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2008:230 CIV

M A S T E R ' S T H E S I S

Low-Noise Amplifier Designand Optimization

Marcus Edwall

Lule University of Technology

MSc Programmes in Engineering

Electrical EngineeringDepartment of Computer Science and Electrical EngineeringDivision of EISLAB

2008:230 CIV - ISSN: 1402-1617 - ISRN: LTU-EX--08/230--SE

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ABSTRACT

Low-Noise Ampliers are key components in the receiving end of nearly every communi-cations system. The wanted input signal of these systems is usually very weak and theprimary purpose of the LNA is consequently to amplify the signal while at the same timeadding as little additional noise as possible. Its performance is measured in a numberof gures of merit among which gain and noise gure are most notable while dynamicrange, return loss and stability are examples of others.

In May 2005 a four year design study entitled EISCAT 3D was initialized. Its purposewas to investigate the feasibility of a next-generation incoherent scatter radar system.One of the responsibilities of EISLAB at Lulea University of Technology is to design areceiver front-end, which include an LNA with extremely high performance requirements.For that reason a MATLAB Particle Swarm Optimization implementation was developedto iteratively nd a solution to optimal component values for a user denable LNAtopology.

In this masters thesis, the radio frequency concepts essential to traditional LNA designas well as the design procedure itself are explained. A description to the optimizer isthen given, including a chapter on 2-port noise calculations.

With the objective to nd an LNA design with even higher performance than thepreviously designed EISCAT 3D LNA, four topologies are evaluated using the optimizerwhile consistently targeting the EISCAT 3D specications. These topologies include theoriginal reference design and one that employs the inductive source degeneration design

technique. The latter showed signicantly improved performance with an approximate 2dB gain increase and 0.1 dB noise gure reduction while still maintaining the return lossand stability requirements.

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P REFACE

This masters thesis was carried out at the Embedded Internet Systems Laboratory (EIS-LAB), Department of Computer Science and Electrical Engineering, Lulea University of Technology and was descendant to the EISCAT 3D design study.

Among the people that I would like to thank for their contribution to my masters thesisare Dr. Jonny Johansson for his encouragement and insightful help, and my supervisor,Ph.D. student Johan Borg for his invaluable expertise.

On a personal level, I wish to express gratitude to family and friends who have sup-ported me throughout this journey.

Marcus Edwall

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C ONTENTS

Chapter 1: Introduction 1

Chapter 2: Radio Frequency Concepts 32.1 Reection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Scattering Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 The Smith Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 The Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Impedance Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 7Chapter 3: Low-Noise Amplifier Design Strategy 9

3.1 Target Specications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Active Device Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 DC Bias Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Matching Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Noise Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Low-Noise Amplier Topologies . . . . . . . . . . . . . . . . . . . . . . . 173.7 Determining S and L . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 4: 2-ports and Noise 234.1 Method of Linear 2-port Noise Analysis . . . . . . . . . . . . . . . . . . . 23

Chapter 5: The Optimizer 275.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Notable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Chapter 6: LNA Topology Evaluation 316.1 Circuit 1: Original . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.2 Circuit 2: Simplied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.3 Circuit 3: Without Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 336.4 Circuit 4: Inductive Source Degeneration . . . . . . . . . . . . . . . . . . 34

Chapter 7: Conclusion 39

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C HAPTER 1

Introduction

The European Incoherent Scatter Association (EISCAT) is an international researchorganization that operates four incoherent scatter radars located in northern Scandinavia.These are used to provide ionospheric radar observations for geophysical environmentalmonitoring, modelling and forecasting as well as plasma physics research.

To meet increasing demands and maintain world leadership in this eld a four yeardesign study, EISCAT 3D [1], is underway since May 2005. This next-generation inco-herent scatter radar system is based on large phased array VHF antennas. The potentialof the system will exceed all other similar facilities both existing and under construction.

Among the involved parties is Lulea University of Technology (LTU) whose main re-sponsibility is the phased array receivers, including a low-noise amplier (LNA). Becauseof the extreme performance requisites, no commercially available pre-built product ex-hibit specications near those required. Particular emphasis has therefore been put ondesign and optimization of these critical components.

The work presented in this masters thesis includes a description of conventional designand optimization techniques for an LNA. In addition, a MATLAB particle swarm opti-mization implementation, a tool developed within the EISCAT 3D framework, has beenused to evaluate variations of an LNA design and optimize its associated performanceparameters. A signicant portion of the work was founded in rstly the recognition of underlying principles of LNA design and secondly understanding of the optimizer.

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C HAPTER 2

Radio Frequency Concepts

For a fundamental understanding of the Low-Noise Amplier design procedure, it is nec-essary to introduce a series of underlying concepts. Covered topics include reection,scattering parameters, the Smith Chart, the quality factor, and impedance transforma-tion. Further information on this topic can be found in [2] and [3] on which this and thenext chapter are based.

2.1 Reection

When a power wave travels through an impedance discontinuity, at that junction (Fig-ure 2.1), a fraction of the wave will be reected. As a consequence, the counterpart (theincident wave) will lose some of its magnitude. Naturally, this is an undesirable phe-nomenon in any application where power conservation is critical. The extent of incidentpower loss is related to the similarity of the impedances as seen in both directions fromthe junction. So the objective, in order to maximize the power transfer, is to optimizethe impedance match. Further information on that subject follows in Chapter 3.

There are a number of performance parameters that show to what extent the impedancesare matched. Firstly, the Reection Coefficient which by denition is the ratio of thereected wave to the incident wave (Equation 2.1), but can also be expressed in terms of impedances. It is a complex entity that describes not only the magnitude of the reection,but also the phase shift.

L = Reflected wave

Incident wave =

Z L Z S Z L + Z S

(2.1)

Note that this is the load reection coefficient with respect to the source impedance. Itis also commonly expressed with respect to the characteristic impedance ( Z 0 ). When theload is short-circuited, maximum negative reection occurs and the reection coefficientassumes minus unity. In contrast, when the load is open-circuited, maximum positive

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4 RF Concepts

Z S

Z L

L

Figure 2.1: Simple circuit showing the impedance discontinuity junction and measurement lo-

cation of L .

Figure 2.2: Incident wave (solid), reected wave (dashed) and standing wave (dotted).

reection occurs and the reection coefficient assumes plus unity. In the ideal case, whenZ L is perfectly matched to Z S , there is no reection and the reection coefficient isconsequently zero.

A closely related parameter is the Voltage Standing Wave Ratio (VSWR), which is

commonly talked about in transmission line applications. As the incident and reectedwave travel in opposite directions the addition of the two generates a standing wave, seeFigure 2.2. The VSWR is dened as the ratio of the maximum voltage to the adjacentminimum voltage of that standing wave (Equation 2.2). Knowing the domain of thereection coefficient, it follows that when there is no reection as in a perfectly matchedsystem; VSWR assumes its minimum and ideal value of 1.0:1.

V S W R = |V |max|V |min

= 1 + |L |1 | L |

(2.2)

The Return Loss (RL) is simply the magnitude of the reection coefficient in decibels

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2.2. Scattering Parameters 5

2-porta 2a 1

b 2b 1

Figure 2.3: A 2-port with incident waves a 1 and a 2 , and reected waves b1 and b2 .

(Equation 2.3). At times it is specied whether the return loss is measured on the input-or output side of the Device Under Test (DUT), with corresponding IRL and ORLnaming. It should be mentioned that the return loss is occasionally expressed withoutthe leading minus sign ending up with a negative RL. This is incorrect however as RLshould be positive.

RL = 20log || (2.3)

2.2 Scattering ParametersScattering Parameters or S-parameters are complex numbers that exhibit how voltagewaves propagate in the radio-frequency (RF) environment. In matrix form they charac-terize the complete RF behaviour of a network.

At this point it is necessary to introduce the concept of 2-ports. It is fundamental in RFcircuit analysis and simulation as it enables representation of networks by a single device.As the properties of the individual components and those of the physical structure of thecircuit are effectively taken out of the equation, circuit analysis is greatly simplied. Thecharacteristics of the 2-port is represented by a set of four S-parameters: S 11 , S 12 , S 21and S 22 , which correspond to input reection coefficient, reverse gain coefficient, forwardgain coefficient and output reection coefficient respectively. The concept of 2-ports isfurther described in Chapter 4 where noise calculations of linear 2-ports are addressed.

There are alternative descriptive parameters for 2-ports, such as impedance param-eters, admittance parameters, chain parameters and hybrid parameters. These are allmeasured on the basis of short- and open circuit tests which are hard to carry out accu-rately at high frequencies. S-parameters, on the other hand, are measured under matchedand mismatched conditions. This is why S-parameters are favoured in microwave appli-cations. S-parameters are both frequency- and system impedance dependent so althoughmanufacturers typically supply S-parameter data with their devices it is not always ap-

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6 RF Concepts

plicable. Under such circumstances, it becomes necessary to measure the parameters.Referring to Figure 2.3, these measurements are carried out by measuring wave ratios

while systematically altering the termination to cancel either forward gain or reverse gainaccording to the following equations:

S 11 = b1a 1 a 2 =0

(2.4)

S 12 = b1a 2 a 1 =0

(2.5)

S 21 = b2a 1 a 2 =0

(2.6)

S 22 = b2

a 2 a 1 =0. (2.7)

Conclusively, the S-parameters relate the four waves in the following fashion:

b1 = S 11 a 1 + S 12 a 2 (2.8)

b2 = S 21 a 1 + S 22 a 2 . (2.9)

2.3 The Smith ChartThe Smith Chart is a classic tool in RF engineering that has many uses and remainswidely used although computers have become a convenient alternative. It is a fundamen-tal aid in impedance matching network design and also serves as a standard for graphicalpresentation of impedance, reectance, stability circles, gain circles, noise circles etc.

In its most common form, the chart is made up out of two overlaid grids: the constantresistance circles and the constant reactance circles. The Cartesian coordinate systemwithin the Smith Chart is used to plot the reection coefficient. Furthermore thereare three varieties of the Smith Chart: with impedance grid (Z Smith Chart), with

admittance grid (Y Smith Chart) and the two combined (ZY Smith Chart).As the radius of the chart is unity, it is implied that all plotted values, whether theyare impedances or admittances, must be normalized with respect to a reference (Equa-tions 2.10 and 2.11). This reference is usually the characteristic impedance of the systemwhich usually is 50 .

z = Z Z 0

(2.10)

y = Y Y 0

(2.11)

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2.4. The Quality Factor 7

2.4 The Quality Factor

The Quality Factor (Q) is a descriptive parameter of the rate of energy loss in completeRLC networks or simply in individual inductors or capacitors. For the latter, Q is ameasurement of how lossy the component is, that is how much parasitic resistance thereis. So it follows that in applications where loss is undesirable, high Q components areadvantageous. Additionally the Q factor is directly related to the bandwidth, wherehigher Q corresponds to narrower bandwidth. The equations for calculating Q are:

QRLC = E totP avg

(2.12)

BW =

0

Q RLC (2.13)

QL = X L

R =

LR

(2.14)

QC = |X C |

R =

1CR

. (2.15)

2.5 Impedance Transformation

As previously stated, in order to maximize power transfer from source to load, matchingimpedances is required. Specically, in a circuit as seen in Figure 2.4 where the source-and load impedances are xed, the objective is to design the input matching networkso that Z S matches Z 1 and the output matching network so that Z L matches Z 2 . Inother words Z 1 and Z 2 respectively, are transformed to perceptually match the inputand output impedances of the transistor. According to the Maximum Power Theorem ,the maximum power transfer will occur when the reactive components of the impedancescancel each other, that is when they are complex conjugates. This is suitably calledconjugate matching.

To achieve the conversion with an impedance matching network of passive components,there are primarily three options. Firstly, there is the L-match. Its advantage is the sim-plicity, but that is simultaneously its downside as well because it has only two degreesof freedom. Since there are only two component values to set, the L-match is restrictedto determining only two out of the three associated parameters: impedance transforma-tion ratio, centre frequency and Q. To acquire a third degree of freedom, it is thereforedesired to cascade another L-match stage. By doing so, another two types of impedancetransformation matches are encountered: the -match and the T-match (Figure 2.5).

The advantages with the T- and -match congurations do not end with an additionaldegree of freedom. But because of their topology they can absorb parasitic reactancepresent in source or load. Specically the T-match will absorb parasitic inductance

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8 RF Concepts

Inputmatchingnetwork

Outputmatchingnetwork

Transistor

Inputmatchingnetwork

Outputmatchingnetwork

TransistorZ2Z1

ZS ZL

Figure 2.4: Matching networks in a microwave amplier.

Z 1

Z 2

Z 3 Z T2

Z T1 Z T3

Figure 2.5: Cascaded - and T-matching networks.

whereas the -match will absorb parasitic capacitance. In addition it is also possible toachieve signicantly higher Q compared to an L-match conguration. Another notewor-thy impedance transformation option is bandpass ltering where the port impedancesare unequal.

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C HAPTER 3

Low-Noise Amplier DesignStrategy

In a system with a series of cascaded devices, where each stage adds additional noisethat is potentially amplied along the way, it becomes evident that the very rst stageand its noise and gain characteristics are critical. This is particularly true if the inputsignal is weak and has relatively large amounts of noise added to it. Hence, under theseconditions, a Low-Noise Amplier is applicable. It is, as the name suggests, an amplier

where particular emphasis has been put on its noise characteristics.

F tot = F 1 + F 2 1

G 1+

F 3 1G 1 G 2

+ F 4 1G 1 G 2 G 3

+ ... (3.1)

Friiss formula (Equation 3.1), with which the total noise factor of a system withcascaded stages is calculated, shows how F 1 and G1 , the Noise Factor and gain of therst stage, dominate the overall Noise Factor. And so, it could be explicitly specied thatthe function of the Low-Noise Amplier is to supply sufficient signal gain to overcome

the noise of the succeeding stages while at the same time producing as little noise aspossible itself.

3.1 Target Specications

As the rst order of business when designing an LNA it seems appropriate to estab-lish what the target specications are. This is done in terms of a number of variousparameters.

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10 LNA Design Strategy

3.1.1 Gain

The gain of the device is its ability to amplify the amplitude or the power of the inputsignal. It is dened as the ratio of the output- to the input signal and is often referredto in terms of decibels (Equation 3.2).

V oltage Gain = 10log(V 2outR outV 2inR in

) = 20 log(V outV in

) (3.2)

Power gain is generally dened as the ratio of the power actually delivered to the loadto the power actually delivered by the source. However, as simple as that may seem, thisdenition is not entirely relevant and is difficult to quantify since the source impedance

in turn is difficult to specify. For that reason, a number of specic and therefore moreuseful denitions have evolved. Most notable are perhaps Transducer Gain - the ratioof average power delivered to the load to maximum available average power from thesource, Available Power Gain - the ratio of maximum available average power at the loadto maximum available average power from the source. As previously discussed, maximumpower is only obtained when an amplier is has complex conjugate terminations.

3.1.2 Noise Performance

The fundamental noise performance parameter is the Noise Factor (F), which is denedas the ratio of the total output noise power to the output noise due to input source.If the Noise Factor is expressed in decibels it is called the Noise Figure (NF) (Equa-tion 3.3). Another related and often talked about parameter in RF applications is theSignal-to-Noise Ratio (SNR), which is the ratio of the signal power and the noise power(Equation 3.4). The Noise Factor is equivalent to the ratio of the SNR at the input andthat at the output of the LNA (Equation 3.5). Hence, the Noise Factor is a measure of towhat extent the LNA degrades the SNR. An alternative way to express just that is theNoise Temperature (T N ), which is particularly useful with cascaded amplier systems orin applications where the Noise Figure is extremely low, as it allows greater resolution.The Noise Temperature is calculated with a reference temperature ( T ref ), that is nor-mally 290 K. By denition T N is the temperature increase that is required in the sourceresistance, so that it alone produces the noise that corresponds to the output noise atT ref . Consequently, if there is no additional noise at the output of the amplier, then T N is 0 K.

NF = 10log(F ) (3.3)

SN R = P signalP

noise

(3.4)

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3.1. Target Specifications 11

P out

P in

1dB

IP3

Figure 3.1: 1st-order output (solid), 3rd-order IM product (dotted).

F = SN R inSN R out

= 1 + T N T ref

(3.5)

3.1.3 Linearity

The linearity of the LNA is another concern that must be taken into account. Linearoperation is crucial, particularly when the input signal is weak with a strong interferingsignal in close proximity. This is because in such a scenario there is a possibility forundesired intermodulation distortion such as blocking and crossmodulation.

Third-order intercept (IP3) and 1-dB compression point (P 1 dB ) are two measures of linearity. IP3 shows at what power level the third-order intermodulation product isequal to the power of the rst-order output. IIP3 and OIP3 are the input power andoutput power respectively, that corresponds to IP3. P1dB shows at what power levelthe output power drops 1 dB, as a consequence of non-linearities, relative the theoreticallinear power gain, Figure 3.1. By knowing either IP3 or P1dB the other can be estimatedwith the following rule-of-thumb formula:

IP 3 = P 1 dB + 10dB. (3.6)

Both measurements indicate an upper distortion limit for the tolerable input power,whereas the noise gure sets a lower limit. The ratio of the two determines the dynamicrange of the amplier. Another similar measurement is the Spurious-Free Dynamic Range (SFDR), which in the LNA context usually relates to the greatest possible differential be-tween the output signal power and the power of the third-order intermodulation product.This occurs at the point where the latter emerges above the noise oor, Figure 3.2.

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P out

P inA BC

P noise

Figure 3.2: 1st-order output (solid), 3rd-order IM product (dotted). P1dB (A), IIP3 (B) and SFDR (C).

3.1.4 Stability

In a stability perspective, an LNA can be either unconditionally stable or potentiallyunstable. Given the former condition, the LNA will not oscillate regardless of whatpassive source- and load impedance it is connected to. In a 2-port network, as seenin Figure 3.3, oscillation may occur when some load and source termination cause theinput- and output impedance to have a negative real part. There are three main causesfor this scenario: internal feedback, external feedback and excessive gain at out-of-bandfrequencies. To prevent instability, the aim is to place S and L in the stable region of the Smith Chart. In practice, this is done with ltering and resistive loading to attenuategain. The condition for unconditional stability, in terms of S-parameters is

K = 1 | S 11 |2 | S 22 | 2 + | |2

2|S 12 S 21 | > 1, (3.7)

where

= S 11 S 22 S 12 S 21 . (3.8)

It is common practice to graphically present the region for which the LNA is uncondi-tionally stable with stability circles in the Smith Chart. That include the input stabilitycircle in the S -plane and the output stability circle in the L -plane. The circumstancesdetermine whether the stable region is inside or outside the stability circle according toFigures 3.4 and 3.5

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3.1. Target Specifications 13

2-portZ LZS

Z IN ZOUT

IN OUT S L

Figure 3.3: Stability of 2-port networks.

C S

r s

C S

r s

Figure 3.4: Smith chart il lustrating grey stable region in the L plane. Left: |S 11 | < 1 , right:|S 11 | > 1 .

C L C Lr L r L

Figure 3.5: Smith chart illustrating grey stable region in the S plane. Left: |S 22 | < 1 , right:|S 22 | > 1 .

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-3dB f

f 0

Figure 3.6: Illustration of centre frequency and bandwidth.

3.1.5 Centre Frequency and Bandwidth

As the LNA will operate with input signals of a particular frequency band, it is desiredto design it with a centre frequency and bandwidth accordingly. Looking at the transferfunction of the LNA, the differential of the two points around the centre frequency f 0 ,where the power gain is halved, is the bandwidth, denoted f in Figure 3.6. Althoughthe target bandwidth should be specied numerically, by naming convention there aretwo options: narrowband and wideband.

3.1.6 Return Loss

The Return Loss is a measure of how well the input impedance is matched to the referenceimpedance or how well the output impedance is matched to the load impedance in apower transfer perspective. Strictly speaking it signies how much power is reected dueto impedance mismatch relative the transmitted power. Return loss is typically speciedas IRL, which corresponds to the return loss at the input port.

3.2 Active Device Selection

The properties of the active device are in many ways the nal limiting factor for manyparameters of the LNA. It is therefore good practice to select an active device withparameters (in terms of noise gure, gain and linearity) that correspond to and preferablyexceed those of the target specications.

There are various active devices that are well suited for LNA applications including,but not limited to, the Heterojunction Bipolar Transistor (HBT), the Metal EpitaxialSemiconductor Field Effect Transistor (MESFET), the Modulation-Doped Field EffectTransistor (MODFET) and the High Electron Mobility Transistor (HEMT). In additionthere are a multitude of varied semiconductor compounds that further extend the op-

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3.3. DC Bias Network Design 15

Figure 3.7: A passive bias network [4].

tions. In the interest of limiting the scope of this thesis however, the device of choiceis the GaAs Enhancement Mode Pseudomorphic High Electron Mobility Transistor (E-PHEMT): Avago Technologies ATF-541M4 [4].

3.3 DC Bias Network Design

The purpose of the bias network is to set the quiescent point. That is the V gs and I dsfor a FET that causes it to operate in the preferred region. In a general perspectivethere are several types of biasing networks, although in LNA applications low complexityis desired and often sufficient. Typical passive and active bias networks can be seen inFigures 3.7 and 3.8.

3.4 Matching Network DesignThe concepts of impedance matching and impedance transformation have already beendescribed in Chapter 2. To display how impedance matching is carried out in practice,it will be illustrated with an example [5] using the Smith Chart. Observe however, thatthere are other means, such as with computer software. Consider the circuit in Figure 3.9.

The objective here is to maximize power gain, and therefore transform Z L into Z S .Assume that the matching network topology is yet unknown. What is known howeveris that that Z L > Z S , for that reason a downward impedance transformer L-match isused. Should a single L-match not be sufficient, additional shunt and series impedances

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Figure 3.8: An active bias network [4].

Z S

Z L

Z S*

C

L

Figure 3.9: Matching network design example circuit. Z S =25-j15 and Z L =100-j15 .

can be added incrementally until the target impedance is reached. Graphically, as it isbe shown in the Smith Chart in Figure 3.10, the objective is to nd a route to link thepoints corresponding to the normalized Z L and Z S .

First normalize Z L and Z S with the system impedance. If it is unknown, use anarbitrary value in the same range as those of the load and source. In this exampleZ 0 = 50 is appropriate.

z S = Z S Z 0

= 0 .5 + j 0.3 (3.9)

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3.5. Noise Optimization 17

z L = Z LZ 0

= 2 + j 0.5 (3.10)

Next, mark z S and z L in the Smith Chart, then consider the rst component, the shuntcapacitor. Because it is a shunt component it is preferable to convert to admittance.Hence, nd the point denoted A by rotating 180 degrees from z L . Next, go clockwisealong the constant conductance circle to nd point B . Because the value of the shuntcapacitor is unknown, the length of the arc from A to B is also unknown. In this examplehowever, as the arc for the series inductor has to be on the same constant resistance circleas z S it is possible to geometrically nd B . Finally, since the next component is in series,convert back to impedance to point D and from there go clockwise to z S .

The length of the arc A through B (b=0.78) is the normalized susceptance of C , whereas

the length of the arc D through z

S (x=1.2) is the normalized reactance of L. To nalize,their respective values are calculated for a specied operating frequency of 60 MHz:

B = bY 0 = C C = B2f

= 41.4 pF (3.11)

X = xZ 0 = L L = X 2f

= 159.9nH. (3.12)

3.5 Noise Optimization

Since the match for optimum noise normally does not coincide with the optimum gain-, linearity- or input match the objective is usually to nd a satisfactory compromise.However, rstly to optimize the LNA exclusively for low noise, consider the generalexpression for noise in an LNA 2-port:

F = F min + RnG s

[(G s G opt )2 (B s B opt )

2 ]. (3.13)

Where F min is the minimum noise factor, Rn is the noise resistance, Gs is the sourceconductance, Bs is the source susceptance, Gopt is the optimum conductance and Boptis the optimum susceptance. It becomes evident that for minimum noise the source

admittance or impedance equivalent must appear to the LNA as Gopt + jB opt .To obtain a simultaneous noise and input match, IN (that yields best gain and inputmatch) has to be shifted graphically in the Smith Chart in closer proximity to opt (thatyields minimum noise). In practice this is done with inductive source degeneration orseries feedback.

3.6 Low-Noise Amplier Topologies

The properties of an LNA are not only determined by the active device and the matchingnetworks around it, but also its topology. A couple of commonplace examples are seen

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z L

z S *

A

B

D

Figure 3.10: Smith Chart for matching network design example circuit.

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3.7. Determining S and L 19

Figure 3.11: Left: narrowband LNA with inductive source degeneration, right: common source

LNA with parallel feedback.

in Figure 3.11. Firstly a narrowband LNA with inductive source degeneration and gateinductance for an additional degree of freedom. Secondly a common source LNA withparallel RC feedback for stability purposes. One might also consider a cascade topologywith a common base to common source conguration, this improves input-output isola-tion which simplies matching. It also improves gain, however the downside with thistopology is reduced noise performance.

3.7 Determining S and LWhen the target specications are determined, an active device and its working pointhave been selected, the LNA design [3] then begins with a plot of source stability-,constant noise- and constant gain circles as depicted in Figure 3.12. The required circleequations derivations follow.

With the equation for noise in a 2-port amplier slightly altered with normalized noiseresistance and source admittance, it becomes:

F = F min + rngs|ys yopt |2 . (3.14)

Expressing ys and yopt in terms of reection coefficients, with these relations

ys = 1 s1 + s

(3.15)

yopt = 1 opt1 + opt

(3.16)

yields the following expression:

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F = F min + 4r n |s opt | 2

(1 | s|2 )|1 +

opt| 2

. (3.17)

From this equation it can be seen that the noise gure depends on the variable s andthree quantities known as the noise parameters: F min , rn and opt . These quantities aregiven by the transistor manufacturer but can also be determined experimentally.

For a specied noise gure F i the equation becomes:

s opt1 + N i

2

= N 2i + N i (1 | opt |

2 )(1 + N i )2

(3.18)

The centre of the circle is at

C F i = opt1 + N i(3.19)

and its radius is

r F i = 1

1 + N i N 2i + N i (1 | opt |2 ). (3.20)The derivations for the constant available gain circles are

GA = |S 21 |2 1 | s | 2

1 | S 22 s1 S 11 s |2 |1 S 11 s |2

= |S 21 |2 ga . (3.21)

The centre of the circle is at

C a = ga C

1

1 + ga (|S 11 |2 | |2 ), (3.22)

where

C 1 = S 11 S 22 . (3.23)

The radius of the circle is

r a = 1 2K |S 12 S 21 |ga + |S 12 S 21 | 2 g2a|1 + ga (|S 11 |2 | |2 )| . (3.24)Conclusively, the output- and input stability circles are plotted with the following

equations

|L (S 22 S 11 )

|S 22 |2 | |2 | = |

S 12 S 21|S 22 |2 | |2

| , |IN | = 1 (3.25)

|L (S 11 S 22 )

|S 11 |2 | |2 | = |

S 12 S 21|S 11 |2 | |2

| , |OUT | = 1 . (3.26)

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3.7. Determining S and L 21

The centres and radii respectively are

C L = (S 22 S 11 )

|S 22 |2 | |2 (3.27)

C S = (S 11 S 22 )

|S 11 |2 | |2 (3.28)

r L =S 12 S 21

|S 22 |2 | |2(3.29)

r S =S 12 S 21

|S 11 |2 | |2. (3.30)

In the example plot of Figure 3.12 it can be seen that the centre of the constant noise-and constant available gain circles that represent optimum s for minimum noise andavailable gain respectively, do not coincide. It is therefore not possible to achieve a simul-taneous optimum match. However, it is possible to shift s for available gain by meansof inductive source degeneration - without degrading the noise performance signicantly.This popular narrowband LNA technique effectively reduces the compromising betweennoise and available gain when selecting and designing s presented to the LNA input.Naturally, if input unconditional stability is desired, s has to be situated in the stableregion of the Smith Chart.

When s is selected, the proceedings continue by determining the output reection

coefficient (out ) of the LNA and then plot load stability circles. If then

out is in thestable region, L is set to out for a complex conjugate matched output. Should that notbe the case, transducer gain circles are drawn to nd a L in the stable region that leadsto reasonably high transducer gain.

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Figure 3.12: Smith Chart with constant noise circles (solid) constant available gain circles (dashed) and input stability circle (dotted).

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C HAPTER 4

2-ports and Noise

In [6] a method for computer aided linear 2-port noise analysis is presented. Followingtraditional 2-port noise analysis, the 2-port is equivalently interpreted as a noiseless 2-port coupled with two external noise sources. Depending on the orientation and typeof the noise sources, there are six representations of equivalent circuits. However, threerepresentations are sufficient for general applications. These representations are admit-tance, impedance and chain/cascade as shown in Table 4. The method in [6] introducesthat the noise in linear circuits be characterized by correlation matrices as opposed to

voltages and currents. Specically, the correlation matrix of a 2-port consists of fourelements that are the power spectrum densities of the cross- and auto-correlations of thenoise sources. The noise sources no longer need to be fully correlated or uncorrelated butcan also be partially correlated as would be the case with transistor 2-ports. Anotheradvantage is that the analysis explicitly provides data of the noise parameters NF min ,R n and Y opt .

4.1 Method of Linear 2-port Noise Analysis

4.1.1 Decomposition

If the noise analysis is applied to a complete circuit, the initial step is to disassembleit into single component 2-ports. This process is performed in a manner so that onecomponent at a time is isolated from the rest of the circuit, while identifying its relativeconnection. The disassembly procedure is shown in Figure 4.1. It can be seen that instage A, the rightmost resistor is isolated and is cascade connected to the remainingcircuit. In stage B , the bottommost resistor is isolated and is series connected. Lastly, instage C , the feedback resistor is isolated from the transistor and is connected in parallel.

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24 2-ports and Noise

2-port

Noiseless

i 1 i 2 2-port

Noiselessu 1 u 2

2-port

Noiselessu

i

Admittance Impedance Cascade

C Y =C i 1 i 1 C i 1 i 2C i 2 i 1 C i 2 i 2

C Z =C u 1 u 1 C u 1 u 2C u 2 u 1 C u 2 u 2

C A =C u 1 u 1 C u 1 i 2C i 2 u 1 C i 2 i 2

Y =y11 y12y21 y22

Z =z 11 z 12z 21 z 22

A =a 11 a12a 21 a22

Table 1: Correlation matrices and electrical matrices of Y, Z and A representations.

4.1.2 Determination of Correlation Matrix

Essential to the noise analysis is that the correlation matrix of each basic 2-port is known.That can be determined either theoretically or by noise measurements. Examples of 2-ports for which the correlation matrix can be determined theoretically are those thatonly consist of passive elements. The correlation matrices in impedance and admittancerepresentation are

C Z = 2kT { Z } (4.1)

C Y = 2kT { Y } (4.2)

where k is the Boltzmann constant, T is the absolute temperature, Z and Y are theelectrical matrices in impedance and admittance representation respectively. For ac-tive device 2-ports it may be necessary to measure the noise parameters and thereaftercalculate the correlation matrix in cascade representation:

C A = 2kT Rn NF min

12 R n Y

optNF min 1

2 R n Y opt Rn |Y opt |2 (4.3)

4.1.3 Interconnection

Once the correlation matrix of each basic 2-port is known, they can be successivelyinterconnected. With each interconnection the resulting correlation matrix is calculatedwith respect to the representation according to the following:

C Z = C Z 1 + C Z 2 (4.4)

C Y = C Y 1 + C Y 2 (4.5)

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4.1. Method of Linear 2-port Noise Analysis 25

A

B

C

Figure 4.1: 2-port amplier disassembly illustration.

C A = A 1 C A 2 A

1 + C A 1 (4.6)

In the event that the 2-ports are in unmatched representations, e.g. when a series2-port is connected with one that is in parallel, one has to be converted into the other.Naturally and if applicable, it is preferred to convert to that representation of the next2-port to avoid redundant calculations. Table 4.1.3 shows a the different transformationmatrices.

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26 2-ports and Noise

FromY Z A

Y 1 00 1 y11 y12y21 y22 y

11 1 y21 0

To Zz 11 z 12z 21 z 22 1 00 1 1 z 110 z 21

A 0 a12

1 a22 1 a 110 a 21 1 00 1Table 2: Y, Z and A transformation matrices.

4.1.4 Noise Parameter Calculation

In conclusion, when the entire original 2-port is rebuilt its noise parameters can beextracted from the equity

C A = 2kT Rn NF min

12 R n Y

optNF min 1

2 R n Y opt Rn |Y opt |

.

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C HAPTER 5

The Optimizer

A classic approach to LNA design using the Smith Chart has been shown. However, asperformance requirements get higher it becomes increasingly difficult to carry out thistask manually. Computer aided optimization provides higher efficiency under such cir-cumstances. For that reason a MATLAB implementation of the method described in[6] has been developed at LTU by Johan Borg. It uses stochastical particle swarm opti-mization (PSO) and has features that enhance the transition to real world applications,including non ideal inductors and real component values. This optimizer has been usedto design the EISCAT 3D LNA.

5.1 Description

The optimizer takes a user denable LNA topology along with matching networks, whereeach component is a 2-port with dened properties. Then a corresponding s-parameterand correlation matrix for the entire circuit is computed and passed on to the PSOimplementation.

Initially in the PSO process, the dimensions of the search space and the solution criteriaare dened. Thereafter the population is initialized, that is particle starting positions areeither randomized or set by user. Then the swarm is launched to search for an optimum.The velocity and position of the individual particles are updated based on discrete time,current position and velocity, and the particles personal best position as well as that of the entire swarm, the global best. The efficiency of PSO is founded in this inter-particlecollaboration. This iterative process proceeds until a position has been found that to thegreatest possible extent qualify with the predetermined solution criteria. The criteriaof this application requires input stability and that noise gure, return loss and gainmatch or exceed the dened target specications seen in Table 1. In addition, a list of values is set that specify the desired transfer function. This sets the centre frequency

27

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28 The Optimizer

Gain 18.0 dBNF 0.55 dB

IRL 17.0 dBCentre 225 MHzBW 30 MHz

Table 1: Target requirements.

and bandwidth of the LNA. As the optimization halts, the applied component values aregiven.

5.2 Notable functionsThe optimizer features a set of functions for inter-representation transformation basedon on the transormation matrices in Table 2 in Chapter 4. These are used when tworepresentations of different representations are connected.

A_to_s A_to_Y A_to_Zs_to_A s_to_Y s_to_ZY_to_A Y_to_ZZ_to_A Z_to_Y

The A-, Z- and Y-representations include electrical- and correlation matrices whereasthe s-representation has s-parameters and noise parameters. To attain compliance, thenoise parameters are therefore converted (Equation 4.3) into correlation matrix formwhen the s to A function is used. As for the functions s to Y and s to Z , a passive 2-port is implied and the correlation matrices are computed solely based on temperatureaccording to Equations 4.1 and 4.2.

In addition there are three functions to connect 2-ports, based on Equations 4.4 - 4.6:

A_connect Y_connect Z_connect

To initialize an A-representation matrix of individual components while differentiatingbetween parallel and series orientation, the following functions are used:

A_s_capacitor A_s_inductor A_s_resistorA_p_capacitor A_p_inductor A_p_resistor

5.3 Usage

As previously stated, the optimizer is based on the noise analysis method of [6]. Usageis therefore similar to that described in Chapter 4, except labour intensive tasks are

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5.3. Usage 29

conveniently carried out by machine. The one critical step the optimizer will not dohowever, is to nd an appropriate matching network- and amplier topology. This subject

will be addressed in the next chapter. After the LNA circuit to be optimized is known,it has to be loaded into the target function that is subsesequently passed on to PSO.A general case with the inductive source degeneration LNA, displayed in Figure 3.12,would be loaded in the following manner;

A_tran=s_to_A(s); % define transistorA_Rs=A_s_resistor(Rs,w); % define series resistor RsA_Lg=A_s_inductor(Lg,w,IM1); % define series inductorA_Ls=A_p_inductor(Ls,w,IM1); % define parallel inductor

A=A_connect(A_Rs,A_Lg); % connect Rs and LgA=A_connect(A,A_tran); % connect transistorY=Y_connect(A_to_Y(A),A_to_Y(A_Ls)); % connect LsA=Y_to_A(Y); % convert to A representation

where s is a merged matrix of s- and noise parameters derived from measured data andmanufacturer data, w contains frequency domain data and IM 1 is one of four inductorcomponent libraries available.

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C HAPTER 6

LNA Topology Evaluation

Given the complexity of LNA design, it is often times challenging to nd a satisfyingdesign topology. It may also have to comply with strict requirements which furtherincrease the difficulty level of this task. In this chapter the single transistor LNA seen inFigure 6.2, that was built for the EISCAT 3D project, is evaluated using the optimizerdescribed in Chapter 5. Then, a simplied version, seen in Figure 6.3 is optimized. Afterthat a version without feedback is evaluated, seen in Figure 6.4. Finally a version withinductive source degeneration, Figure 6.5, is optimized. All variations of the nal designhave been optimized using the same solution criteria. Furthermore, all circuits have abandpass lter, Figure 6.1, connected to the output to achieve the required narrowbandoperation.

Lf1

Lf2

Lf3

Lf4

Cf1

Cf2

Cf3

Cf4

Figure 6.1: Bandpass lter.

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32 LNA Topology Evaluation

Cc

Cl4

Ls2Ls1Lp1

RcLs3Cs2Cs4Cplr

Cl5 Cl3

Rd1 Cs1

Rd2

Figure 6.2: Circuit 1: Original.

R c 87.97 C c 0.1410 C l4 14.06 C s 1 246.1 C s 2 144.0 Ls 1 55.03L s 2 330.0 Lf 1 68.00 Lf 2 1.000 Lf 3 33.00 Lf 4 10.00 C f 1 15.48C f 2 22.40 C f 3 25.19 C f 4 373.6 Ls 3 33.00 C l3 1.773 Rd1 37.13R d2 469.5 L p1 150.0 C s 4 22.00 Ls -

Table 1: Component values for circuit 1. Capacitances in pF, inductances in nH and resistances in .

6.1 Circuit 1: Original

The original EISCAT 3D design, includes the following components: a protective shuntinductor: L p1 , a directional coupler: Cplr , a protective diode modelled as a capacitor:C l5 , signal coupling capacitors: C s 1 , C s 4 and C s 2 , bias network decoupling inductors: Ls 1and L s 2 , an input matching network: Ls 3 and C l3 , parallel negative feedback: Rc and C c,a stabilizing shunt capacitor: C l4 , series and shunt stabilizing resistors: Rd1 and Rd2 .

This circuit performed as expected and the performance parameters are within speci-cations. Figures 6.6 and 6.7 illustrate the frequency domain behaviour and the optimizedcomponent values are shown in Table 1.

6.2 Circuit 2: Simplied

In the simplied circuit, a signicant number of components are removed from the originalcircuit. With the exception of those with protective purposes, three signal couplingcapacitors and the bias network decoupling inductors. In Figures 6.6 and 6.7, it can beseen that this circuit is unstable for some frequencies. Furthermore, as it can be seen inTable 5, neither NF nor IRL is satisfactory.

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6.3. Circuit 3: Without Feedback 33

Figure 6.3: Circuit 2: Simplied.

R c - C c - C l4 - C s 1 283.4 C s 2 353.5 Ls 1 339.4L s 2 470.0 Lf 1 33.00 Lf 2 33.00 Lf 3 12.00 Lf 4 12.00 C f 1 5.243C f 2 33.08 C f 3 20.45 C f 4 952.3 Ls 3 - C l3 - Rd1 -R d2 - L p1 68.00 C s 4 22.00 Ls -


6.3 Circuit 3: Without Feedback

Circuit 3 is a copy of the original circuit, short of the parallel feedback. The circuit showssimilar in-band characteristics, see Figures 6.6 and 6.7, as that of the original circuit, hasequivalent noise- and return loss gures according to Table 5, and roughly 1 dB highergain.

R c - C c - C l4 10.82 C s 1 260.2 C s 2 41.39 Ls 1 43.89L s 2 220.0 Lf 1 68.00 Lf 2 2.200 Lf 3 33.00 Lf 4 10.00 C f 1 41.93C f 2 15.21 C f 3 42.87 C f 4 254.7 Ls 3 33.00 C l3 1.836 Rd1 45.06R d2 693.5 L p1 150.0 C s 4 22.00 Ls -


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Figure 6.4: Circuit 3: Without feedback.

Figure 6.5: Circuit 4: Inductive source degeneration.

6.4 Circuit 4: Inductive Source Degeneration

This circuit employs inductive source degeneration that should shift

IN closer to

OP T

and thereby render a better noise-gain match possible. According to [2], this methodcreates a resistive input impedance without the noise of a real resistor. The resistanceis controlled by choice of inductance. Since the impedance is only purely resistive atresonance, this method offers a narrowband impedance match. As it can be seen inFigure 6.5, there is no parallel feedback in this circuit. Figures 6.6 and 6.7 indicate thatthe circuit is unconditionally stable and has a at in-band gain, much like circuits 1 and3. In addition, in comparison with the original circuit, this circuit has a notably highergain and lower noise gure.

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6.4. Circuit 4: Inductive Source Degeneration 35

a 10GHz

20

10

0

-10a 10GHz

-150

-100

-50

0

a 10GHz

20

10

0

-10a 10GHz

-150

-100

-50

0

a 10GHz

20

10

0

-10a 10GHz

-150

-100

-50

0

a 10GHz

20

10

0

-10a 10GHz

-150

-100

-50

0

(dB) (dB)

(dB) (dB)

(dB)

(dB)(dB)

(dB)

Circuit 1: Original

Circuit 2: Simplied

Circuit 3: Without feedback

Circuit 4: Inductive source degeneration

Figure 6.6: Input return loss (left) and gain (right) of all circuits. A denotes the centre frequency of 225 MHz.

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bc 1GHz

20

10

0

-10bc 1GHz

-100

-50

0

bc 1GHz

20

10

0

-10bc 1GHz

-100

-50

0

bc 1GHz

20

10

0

-10bc 1GHz

-100

-50

0

bc 1GHz

20

10

0

-10bc 1GHz

-100

-50

0

(dB)

(dB)

(dB)

(dB)(dB)

(dB)

(dB)

(dB)

Circuit 1: Original

Circuit 2: Simplied

Circuit 3: Without feedback

Circuit 4: Inductive source degeneration

Figure 6.7: Input return loss (left) and gain (right) of all circuits. B and C denote the passband 210-240 MHz.

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6.4. Circuit 4: Inductive Source Degeneration 37

R c - C c - C l4 12.75 C s 1 160.0 C s 2 18.70 Ls 1 68.00L s 2 220.0 Lf 1 68.00 Lf 2 2.200 Lf 3 22.00 Lf 4 12.00 C f 1 30.94C f 2 19.33 C f 3 38.79 C f 4 280.6 Ls 3 22.00 C l3 0.1002 Rd1 22.53R d2 517.8 L p1 220.0 C s 4 22.00 Ls 1.646


Target 1 2 3 4Gain 18.0 18.16 18.47 19.33 20.07NF 0.55 0.5480 0.6436 0.5411 0.4609

IRL 17.0 16.95 4.796 16.96 17.04Table 5: Gain, Noise Figure and IRL of all circuits.

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C HAPTER 7

Conclusion

To design a low-noise amplier using traditional methods is an involved process thatrequires a solid understanding of underlying principles. As one performance parameteris affected negatively by the optimization of another it is clear that some degree of compromising is necessary. With high overall performance requirements, little freedomfor compromising is allowed and the design process becomes increasingly complex.

The computer aid given by the particle swarm MATLAB optimizer has therefore beenan invaluable asset. Given an LNA topology and a tness function to evaluate possiblesolutions, it outputs component values and performance parameters in terms of returnloss, gain and noise gure. The frequency dependance of the former two are also givenin graphical format.

Since the objective was to nd an LNA topology with performance which exceededthat of the EISCAT 3D design, it was natural to use that as a starting point and withthe use of the optimizer see how slight changes to it affected performance. An initialrun on the basis of the the original circuit showed that it had adequate performace withrespect to the predetermined specications. A simplied and over-constrained toplogythen showed insufficient performance in terms of noise gure and stability although gainwas reasonable. In a third run, the effect of the stabilizing feedback stage was examined byremoving it from the circuit. As expected this circuit showed higher gain, but could alsomeet the stability requirements. The conclusion must be that with the existing criteria,feedback is redundant. The nal circuit employed inductive source degeneration, wherean inductor is connected to the source terminal. The desired effect of this is to decreasethe performance loss associated with gain-noise gure compromising. This led to anapproximate 2 dB gain increase and 0.1 dB noise gure reduction. The computed valueof the source inductor ( L s ) was 1.6 nH, which is low enough to be practically realizablewith a micro strip solution.

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Bibliography

[1] EISCAT, EISCAT 3D the Next Generation European Incoherent Scatter Radar Sys-tem. https://e7.eiscat.se/groups/EISCAT_3D\_info/Introduction%20and%20Brief%20Background.pdf , Nov 2008.

[2] T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits . Cambridge:Cambridge University Press, 2nd ed., 2004.

[3] G. Gonzalez, Microwave Transistor Ampliers: Analysis and Design . New Jersey:Prentice Hall, 2nd ed., 1996.

[4] Avago, ATF-541M4 Low Noise Enhancement Mode Pseudomorphic HEMT ina Miniature Leadless Package: Data Sheet. http://www.avagotech.com/docs/AV02-0924EN , Nov 2008.

[5] Maxim, Impedance Matching and the Smith Chart: The Fundamentals. http://www.maxim-ic.com/appnotes.cfm/an_pk/742 , Nov 2008.

[6] H. Hillbrand and P. H. Russer, An efficient method for computer aided noise analysisof linear amplier networks, IEEE Transactions on Circuits and Systems , vol. CAS-23, no. 4, pp. 235238, 1976.

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