB.

31

In the example given in Fig. 3-6, the ASN optimizes gain according to operating

frequency, but in principle, these adaptive stabilization networks can also adjust to

produce the highest amount of gain over temperature, device, etc. When compared to a

fixed stabilization network, the potential increase in gain with the use of an ASN is

highly dependent on the device and stabilization network configuration, but simulations

have shown an increase of approximately 3 dB.

Simulations

Simulations with adaptive stabilization networks were performed from 0 GHz to 50 GHz

using the Agilent RF simulator, Advanced Design System (ADS). A proprietary GaAs

four-finger, 200-Jlm small-signal HEMT model was provided by Northrop Grumman

Space Technologies (NGST, formerly TRW) and was the primary model used in

simulating the effect of the frequency-dependent stabilization networks. Initial

characterization of the HEMT revealed that the device was not unconditionally stable

10

30 .--~-.,----,.-----,.-~------,

o 5 10 15 20 25 30 35 40 45 50

25-{,,······.········································.· ....

freq, GHz

(b)

~

ffi 20-t,,···················Gi{j 15-t····· ...•..... ""'<" .

:;:E

5 10 15 20 25 30 35 40 45 50

freq, GHz

(a)

0.8

1.0 ,-~-,.-~--,.--,.-~--.,---,.--,

~0.6

::J:;:E

0.4

0.2

0.0

0

Fig. 3-7: (a) Stabilization characteristics of the simulated HEMT model showing that the device is notunconditionally stable from 0 - 50 GHz. (b) GMSG of the HEMT.

32

over the simulated frequency range, as shown in Fig. 3-7a. Additionally, Mu revealed

that the device was more unstable at lower frequencies, which is a logical conclusion

since the device has more available gain at the low frequencies, as shown in Fig. 3-7b.

Stability simulations were first performed with resistive loading networks (see

Fig. 3-4) to test which configurations were able to achieve unconditional stability. This

was done because the resistive component of any stabilization network controls the

overall stability. If a particular resistive loading configuration is not able to provide

unconditional stability to a device, then an ASN in the same configuration will not

provide unconditional stability either. Figs. 3-8 through 3-12 below show the

stabilization characteristics of each type of resistor loading configuration.

Figs. 3-8 and 3-9 show the stabilizing effect of a resistor placed in series with the

gate and drain of the HEMT, respectively. In both instances, the stabilization

R=10n

R=100n

R= 1000 n

50

...........

4020 30

Freq, GHz(c)

............... • .. 'Ow

10

(a)

40

in20~

c:'mCl 0=::E

-2050 020 30 40

Freq, GHz(b)

10

-----....................................... '" ..

o ..,e:.--r---,----r-----,r----l

o

0.5 -I f ......---

1.5

2 -..-------------..,

Fig.3-8: (a) Resistive loading stabilization network consisting of a resistor in series with the gate ofthe HEMT. (b) Mu and (c) Maxgain.

33

R=lonR= loonR=looon

2 -r-------------,

1.5

0.5

•• _ M .

• ~ <II •

50403020105040302010o -,llO<==---.---.....---+-----,---l

oFreq, GHz

(b)

Freq, GHz

(c)

Fig. 3-9: (a) Resistive loading stabilization network consisting of a resistor in series the drain of theHEMT. (b) Mu and (c) Maxgain of the HEMT for varying values of resistance.

characteristics (Figs. 3-8b and 3-9b) are very similar to one another, showing that the

resistor does not provide unconditional stability over the desired frequency range.

Although it comes close in both instances with a 1000-0 resistor, the overall gain

decreases to unacceptable levels (Figs. 3-8c and 3-9c). Even with a 100-0 resistor, the

higher frequencies have too much loss and do not exhibit any gain. Therefore, resistive

loading in series with the HEMT did not prove to be a good match with the NGST

device.

Fig. 3-10 and 3-11 show the stabilizing effect of a resistor placed in shunt with the

gate and drain of HEMT, respectively. Unlike series resistive loading, the stabilization

characteristics of a shunt resistor are dependent on its placement at the gate or drain of

the device. As the pink line in Fig. 3-10b and 3-10c shows, a 38-0 resistor in shunt with

34

R

-R=38n

- -' R=100n

........ R=1000n

(a)

1.2 ,--------------,

"" .... -"""'"-..­--------

30 ....-----------...,

504020 30

Freq, GHz(c)

10

,'.

" ,

o -I------r------,------,r---...----l

o

iii'~ 20c

.~ 10 i=-=-=~=;=.,'=~=-....,=~=.'=~=-='-....,=.-:I:~:=.::=..::::::.::==:::;''''i.~C;;';'':E

5040

..... ~- , , ~

10 20 30

Freq, GHz

(b)

~ 0.8~

0.6

0.40

Fig.3-10: (a) Resistive loading stabilization network consisting of a resistor in shunt with the gate ofthe HEMT. (b) Mu and (c) Maxgain of the HEMT for varying values of resistance.

the gate of the device provides unconditional stability over the entire frequency range

with a maxgain of approximately 8 dB. In Fig. 3-10c, the abrupt cusps in the maxgain

graph are due to the transition from GMSG to GTmax as the device becomes unconditionally

stable.

A resistor in shunt with the drain of the device cannot provide unconditional

stability at low frequencies, and cancels out most of the high frequency gain of the

device. In this case, a resistor in shunt with the gate (Fig. 3-10a) exhibits a good balance

between stability and gain, and has the necessary requirements to be evaluated with an

ASN.

Fig. 3-12 shows the effect of a resistor connected to the gate and drain of the

HEMT. In this configuration, lower resistances cause more negative feedback, thereby

stabilizing the transistor. Here, a resistance of 450 n stabilizes the device over the

35

Fig.3-11: (a) Resistive loading stabilization network consisting ofa resistor in shunt with the drain ofthe HEMT. (b) Mu and (c) Maxgain of the HEMT for varying values of resistance.

desired frequency range with a maximum gain of approximately 8 dB. These results are

very similar to those obtained with the resistor shunted to the gate of the device, and as

such was also chosen for implementation in an ASN.

The resistive loading in Fig. 3-10 and 3-12 proved to be the most effective

configurations for stability of the HEMT, and therefore ASN simulations were conducted

with them. The characteristics of the lumped elements used in the simulations were

dictated by those that were in prototype production at NGST. The variable capacitors

were limited to a 5:1 max-min ratio, meaning that the maximum value of the variable

capacitor could not exceed five times the minimum value. Although there did not exist a

minimum absolute capacitance, the maximum capacitance value was limited to 10 pF

because of size constraints. Variable resistors were also limited to a 5:1 max-min ratio,

with no restrictions on minimum or maximum values. At the time of design, variable

36

R

R=450n

R=500n

R= 1000n

(a)

14 ~------------,

504030Freq, GIz

(c)

20106-l---,...--.,------r---.,..----l

o

.'.,. ....m12~

.5 10 -+ .

"'Cl ,._ ".>< . -~'- '."' 8 I-----~-~--~.- ~""~'~,...:..:E r-....;".;";....;".;,,;~....;,,.;,,; ....................................---:......:::;,:j

1.6

1.4

~1.2

:E 1

0.8

0.60 10 20 30 40 50

Freq, GIz(b)

Fig.3-12: (a) Resistive loading stabilization network consisting ofa resistor connected to the gate anddrain of the HEMT. (b) Mu and (c) Maxgain of the HEMT for varying values of resistance.

inductors were not available, but static inductors with values ranging from 0.1 nH to 5 nH

were available.

Figs. 3-13 and 3-14 demonstrate the capacitive adaptive stabilization network

configurations. Simulations were performed by varying one component while keeping

the other constant. This way, the stabilization effect of the resistor or capacitor could be

clearly observed.

In both configurations, the capacitive ASN works as a high pass filter with a lossy

passband that stabilizes the high frequency regime. When the capacitor is introduced, the

resistor is effectively canceled out at low frequencies, and the device becomes unstable

below 1 GHz. The effect of the capacitor can be seen in Figs. 3-13b and 3-14b. When

37

the resistance is kept constant and the capacitance varies, the stabilization levels stay

relatively constant since the capacitor is a lossless element.

(a)

5040

C =0.5 pF

C=lpF

C=5pF

30

Freq, GHz

20o 10

.: 20I'llCl

=10:IE

0+----.---..------r--~-____1

50

Constant R = 100040 -r------------...,

iii' 30~

4020 30

Freq, GHz

10O+----;---..,..-----r---.,...----!

o

1.2 ,~

'f .....0.8 r····· ,

~

:::E0.4

(b)

1.3 -.-------------...,

R=500

R=2000

R=5000-~ --"-.--.

.: 20 -n···c·········,········································I'll

=10:IE

.. ,

Constant C = 1 pF40 ,...------------,

iii' 30't:l....

..............

........ ~-----------0.7

.t., ......

~ ..::!!

5040302010o0+---.------r--...----.-----1

50403020100.4 +--.....,....--,....---r-----,---l

oFreq, GHz Freq, GHz

(c)

Fig.3-13: (a) Adaptive stabilization network consisting of a series resistor and capacitor shunted to thegate of the HEMT. (b) Mu and rnaxgain of the HEMT for a static resistor of 1000 and varyingvalues of capacitance. (c) Mu and rnaxgain of the HEMT for a constant capacitance of 1 pF andvarying values of resistance.

38

-- -(a)

Constant R = 450 0

1.5 17..... C=2.0 pFa:I't:I 13- C=1 pF

:;, .5:e III C =0.5pF0.5 CJ 9><

III:e

0 5

0 10 20 30 40 50 0 10 20 30 40 50Freq, GHz Freq, GHz

(b)

5040

R= 10000

R=5000

R= 100 0

20 30

Freq, GHz

10

-- - ,- - - .-.... - - - - - - - - - -

.....a:I 10't:I-.5 5IIICJ>< 0III:e

-5

50 0

(c)

Constant C = 1 pF15 -y-------,.---

4020 30

Freq, GHz

10

,.i

(

~ , .I

2

0+---,---,----.---,..----/o

3-r---.......---=-:------...,-- -._--

Fig.3-14: (a) Adaptive stabilization network consisting of a series resistor and capacitor connectingthe gate and drain of the HEMT. (b) Mu and maxgain of the HEMT for a static resistor of 450 n andvarying values ofcapacitance. (c) Mu and maxgain of the HEMT for a constant capacitance of 1 pFand varying values ofresistance.

39

Although the capacitance does not have an effect on the stabilization levels, it has

a significant effect on the passband frequency. In this case, the passband frequency is

defined as the frequency where the stabilization network has enough of an effect (i.e., lets

enough of the signal pass through the network) so that Mu = 1. This passband frequency

has a linear inverse relationship with the capacitor value, which is related to the RC time

constant of the stabilization network. Therefore, larger values of capacitance cause a

decrease in the frequency of the passband, as demonstrated in the simulations of Fig. 3-

15. This graph shows that an increase in the capacitance decreases the passband

proportionally, allowing the resistor to stabilize the device at lower frequencies.

The resistor has a similar effect on the passband of the stabilization circuit due to

the RC time constant. This causes the device to stabilize at lower frequencies when a

higher resistance is introduced, as shown in Figs. 3-13c and 3-14c.

With respect to stabilization, a resistor will have a different effect depending on

its connection to the device. In the setup ofFigs. 3-13 and 3-14, however, the resistor has

32

Constant R = 4500

C =O.5pF

C=l pF

C=2pF

0.5

~ 0:E

-0.5

-1

-1.5

0

1.5 .------~-------:-:~~~~~....__,... '" ...... :-: :..: =-'" ;,......:. .....:. --= :..: ...:..,.. ...:,......:.. ..-:, "'-= :..: ::-... .::--- - ~---------

Freq,GHz

Fig. 3-15: Effect of varying capacitance on the stabilization of circuit 2-14a.

40

the same effect. As the resistance is decreased, less gain is available and the stability

increases, and vice versa. Overall, these simulations show that the addition of a capacitor

to the chosen resistive stabilization network does not significantly enhance the stability

and gain of the device.

Inductor and resistor stabilization networks were also simulated in the same

manner as the capacitor/resistor networks. The inductive ASNs that were tested work as

a low-pass filter with a lossy passband that stabilizes the low-frequency regime. This

causes the device to become unstable at high frequencies when the inductor is introduced.

With respect to the passband of the stabilization network, the effect of the

inductor is similar to that of the capacitor because the inductance is also inversely

proportional to its passband. When larger values of inductance are used, the passband

decreases, and the device becomes more unstable at high frequencies, and vice versa.

The effect of the resistor is a little more complicated, since it affects both the

stability and the passband. A decrease in resistance will increase the stabilization of the

device while decreasing the passband. Fig. 3-16 and 3-17 show the effect of a static

inductor with varying resistance. Although there is some stabilization control with the

inclusion of an inductor, it does not enhance the stability performance of the device.

41

5040

R= 1000 0

R= 1000

R=380

302010o5+-----,---+-----,---,.----,'

-a:1 20~

.!: 15IIICll

~ 10:E

50

(a)

Constant L = 1 nB25~-~----

40302010

1.3

::;,:E

0.7

0.4

0

Freq, GHz Freq, GHz

(b)

Fig.3-16: (a) Adaptive stabilization network consisting of a series resistor and inductor shunting thegate of the HEMT. (b) Mu and maxgain of the HEMT for a static resistor of 450 0 and varyingvalues of capacitance. (c) Mu and maxgain of the HEMT for a constant capacitance of I pF andvarying values of resistance.

Since the four ASN configurations examined in Fig. 3-11 through 3-15 did not

prove to be effective, multiple networks were combined together and simulated. This

gave greater control of the stabilization by combining an ASN that affects low-frequency

stabilization with an ASN that affects high-frequency stabilization. The following four

configurations in Fig. 3-18, which combines the low-and high-frequency stabilization

networks, were simulated to observe the effectiveness of each.

42

'.

= =

6-l----,---,....---T-----r-----i

5040

R=1000n

R=500n

R=450n

302010

.oo ........ - ......-~-------~-

o

.....m~ 12.S:I'Cl

~ 9I'Cl:s

50

(a)

Constant L = 1 nB

15 -.--------

40302010

1.5 ,...--------------,

1.25

0.75 -I ~ ~~=.;:==~

0.5 +----"T"'----,_-_r_----r-----i

oFreq, GHz Freq, GHz

(b)

Fig,3-17: (a) Adaptive stabilization network consisting of a series resistor and inductor connecting thegate and drain of the HEMT. (b) Mu and maxgain of the HEMT for a constant inductor of 1 nH andvarying values of resistance.

~··········i

i.... L

~·I··J···HEMIfJ X1_

-==- '=-'"-==-

Fig,3-18: The four simulated combination adaptive stabilization networks that utilize networks thataffect both low and high frequencies.

43

Initial simulations showed that greater control was indeed possible by combining

multiple networks. However, some configurations Were more effective then others. The

stabilization network that preserved the most gain with unconditional stability was that of

Fig. 3-19a, where a CR and LR network was placed in shunt with the gate of the device.

This could be due to the negative feedback loop from the drain to the gate of the device

that the other three configurations have. This diverts drain current away from the output,

thereby decreasing the overall power and gain at the output.

The performance of the most effective stabilization network is shown in Fig. 3-

19b and Table 3-3. The increase in gain of the device when compared to a conventional

+ L

R2

TR _HEMTX1

+

(a)

20 -.--------------------...,

15

.~ 10C)

><cu 5:E

o

--10 GHz optimized

- - - - 30 GHz optimized

....... 50 GHz optimized

(b)5040302010

-5 ~--___,.---_r_---.,.._--__r_----I

oFreq, GHz

Fig.3-19: (a) Configuration of the most effective ASN in providing unconditional stability and gainoptimization. (b) Resulting gain of the ASN when optimized for operation at various frequencies.

44

resistor network is substantial. Increases in potential gain of 6.5 dB, 2.3 dB, and 1 dB

were recorded at 10-GHz, 30-GHz, and 50-GHz, respectively. The ASN has a more

significant contribution at the lower frequencies where there is more instability (Mu

lower at low frequencies), since these regions are affected more by the stabilization

network. At frequencies where the device is more stable or already stable, the increase in

gain is lower since the stabilization network plays less of a role.

Table 3-3: Component values and maximum gain for the best performing ASN in order to optimizefor gain at various frequencies.

ASN NetworksResistor

Values Gain Optimized Frequency Network10 GHz 30 GHz 50GHz

C rpF] 0.25 0.204 0.206 N/AL [nH] 0.5 0.5 0.5 N/AR1 [Q] 33.1 31.07 32.5 38R2[m 2.1 9.1 10.37 N/A

10-GHz Gain [dB] 14.2 11.1 10.6 7.730-GHz Gain [dB] 9 10 9.9 7.750-GHz Gain [dB] 7.6 7.7 8.6 7.6

45

CHAPTER 4CHARACTERIZATION OF RECONFIGURABLE MATCIDNG NETWORKS

In Chapter 2, a reconfigurable matching network application was investigated that

allowed for the development of an autonomous, remote self-reconfigurable amplifier. To

ensure proper operation of these remote reconfigurable systems, however, additional

considerations must be taken compared to their operator-assisted counterparts. In remote

locations, an amplifier does not have the benefit of a technician or repairman to monitor

its performance for signs of failure or degradation. This fail-safe monitoring is of

concern especially when dealing with MEMS-based reconfigurable matching networks

(RMN), which have unresolved long-term performance issues [14]. The ability to

monitor the operation of the RMN is the basis of this chapter, and will investigate

different ways for an autonomous amplifier to accurately characterize its RMNs so that it

will be able to identify problems that may occur.

During the research collaboration between NGST and the University of Hawaii

Microwave and Millimeter-Wave Research Lab, NGST showed an interest in the ability

for an autonomous reconfigurable amplifier to characterize its matching networks for self

diagnosis. This is because NGST is in the process of building a prototype reconfigurable

matching network that utilizes transmission-line capacitive loading techniques [15].

On a capacitively loaded transmission line, periodic capacitors can be used to

change the characteristic impedance and phase velocity of a transmission line according

to the following relationship:

(4.1)

46

If these capacitors are loaded onto a transmission line through MEMS switches, the

localized impedance around the region of the capacitor can then be altered. A schematic

of this variable capacitive loaded transmission line is shown in Fig. 4-1. With the MEMS

switches in the OFF state, no additional capacitance is added and the transmission line

has a characteristic impedance of 50 Q (or whatever impedance the transmission line is

designed for). When one of the switches is placed in the ON state, the capacitance

increases in the area of the switch and the localized impedance decreases. With more

MEMS switches in the ON state, the localized characteristic impedance of the

transmission line will decrease accordingly, transforming a 50 Q transmission line into

one with lower characteristic impedance. Similarly, a transmission line that converts to a

higher characteristic impedance can be made by designing the chracteristic impedance for

L=66 J.lrnI I

Transmission Line IW=40J.lrn

Connecting Switches I I

N=l N=2 N=30

"'I \

'I \son I I• • • • \ I

I'-

MEMS Switch Capacitance TITL Switch(CTITL)

Fig.4-1: Schematic of the NGST TITL using MEMS switches and a capacitively loaded.

47

(a) (b)

Fig. 4-2: Smith chart coverage ofa sample variable capacitive loaded line using 8 MEMS switchesat (a) 20.2 GHz and (b) 60 GHz [15].

50 n when all the MEMS switches are in the ON state. Reference [15] has shown that a

variable capacitive loaded line with 8 switches can be used to obtain a relatively wide

coverage of the Smith Chart over various frequencies as shown in Fig. 4-2.

NGST has designed a similar reconfigurable transmission line model using 30

MEMS switches and capacitors (see Fig. 4-1), which they call a tunable impedance

transmission line (TITL). The TITL allows the overall capacitance (Ceq) on the line and

the electrical length between the capacitors to be tuned, resulting in 230 different

combinations that can be presented to the transistor.

For an autonomous amplifier, presenting the right source and load impedance, Zs

and ZL, to a transistor is of utmost importance, and is highly dependent on an accurate

characterization of the TITL. For an amplifier in a controlled environment, the

impedance looking into a reconfigurable matching network configuration can be

measured to ensure the right load is being presented to the device. However, remote

48

autonomous amplifiers do not have this luxury due to size and cost constraints. One

solution to this problem is characterizing the reconfigurable matching network through a

circuit model. If an accurate TITL model can be designed, the impedance of a particular

configuration can be found without the use of expensive and bulky measurement

equipment. However, because the performance of MEMS switches can change over

time, it is not only important to create an accurate model, but to update the model during

the course ofoperation. This can be done by characterizing the matching networks.

To simplify the problem of characterizing the TITL, several assumptions can be

made. The first assumption is that all TITL switches are identical. This allows for a full

characterization of the entire reconfigurable matching network with the analysis of a

single TITL switch. Since the difference between TITL switches on a single amplifier is

minimal, this assumption will not invalidate the analysis.

The effect of the interconnecting transmission lines between successive TITL

switches is also ignored in our analysis. Although the transmission line introduces a

slight phase delay and a minimal amount of loss, the actual length of the line between

switches is 66 /lm, which is approximately 0.0211. at 10 GHz. This relatively minimal

length allows us to ignore transmission-line effects for our initial set of calculations.

However, subsequent revisions to the process should include the effects of the

transmission line. Lastly, the model of the TITL switch can be simplified. The MEMS

switch can be modeled as a short circuit in the ON state and an open circuit in the OFF

state, as shown in Fig. 4-3. In actuality, the MEMS switch has a finite resistance and

capacitance in series in the ON and OFF state. In the OFF state, this resistance and

49

TITL Switcb ModelTITL SwitcbScbematic

iT

ON State OFF State

Fig.4-3: Schematic of the NGST TITL model in the ON and OFF state that will be used in theinvestigation.

capacitance is relatively insignificant, but in the ON state, CON = 20 pF and RoN = 2 Q.

However, adding these parasitic elements complicates the TITL model, and will be

ignored initially. In addition, the parasitic resistance of the capacitor will also be ignored,

thereby allowing for the complete characterization of the entire TITL with a single

capacitance, CSwitch. This capacitance is the parameter that will be solved for in the

following investigation.

These three assumptions are made to minimize the complexity of the problem and

obtain an initial solution. After completing this simplified analysis, an in-depth analysis

of the problem can be continued by eliminating one assumption at a time.

50

TITL Characterization Theory

The assumptions described earlier allow for a complete characterization of the

reconfigurable matching network with a single capacitance, CSwitch. In the lab, CSwitch can

be easily calculated using full 2-port S-parameter measurements. However, for a remote

autonomous reconfigurable amplifier, limited resources do not allow for a full 2-port

analysis. Oftentimes, the only measurement available is a single port analysis, such as

the reflection coefficient of the amplifier. Therefore, the following analysis will attempt

to calculate CSwitch using reflection coefficient measurements. Although many different

data points can be measured, no knowledge of the device or TITL parameters will be

available to calculate CSwitch.

To find the value of this capacitance, ABCD parameters are utilized since the

parameters of a cascaded TITL network can be calculated by simple matrix

multiplication.

In the OFF state, a TITL switch is characterized by an open circuit, which is

simply an identity matrix in ABCD parameters. In the ON state, the ABCD parameters

of the TITL switch are

(4.1)

Matrix (4.1) allows us to create an ABCD matrix of the entire TITL by the following,

( 1 0)TITL== .

N'j'O).CSwitch 1

51

(4.2)

where N is the number of TITL switches in the ON state. From here on out, CSwitch will

be the only capacitance we will deal with, and will be referred to as C. Conversion of

(4.2) into 8-parameters yields

(4.3)

where Z is the characteristic impedance of the network. 811 and 822, as well as 821 and 812

are similar since the TITL network looks the same when looking into the input or the

output. With these TITL 8-parameters, a solution for the TITL capacitance can be solved

by relating the reflection coefficient of an amplifier to its matching networks. In this way

we can then substitute (4.3) and solve for the capacitance.

An easy way to relate the reflection coefficient to the TITL is to use Mason's rule

in signal flow-graph theory [16], which provides a systematic analysis of the path of any

given signal flow graph. Figure 4-4 shows the signal flow graph of an amplifier with r L

set to zero (e.g. for TITL, all output TITL switches are in the OFF state). Therefore only

Inpu.t TI.TL(A)

SUB

Device(8)

Fig. 4-4: Signal flow graph of an input reconfigurable matching network and a unilateral amplifier thatwas used to solve for the overall gain.

52

the input TlTL has an effect on the reflection coefficient, which simplifies the equation.

The resulting reflection coefficient of the reconfigurable amplifier, r, is

(4.4)

Substituting (4.3) into (4.4) produces a relationship between the TITL capacitance and [

-(j-oo.GZ.N + j.oo·GZ.N·S1IB - 2.S1IB)f =~--------~--....!..

2 + j-oo·C.Z·N + j-oo·C.Z·N·SIIB (4.5)

With two measured values off (e.g. [1 and [2) at two distinct TITL combinations, 8UB

and C can be solved for. The values of 8118 and the TlTL switch capacitance then reduce

to the following

Nr f r f 2 + Nr f 2 - f r N2·f2 - f rN2SIIB= --=--------....:....---­

Nr f 1 + N1 - N2·f2 - N2

-f2 + f 1C= -2·------------------------

(Nr f 1 + N1 - N/.'f7. - N7 + Nr f 1·f7 + N1·f7 - f 1·N7·f7 - f r N7)·j-oo.Z

(4.6)

where N1 and N2 correspond to the number of switches in the ON state when [1 and [2

are simulated, respectively.

In keeping with the single port measurement analysis, a technique to solve for

CSwitch using gain measurements was also conducted. Mason's signal flow graph analysis

was used to obtain a relation between the gain of a reconfigurable amplifier and the

TITL, but mathematically a solution could not be solved for. This analysis is included in

Appendix 2 for further review.

53

Simulation and Calculations

Verification of (4.6) was conducted with simulations in ADS. A small-signal S-

parameter file of Fujitsu's FHX35LG HEMT, biased at VDS = 3V and IDS = lamA, was

used as the active device. To represent the TITL switches, 10-pF shunt capacitors

representing CSwitch (refer to Fig. 4-1) were placed at the input of the device when in the

ON state, and were removed when in the OFF state, as shown in Fig. 4-5.

Two separate reflections were simulated; one with a single TITL switch turned on

and another with two TITL switches in the ON state. This provided us with our

"measured" ['1 and ['2, which was used to solve for the capacitance of the TITL. Fig. 4-6

shows the accuracy of the derivation in calculating both SllB and CTlTL of the simulation.

This data in Fig. 4-6 shows that an accurate characterization of NGST's reconfigurable

matching network can be done with two reflection measurements. In a properly working

reconfigurable amplifier, this characterization will be the same as the characterization

done in a controlled environment. Any differences between the calculated and measured

Yinerm

Term1Num=1Z=50 Ohm

"ON"State

r' ". ,/,//1i ····G,5 / :

~i ,/"S~:~:·~·.~.1 .:~: /' mm•• m ••••m ••••'. FHX35LG.......:.,.

"OFF"State

TermTerm2Num=2Z=50 Ohm

Fig. 4-5: Advanced Design System simulation setup to verify the iterative TITL characterizationprocedure.

54

10.05,....-------------,

10.025 +-------------1

10 -1-------_-_......-19.975 +-------11 - - - - Calculated

1--Simulated

9.95 +--.-----r---,-,-,--,----,-----,;--T

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

GHz

(a)

180 t---t---t---t---t--l----1 0

170

(b)

Fig.4-6: Simulated and calculated (a) capacitance and (b) SUB of the setup in Fig. 4-4.

TITL capacitance will indicate a defective system. This demonstrates that a proper self-

diagnosis of a reconfigurable amplifier is possible with data that can be measured

internally to the device.

55

CHAPTERS

CONCLUSION

This thesis discussed the benefits of using reconfigurable amplifiers in remote

systems and presented three new design techniques to fully utilize the capabilities of this

new technology.

Up to this point, published reconfigurable amplifier technology has focused on the

development of non-autonomous prototypes that must be controlled by an operator.

However, the next generation of reconfigurable amplifiers will have to be fully

autonomous, and be able to self-reconfigure to any type of situation. Chapter 3 presented

a method to make this a reality. An algorithm was created so that an autonomous self-

reconfigurable maximum gain amplifier could be designed without any previous

knowledge of the active device. Simulations with a non linear FET model have shown

that the maximum transducer gain of a device can be achieved at any frequency with only

seven different output power measurements. Measurements were also conducted to

verify the technique. Although there was a definite correlation between the calculated

and measured data, it was not as accurate as the simulated results due to measurement

errors introduced in the setup. However, the results shown here definitely substantiate

the accuracy of the proposed technique.

Chapter 3 addressed the problem of stability in reconfigurable amplifiers. Current

stabilization techniques for static amplifiers do not work with reconfigurable amplifiers

because they are designed to maximize gain at a single frequency. However,

reconfigurable amplifiers must maintain a high level of gain over a large frequency band,

so new stabilization techniques were investigated using tunable resistors and capacitors.

56

Simulations show that a combination of tunable shunt stabilization networks at the gate

of a HEMT is effective in maximizing gain from 10 GHz - 50 GHz. With this novel

tunable stabilization network, an added potential gain of 6.5 dB, 2.3 dB, and 1 dB were

recorded at 10-GHz, 30-GHz, and 50-GHz, respectively, when compared to conventional

resistive stabilization networks.

Chapter 4 dealt with the reliability issues associated with the MEMS

reconfigurable matching networks. For an autonomous amplifier, proper operation of the

reconfigurable matching network is of the utmost importance. To ensure that the

matching networks are functioning, a quick and simple method to calculate the network

model while in operation was created to compare it to a functional pre-measured network

model. Simulated measurements on a reconfigurable amplifier have shown that full

characterization of the tunable matching network can be achieved in as little as two

reflection measurements, which would allow a reconfigurable system to detect

degradations to the system.

SUGGESTIONS FOR FUTURE WORK

There are many areas in the thesis where the research can be extended. For the

investigation in Chapter 2, future measurements should be performed with a manual or

electronic (more accurate, but expensive) tunable transmission line offered by an RF

equipment supplier such as Maury Microwave [17]. With these tunable systems, accurate

and reliable measurements can be performed on a network analyzer. This should increase

the accuracy of the calculations. With these tunable systems, a reconfigurable prototype

57

like that of Fig. 2-1 can be fabricated. A system that would be able to receive or transmit

at multiple frequencies would prove that the reconfigurability concept can be a reality.

In Chapter 3, an adaptive stabilization network was created for use on a single

NGST GaAs FET. However, for use with other types of devices, a general

comprehensive analysis of the interaction between the adaptive stabilization network and

device should be studied. This would explain why certain resistive loading

configurations work better than others, and would allow one to design an adaptive

stabilization network without an analysis by trial and error.

Fabricating and testing the proposed adaptive stabilization network should also be

performed, but this would be contingent on NGST's development of the tunable resistors

and capacitors.

In Chapter 4, future work could involve reducing the amount of assumptions in

the analysis. Reducing the model of the TITL to a single capacitance required that the

switch and transmission line parasitics be ignored. Including the parasitics into the

analysis would make the analysis more complex, but would increase the accuracy.

58

LIST OF PUBLICATIONS

1. K.S. Ching, RY. Miyamoto, W.A. Shiroma, "Scattering-Parameter ExtractionTechnique for Designing Self-Reconfigurable Amplifiers," To be submitted June2005.

2. KS. Ching, RY. Miyamoto, W.A. Shiroma, "Unilateral Amplifier S-ParameterExtraction Technique," 2005 IEEE/ACES International Conference, Honolulu,Hawaii, Apr. 2005.

3. KS. Ching, G.S. Shiroma, RT. Murakami, RY. Miyamoto, W.A. Shiroma, "ActiveAntennas for Picosatellite Communication," Proceedings ofthe 2004 InternationalSymposium on Antennas and Propagation, Sendai, Japan, p. 405 - 408, Aug. 2004.

4. W. A. Shiroma, R T. Murakami, K. S. Ching, M. A. Tamamoto, and A. T. Ohta, "AProject-Based Undergraduate Curriculum in High-Frequency Electronics andAntennas," presented at the USNCIURSI 2004 National Radio Science Meeting,Boulder, CO, paper D2-3, p. 107, Jan. 2004.

5. C. S. Suh, l M. Bell, K S. Ching, T. A. Heffner, W. W. G. Hui, G. S. Shiroma, C.Song, R K Sorenson, and W. A. Shiroma, "An Investigation of GroundingTechniques in Microwave Amplifiers," presented at the 2003 IEEE TopicalConference on Wireless Communication Technology, Honolulu, HI, Oct. 2003.

6. D. S. Goshi, A. T. Ohta, M. A. Tamamoto, K S. Ching, H. L. Caraang, Jr., N. H.Phan, G. S. Shiroma, M. C. Binonwangan, and W. A. Shiroma, "An undergraduatewireless transceiver," presented at the United States National Committee /International Union ofRadio Science (USNCIURSI) 2002 National Radio ScienceMeeting, Boulder, CO, paper B7-6, p. 102, Jan. 2002.

7. KY. Sung, D. M. K Ah Yo, R Elamaran, J.A. Mazotta, KS. Ching and W.A.Shiroma, "An Omnidirectional Quasi-Optical Source," IEEE Transactions onMicrowave Theory and Techniques, vol. 47, pp. 2586-2590, Dec. 1999.

8. R Elamaran, KY. Sung, D.M.K Ah Yo, KS. Ching, and W.A. Shiroma, "A Three­Dimensional Quasi-Optical Self-Oscillating Mixer," IEEE Transactions onMicrowave Theory and Techniques, vol. 47, pp. 2144-2147, Nov. 1999.

9. W.A. Shiroma, lA. Mazotta and KS. Ching, "An Omnidirectional Active Antenna"(invited), presented at the XXVIth General Assembly, International Union ofRadioScience, Toronto, Ontario, Canada, paper BD1.9, p. 679, Aug. 1999.

59

10. lA. Mazotta, K.S. Ching and W.A. Shiroma, "A Three-Dimensional Quasi-OpticalSource," in 1999 IEEE MTT-S International Microwave Symposium Digest,Anaheim, CA, pp. 547-550, June 1999.

60

APPENDIXlMATLAB CODE USED TO CALCULATE 81h 822, AND 821812

Description

The following code, "Extraction.m" is a matlab program that extracts 811, 822, and 821812 from 7different output power measurements. For the program to work, the following 11 files below are needed inthe same directory as Extraction.m.

1. ipn_stubl.s2p: This file represents the "power" at the output of the amplifier with stub1connected to the input of the device. The program extracts 821 "power" from the fourthcolumn of the file

2. ipn_stub2.s2p: This file represents the "power" at the output of the amplifier with stub2connected to the input of the device. The program extracts 821 "power" from the fourthcolumn of the file

3. ipn_stub3.s2p: This file represents the "power" at the output of the amplifier with stub3connected at input and stub1 connected to the output of the device. The program extracts 821"power" from the fourth column of the file.

4. ipn_stub4.s2p: This file represents the "power" at the output of the amplifier with stub4connected at input and stub1 connected to the output of the device. The program extracts 821"power" from the fourth column of the file.

5. opn_stubl.s2p: This file represents the "power" at the output of the amplifier with stub1connected to the output of the device. The program extracts 821 "power" from the fourthcolumn of the file

6. opn_stub2.s2p: This file represents the "power" at the output of the amplifier with stub2connected to the output of the device. The program extracts 821 "power" from the fourthcolumn of the file

7. sparameters.s2p: This file represents the "power" at the output of the amplifier without anytype of matching networks connected to the device. The program extracts 821 "power" fromthe fourth column of the file

8. stubl.s2p: This file represents the reflection coefficient of stubl. The program extracts 811magnitude and phase from the 2nd and 3rd columns of the file.

9. stub2.s2p: This file represents the reflection coefficient of stub2. The program extracts 811magnitude and phase from the 2nd and 3rd columns of the file.

10. stub3.s2p; This file represents the reflection coefficient of stub3. The program extracts 811magnitude and phase from the 2nd and 3rd columns of the file.

11. stub4.s2p: This file represents the reflection coefficient of stub4. The program extracts S 11magnitude and phase from the 2nd and 3rd columns of the file.

All files need to have 91 points. Ifnot, the program # ofpoints needs to be changed in the matlab code. Todo this, look for "freqpoints=9l;" in the beginning of the code. Also, the .s2p files should be in dB/deg.and the headers need to be eliminated (see copy of *.s2p files). The stub..s2p files must be designed so thatthe reflection coefficient doesn't provide a magnitude of 1 or 0 at any frequency.

61

Matlab Code

%% Matlab code: Extraction.m%% Author: Kendall Ching%% University of Hawaii at Manoa%% Date: Nov. 15, 2004 (revised May 3, 2005)%% Program calculates 811, 822, and 812*821 over frequency range from 11%% input files (see readme text for description ofprogram and input files)%%%% Device = FHX35LG (Vgs=-0.2, Vds=3V) wi 3-ohm series resistor and 4000hm%% shunt resistor at gate for stabilization%-------------------------------------------------%Initialize parametersclear allclose allfreqpoints=91 ;s=zeros(freqpoints,3); % matrix for 822 and 811 and 812821 (3 columns)Zo=50;%%-------------------------------------------------%Input data through files (should be in same directory as program)load sparameters.s2pload opn_stub l.s2pload opn_stub2.s2pload ipn_stub l.s2pload ipn_stub2.s2pload ipn_stub3.s2pload ipn_stub4.s2pload stub l.s2pload stub2.s2pload stub3.s2pload stub4.s2p%%-------------------------------------------------% Set Stub-Network Parameters%Gamrna 1gamma1=10."(stub1(:,2)120).*exp(j.*stub1(:,3)./180*pi); %gamrna of stub 1gamrnar1=real(gamrna1);gamrnai1=imag(gamma1);% Gamma 2gamrna2=10."«stub2(:,2))/20).*exp(j.*stub2(:,3).I180*pi); %gamma of stub 2gamrnar2=real(gamrna2);gamrnai2=imag(gamma2);%Gamrna3gamrna3=10."«stub3(:,2))/20).*exp(j.*stub3(:,3).1180*pi); %gamrna of stub 3gamrnar3=real(gamrna3);gamrnai3=imag(gamrna3);% Gamrna4gamrna4=1 O."«stub4( :,2))/20).*exp(j.*stub4(:,3)./180*pi); %gamrna of stub 4gamrnar4=real(gamma4);gamrnai4=imag(gamrna4);%

62

%--------------------------------------------------% Initial Power Calculations (50 Ohm)%output power ofdevice without matching in dBmpoutO_dBm=sparameters(:,4);%convert power into WartspoutO=O.001*10."(poutO_dBm!10);%%--------------------------------------------------% S22 Calculationpout=O.OOl *10."(opn_stub1(:,4)/10); %power with Gamma 1A=(1-abs(gamma1 )."2).*poutO.lpout;pout=O.OOl *10."(opn_stub2(:,4)/10); %power with Gamma 2B=(1-abs(gamma2)."2).*poutO.lpout;s22r1=(2*gammail.*gammai2."5.*gammar1."2+(-1+A).*gammai2."6 ...

.*gammar1."2+gammai1."6.*gammar2."2-B.*gammai1."6 ...

.*gammar2."2 + 3.*gammai1."4.*gammar1."2.*gammar2."2 ...-3.*B.*gammai1."4.*gammar1."2.*gammar2."2 ...+ 4.*gammai1.*gammai2."3.*gammar1."2.*gammar2."2 ...+ 3.*gammai1."2. *gammar1."4.*gammar2."2 ...- 3.*B.*gammai1."2.*gammar1."4.*gammar2."2 ...+gammar1."6.*gammar2."2 - B.*gammar1."6.*gammar2."2 ...- 2.*gammai1."4.*gammar1.*gammar2."3 + .

2.*B.*gammai1."4.*gammarl.*gamrnar2."3 .- 4.*gammai1."2.*gammar1."3.*gammar2."3 + .

4.*B.*gamrnai1."2.*gammar1."3.*gammar2."3 .- 2.*gamrnar1."5.*gammar2."3 + 2.*B.*gamrnarl."5.*gamrnar2."3 ...+ gammail."4.*gammar2."4 + A.*gamrnail."4.*gamrnar2."4 ...+ gamrnail."2.*gammarl."2.*gammar2."4 ...+ 2.*A.*gammail."2.*gamrnarl."2.*gamrnar2."4 ...- B.*gammail."2.*gammarl."2.*gammar2."4 ...+ A.*gamrnar1."4.*gammar2."4 - B.*gamrnarl."4.*gammar2."4 ...- 2.*A.*gamrnail."2.*gammarl.*gamrnar2."5 + ...

2.*gamrnar1."3.*gamrnar2."5 - 2.*A.*gammarl."3.*gammar2."5 ...- gammarl."2.*gammar2."6 + A.*gammarl."2.*gammar2."6 ...+ gammai2."2.*gamrnar2.*(2.*B.*gamrnarl.*(gammai1."2 + gamrnarl."2).* ...

(gamrnail."2 + gammarl.*(gamrnar1 - gamrnar2» + ...«1 + A).*gamrnail."4 + 2.*A.*gamrnail."2.*gammar1.*(gammarl - 2.*gammar2) + ...

(-1 + A).*gammar1."2.*(gamrnarl ...- 3.*gamrnar2).*(gammar1 - gammar2».*gammar2) ...

+gammai2."4.*gamrnarl.*(-«1 + B).*gamrnar1.*(gammai1."2 + gamrnarl."2» ...-2.*(A.*gamrnail."2 + (-1 + A).*gammarl."2).*gammar2 ...

+ 3.*(-1 + A). *gammar1.*gamrnar2."2) ...+ 2.*gamrnail.*gamrnai2.*gammar2."2.*(-(gamrnail."2 + gamrnarl."2)."2 ...

+ gamrnarl."2. *gammar2."2) -gamrnail."2. *gammai2 ....*sqrt(-«gammai2."2.*gamrnar1 - (gamrnail."2 + gammarl.*(gammarl ...

- gamrnar2».*gamrnar2)."2.*«-1 + B)."2.*gammail."4 + 4.*(-1 ...+ B).*gammail."3.*gamrnai2 +(-1 + A)."2.*gammai2."4 ...+ (-1 + B)."2.*gammarl."4 + 4.*(-1 + B).*gamrnarl."3.*gamrnar2 ...

-2.*(-3 + A + B + A.*B).*gamrnarl."2.*gammar2."2 + 4.*(-1 ..+ A).*gamrnarl.*gammar2."3 + (-1 + A)."2.*gammar2."4 .+ 4.*gammail.*gammai2.*«-1 + A).*gammai2."2 + (-1 + B).*gammarl."2 + ...

2.*gamrnarl.*gamrnar2 + (-1 + A).*gammar2."2) + 2.*gammai2."2.* ...(-«-1 + A + B + A.*B).*gamrnarl."2) + 2.*(-1 + A).* ...

63

gammarl.*gammar2 + ...(-1 + A).1\2.*gammar2.1\2) + 2.*gammail.I\2.*(-«-3 + A ...

+ B + A*B).*gammai2.1\2) + (-1 + B).1\2.*gammarl.I\2 + 2.*(-1 ...+ B).*gammarl.*gammar2 - (-1 + A + B + A*B).*gammar2.1\2)))) .

+ gammail.*gammai2.1\2.*sqrt(-«gammai2.1\2.*gammarl - (gammail.1\2 .+ gammarl.*(gammarl - gammar2)).*gammar2).1\2.* ...

«-1 + B).1\2.*gammail.I\4 + 4.*(-1 + B).*gammail.I\3.*gammai2 ...+ (-1 + A).1\2.*gammai2.1\4 + ...

(-1 + B).1\2.*gammarl.I\4 + 4.*(-1 + B).*gammarl.I\3.*gammar2 ...- 2.*(-3 + A + B + A*B).*gammarl.I\2.* ...

gammar2.1\2 + 4.*(-1 + A). *gammar1.*gammar2.1\3 ...+ (-1 + A).1\2.*gammar2.1\4 + ...

4.*gammail.*gammai2.*«-1 + A).*gammai2.1\2 + (-1 + B).*gammarl.I\2 ...+ 2.*gammarl.*gammar2 + ...

(-1 + A).*gammar2.1\2) ...+ 2.*gammai2.1\2.*(-«-1 + A + B + A*B).*gammarl.I\2) + ...

2.*(-1 + A). *gammarl.*gammar2 + (-1 + A).1\2.*gammar2.1\2) + ...2.*gammail.I\2.*(-«-3 + A + B + A*B).*gammai2.1\2) ...

+ (-1 + B).1\2.*gammarl.I\2 + ...2.*(-1 + B).*gammarl.*gammar2 - (-1 + A + B ...

+ A*B).*gammar2.1\2)))) - ...gammai2.*gammarl.I\2.*sqrt(-«gammai2.1\2.*gammarl - (gammail.I\2 ...

+ gammarl.*(gammarl - gammar2)).* .gammar2).1\2.*«-1 + B).1\2.*gammail.I\4 + 4.*(-1 .

+ B).*gammail.I\3.*gammai2 + ...(-1 + A).A2. *gammai2.1\4 + (-1 + B).1\2. *gammarl.l\4 ..

+ 4.*(-1 + B).*gammarl.I\3.*gammar2 - ..2.*(-3 + A + B + A *B).*gammarl.I\2.*gammar2.1\2 + 4.*(-1 .

+ A).*gammarl.*gammar2.1\3 + ..(-1 + A).1\2.*gammar2.1\4 + 4.*gammail.*gammai2.*«-1 .

+ A).*gammai2.1\2 + (-1 + B).*gammarl.I\2 + ..2.*gammar1.*gammar2 + (-1 + A).*gammar2.1\2) + 2.*gammai2.1\2.* ...

(-«-1 + A + B + A*B).*gammarl.I\2) + 2.*(-1 + A).*gammarl.*gammar2 + ...(-1 + A).1\2.*gammar2.1\2) + 2.*gammail.I\2.*(-«-3 + A + B ..

+ A*B).*gammai2.1\2) + (-1 + B).1\2.*gammarl.I\2 + 2.*(-1 .+ B).*gammarl.*gammar2 - (-1 + A + B + A*B).* ..

gammar2.1\2)))) + gammail.*gammar2.1\2.* ...sqrt(-«gammai2.1\2.*gammarl - (gammail.l\2 + gammarl.*(gammarl ...

- gammar2)).*gammar2).1\2.* ...«-1 + B).1\2.*gammai1.1\4 + 4.*(-1 + B).*gammail.I\3.*gammai2 ...

+ (-1 + A).1\2.*gammai2.1\4 + ...(-1 + B).1\2.*gammarl.I\4+4.*(-1 + B).*gammarl.I\3.*gammar2 - 2.*(-3 ...

+ A + B + A*B).*gammarl.I\2.* ...gammar2.1\2 + 4.*(-1 + A). *gammarl.*gammar2.1\3 ...

+ (-1 + A).1\2.*gammar2.1\4 + ...4.*gammail.*gammai2.*«-1 + A).*gammai2.1\2 + (-1 ...

+ B).*gammarl.I\2 + 2.*gammarl.*gammar2 + .(-1 + A).*gammar2.1\2) + 2.*gammai2.1\2.*(-«-1 + A + B .

+A *B).*gammarl.I\2) + '"2.*(-1 + A).*gammarl.*gammar2 + (-1 + A).1\2.*gammar2.1\2) + ...

2.*gammail.I\2. *(-«-3 + A + B + A. *B).*gammai2.1\2) ...+ (-1 + B).1\2.*gammarl.I\2 + ...

2.*(-1 + B).*gammarl.*gammar2 - (-1 + A + B ...

64

+ A.*B).*gammar2/\2))))).1 ...(2.*(gammail/\2 + gammarLA2).*«gammail - gammai2)."2 ...

+ (gammarl - gammar2)."2).* .(-(gammai2."2.*gammarl) + (gammail."2 + gammarl.*(gammarl .

- gammar2)).*gammar2).* ...(gammai2."2 + gammar2."2));

s22il=(gammai2.*(gammail.*(gammail-gammai2).*«-1 + B).*gammail."2 ...- (-1 + A).*gammai2."2) - ...

(-2.*(-1 + B).*gammail."2 + (-1 + B).*gammail.*gammai2 '"+ (1 + A).*gammai2."2).*gammar1."2 + ...

(-1 + B).*gammarl."4) + 2.*gammai2.*gammarl.*(gammail.*(gammail ...+ gammai2) + gammarl."2).* ...

gammar2 - (gammail.*«1 + B).*gammail."2 + (-1 ...+ A).*gammail.*gammai2 - ...

2.*(-1 + A).*gammai2."2) + (gammail + B.*gammail ...+ gammai2 + A.*gammai2).*gammarl."2).* .

gammar2."2 + 2.*gammail.*gammarl.*gammar2."3 + (-1 .+ A).*gammail.*gammar2."4 - ...

sqrt(-«gammai2."2.*gammarl - (gammail."2 + gammarl.*(gammarl ...- gammar2)).*gammar2)."2.* ...

«-1 + B)."2.*gammail."4 + 4.*(-1 + B).*gammail."3.*gammai2 ...+ (-1 + A)."2.*gammai2."4 + ...

(-1 + B)."2.*gammarl."4 + 4.*(-1 + B).*gammarl."3.*gammar2 ...- 2.*(-3 + A + B + A.*B).*gammarl."2.* ...

gammar2."2 + 4.*(-1 + A).*gammarl.*gammar2."3 + (-1 ...+ A)."2.*gammar2."4 + ...

4.*gammail.*gammai2.*«-1 + A).*gammai2."2 + (-1 ...+ B).*gammarl."2 + 2.*gammarl.*gammar2 + .

(-1 + A).*gammar2."2) + 2.*gammai2."2.*(-«-1 + A + B ..+ A.*B).*gammarl."2) + ...

2.*(-1 + A).*gammar1.*gammar2 + (-1 ...+ A)."2.*gammar2."2) + ...

2.*gammail."2.*(-«-3 + A + B + A.*B).*gammai2."2) ...+ (-1 + B)."2.*gammarl."2 + ...

2.*(-1 +B).*gammarl.*gammar2-(-1 +A+B .+ A.*B).*gammar2."2))))).1 .

(2.*(gammail."2 + gammarl."2).*«gammail - gammai2)."2 ...+ (gammarl - gammar2)."2).* ...

(gammai2."2 + gammar2."2));s22r2=(2.*gammail.*gammai2."5.*gammarl."2 + (-1 + A).*gammai2."6.*gammar1."2 ...

+ gammail."6.*gammar2."2 - ...B.*gammail."6.*gammar2."2 + 3.*gammail."4.*gammar1."2.*gammar2."2 - ...3.*B.*gammail."4.*gammarl."2.*gammar2."2 ...

+ 4.*gammail.*gammai2."3.*gammarl."2.*gammar2."2 + ...3.*gammail."2.*gammarl."4.*gammar2."2 ...

- 3.*B.*gammail."2.*gammarl."4.*gammar2."2 + ...gammarl."6.*gammar2."2 - B.*gammarl."6.*gammar2."2 ...- 2.*gammail."4.*gammarl.*gammar2."3 + ..

2.*B.*gammail."4.*gammarl.*gammar2."3 .- 4.*gammail."2.*gammarl."3.*gammar2."3 + ...

4.*B.*gammai1."2.*gammarl."3.*gammar2."3 - 2.*gammarl."5.*gammar2."3 ...+ 2.*B.*gammarl."5.*gammar2."3 + ...

gammail."4.*gammar2."4 + A.*gammail."4.*gammar2."4 ...

65

+ gammai1."2.*gammar1."2.*gammar2."4 + .2.*A *gammail."2.*gammar1."2.*gammar2."4 .

- B.*gammail."2.*gammarl."2.*gammar2."4 + .A*gammar1."4.*gammar2."4 - B.*gammar1."4.*gammar2."4 ...

- 2.*A*gammai1."2.*gammar1.*gammar2."5 + ...2.*gammar1."3.*gammar2."5 - 2.*A*gammar1."3.*gammar2."5 .. ,

- gammar1."2.*gammar2."6 +A*gammar1."2.*gammar2."6 '"+ gammai2."2.*gammar2.*(2.*B.*gammar1.*(gammai1."2 ...

+ gammar1."2).*(gammai1."2 + gammar1.*(gammarl - gammar2» + ...«1 + A).*gammai1."4 + 2.*A*gammai1."2.*gammar1.*(gammar1 ...

- 2.*gammar2) + (-1 + A). *gammar1."2.*(gammar1 - 3.*gammar2).*(gammarl ...- gammar2».*gammar2) + gammai2."4.*gammar1.*(-«1 ...+ B).*gammar1.*(gammai1."2 + gammar1."2» - ...

2.*(A*gammai1."2 + (-1 + A).*gammar1."2).*gammar2 + 3.*(-1 ...+ A).*gammar1.*gammar2."2) + ...

2.*gammai1.*gammai2.*gammar2."2.*(-(gammai1."2 + gammar1."2)."2 ...+ gammar1."2.*gammar2."2) + ...

gammai1."2.*gammai2.*sqrt(-«gammai2."2.*gammar1 - (gammai1."2 ...+ gammar1.*(gammar1 - gammar2».* ...

gammar2)."2.*«-1 + B)."2.*gammai1."4 + 4.*(-1 ...+ B).*gammai1."3.*gammai2 + ...

(-1 + A)."2.*gammai2."4 + (-1 + B)."2.*gammar1."4 + 4.*(-1 ...+ B).*gammar1."3.*gammar2 - ...

2.*(-3 + A + B + A*B).*gammar1."2.*gammar2."2 + 4.*(-1 ...+ A).*gammar1.*gammar2."3 + ...

(-1 + A)."2.*gammar2."4 + 4.*gammai1.*gammai2. *«-1 .+ A).*gammai2."2 + (-1 + B).*gammar1."2 + .

2.*gammar1.*gammar2 + (-1 + A).*gammar2."2) + 2.*gammai2."2.* ...(-«-1 + A + B + A*B).*gammar1."2) + 2.*(-1 + A).*gammar1.*gammar2 + ...

(-1 + A)."2.*gammar2."2) + 2.*gammai1."2.*(-«-3 + A + B ...+ A*B).*gammai2."2) + ...

(-1 + B)."2.*gammar1."2 + 2.*(-1 + B).*gammar1.*gammar2 ...- (-1 + A + B + A*B).* ...

gammar2."2»» - gammai1.*gammai2."2.* .. ,sqrt(-«gammai2."2.*gammar1 - (gammai1."2 + gammar1.*(gammar1 ...

- gammar2».*gammar2)."2.* '"«-1 + B)."2.*gammai1."4 + 4.*(-1 + B).*gammai1."3.*gammai2 ...

+ (-1 + A)."2.*gammai2."4 + ...(-1 + B)."2.*gammar1."4 + 4.*(-1 + B).*gammar1."3.*gammar2 .. ,

- 2.*(-3 + A + B + A*B).*gammar1."2.* ...gammar2."2 + 4.*(-1 + A).*gammar1.*gammar2."3 + (-1 ...

+ A)."2.*gammar2."4 + ...4.*gammai1.*gammai2.*«-1 + A).*gammai2."2 + (-1 ...

+ B).*gammar1."2 + 2.*gammar1.*gammar2 + ...(-1 + A).*gammar2."2) + 2.*gammai2."2.*(-«-1 + A + B '"

+ A *B).*gammar1."2) + ...2.*(-1 + A). *gammar1.*gammar2 + (-1 + A)."2.*gammar2."2) + ...

2.*gammai1."2.*(-«-3 + A + B + A*B).*gammai2."2) ...+ (-1 + B)."2.*gammar1."2 + ...

2.*(-1 + B).*gammar1.*gammar2 - (-1 + A + B + A*B).*gammar2."2»» + ...gammai2.*gammar1."2.*sqrt(-«gammai2."2.*gammar1 - (gammai1."2 ...

+ gammar1.*(gammar1 - gammar2».* ...gammar2)."2.*«-1 + B)."2.*gammai1."4 + 4.*(-1 ...

66

+ B).*gammail.I\3.*gammai2 + ...(-1 + A).1\2.*gammai2.1\4 + (-1 + B).1\2.*gammarl.I\4 + 4.*(-1 ...

+ B).*gammarl.I\3.*gammar2 - ...2.*(-3 + A + B + A.*B).*gammarl.I\2.*gammar2.1\2 + 4.*(-1 ...

+ A). *gammar1.*gammar2.1\3 + ...(-1 + A).1\2.*gammar2.1\4 + 4.*gammail.*gammai2.*«-1 ..

+ A).*gammai2.1\2 + (-1 + B).*gammarl.I\2 + .2.*gammarl.*gammar2 + (-1 + A).*gammar2.1\2) + 2.*gammai2.1\2.* ...

(-«-1 + A + B + A.*B).*gammarl.I\2) + 2.*(-1 + A).*gammarl.*gammar2 + ...(-1 + A).1\2.*gammar2.1\2) + 2.*gammail."2.*(-«-3 + A + B .

+ A.*B).*gammai2.1\2) + (-1 + B).1\2.*gammarl.I\2 + 2.*(-1 .+ B).*gammarl.*gammar2 - (-1 + A + B + A.*B).* ...

gammar2.1\2))))- gammail.*gammar2.1\2.* ...sqrt(-«gammai2.1\2.*gammar1 - (gammail.l\2 + gammarl.*(gammar1 ...

- gammar2)).*gammar2).1\2.* ...«-1 + B).1\2.*gammail.I\4 + 4.*(-1 + B).*gammail.I\3.*gammai2 ...

+ (-1 + A).1\2.*gammai2.1\4 + ...(-1 + B).1\2.*gammarl.I\4 + 4.*(-1 + B).*gammarl.I\3.*gammar2 ...

- 2.*(-3 + A + B + A.*B).*gammarl.I\2.* ...gammar2.1\2 + 4.*(-1 + A).*gammarl.*gammar2.1\3 + (-1 ...

+ A).1\2.*gammar2.1\4 + ...4.*gammail.*gammai2.*«-1 + A).*gammai2.1\2 + (-1 + B).*gammarl.I\2 ...

+ 2.*gammar1.*gammar2 + ...(-1 + A).*gammar2.1\2) + 2.*gammai2.1\2.*(-«-1 + A + B '"

+ A.*B).*gammarl.I\2) + ...2.*(-1 + A). *gammarl.*gammar2 + (-1 + A).1\2. *gammar2.1\2) + ...

2.*gammail.I\2.*(-«-3 + A + B + A.*B).*gammai2.1\2) + (-1 ...+ B).1\2.*gammarl.I\2 + ...

2.*(-1 + B).*gammarl.*gammar2 - (-1 + A + B ...+ A. *B).*gammar2.1\2)))))./ ...

(2.*(gammail.I\2 + gammarl.I\2).*«gammai1 - gammai2).1\2 + (gammarl ...- gammar2).1\2).* ...

(-(gammai2.1\2.*gammarl) + (gammail.l\2 + gammarl.*(gammarl ...- gammar2)).*gammar2).* ...

(gammai2.1\2 + gammar2.1\2));s22i2=(gammai2.*(gammail.*(gammai1 - gammai2).*«-1 + B).*gammail.I\2 ...

- (-1 + A). *gammai2.1\2) - ...(-2.*(-1 + B).*gammail.I\2 + (-1 + B).*gammai1.*gammai2 + (1 ...

+ A).*gammai2.1\2).*gammarl.I\2 + ...(-1 + B).*gammarl.I\4) + 2.*gammai2.*gammarl.*(gammail.*(gammai1 ...

+ gammai2) + gammarl.I\2).* ...gammar2 - (gammail.*«1 + B).*gammail.I\2 + (-1 ...

+ A).*gammail.*gammai2 - ...2.*(-1 + A).*gammai2.1\2) + (gammail + B.*gammail + gammai2 ...

+ A.*gammai2).*gammarl.I\2).* ...gammar2.1\2 + 2.*gammail.*gammarl.*gammar2.1\3 + (-1 ...

+ A). *gammail.*gammar2.1\4 + ...sqrt(-«gammai2.1\2.*gammar1 - (gammail.l\2 + gammarl.*(gammarl ...

- gammar2)).*gammar2).1\2.* ...«-1 + B).1\2.*gammail.I\4 + 4.*(-1 + B).*gammail.I\3.*gammai2 + (-1 ...

+ A).1\2.*gammai2.1\4 + ...(-1 + B).1\2.*gammarl.I\4 + 4.*(-1 + B).*gammarl.I\3.*gammar2 ...

- 2.*(-3 + A + B + A.*B).*gammar1.1\2.* ...

67

gammar2.A2 + 4.*(-1 + A).*gammarl.*gammar2.A3 + (-1 ...+ A).A2.*gammar2.A4 + ...

4.*gammail.*gammai2.*«-1 + A).*gammai2.A2 + (-1 + B).*gammarl.A2 ...+ 2.*gammarl.*gammar2 + ...

(-1 + A).*gammar2.A2) + 2.*gammai2.A2.*(-«-1 + A + B ...+ A*B).*gammarl.A2) + ...

2.*(-1 + A).*gammarl.*gammar2 + (-1 + A).A2.*gammar2.A2) + ...2.*gammail.A2.*(-«-3 + A + B + A*B).*gammai2.A2) ...

+ (-1 + B).A2.*gammarl.A2 + ...2.*(-1 + B).*gammarl.*gammar2 - (-1 + A + B .

+ A*B).*gammar2.A2»»).I .(2.*(gammail.A2 + gammarl.A2).*«gammail - gammai2).A2 ...

+ (gammarl - gammar2).A2).* ...(gammai2.A2 + gammar2.A2»;

s22a=s22r1+j*s22i1;s22b=s22r2+j*s22i2;for n=1:freqpointsif abs(s22a(n)»abs(s22b(n»

s(n,2)=s22b(n);elses(n,2)=s22a(n);

endenddisp('IS221 is')s22_db=20*log1O(abs(s(:,2»)disp('Angle(S22) is')s22_ang=angle(s(:,2»/pi*180%%-------------------------------------------------% S11 Calculation%% Initialize gammalpout=O.OOI *10.A(ipn_stubl(:,4)/10); %power with Gamma 1A=(l-abs(gammal).A2).*poutO.lpout;pout=O.OOI *1O.A(ipn_stub2(:,4)/1O); %power with Gamma 2B=(I-abs(gamma2).A2).*poutO.lpout;sI1rl=(2*gammail.*gammai2.A5.*gammarl.A2+(-1 +A).*gammai2.A6 ...

.*gammarl.A2+gammail.A6.*gammar2.A2-B.*gammail.A6 ...

.*gammar2.A2 + 3.*gammail.A4.*gammarl.A2.*gammar2.A2 ...-3.*B.*gammail.A4.*gammarl.A2.*gammar2.A2 ...+ 4.*gammail.*gammai2.A3.*gammarl.A2.*gammar2.A2 ...+ 3.*gammail.A2.*gammarl.A4.*gammar2.A2 ...- 3.*B.*gammail.A2.*gammarl.A4.*gammar2.A2 ...+gammarl.A6.*gammar2.A2 - R*gammarl.A6.*gammar2.A2 ...- 2.*gammail.A4.*gammarl.*gammar2.A3 + ..

2.*B.*gammail.A4. *gammarl.*gammar2.A3 .- 4.*gammail.A2.*gammarl.A3.*gammar2.A3 + ..

4.*B.*gammail.A2. *gammarl.A3.*gammar2.A3 .- 2.*gammarl.A5.*gammar2.A3 + 2.*B.*gammarl.A5.*gammar2.A3 ...+ gammail.A4.*gammar2.A4 + A*gammail.A4.*gammar2.A4 ...+ gammail.A2.*gammarl.A2.*gammar2.A4 ...+ 2.*A*gammail.A2.*gammarl.A2.*gammar2.A4 ...

68

- B.*gammai1."2.*gammar1."2.*gammar2."4 ...+ A.*gammar1."4.*gammar2."4 - B.*gammar1."4.*gammar2."4 ...- 2.*A.*gammai1."2.*gammar1.*gammar2."5 + ...

2.*gammar1."3.*gammar2."5 - 2.*A.*gammar1."3.*gammar2."5 ...-gammar1."2.*gammar2.A 6 + A. *gammar1."2.*gammar2.A 6 ...+ gammai2."2.*gammar2.*(2.*B.*gammar1.*(gammai1."2 + gammar1."2).* ...

(gammai1."2 + gammar1.*(gammar1 - gammar2» + ...«(1 + A).*gammai1."4 + 2.*A.*gammai1."2.*gammar1.*(gammarl - 2.*gammar2) + ...

(-1 + A).*gammar1."2.*(gammarl ...- 3. *gammar2). *(gammar1 - gammar2».*gammar2) ...

+gammai2."4.*gammarl.*(-«1 + B).*gammarl.*(gammail."2 + gammarl."2» ...-2.*(A.*gammai1."2 + (-1 + A).*gammar1."2).*gammar2 ...

+ 3.*(-1 + A).*gammar1.*gammar2."2) ...+ 2.*gammai1.*gammai2.*gammar2.A 2.*(-(gammai1."2 + gammar1.A 2)."2 '"

+ gammar1."2.*gammar2."2) -gammai1."2.*gammai2 ....*sqrt(-«gammai2.A 2.*gammarl - (gammai1."2 + gammar1.*(gammar1 ...

- gammar2».*gammar2)."2.*«-1 + B)."2.*gammai1."4 + 4.*(-1 ...+ B).*gammai1."3.*gammai2 +(-1 + A)."2.*gammai2."4 ...+ (-1 + B)."2.*gammar1."4 + 4.*(-1 + B).*gammar1."3.*gammar2 ...

-2.*(-3 + A + B + A.*B).*gammar1.A 2.*gammar2."2 + 4.*(-1 .+ A).*gammar1.*gammar2."3 + (-1 + A)."2.*gammar2."4 .+ 4.*gammai1.*gammai2.*«-1 + A).*gammai2."2 + (-1 + B).*gammar1."2 + ...

2.*gammar1.*gammar2 + (-1 + A).*gammar2."2) + 2.*gammai2."2.* ...(-«-1 + A + B + A.*B).*gammar1."2) + 2.*(-1 + A).* ...

gammar1.*gammar2 + ...(-1 + A)."2.*gammar2."2) + 2.*gammai1."2.*(-«-3 + A ...

+ B + A.*B).*gammai2."2) + (-1 + B)."2.*gammar1."2 + 2.*(-1 ...+ B).*gammar1.*gammar2 - (-1 + A + B + A.*B).*gammar2."2»» ..

+ gammai1.*gammai2."2.*sqrt(-«gammai2."2.*gammar1 - (gammai1."2 .+ gammar1.*(gammarl - gammar2».*gammar2)."2.* ...

«-1 + B)."2.*gammai1."4 + 4.*(-1 + B).*gammai1."3.*gammai2 ...+ (-1 + A)."2.*gammai2."4 + ...

(-1 + B)."2.*gammar1."4 + 4.*(-1 + B).*gammar1."3.*gammar2 ...- 2.*(-3 + A + B + A. *B).*gammar1."2.* ...

gammar2."2 + 4.*(-1 + A).*gammar1. *gammar2."3 ...+ (-1 + A)."2.*gammar2."4 + ...

4.*gammai1.*gammai2.*«-1 + A).*gammai2."2 + (-1 + B).*gammar1."2 '"+ 2.*gammar1.*gammar2 + ...

(-1 + A).*gammar2."2) ...+ 2.*gammai2."2.*(-«-1 + A + B + A.*B).*gammar1."2) + ...

2.*(-1 + A).*gammar1.*gammar2 + (-1 + A)."2.*gammar2."2) + ...2.*gammai1."2. *(-«-3 + A + B + A. *B). *gammai2."2) ...

+ (-1 + B)."2.*gammar1."2 + ...2.*(-1 + B).*gammar1.*gammar2 - (-1 + A + B ...

+ A.*B).*gammar2."2»» - ...gammai2.*gammar1."2.*sqrt(-((gammai2."2.*gammar1 - (gammai1."2 ...

+ gammar1.*(gammarl - gammar2».* ..gammar2)."2.*«-1 + B)."2.*gammai1."4 + 4.*(-1 .

+ B).*gammai1."3.*gammai2 + ...(-1 + A)."2.*gammai2."4 + (-1 + B)."2.*gammar1."4 ..

+ 4.*(-1 + B).*gammar1."3.*gammar2 - .2.*(-3 + A + B + A.*B).*gammar1."2.*gammar2."2 + 4.*(-1 .

+ A).*gammar1.*gammar2."3 + .

69

(-1 + A). A2. *gammar2.A4 + 4.*gammail.*gammai2. *«-1 .+ A).*gammai2.A2 + (-1 + B).*gammarl.A2 + .

2.*gammarl.*gammar2 + (-1 + A).*gammar2.A2) + 2.*gammai2.A2.* ...(-«-1 + A + B + A*B).*gammarl.A2) + 2.*(-1 + A).*gammarl.*gammar2 + ...

(-1 + A).A2.*gammar2.A2) + 2.*gammail.A2.*(-«-3 + A + B .+ A*B).*gammai2.A2) + (-1 + B).A2.*gammarl.A2 + 2.*(-1 ..

+ B).*gammarl.*gammar2 - (-1 + A + B + A.*B).* .gammar2.A2)))) + gammai1.*gammar2.A2.* ...

sqrt(-«gammai2.A2.*gammarl - (gammail.A2 + gammarl.*(gammarl ...- gammar2)).*gammar2).A2.* ...

«-1 + B).A2.*gammail.A4 + 4.*(-1 + B).*gammail.A3.*gammai2 ...+ (-1 + A).A2.*gammai2.A4 + ...

(-1 + B).A2.*gammarl.A4+4.*(-1 + B).*gammarl.A3.*gammar2 - 2.*(-3 ...+ A + B + A*B).*gammarl.A2.* ...

gammar2.A2 + 4.*(-1 + A).*gammarl.*gammar2.A3 ...+ (-1 + A).A2.*gammar2.A4 + ...

4.*gammail.*gammai2.*«-1 + A).*gammai2.A2 + (-1 ...+ B).*gammarl.A2 + 2.*gammarl.*gammar2 + ..

(-1 + A).*gammar2.A2) + 2.*gammai2.A2.*(-«-1 + A + B .+ A *B).*gammarl.A2) + '"

2.*(-1 + A).*gammarl.*gammar2 + (-1 + A).A2.*gammar2.A2) + ...2.*gammail.A2.*(-«-3 + A + B + A*B).*gammai2.A2) ...

+ (-1 + B).A2.*gammarl.A2 + ...2.*(-1 + B).*gammar1.*gammar2 - (-1 + A + B .

+ A *B).*gammar2.A2))))).I .(2.*(gammail.A2 + gammarl.A2).*«gammail - gammai2).A2 ...

+ (gammarl - gammar2).A2).* .(-(gammai2.A2.*gammarl) + (gammail.A2 + gammarl.*(gammarl .

- gammar2)).*gammar2).* ...(gammai2.A2 + gammar2.A2));

sllil=(gammai2.*(gammail.*(gammail-gammai2).*«-1 + B).*gammail.A2 ...- (-1 + A).*gammai2.A2) - ...

(-2.*(-1 + B).*gammail.A2 + (-1 + B).*gammail.*gammai2 ...+ (1 + A).*gammai2.A2).*gammarl.A2 + ...

(-1 + B).*gammar1.A4) + 2.*gammai2.*gammarl.*(gammail.*(gammail ...+ gammai2) + gammarl.A2).* ...

gammar2 - (gammail.*«1 + B).*gammail.A2 + (-1 ...+ A).*gammail.*gammai2 - ...

2.*(-1 + A).*gammai2.A2) + (gammail + B.*gammail ...+ gammai2 + A *gammai2).*gammarl.A2).* .

gammar2.A2 + 2.*gammai1.*gammar1.*gammar2.A3 + (-1 .+ A). *gammail.*gammar2.A4 - ...

sqrt(-«gammai2.A2.*gammarl - (gammail.A2 + gammarl.*(gammarl ...- gammar2)).*gammar2).A2.* ...

«-1 + B).A2.*gammail.A4 + 4.*(-1 + B).*gammaiI.A3.*gammai2 ...+ (-1 + A).A2.*gammai2.A4 + ...

(-1 + B).A2.*gammarl.A4 + 4.*(-1 + B).*gammarl.A3.*gammar2 ...- 2.*(-3 + A + B + A*B).*gammarl.A2.* ...

gammar2.A2 + 4.*(-1 + A). *gammarl.*gammar2.A3 + (-1 ...+ A).A2.*gammar2.A4 + ...

4.*gammail.*gammai2.*«-1 + A).*gammai2.A2 + (-1 ...+ B).*gammarl.A2 + 2.*gammarl.*gammar2 + ...

(-1 + A).*gammar2.A2) + 2.*gammai2.A2.*(-«-1 + A + B '"

70

+ A *B). *gammar1.A2) + ...2.*(-1 + A).*gammar1.*gammar2 + (-1 ...

+ A).A2.*gammar2.A2) + ...2.*gammai1.A2.*(-«-3 + A + B + A*B).*gammai2.A2) ...

+ (-1 + B).A2.*gammar1.A2 + ...2.*(-1 + B).*gammar1.*gammar2 - (-1 + A + B .

+ A *B).*gammar2.A2»»).1 .(2.*(gammai1.A2 + gammar1.A2).*«gammai1 - gammai2).A2 ...

+ (gammar1 - gammar2).A2).* ...(gammai2.A2 + gammar2.A2»;

sllr2=(2.*gammai1.*gammai2.A5.*gammar1.A2 + (-1 + A).*gammai2.A6.*gammar1.A2 ...+ gammai1.A6. *gammar2.A2 - ...

B.*gammai1.A6.*gammar2.A2 + 3.*gammai1.A4.*gammar1.A2.*gammar2.A2 - ...3.*B.*gammai1.A4.*gammar1.A2.*gammar2.A2 ...+ 4.*gammai1.*gammai2.A3.*gammar1.A2.*gammar2.A2 + ...

3.*gammai1.A2.*gammarl.A4.*gammar2.A2 ...- 3.*B.*gammai1.A2.*gammar1.A4.*gammar2.A2 + ...

gammar1.A6.*gammar2.A2 - B.*gammar1.A6.*gammar2.A2 ...- 2.*gammai1.A4. *gammar1.*gammar2.A3 + .

2.*B.*gammai1.A4.*gammar1.*gammar2.A3 .- 4.*gammai1.A2.*gammar1.A3.*gammar2.A3 + ...

4.*B.*gammai1.A2.*gammar1.A3.*gammar2.A3 - 2.*gammar1.A5.*gammar2.A3 ...+ 2.*B.*gammar1.A5.*gammar2.A3 + ...

gammai1.A4.*gammar2.A4 + A *gammai1.A4.*gammar2.A4 ...+ gammai1.A2.*gammar1.A2.*gammar2.A4 + .

2.*A *gammai1.A2. *gammar1.A2. *gammar2.A4 .- B.*gammai1.A2.*gammar1.A2.*gammar2.A4 + .

A*gammar1.A4.*gammar2.A4 - B.*gammar1.A4.*gammar2.A4 ...- 2.*A*gammai1.A2.*gammar1.*gammar2.A5 + ...

2.*gammar1.A3.*gammar2.A5 - 2.*A*gammar1.A3.*gammar2.A5 ...- gammarl.A2.*gammar2.A6 +A*gammarl.A2.*gammar2.A6 .+ gammai2.A2.*gammar2.*(2.*B.*gammarl.*(gammail.A2 .

+ gammar1.A2).*(gammai1.A2 + gammar1.*(gammar1 - gammar2» + ...«1 + A).*gammail.A4 + 2.*A*gammail.A2.*gammar1.*(gammarl ...

- 2.*gammar2) + (-1 + A).*gammar1.A2.*(gammar1 - 3.*gammar2).*(gammar1 ...- gammar2».*gammar2) + gammai2.A4.*gammarl.*(-«1 ...+ B).*gammarl.*(gammai1.A2 + gammarl.A2» - ...

2.*(A*gammai1.A2 + (-1 + A).*gammar1.A2).*gammar2 + 3.*(-1 ...+ A).*gammar1.*gammar2.A2) + ...

2.*gammail.*gammai2.*gammar2.A2.*(-(gammail.A2 + gammarl.A2).A2 ...+ gammar1.A2.*gammar2.A2) + ...

gammai1.A2.*gammai2.*sqrt(-«gammai2.A2.*gammar1 - (gammai1.A2 ...+ gammar1.*(gammarl - gammar2».* ...

gammar2).A2.*«-1 + B).A2.*gammail.A4 + 4.*(-1 ...+ B).*gammai1.A3.*gammai2 + ...

(-1 + A).A2.*gammai2.A4 + (-1 + B).A2.*gammarl.A4 + 4.*(-1 ...+ B).*gammar1.A3.*gammar2 - ...

2.*(-3 + A + B + A *B).*gammar1.A2.*gammar2.A2 + 4.*(-1 ...+ A).*gammar1.*gammar2:"3 + ...

(-1 + A).A2.*gammar2.A4 + 4.*gammail.*gammai2.*«-1 .+ A).*gammai2.A2 + (-1 + B).*gammarl.A2 + .

2.*gammar1.*gammar2 + (-1 + A).*gammar2.A2) + 2.*gammai2.A2.* ...(-«-1 + A + B + A*B).*gammar1.A2) + 2.*(-1 + A).*gammar1.*gammar2 + ...

71

(-1 + A)."2.*gammar2/'2) + 2.*gammai1."2.*(-«-3 + A + B ...+ A*B).*gammai2."2) + ...

(-1 + B)."2.*gammar1."2 + 2.*(-1 + B).*gammar1.*gammar2 ...- (-1 + A + B + A*B).* ...

gammar2."2)))) - gammai1.*gammai2."2.* ...sqrt(-«gammai2."2.*gammar1 - (gammai1."2 + gammar1.*(gammar1 ...

- gammar2)).*gammar2)."2.* ...«-1 + B)."2.*gammai1."4 + 4.*(-1 + B).*gammai1."3.*gammai2 ...

+ (-1 + A)."2.*gammai2."4 + ...(-1 + B)."2.*gammar1."4 + 4.*(-1 + B).*gammar1."3.*gammar2 ...

- 2.*(-3 + A + B + A*B).*gammar1."2.* ...gammar2."2 + 4.*(-1 + A).*gammar1.*gammar2."3 + (-1 ...

+ A)."2.*gammar2."4 + ...4.*gammai1.*gammai2.*«-1 + A).*gammai2."2 + (-1 ...

+ B).*gammar1."2 + 2.*gammar1.*gammar2 + .(-1 + A).*gammar2."2) + 2.*gammai2."2.*(-«-1 + A + B .

+ A *B).*gammar1."2) + ...2.*(-1 + A). *gammar1.*gammar2 + (-1 + A)."2.*gammar2."2) + ...

2.*gammai1."2.*(-«-3 + A + B + A*B).*gammai2."2) ...+ (-1 + B)."2.*gammar1."2 + ...

2.*(-1 + B).*gammar1.*gammar2 - (-1 + A + B + A.*B).*gammar2."2)))) + ...gammai2.*gammar1."2.*sqrt(-«gammai2."2.*gammarl - (gammai1."2 ...

+ gammar1.*(gammar1 - gammar2)).* ...gammar2)."2.*«-1 + B)."2.*gammai1."4 + 4.*(-1 ...

+ B).*gammai1."3.*gammai2 + ...(-1 + A)."2.*gammai2."4 + (-1 + B)."2.*gammar1."4 + 4.*(-1 ...

+ B).*gammar1."3.*gammar2 - ...2.*(-3 + A + B + A*B).*gammar1."2.*gammar2."2 + 4.*(-1 ...

+ A).*gammar1.*gammar2."3 + ...(-1 + A)."2.*gammar2."4 + 4.*gammai1.*gammai2.*«-1 ..

+ A). *gammai2."2 + (-1 + B).*gammar1."2 + .2.*gammar1.*gammar2 + (-1 + A).*gammar2."2) + 2.*gammai2."2.* ...

(-«-1 + A + B + A*B).*gammar1."2) + 2.*(-1 + A).*gammar1.*gammar2 + ...(-1 + A)."2.*gammar2."2) + 2.*gammai1."2.*(-«-3 + A + B ...+ A*B).*gammai2."2) + (-1 + B)."2.*gammar1."2 + 2.*(-1 .+ B).*gammar1.*gammar2 - (-1 + A + B + A*B).* ...

gammar2."2))))- gammai1.*gammar2."2.* ...sqrt(-«gammai2."2.*gammarl - (gammai1."2 + gammar1.*(gammarl ...

- gammar2)).*gammar2)."2.* ...«-1 + B)."2.*gammai1."4 + 4.*(-1 + B).*gammai1."3.*gammai2 ...

+ (-1 + A)."2.*gammai2."4 + ...(-1 + B)."2.*gammar1."4 + 4.*(-1 + B).*gammar1."3.*gammar2 ...

- 2.*(-3 + A + B + A*B).*gammar1."2.* ...gammar2."2 + 4.*(-1 + A). *gammar1.*gammar2."3 + (-1 ...

+ A)."2.*gammar2."4 + ...4.*gammai1.*gammai2.*«-1 + A).*gammai2."2 + (-1 + B).*gammar1."2 ...

+ 2.*gammar1.*gammar2 + ...(-1 + A).*gammar2."2) + 2.*gammai2."2.*(-«-1 + A + B ...

+ A *B).*gammar1."2) + ...2.*(-1 + A).*gammar1.*gammar2 + (-1 + A)."2.*gammar2."2) + ...

2.*gammai1."2.*(-«-3 + A + B + A*B).*gammai2."2) + (-1 ...+ B)."2.*gammar1."2 + ...

2.*(-1 + B).*gammar1.*gammar2 - (-1 + A + B ...

72

+ A *B).*gammar2."2»»).1 ...(2.*(gammail."2 + gammarl."2).*«gammai1 - gammai2)."2 + (gammar1 ...

- gammar2)."2).* ...(-(gammai2."2.*gammarl) + (gammail."2 + gammarl.*(gammarl ...

- gammar2».*gammar2).* ...(gammai2."2 + gammar2."2»;

slli2=(gammai2.*(gammail.*(gammail - gammai2).*«-1 + B).*gammail."2 ...- (-1 + A).*gammai2."2) - ...

(-2.*(-1 + B).*gammail."2 + (-1 + B).*gammai1.*gammai2 + (1 ...+ A).*gammai2."2).*gammarl."2 + ...

(-1 + B).*gammarl."4) + 2.*gammai2.*gammarl.*(gammai1.*(gammai1 ...+ gammai2) + gammarl."2).* ...

gammar2 - (gammail.*«l + B).*gammail."2 + (-1 ...+ A).*gammail.*gammai2 - ...

2.*(-1 + A).*gammai2."2) + (gammail + B.*gammai1 + gammai2 ...+ A *gammai2).*gammarl."2).* ...

gammar2."2 + 2.*gammail.*gammarl.*gammar2."3 + (-1 ...+ A).*gammail.*gammar2."4 + ...

sqrt(-«gammai2."2.*gammar1 - (gammail."2 + gammarl.*(gammarl ...- gammar2».*gammar2)."2.* ...

«-1 + B)."2.*gammail."4 + 4.*(-1 + B).*gammail."3.*gammai2 + (-1 ...+ A)."2.*gammai2."4 + ...

(-1 + B)."2.*gammar1."4 + 4.*(-1 + B).*gammarl."3.*gammar2 ...- 2.*(-3 + A + B + A *B).*gammarl."2.* ...

gammar2."2 + 4.*(-1 + A).*gammarl.*gammar2."3 + (-1 ...+ A)."2.*gammar2."4 + ...

4.*gammail.*gammai2.*«-1 + A).*gammai2."2 + (-1 + B).*gammarl."2 ...+ 2.*gammar1.*gammar2 + ...

(-1 + A).*gammar2."2) + 2.*gammai2."2.*(-«-1 + A + B ...+ A*B).*gammarl."2) + ...

2.*(-1 + A).*gammarl.*gammar2 + (-1 + A)."2.*gammar2."2) + ...2.*gammail."2. *(-«-3 + A + B + A *B).*gammai2."2) ...

+ (-1 + B)."2.*gammarl."2 + ...2.*(-1 + B).*gammarl.*gammar2 - (-1 + A + B .

+ A *B).*gammar2."2»»).1 .(2.*(gammail."2 + gammarl."2).*«gammai1 - gammai2)."2 ...

+ (gammarl - gammar2)."2).* ...(gammai2."2 + gammar2."2»;

slla=sllr1+j*sllil ;sllb=sllr2+j*slli2;for n=1:freqpointsif abs(slla(n»>abs(sllb(n»

s(n,l)=sllb(n);else

s(n,l)=slla(n);endenddisp('IS 111 is')sll_db=20*loglO(abs(s(:,1»)disp('Angle(S11) is')sll_ang=angle(s(:,1»/pi*180

73

%-------------------------------------------------% 821812 Calculation%% Initialize paramterspout3=O.OOl *10."(ipn_stub3(:,4)./10); %power with Gamma 1A=(poutO./pout3).*«(I-abs(gamma3)."2).*(I-abs(gammal)."2»./(abs(I-(s(:,2).*gammal»."2»;pout4=O.OOl *1O."(ipn_stub4(:,4)./IO); %power with Gamma 2B=(poutO./pout4).*«(I-abs(gamma4)."2).*(I-abs(gammal)."2»./(abs(I-(s(:,2).*gamma1»."2»;sIIri=(2*gammai3.*gammai4."5.*gammar3."2+(-1+A).*gammai4."6 ...

.*gammar3."2+gammai3."6.*gammar4."2-B.*gammai3."6 ...

.*gammar4."2 + 3.*gammai3."4.*gammar3."2.*gammar4."2 ...-3.*B.*gammai3."4.*gammar3."2.*gammar4."2 ...+4.*gammai3.*gammai4."3.*gammar3."2.*gammar4."2 ...+ 3.*gammai3."2.*gammar3."4.*gammar4."2 ...- 3.*B.*gammai3."2.*gammar3."4.*gammar4."2 ...+gammar3."6.*gammar4."2 - B.*gammar3."6.*gammar4."2 ...- 2.*gammai3."4.*gammar3.*gammar4."3 + .

2.*B.*gammai3."4.*gammar3.*gammar4."3 .- 4.*gammai3."2.*gammar3."3.*gammar4."3 + .

4.*B.*gammai3."2. *gammar3."3.*gammar4."3 .- 2.*gammar3."5.*gammar4."3 + 2.*B.*gammar3."5.*gammar4."3 .. ,+ gammai3."4.*gammar4."4 + A.*gammai3."4.*gammar4."4 ...+ gammai3."2.*gammar3."2.*gammar4."4 ...+ 2.*A.*gammai3."2.*gammar3."2.*gammar4."4 ...- B.*gammai3."2.*gammar3."2.*gammar4."4 ...+ A. *gammar3."4.*gammar4."4 - B.*gammar3."4.*gammar4."4 ...- 2.*A.*gammai3."2.*gammar3.*gammar4."5 + ...

2.*gammar3."3.*gammar4."5 - 2.*A.*gammar3."3.*gammar4."5 ...- gammar3."2.*gammar4."6 + A. *gammar3."2.*gammar4."6 ...+ gammai4."2.*gammar4.*(2.*B.*gammar3.*(gammai3."2 + gammar3."2).* ...

(gammai3."2 + gammar3.*(gammar3 - gammar4» + ...«1 + A).*gammai3."4 + 2.*A.*gammai3."2.*gammar3.*(gammar3 - 2.*gammar4) + ...

(-1 + A).*gammar3."2.*(gammar3 ...- 3.*gammar4).*(gammar3 - gammar4».*gammar4) ...

+gammai4."4.*gammar3.*(-«I + B).*gammar3.*(gammai3."2 + gammar3."2» ...-2.*(A.*gammai3."2 + (-1 + A).*gammar3."2).*gammar4 ...

+ 3.*(-1 + A). *gammar3.*gammar4."2) ...+ 2.*gammai3.*gammai4.*gammar4."2.*(-(gammai3."2 + gammar3."2)."2 ...

+ gammar3."2.*gammar4."2) -gammai3."2.*gammai4 ....*sqrt(-«gammai4."2.*gammar3 - (gammai3."2 + gammar3.*(gammar3 ...

- gammar4».*gammar4)."2.*«-I + B)."2.*gammai3."4 + 4.*(-1 ...+ B).*gammai3."3.*gammai4 +(-1 + A)."2.*gammai4."4 ...+ (-1 + B)."2.*gammar3."4 + 4.*(-1 + B).*gammar3."3.*gammar4 ...

-2.*(-3 + A + B + A. *B).*gammar3."2.*gammar4."2 + 4.*(-1 .+ A).*gammar3.*gammar4."3 + (-1 + A)."2.*gammar4."4 ..+ 4.*gammai3.*gammai4.*«-1 + A).*gammai4."2 + (-1 + B).*gammar3."2 + ...

2.*gammar3.*gammar4 + (-1 + A).*gammar4."2) + 2.*gammai4."2.* ...(-«-1 + A + B + A.*B).*gammar3."2) + 2.*(-1 + A).* ...

gammar3.*gammar4 + ...(-1 + A)."2.*gammar4."2) + 2.*gammai3."2.*(-«-3 + A ...

+ B + A.*B).*gammai4."2) + (-1 + B)."2.*gammar3."2 + 2.*(-1 ...+ B).*gammar3.*gammar4 - (-1 + A + B + A. *B).*gammar4."2»» '"

+ gammai3.*gammai4."2.*sqrt(-«gammai4."2.*gammar3 - (gammai3."2 ...

74

+ gammar3.*(gammar3 - gammar4)).*gammar4).A2.* ...«-1 + B).A2.*gammai3.A4 + 4.*(-1 + B).*gammai3.A3.*gammai4 ...

+ (-1 + A).A2.*gammai4.A4 + ...(-1 + B).A2.*gammar3.A4 + 4.*(-1 + B).*gammar3.A3.*gammar4 ...

- 2.*(-3 + A + B + A*B).*gammar3.A2.* ...gammar4.A2 + 4.*(-1 + A). *gammar3.*gammar4.A3 ...

+ (-1 + A).A2.*gammar4.A4 + ...4.*gammai3.*gammai4.*«-1 + A).*gammai4.A2 + (-1 + B).*gammar3.A2 ...

+ 2.*gammar3.*gammar4 + ...(-1 + A).*gammar4.A2) ...

+ 2.*gammai4.A2.*(-«-1 + A + B + A*B).*gammar3.A2) + ...2.*(-1 + A).*gammar3.*gammar4 + (-1 + A).A2.*gammar4.A2) + ...

2.*gammai3.A2.*(-«-3 + A + B + A*B).*gammai4.A2) ...+ (-1 + B).A2.*gammar3.A2 + ...

2.*(-1 + B).*gammar3.*gammar4 - (-1 + A + B ...+ A*B).*gammar4.A2)))) - ...

gammai4.*gammar3.A2.*sqrt(-«gammai4.A2.*gammar3 - (gammai3.A2 .. ,+ gammar3.*(gammar3 - gammar4)).* .. ,

gammar4).A2.*«-1 + B).A2.*gammai3.A4 + 4.*(-1 ...+ B).*gammai3.A3.*gammai4 + ...

(-1 + A).A2.*gammai4.A4 + (-1 + B).A2.*gammar3.A4 .+ 4.*(-1 + B).*gammar3.A3.*gammar4 - .

2.*(-3 + A + B + A *B).*gammar3.A2.*gammar4.A2 + 4.*(-1 .+ A).*gammar3.*gammar4.A3 + .

(-1 + A).A2.*gammar4.A4 + 4.*gammai3.*gammai4.*«-1 ..+ A). *gammai4.A2 + (-1 + B).*gammar3.A2 + .

2.*gammar3.*gammar4 + (-1 + A).*gammar4.A2) + 2.*gammai4.A2.* ...(-«-1 + A + B + A*B).*gammar3.A2) + 2.*(-1 + A). *gammar3.*gammar4 + ...

(-1 + A).A2.*gammar4.A2) + 2.*gammai3.A2.*(-«-3 + A + B ..+ A*B).*gammai4.A2) + (-1 + B).A2.*gammar3.A2 + 2.*(-1 .

+ B).*gammar3.*gammar4 - (-1 + A + B + A*B).* .gammar4.A2)))) + gammai3.*gammar4.A2.* ...

sqrt(-«gammai4.A2.*gammar3 - (gammai3.A2 + gammar3.*(gammar3 ...- gammar4)).*gammar4).A2.* ...

«-1 + B).A2.*gammai3.A4 + 4.*(-1 + B).*gammai3.A3.*gammai4 ...+ (-1 + A).A2.*gammai4.A4 + ...

(-1 + B).A2.*gammar3.A4+4.*(-1 + B).*gammar3.A3.*gammar4 - 2.*(-3 ...+ A + B + A*B).*gammar3.A2.* ...

gammar4.A2 + 4.*(-1 + A).*gammar3.*gammar4.A3 ...+ (-1 + A).A2.*gammar4.A4 + ...

4.*gammai3.*gammai4.*«-1 + A).*gammai4.A2 + (-1 ...+ B).*gammar3.A2 + 2.*gammar3.*gammar4 + .

(-1 + A).*gammar4.A2) + 2.*gammai4.A2.*(-«-1 + A + B .+ A*B).*gammar3.A2) + ...

2.*(-1 + A).*gammar3.*gammar4 + (-1 + A).A2.*gammar4.A2) + ...2.*gammai3.A2.*(-«-3 + A + B + A*B).*gammai4.A2) ...

+ (-1 + B).A2.*gammar3.A2 + ...2.*(-1 + B).*gammar3.*gammar4 - (-1 + A + B ...

+ A *B).*gammar4.A2)))))./ .. ,(2.*(gammai3.A2 + gammar3.A2).*«gammai3 - gammai4).A2 .. ,

+ (gammar3 - gammar4).A2).* ...(-(gammai4.A2.*gammar3) + (gammai3.A2 + gammar3.*(gammar3 '"

- gammar4)).*gammar4).* ...

75

(gammai4."2 + gammar4."2»;slli1=(gammai4.*(gammai3.*(gammai3-gammai4).*«-1 + B).*gammai3."2 ...

- (-1 + A).*gammai4."2) - ...(-2.*(-1 + B).*gammai3."2 + (-1 + B).*gammai3.*gammai4 ...

+ (l + A).*gammai4."2).*gammar3."2 + ...(-1 + B).*gammar3."4) + 2.*gammai4.*gammar3.*(gammai3.*(gammai3 ...

+ gammai4) + gammar3/'2).* ...gammar4 - (gammai3.*«l + B).*gammai3."2 + (-1 ...

+ A).*gammai3.*gammai4 - ...2.*(-1 + A).*gammai4."2) + (gammai3 + B.*gammai3 ...

+ gammai4 + A*gammai4).*gammar3."2).* .gammar4."2 + 2.*gammai3.*gammar3.*gammar4."3 + (-1 .

+ A).*gammai3.*gammar4."4 - ...sqrt(-«gammai4."2.*gammar3 - (gammai3/'2 + gammar3.*(gammar3 ...

- gammar4».*gammar4)/'2.* ...«-1 + B)."2.*gammai3."4 + 4.*(-1 + B).*gammai3."3.*gammai4 ...

+ (-1 + A)."2.*gammai4."4 + ...(-1 + B)."2.*gammar3."4 + 4.*(-1 + B).*gammar3."3.*gammar4 ...

- 2.*(-3 + A + B + A*B).*gammar3."2.* ...gammar4."2 + 4.*(-1 + A).*gammar3.*gammar4."3 + (-1 ...

+ A)."2.*gammar4."4 + ...4.*gammai3.*gammai4.*«-1 + A).*gammai4."2 + (-1 ...

+ B).*gammar3."2 + 2.*gammar3.*gammar4 + .(-1 + A).*gammar4."2) + 2.*gammai4."2.*(-«-1 + A + B .

+ A*B).*gammar3."2) + ...2.*(-1 + A).*gammar3.*gammar4 + (-1 ...

+ A)."2.*gammar4."2) + ...2.*gammai3."2.*(-«-3 + A + B + A*B).*gammai4."2) ...

+ (-1 + B)."2.*gammar3."2 + ...2.*(-1 + B).*gammar3.*gammar4 - (-1 + A + B .

+ A.*B).*gammar4."2»»).I .(2.*(gammai3."2 + gammar3."2).*«gammai3 - gammai4)."2 ...

+ (gammar3 - gammar4)."2).* ...(gammai4."2 + gammar4."2»;

s1lr2=(2.*gammai3.*gammai4."5.*gammar3."2 + (-1 + A).*gammai4."6.*gammar3."2 ...+ gammai3."6.*gammar4."2 - ...

B.*gammai3."6.*gammar4."2 + 3.*gammai3."4.*gammar3."2.*gammar4."2 - ...3.*B.*gammai3."4.*gammar3."2.*gammar4."2 ...+ 4.*gammai3.*gammai4."3.*gammar3."2.*gammar4."2 + ...

3.*gammai3."2.*gammar3."4.*gammar4."2 ...- 3.*B.*gammai3."2.*gammar3."4.*gammar4."2 + ...

gammar3."6.*gammar4."2 - B.*gammar3."6.*gammar4."2 ...- 2.*gammai3."4.*gammar3.*gammar4."3 + ..

2.*B.*gammai3."4.*gammar3.*gammar4."3 .- 4.*gammai3."2.*gammar3."3.*gammar4."3 + ...

4.*B.*gammai3."2.*gammar3."3.*gammar4."3 - 2.*gammar3."5.*gammar4."3 ...+ 2.*B.*gammar3."5.*gammar4."3 + ...

gammai3."4.*gammar4."4 + A*gammai3."4.*gammar4."4 ...+ gammai3."2.*gammar3."2.*gammar4."4 + .

2.*A*gammai3."2.*gammar3."2.*gammar4."4 .- B.*gammai3."2.*gammar3."2.*gammar4."4 + .

A*gammar3."4.*gammar4."4 - B.*gammar3."4.*gammar4."4 ...- 2.*A*gammai3."2.*gammar3.*gammar4."5 + ...

76

2.*gammar3/'3.*gammar4."5 - 2.*A*gammar3."3.*gammar4."5 ...- gammar3."2.*gammar4."6 +A*gammar3."2.*gammar4."6 .+ gammai4."2.*gammar4.*(2. *B.*gammar3.*(gammai3."2 .

+ gammar3."2).*(gammai3."2 + gammar3.*(gammar3 - gammar4)) + ...«1 + A).*gammai3/'4 + 2.*A*gammai3."2.*gammar3.*(gammar3 ...

- 2.*gammar4) + (-1 + A).*gammar3."2.*(gammar3 - 3.*gammar4).*(gammar3 ...- gammar4)).*gammar4) + gammai4."4.*gammar3.*(-«1 ...+ B).*gammar3.*(gammai3."2 + gammar3."2)) - ...

2.*(A*gammai3."2 + (-1 + A).*gammar3."2).*gammar4 + 3.*(-1 ...+ A).*gammar3.*gammar4."2) + ...

2.*gammai3.*gammai4.*gammar4."2.*(-(gammai3."2 + gammar3."2)."2 ...+ gammar3."2.*gammar4."2) + ...

gammai3."2.*gammai4.*sqrt(-«gammai4."2.*gammar3 - (gammai3."2 ...+ gammar3.*(gammar3 - gammar4)).* .. ,

gammar4)."2.*«-1 + B)."2.*gammai3."4 + 4.*(-1 ...+ B).*gammai3."3.*gammai4 + ...

(-1 + A)."2.*gammai4."4 + (-1 + B)."2.*gammar3."4 + 4.*(-1 ...+ B).*gammar3."3.*gammar4 - ...

2.*(-3 + A + B + A*B).*gammar3."2.*gammar4."2 + 4.*(-1 ...+ A).*gammar3.*gammar4."3 + ...

(-1 + A)."2.*gammar4."4 + 4.*gammai3.*gammai4.*«-1 .+ A).*gammai4."2 + (-1 + B).*gammar3."2 + .

2.*gammar3.*gammar4 + (-1 + A).*gammar4."2) + 2.*gammai4."2.* ...(-«-1 + A + B + A*B).*gammar3."2) + 2.*(-1 + A).*gammar3.*gammar4 + ...

(-1 + A)."2.*gammar4."2) + 2.*gammai3."2.*(-«-3 + A + B ...+ A *B).*gammai4."2) + ...

(-1 + B)."2.*gammar3."2 + 2.*(-1 + B).*gammar3.*gammar4 ...- (-1 + A + B + A*B).* ...

gammar4."2)))) - gammai3.*gammai4."2.* ...sqrt(-«gammai4."2.*gammar3 - (gammai3."2 + gammar3.*(gammar3 ...

- gammar4)).*gammar4)."2.* .. ,«-1 + B)."2.*gammai3."4 + 4.*(-1 + B).*gammai3."3.*gammai4 ...

+ (-1 + A)."2.*gammai4."4 + ...(-1 + B)."2.*gammar3."4 + 4.*(-1 + B).*gammar3."3.*gammar4 ...

- 2.*(-3 + A + B + A*B).*gammar3."2.* ...gammar4."2 + 4.*(-1 + A).*gammar3.*gammar4."3 + (-1 ...

+ A)."2.*gammar4."4 + ...4.*gammai3.*gammai4.*«-1 + A).*gammai4."2 + (-1 ...

+ B).*gammar3."2 + 2.*gammar3.*gammar4 + .(-1 + A).*gammar4."2) + 2.*gammai4."2.*(-«-1 + A + B .

+ A*B).*gammar3."2) + ...2.*(-1 + A).*gammar3.*gammar4 + (-1 + A)."2.*gammar4."2) + ...

2.*gammai3."2.*(-«-3 + A + B + A.*B).*gammai4."2) ...+ (-1 + B)."2.*gammar3."2 + ...

2.*(-1 + B).*gammar3.*gammar4 - (-1 + A + B + A*B).*gammar4."2)))) + ...gammai4.*gammar3."2.*sqrt(-«gammai4."2.*gammar3 - (gammai3."2 ...

+ gammar3.*(gammar3 - gammar4)).* ...gammar4)."2.*«-1 + B)."2.*gammai3."4 + 4.*(-1 ...

+ B).*gammai3."3.*gammai4 + ...(-1 + A)."2.*gammai4."4 + (-1 + B)."2.*gammar3."4 + 4.*(-1 ...

+ B).*gammar3."3.*gammar4 - ...2.*(-3 + A + B + A*B).*gammar3."2.*gammar4."2 + 4.*(-1 ...

+ A). *gammar3.*gammar4."3 + ...

77

(-1 + A)."2. *gammar4."4 + 4.*gammai3. *gammai4.*«-1 '"+ A).*gammai4."2 + (-1 + B).*gammar3."2 + ...

2.*gammar3.*gammar4 + (-1 + A).*gammar4."2) + 2.*gammai4."2.* ...(-«-1 + A + B + A*B).*gammar3."2) + 2.*(-1 + A).*gammar3.*gammar4 + ...(-1 + A)."2.*gammar4."2) + 2.*gammai3."2.*(-«-3 + A + B .+ A*B).*gammai4."2) + (-1 + B)."2.*gammar3."2 + 2.*(-1 .+ B).*gammar3.*gammar4 - (-1 + A + B + A*B).* ...

gammar4."2»»- gammai3.*gammar4."2.* ...sqrt(-«gammai4."2.*gammar3 - (gammai3."2 + gammar3.*(gammar3 ...

- gammar4».*gammar4)."2.* ...«-1 + B)."2.*gammai3."4 + 4.*(-1 + B).*gammai3."3.*gammai4 ...

+ (-1 + A)."2. *gammai4."4 + ...(-1 + B)."2.*gammar3."4 + 4.*(-1 + B).*gammar3."3.*gammar4 ...

- 2.*(-3 + A + B + A*B).*gammar3."2.* ...gammar4."2 + 4.*(-1 + A).*gammar3.*gammar4."3 + (-1 ...

+ A)."2.*gammar4."4 + ...4.*gammai3.*gammai4.*«-1 + A).*gammai4."2 + (-1 + B).*gammar3."2 ...

+ 2.*gammar3.*gammar4 + ...(-1 + A).*gammar4."2) + 2.*gammai4."2.*(-«-1 + A + B ...

+ A*B).*gammar3."2) + ...2.*(-1 + A).*gammar3.*gammar4 + (-1 + A)."2.*gammar4."2) + ...

2.*gammai3."2.*(-«-3 + A + B + A*B).*gammai4."2) + (-1 ...+ B)."2.*gammar3."2 + ...

2.*(-1 + B).*gammar3.*gammar4 - (-1 + A + B ...+ A *B).*gammar4."2»»)./ ...

(2.*(gammai3."2 + gammar3."2).*«gammai3 - gammai4)."2 + (gammar3 ...- gammar4)."2).* ...

(-(gammai4."2.*gammar3) + (gammai3."2 + gammar3.*(gammar3 ...- gammar4».*gammar4).* ...

(gammai4."2 + gammar4."2»;slli2=(gammai4.*(gammai3.*(gammai3 - gammai4).*«-1 + B).*gammai3."2 ...

- (-1 + A).*gammai4."2) - ...(-2.*(-1 + B).*gammai3."2 + (-1 + B).*gammai3.*gammai4 + (1 ...

+ A).*gammai4."2).*gammar3."2 + ...(-1 +B).*gammar3."4) + 2.*gammai4.*gammar3.*(gammai3.*(gammai3 ...

+ gammai4) + gammar3."2).* ...gammar4 - (gammai3.*«1 + B).*gammai3."2 + (-1 ...

+ A).*gammai3.*gammai4 - ...2.*(-1 + A).*gammai4."2) + (gammai3 + B.*gammai3 + gammai4 ...

+ A.*gammai4).*gammar3."2).* ...gammar4."2 + 2.*gammai3.*gammar3.*gammar4."3 + (-1 ...

+ A). *gammai3.*gammar4."4 + ...sqrt(-«gammai4."2.*gammar3 - (gammai3."2 + gammar3.*(gammar3 ...

- gammar4».*gammar4)."2.* ...«-1 + B)."2.*gammai3."4 + 4.*(-1 + B).*gammai3."3.*gammai4 + (-1 ...

+ A)."2.*gammai4."4 + ...(-1 + B)."2.*gammar3."4 + 4.*(-1 + B).*gammar3."3.*gammar4 ...

- 2.*(-3 + A + B + A*B).*gammar3."2.* ...gammar4."2 + 4.*(-1 + A). *gammar3.*gammar4."3 + (-1 ...

+ A)."2.*gammar4."4 + ...4.*gammai3.*gammai4.*«-1 + A).*gammai4."2 + (-1 + B).*gammar3."2 ...

+ 2.*gammar3.*gammar4 + ...(-1 + A).*gammar4."2) + 2.*gammai4."2.*(-«-1 + A + B ...

78

+ A.*B).*gammar3."2) + ...2.*(-1 + A).*gammar3.*gammar4 + (-1 + A)."2.*gammar4."2) + ...

2.*gammai3."2.*(-((-3 + A + B + A.*B).*gammai4."2) ...+ (-1 + B)."2.*gammar3."2 + ...

2.*(-1 + B).*gammar3.*gammar4 - (-1 + A + B .+ A.*B).*gammar4."2))))).I .

(2.*(gammai3."2 + gammar3."2).*((gammai3 - gammai4)."2 ...+ (gammar3 - gammar4)."2).* ...

(gammai4."2 + gammar4."2));%------------------------------------------------%Choose correct solution between the two sets of values%Eqla=sllr1+j*sllil;Eq2a=sllr2+j*slli2;for n=1:freqpointsifabs(Eqla(n))>abs(Eq2a(n))s(n,3)=Eq2a(n);

elses(n,3)=Eq1a(n);

endend%%Calculate S21S12 from the above equation set, s(n:3)S21S12r=((1.I(gammail."2 + gammarl."2)).*((gammarl.*((real(s(:,3))-real(s(:,1)))) ...

+ ((imag(s(:,3)) - irnag(s(:,l)))).*((gammail + ((gammail."2 + gammarl."2)) ....*irnag(s(:,2)))) - ((gammail."2 + gammar1."2)).*((real(s(:,3)) - ...real(s(:, 1)))).* real(s(:,2)))));

S21S12i=((1.I(gammail."2 + gammar1."2)).*((gammarl.*((irnag(s(:,3))-irnag(s(:,1)))) ...+ gammail.*(((((-real(s(:,3))) +real(s(:,l)))).*((l + gammail.*irnag(s(:,2)))) ...- gammail.*((irnag(s(:,3)) - irnag(s(:,1)))).*real(s(:,2)))) + gammar1."2.* .(((-real(s(:,3))).*irnag(s(:,2)) + real(s(:,1)).*imag(s(:,2)) - imag(s(:,3)).* ..real(s(:,2)) + irnag(s(:, 1)).*real(s(:,2)))))));

%s(:,3)=S21 S12r+j*S21S12i;disp('IS21S121 is')S21S12_dB=20*10g10(abs(s(:,3)))disp('Angle(SII) is')S21 S12_ang=angle(s(:,3))*(180/pi)%%Find "measured" values ofS21 *S12 from "sparameters" file to compare%calculated values toS21=1 O."((sparameters(:,4).120)).*expG.*(sparameters(:,5).*(pi/180)));S12=1O."((sparameters(:,6).120)).*expG.*(sparameters(:,7).*(pi/180)));S21S12_db_actual=20.*loglO(abs(S2l.*S12));S21 S12_ang_actual=(angle(S21.*S12)).*(180/pi);%%Calculating Stability Factor Mu-----------------­%calculating Mu from calculated Sll, S22, and S21S12s11_inverse=abs(s(:,1)).*exp(-j.*(angle(s(:,1))));delta=(s(:, 1).*s(:,2))-(s(:,3));Mu=(l-abs(s(:, 1))."2).I(abs(s(:,2)-(s II_inverse.*de1ta))+abs(s(:,3)));%calculating Mu from "measured" device parameters (sparameters.s2p)S11=1 O."(sparameters(:,2).I20).*expG.*(sparameters(:,3).*(pi/180)));

79

821=1 O."(sparameters(:,4)./20).*exp(j.*(sparameters(:,5).*(pi/180)));812=1O."(sparameters(:,6)./20).*exp(j.*(sparameters(:,7).*(pi/180)));822=10."(sparameters(:,8)./20).*exp(j.*(sparameters(:,9).*(pi/180)));SII_inverse_a=l O."(sparameters(:,2)./20).*exp(-j.*(sparameters(:,3).*(pi/180)));delta_actua1=(S 11.*822)-(812.*821);Mu_actua1=(1-abs(811)."2)./(abs(S22-(811_inverse_a.*de1ta_actua1))+abs(S12.*S21));%%Calculating Stability Factor K and delta--------­%calcu1ating K and B from calculated Sll, S22, and S21S12de1ta=(s(:, 1).*s(:,2))-(s(:,3));K=(l- abs(s(:,1))."2 - abs(s(:,2))."2 + abs(delta)."2)./(2.*abs(s(:,3)));B=1+abs(s(:, 1))."2-abs(s(:,2))."2-abs(delta)."2;%%calculating K and B from "measured" device parameters (sparameters.s2p)S11=1 O."(sparameters(:,2)./20).*exp(j.*(sparameters(:,3).*(pi/180)));S21=1O."(sparameters(:,4)./20).*exp(j.*(sparameters(:,5).*(pi/180)));S12=10."(sparameters(:,6)./20).*exp(j.*(sparameters(:,7).*(pi/180)));S22=1O."(sparameters(:,8)./20).*exp(j.*(sparameters(:,9).*(pi/180)));de1ta_actua1=(Sll.*S22)-(812.*S21);K_actual=(1-abs(811)."2-abs(822)."2+abs(de1ta_actual)."2)./(2.*abs(S12.*S21));B_actual=l+abs(S 11)."2-abs(S22)."2-abs(delta_actual)."2;%%Plotting K and delta results-------------------------­% subplot(2,1,1)% plot(sparameters(:,1),K,sparameters(:,1),K_actual,'--')% axis([lE9 10E9 02])% subplot(2,1,2)% plot(sparameters(:,1),B,sparameters(:,1 ),B_actual,'--')% axis([lE9 lOE9 -2 2])%%Calculating simultaneous conjugate matching conditions-­%Input matching conditions% de1ta=(s(:,1).*s(:,2))-(s(:,3));% B1=1+abs(s(:, 1))."2-abs(s(:,2))."2-abs(delta)."2;% B2=1+abs(s(:,2))."2-abs(s(:,1))."2-abs(delta)."2;% C1 =s(:, l)-(delta.*(abs(s(:,2)).*exp(-j.*angle(s(:,2)))));% C2=s(:,2)-(delta.*(abs(s(:,1)).*exp(-j.*angle(s(:,1)))));% Gamma_i=(B l-sqrt(B 1."2-4.*abs(C1)."2))./(2.*C1);% Z_i=(Zo.*(1+Gamma_i))./(l-Gamma_i);% Gamma_o=(B2-sqrt(B2."2-4.*abs(C2)."2))./(2.*C2);% Z_o=(Zo.*(1+Gamma_0))./( I-Gamma_0);%%Plotting 821 S12 results-------------------------­% subplot(3,1,1)% plot(sparameters(:,1),S21 812_dB,sparameters(:, 1),S21 812_db_actual,'--')% axis([lE9 lOE9 -50 50])% subplot(3,1,2)% plot(sparameters(:,1),S21S12_ang,sparameters(:, 1),S21 S12_ang_actual,'--')% axis([IE9 lOE9 -200200])% subplot(3,1,3)% plot(sparameters(:,1),Mu,sparameters(:,1),Mu_actual, '--')%%Plotting S11 and S22 results--------------------­%Plot results with correct solution between a and b

80

% subplot(4,1,1)% plot(sparameters(:, 1),sll_db,sparameters(:, 1),sparameters(:,2),'--')% axis([IE9 IOE9 -20 20))% subplot(4,1,2)% plot(sparameters(:, 1),s11_ang,sparameters(:, 1),sparameters(:,3),'--')% axis([IE9 10E9 -200 200))% subplot(4,1,3)% plot(sparameters(:,1),s22_db,sparameters(:,1),sparameters(: ,8),'--')% axis([IE9 IOE9 -20 20))% subplot(4,1,4)% plot(sparameters(:, 1),s22_ang,sparameters(:, 1),sparameters(:,9),'--')% axis([IE9 10E9 -200 200))%%Plot 811 and 822 results with both a and b solutions---c­% subplot(4,1,1)% plot(sparameters(:,I),20*logI0(abs(slla», sparameters(:, 1),20*loglO(abs(sllb» ...% ,sparameters(:, 1),sparameters(:,2),'--')% axis([2E9 5E9 -20 20))% subplot(4,1,2)% plot(sparameters(:, I),angle(s lla)/pi* 180,sparameters(:, I),angle(sllb)/pi* 180,...% sparameters(:, 1),sparameters(:,3),'--')% subplot(4,1,3)% plot(sparameters(:,I),20*logI0(abs(s22a», sparameters(:,I),20*logI0(abs(s22b»,...% sparameters(:, 1),sparameters(: ,8), '--')% axis([2E9 5E9 -20 20))% subplot(4,1,4)% plot(sparameters(:, 1),angle(s22a)/pi*180,sparameters(:,1),angle(s22b)/pi*180,...% sparameters(:, 1),sparameters(:,9),'--')%%Plot 811 and 822 results in polar coordinates---­%subplot(2,2,1)%polar«sparameters(:,3))*(pi/180), 1O.A(sparameters(:,2)/20»%subplot(2,2,2)%polar((sparameters(:,9))*(pi/180),1O.A(sparameters(:,8)/20»%subplot(2,2,3)%polar(angle(s(:, 1»,(abs(s(:, 1»),'--')%subplot(2,2,4)%polar(angle(s(:,2»,(abs(s(:,2»),'--')

81

Files needed in same directory as Matlab code

These files are examples of the format needed for the .s2p files without the header. These files are all indB/deg and are in the following format:

ipn_stubl.s2p1000000000 -2.049 -54.923 12.966 144.635 -39.788 62.146 -5.658 -13.3181100000000 -2.023 -60.240 12.904 141.158 -39.026 59.424 -5.686 -14.6171200000000 -1.994 -65.507 12.838 137.699 -38.342 56.720 -5.717 -15.9061300000000 -1.965 -70.721 12.766 134.258 -37.724 54.036 -5.749 -17.1861400000000 -1.933 -75.879 12.689 130.837 -37.163 51.374 -5.783 -18.4551500000000 -1.901 -80.975 12.606 127.439 -36.652 48.735 -5.819 -19.715

ipn_stub2.s2p1000000000 -0.004 -36.253 -13.565 -29.518 -66.319 -112.006 -5.442 -10.5601100000000 -0.005 -39.941 -11.861 -32.533 -63.792 -114.267 -5.425 -11.6341200000000 -0.008 -43.649 -10.297 -35.567 -61.477 -116.546 -5.405 -12.7141300000000 -0.011 -47.379 -8.849 -38.624 -59.338 -118.846 -5.383 -13.8011400000000 -0.015 -51.136 -7.499 -41.706 -57.350 -121.170 -5.359 -14.8971500000000 -0.020 -54.922 -6.233 -44.815 -55.491 -123.519 -5.331 -16.003

ipn_stub3.s2p1000000000 -1.829 -90.427 12.342 112.548 -40.412 30.060 -5.332 -50.8241100000000 -1.760 -98.994 12.142 105.952 -39.788 24.217 -5.290 -55.7031200000000 -1.687 -107.396 11.922 99.413 -39.258 18.435 -5.243 -60.5271300000000 -1.611 -115.611 11.682 92.943 -38.808 12.721 -5.192 -65.2941400000000 -1.534 -123.619 11.423 86.550 -38.429 7.086 -5.135 -70.0001500000000 -1.456 -131.404 11.145 80.244 -38.113 1.540 -5.072 -74.645

ipn_stub4.s2p1000000000 -0.058 -73.895 -1.804 -65.074 -54.559 -147.563 -5.037 -48.3991100000000 -0.086 -81.718 -0.178 -71.906 -52.109 -153.640 -4.936 -53.1791200000000 -0.123 -89.659 1.293 -78.822 -49.887 -159.801 -4.827 -57.9511300000000 -0.171 -97.727 2.631 -85.827 -47.859 -166.048 -4.709 -62.7161400000000 -0.232 -105.925 3.850 -92.919 -46.001 -172.383 -4.583 -67.4771500000000 -0.309 -114.257 4.963 -100.099 -44.295 -178.803 -4.452 -72.235

82

opn_stubl.s2p1000000000 -2.277 -19.306 13.088 145.757 -39.666 63.269 -5.200 -51.1471100000000 -2.296 -21.227 13.051 142.344 -38.880 60.610 -5.137 -56.1281200000000 -2.317 -23.145 13.010 138.935 -38.170 57.956 -5.068 -61.0701300000000 -2.340 -25.060 12.966 135.530 -37.524 55.308 -4.995 -65.9721400000000 -2.364 -26.972 12.917 132.130 -36.934 52.666 -4.917 -70.8291500000000 -2.390 -28.880 12.865 128.736 -36.393 50.032 -4.836 -75.637

opn_stub2.s2p1000000000 -2.119 -17.091 -12.402 -34.700 -65.156 -117.189 -0.009 -36.2391100000000 -2.105 -18.825 -10.731 -38.246 -62.662 -119.981 -0.013 -39.9191200000000 -2.090 -20.567 -9.203 -41.815 -60.383 -122.793 -0.018 -43.6151300000000 -2.072 -22.319 -7.796 -45.407 -58.286 -125.629 -0.025 -47.3291400000000 -2.053 -24.081 -6.491 -49.026 -56.343 -128.489 -0.034 -51.0631500000000 -2.033 -25.855 -5.275 -52.673 -54.534 -131.377 -0.045 -54.817

sparameters.s2p1000000000 -2.227 -19.391 13.245 161.162 -39.509 78.673 -5.588 -13.5001100000000 -2.236 -21.340 13.242 159.275 -38.689 77.540 -5.603 -14.8571200000000 -2.246 -23.292 13.238 157.386 -37.942 76.408 -5.619 -16.2151300000000 -2.257 -25.247 13.234 155.497 -37.256 75.276 -5.637 -17.5761400000000 -2.269 -27.205 13.229 153.607 -36.622 74.143 -5.656 -18.9391500000000 -2.281 -29.166 13.224 151.715 -36.034 73.011 -5.676 -20.304

stubl.s2p1000000000 -20.36821793 -90.815354741100000000 -19.51100374 -90.945088961200000000 -18.72360546 -91.08794421300000000 -17.99460435 -91.244831021400000000 -17.31516223 -91.416593871500000000 -16.67830919 -91.60400839

stub2.s2p1000000000 -0.02606734681 165.14901471100000000 -0.03784078418 163.57127151200000000 -0.05308372365 161.97197591300000000 -0.07234192825 160.35132721400000000 -0.09616590974 158.71002811500000000 -0.1251000949 157.0492933

83

stub3.s2p1000000000 -13.96959696 -92.797947371100000000 -13.05065127 -93.404054361200000000 -12.20180266 -94.086932161300000000 -11.41288737 -94.847183791400000000 -10.67628641 -95.684294611500000000 -9.986193874 -96.59669608

stub4.s2p1000000000 -0.3636324802 148.53112681100000000 -0.5100211191 145.08278351200000000 -0.6881511786 141.6580171300000000 -0.8980380458 138.29097961400000000 -1.138440023 135.01452081500000000 -1.407019655 131.8579378

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APPENDIX 2CALCULATING CSWITCH WITH GAIN MEASUREMENTS

Description

After solving for CSwitch using reflection coefficients, a technique for calculating

CSwitch using gain was attempted. A similar approach was used to that of the previous

reflection coefficient analysis, where an equation that related the gain of an amplifier to

its matching networks was used.

The transducer gain of an amplifier,

•

(A2.I)

was the first choice, where the various S-parameters are the characteristics of the device

itself, and Is and I L describe the reflection coefficient of the reconfigurable matching

networks. If we set either Is or I L to zero, we can simplify the equation and still solve

for the capacitance. However, the capacitance will still be difficult to solve for even if

we simplify this equation since it includes squared terms (e.g. IIsI2), and will have a real

and imaginary solution set.

Instead, Mason's rule was used to find the resulting gain of Fig. 4-3.

(A2.2)

If we substitute the results from (4.3) into (A2.2), the following equation relating

capacitance to gain is produced,

Gain = (2)(S21B)2 + jOJCZN + jOJCZN(SIIB)

85

(A2.3)

In (A2.3), a total of three unknown parameters are involved; 8 218, 8 11B, and C. Referring

to Fig. 4-3, 821B and 811B are the respective parameters of the device, while C is the

capacitance of the TITL switch.

Equation (A2.3) would seem to imply that C can be calculated with three

equations since there are three unknowns. With the tunable matching networks,

measuring three values of gain at three different TITL configurations would be simple to

do. However, further analysis shows that C cannot be calculated in this manner. In the

process of solving the three equations simultaneously through substitution, the TITL

switch capacitance cancels out and a unique solution cannot be found, as demonstrated in

the step-by-step analysis below:

2·S21BGainl =---------

1. 2 + j-ro·c.z + j-ro·C.Z·SllB Gain with 1 TITL "ON". Solve for 8 218.

2·S21BGai~=----------

2. 2 + j-ro·C.Z·2 + j·ro·C.Z·2·S11B Gain with 2 TITL "ON". Substitute 8 218 and

solve for 811B.

2·S21BGain3 =----------

3. 2 + j-ro·C.Z·N + j-ro·C.Z·N·SIIB Gain with N TITL "ON". Substitute 811B

and 8218 and solve for C.

G1Gai~ =~.-------

4. (N - 1)·G1 - (N - 2).~ . Equation in step 3 reduces to the following,

which is independent of the TITL capacitance. Therefore, a unique solution

cannot be found.

86

Further analysis shows that the inability to calculate a unique solution for the

TITL switch capacitance is a direct result of the independence of 8 218 (e.g. 821 of the

device) with respect to C when using gain measurements. In the above four-step

analysis, the first two equations can be used to solve for 8 218 and 8 118 with respect to C.

Doing this gives us the following,

GainlS21B= Gai~.-----­

-Gainl + 2·Gai~

2·Gainl + Gainrj-m.e-Z - 2.Gai~ - 2.Gai~.j-m.e-Z

Sl1 - ---------;'-----"'7""-----B - j-m.e-Z.(-Gainl + 2'Gai~)

which shows that 8118 is dependent on capacitance while 8218 is not.

(4.7)

The independence of the device 821 on the TITL switch capacitance means that

this can only be solved ifthe device 821 is not directly involved. Since reflection

coefficient measurements do not involve the device 821, the analysis in Chapter 4 works

out.

87

REFERENCES

[1] J.M. Go1io, Microwave MESFET & HEMTs, Boston, MA: Artech House, 1991.[2] R. K. Ackennan, "Communications-electronics command builds in change," Signal

Magazine, July 2002.[3] J. Papapo1ymerou, K. L. Lange, C. L. Goldsmith, A Malczewski, J. Kleber,

"Reconfigurab1e Double-Stub Tuners Using MEMS Switches for Intelligent RFFront-Ends," IEEE Trans. on Microwave Theory and Techniques, vol. 51, no. 1, pp.271-278, January 2003.

[4] C. Adams, "Chips that think: RF design," Avionics Magazine, Feb 1,2003.[5] E. R. Brown, "RF-MEMS switches for reconfigurab1e integrated circuits," IEEE

Trans. on Microwave Theory and Techniques, vol. 46, no. 11, pp. 1868-1880, Nov.1998.

[6] Y Lu, D. Peroulis, S. Mohammadi, and L. P. B. Katehi, "A MEMS ReconfigurableMatching Network for a Class AB Amplifier," IEEE Microwave and WirelessComponents Letters, vol. 13, no. 10, pp. 437-439, October 2003.

[7] T. Vaha-Heikkila, J. Vans, J. Tuovinen, and G. M. Rebeiz, "A 6-20 GHzReconfigurable RF MEMS Impedance Tuner," 2004 IEEE MTT-S Int. MicrowaveSymp. Dig., vol. 2, pp. 729-732, June 2004.

[8] T. Vaha-Heikkila and G. M. Rebeiz, "A 20-50 GHz Reconfigurable MatchingNetwork for Power Amplifier Applications," 2004 IEEE MTT-S Int. MicrowaveSymp. Dig., vol. 2, pp. 717-720, June 2004.

[9] AS. Sedra and K.C. Smith, Microelectronic Circuits, lh ed., Oxford, England:Oxford University Press, 1998.

[10] G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd ed.,Englewood Cliffs, NJ: Prentice Hall, 1997.

[11] D. M. Pozar, Microwave Engineering, r d ed., New York, NY: John Wiley & Sons,Inc, 1998.

[12] M.L. Edwards and J. H. Sinksy, "A New Criteria for Linear 2-Port Stability Using aSingle Geometrically Derived Parameter," IEEE Trans. on Microwave Theory andTechniques, vol. MTT-40, pp. 2803-2811, December 1992.

[13] H.F. Hammad, AP. Freundorfer, and YM.M. Antar, "Comprehensive Study ofMultiband Unconditional Stabilization of Common-Source and Common-GateMESFET Transistors Using Feedback," IEEE Journal o/Solid State Circuits, vol.37, pp. 1260-1270, Oct. 2002.

[14] J. DeNatale and R. Mihailovich, "RF MEMS Reliability," 12th InternationalConference on Solid State Sensors Actuators, and Microsystems, pp. 943-946, June2003.

[15] N.S. Barker and G.M. Rebeiz, "Optimization of distributed MEMS transmissionline phase shifters - U-band and W-band designs," IEEE MTT, vol. 48, pp. 1957­1966, Nov. 2000.

[16] S. J. Mason, "Feedback theory-Some properties of signal flow graphs,"Proc. IRE, vol. 41, Sep. 1953, pp. 1144-1156.

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[17] Maury Microwave Corporation (2005). RF Device Characterization Solutions[online] Available from: http://www.maurymw.com/products/rfdcs/rfdcs.htm[Accessed: 4th May 2005].

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