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ANTENNA LABORATORY

Thnia bped No. 64

ANTENNA IMPEDANCE MATCHINGCq BY MEANS OF ACTIVE NETWORKS

291 7S61 by

Contract AP331637)-SA6Oit) Hitch Elemntw Nv. 62405454

7600-Prolect 4026, Took 402624Aeronautical Systems Dison

Project Eoglw L Turner, ASRNCV.1

NOVEMBER 1962

Sponored bp:AIONAJTCAL SYSTEMS DIVISIO

VAW~N-PATTOSON AIR FORCE BASE, 01410

A S TI IA

DEC 2 71962

TISIA

ELECTRICAL ENGINEERING RESEARCH LABORATORYENGINEERING EXPERIMENT' STATION

UNIVERSITY OF ILLINOISURBANA, ILLINOIS

Antenna Laboratory

Technical Report No. 64

ANTENNA IMPEDANCE MATCHING

BY MEANS Of

ACJ'IYE NETWORKS

by

S. LaxpatiR. Mittra

Contract A133 (657)-8460Hitch Element Kr. 62405454

760D - Project 4028, Task 402624Aeronautical System Division

Project Engineer Z. Turner, ASJWC1-l

November 1962

Aeronautical Systems DivisionWright-Pattearson Air Force Base, Ohio

ELECTRI CAL ENOINEERING 335 AMO LABORATORYENGINEERING UPMENWT STATIONK

UNIVERSITY Or ILLKNOXSURBANA, ILLINOIS

ABSTRACT

The paper introduces several schemes for wideband matching of

impedances using active elements in the matching network. It is shown

that the design procedure is a straightforward one when two active

elements such as negative impedance converters are used. An alternate

scheme using one active element is also discussed and it is shown that

a RC matching circuit using Kinariwalas' synthesis technique usually

obtains a rather complicated and sometimes impractical type of network.

However, a simple . matching network may be designed using one active

element if certain approximation of the load impedance function is

made, Illustrative designs using one and two active elements are

described in thr paper.

The noire performance of an active matching circuit which has an

infinite bandwidth in Atu Ideal sense is compared with a simple active

pudding network. It is found that there is little relative advantage

of one over the other. It is shown, however, that there is a definito

advantage of the former circuit over the latter one if one compares

their power performance,

Experimental verification of the theoretical designs are included

in the puper.

CONT NTS

Page

Abstract ±

1. Introduction 1

2. 0eneral Scheme of Matching 2

3. Design of the System 4

3.1 Determination of Antsnna Equivalent Network 43.2 Design of Matching Network [-2X(S)] 4

3.2.1 Design by Means of Two Active Elements 43.2.2 Jesign by Means of One Active Element 5

4. Examples 7

4.1 Antenna Equivalent Impedance Z(5) 74.2 Synthesis of Z1(S) using RC elements 8

4.2.1 General Procedure 8

4.2.2 Computations for the Z (S) 101

4.3 Synthesis of ZI(0) using LC elements 154.4 Simplification of Network by means of Approximation 19

5. Noise and Power Considerations 24

5.1 General Considirations 24

5.2 Noise Considerations

5.2.1 Noise Figure Of Network of Figure 16(a) 245.2.2 Noise Figure of Network of Figure 16(b) 26

5.3 Power Considerations 27

6. Uxperlmental Work 30

7. Conclusiuns 39

References 40

ILLUSTRATIONS

Figure Page

1 Antenna Equivalent Impedance 1

2 Scheme for Matching Arbitrary Impedance 2

3 Equivalent Load on the Generator 2

4 Scheme for N..tching Simple Load Impedances 3

B Matching Network Using Two Active Elements 4

6 Cascade Matching Network Using One Active Element 5

7 A Simple Equivalent Impedance of an Antenna 7

8 Matching Network for the Impedance of Figure (7) 8

9 Block Diagram for Cascade Synthesis of Y (a) 9

10 a and B Ladders of the Four Terminal Network 131 15

11 Two "erminal Network with Input Impedance YL12 Complete RC Matching Network for the Z1 () 1s

13 Complete LC Matching Network for the Z (8) ls

14 A Second Antenna Equivalont Impedance 19

15 A Simplified LC Match Ang Netwrk 20

1s (a) Negative Impedanca Inverter Matching Network(b) Network Matching Impedance by Means of Padding Resistor 23

17(a) Negative Impedance Inverter Network with all Noise Sources 24

17(b) Network Matching Impedance by Means of Podding ResistorqWith all Noise Sources 26

1$ (a) Negative Impodance Invorter Matching Network for PowerConsiderations(b) Padding Resistor Matching Network for Power Considerations 28

19 Negative Impedance Converter Number 1 31

20 Negative Impedance Converter Number 2 32

21 Negative Impedance Converter Number 3 33

22 Compensation Curve Using NIC Number 1 for Shunt Compens.tion 34

23 Compensation Curve Using NIC Number 2 for Series Compensation 35

24 Compensation Curve Using NIC Nur& er 3 for Series Compensation 36

25 Compensation Curve for Oeries Resonence Equivalent Impedance 37

26 Compensation Using Two NIC a 38

1. INTRODUC'ION

The input impedance of an antenna is a complicated function of

frequency and as such cannot be put into an analytic form for all frequencies.

However it can be approximately represented ovor a frequency band by a lumped

element loasless network terminated by a resistance Rr as shown in Figure 1. V

The complexity of this two terminal pair losle5s network is dependent on

the degree of approximation of the impednace curve of the antenna in the

desired frequency range.

Z(S) R+IX , 1 2 Rr

NETWORK,,

Figure 1. Antenna Equivalent Impedance.

When the antenna is electrically short, it has a high Q and is a highly

frequency sensitive device. It can be approximated by a large capacitive

reactance in series with a resistance. In order to use these antennas

efficiently over a band of frequencies, various impedance matching techniques

are employed.

This problem of matching of antennas using passive networks, based on

different matching criteria has been treated extensively by various authors1' 2.

Fano 3 in his paper, has discussed limitations on passive network, matching

of arbitrary impedance. Not much work has been done on the matching pro-

blem using active networks.

This report is a study of the possibility of using active networks for

antenna impedance matching.

2

2. OZNBRAL SCHIME OF MATCHINO

The proposed scheme (Figure 2) Is for a general type of matching circuit for

an antenna load which admits an equivalent representation of the type shown

in Figure 1. It can be easily shown that the impedance Zin' input impedance of

the cnmposite network is a pure resistive, and is equal to Rr,

ACTIVE

NETWORK

II

I/c___ N TWORK I NETWORK

Z(S) sR+j X

Figure 2. Scheme for Matching Arbitrary Impedance

We can thus represent the equivalent generator load as shown in Figure 3.

If the generator impedance R is taken as Rr, it is obviuus that the power

transferred to the load R is a maximum and constant.

R 9:Re

uZing ra

Figure 3. Equivalent lead on the Generatov

Since the network in Ftgura 2 has input Impedance R r' and has all itu

elements lossless except the load resistance Rrx the power transfer to this

load is constant and maximum. This Rr in cse of the antenna equivalent

network is the radiation resistance of an antenna.

When R , tho even part of Z(J) (real part of Z(W)) is constant a simple

matching scheme of the type shown in Figure 4 may be used.

ACTIVE

NETWORK

-x NETWORK Rr

leo

Z(S) S Rr+X(S)

Figure 4. Schome for Matchifng Simple Load Impedances

The bandwidth of the system in Figure 2 would be dependent on the band-

width of the active network and also the frequency band over which the

equivalent representation of the antenna impedance is employed. Thum the

bandwidth of the system would be lower than either one.

3. DION OF THE SYSTEM

3.1 Dtermination of Antenna Equivalent Network

The process of determining the lumped element equivalent network re-

presentation is largely an empirical one. If the impedance variation as

a function of frequency is available, to obtain ratioaal function of the

form of quotient of polynomials is a standard process It will be assumed

here that either the quotient of polynomials representation of antenna

impedance or a complete equivalent network is available and our discussion

will only pertain to the design of a matching network.

3.2 Design of Matching Network C-2X(S)]

If the form of the Impedance function Z(S) is known, finding X(S) is

just a matter of working out the algebra. Then the major problem is to

synthesize an active network which will produce an impedance of -2X(S)

at its input trrminals. Various schemes for doing thi& are described below.

3.2.1 Design by Means of Two Active Elements

We shall first describe a simple way of synthesizing an impedance function

C-2X(g)]. Figure 5 shows this method wherein it is required to use two active

elements. One of these is a negative Impedance converter while the other is

a negative resistance. This negative resistance can Very well be a negative

impedance converter terminated by resistance Rr, The two four terminal lo.isless

networks vrown are identical and they are the some as the four terminal loesless

networks employed in equivalent antenna impedance representation In Figure 1.

SXl1 X22 Rr

I LCI o j [ NETWORK

N I

-2X(S) 1 2X(S)

X1l X22 -RrLC

NETWORK

Figure 5. Matching Network Using Two Active Elements

3.2.2 Design by Means of One Active Element

We shall now study the possibility of designing this network by means of

a single active element.

4Kinariwala has shown that "any driving point immittance function can bo

realized by a transformerless RC structure tn which is embedded only one

active element. The only restriction on the Immittance function is that it

is a ratio of two polynomials with real coefficients." A more practical method

has been suggested by him in the same paper but with added restrictions on

the immittance function. This results in a cascade network as shown in

Figure 6. The restriction on the immitta'ce is that in addition to its having

real coefficients, the function is positive on some interval of a-axis (J89+jw),

Z2 "N

Z(S) Z1 Z22 R1RC - NETWORK

I l o NETWORK j 2 _jt

Figuro 6. Cascade Matching' Neowork Using One Active Element

Carlin and Youla have shown in their paper that "any arbitrary real,

rational driving point immittance function whose zeros and poles are completely

unrestriczed as to multiplicity and location in the complex S-plane, may be

realized as a lumped network consisting of reciprocal lossless elements and

at most one positive and one negative resistor."6

Rohrer has used a method of synthesis similar to that used by Kinariwala,

but for LC networks. This neework structure In the same as in Figure 6, where

the two terminal network and four terminal network are LC networks.

The function -2X(S) that we wish tn design has indeed tts representation

as a quotient of polynomials of real coefficients. Thus by means of any of

the above techniques one is able to design this impodance uRing either only

RC or LC elements and one active element. The active element is either a NIC

or a single negative resistor. Use of transformers in synthesis of a fouLr

terminal network can be avoided by use of a method suggested by Fialkow

and Gerst7 In case of AC networks, where as the same can be achieved in

case of LC networks by performing an impedance level change similar to

loop impedance level change on a Darlington network.

Examples of synthesis using Kinariwala's synthesis method for RC

networks and synthesis using LC elements are worked out in the next section

of this report.

It should be noted that these methods of synthesis do not always lead

to a practically realizable networx. This is largely due to the arbitrariness

in the choice of certain factors.

7

4. EXAMPLES

4.1 Antenna Equivalent Impedance Z(S).

Coitsider an antenna equivalent impedance as shown in Figure 7.

C

R I

ZIS) L R L:600myH

C 2 2500 pF

Figure 7. A Simple Equivalent' Impedance of an Antenna

Then Z(S) of the network is given by

Z(S) = LCR S2 + IS

WC S 2+CR S + 1

Substituting for R, L, and C and thcn frcqucncy scaling by 10 8 magnitude1

scaling by -. ; the impedance Z(S) is

.125 82 + .5 S

15 S + .25 S + 1

hence,

-2X(S) = 2(negative of odd part of Z(S) )

-14.94 S3 - S

225 S4 + 29.94 S2 + 1

The complete network for matching is shown in Figure 8, where Z1 () = -2X(S)

is still undetermined.

ACTIVE NETWORK

Figure 8. Matching Network'Sor-the Impedance of Figure (7)

4.2 Synthesis of ZI(S) using RC elements

4.2.1 General Procedure

We shall summarize here the synthesis iethod suggested by Kinariwala4 using

RC elements. We shall work with admittance function Y (8) rather than impedance1

function Z (S) ar exemplified in his paper, with relevant modifications in the

procedure.Let us write Y (S) 1 N

et1 Zl() = D

Selecting B P2P4 where B has only negative real roots, we can write

N - _

~P41lS D - D -5- Q 1 Q3()

2 4

where N =P1 P3 D =Q -Q;3 and P., P3 Q; Q all have negative real roots.

Rearranging Equation (1) we have

P3 P4

Y1(S) = P 1 2 2 (2)

Ql P

9 k

Also for cascade structure of Figure 9 we can virite

z LY22 L

CoprnYqatos()ad () we obtain, Y (

P P

-YI 2

NY,(S) Y1 122 Y 2 L L RC

NETWORKl/€ NETWORK

Figure 9. Block Diagram'for"Cnornde Synthesis of Y (S)

Comparing Equations (2) and (3); we obtain,

Y P, Q3

(4)

1 P3 P4

We can rearrange Equation (3) and also Equation (4) to obtain the following

(-Y 12) P 1 , Q111

B ) I Y22" YL = Q1 Q3 P 4(5

Q1 P2

10

Using Xquation (1) in above we have

N DP1 + g2P2Y I(a) a - DQi ..

g 2P 2 = NQ1 - DP1 0 P2R (May) (6)

We now give the procedure to obtain the Y parameters of the four terminal

network and YL; using the above information from Equations(4) and (6)

(a) Choose P1 and Q of degrees equal to the rank of Yl(8) such that p i

a RC admittance.(b) Zvaluate Y1 () at some point on the negative real axis farther away

from the origin thap the root of Q, farthest away from origin.

(c) Make Y (a) w Y 1 " Q- at this point by merely multiplying Y1 1 (5)by au aprropriate coistant.

(d) Determine P 2R and find its roots.

(e) Assign appropriate roots to P2 from steps (c) and (d).

(f) DetermineR. Find 2 n R2 N a NR: D' w DR.

(g) Zxpress D in partial fraction to obtain

(h) All admittances are obtained by noting that-Yl2 = _L12 Ql

4.2." Computations for the Zl(S)

We have from previous computations in Section 4.1;

225 84 + 29.94 S2 P 1

- 14.94 S - S

Let us consider

P1 = K (5+1) (S+3) (S+5) (S4.7)

Q1 = (S+2) (S+4) (S+6) (S+8)

n 1

and equating Y s - to Y (B) at S z -10; we obtain K 61.2375.

11

Note that the above choice of roots of 1,1, Q, and the point 5 w -10

are the ones that effec the resulting networks, It is essential that the

be chosen judiciously.

We now obtain P2R a NQ1 - DP,, which gives after considerable algebra

work,

P2 R a 225(B + .0666)(0 + .53)(J + 2.481)(8 + 4.467)(8 '+6.522)

2(S + 10)(S - .00023 8 + .0671)

select

P2 u 225(S + 2.481)(S + 4.467)(S + 6.522)(S + 10)

R - (S .0666)(S + .53)(S - .00023 S + .0671)

We now obtain from above,

Q1 P2

and then put into its partial fraction expansion which is given by

DO Q3 P4 1 (.34085 8 306.49 S 350U.13 5 1313.62 Sj

Q1 P Q1 P 225 5+ 2 + 4 5++ S+ 8

L 3.151 S 736.58 4120 S 223.554 8H 225 + 2.481 + + 4.487 + 5+ 6.52 + 10

Now we identify

1.237F (S + )(S + 3)(S + 5)( S + 7) (7)11l (S + 2)(8 + 4)(S + 6)(S + 8)

12

(s + .06e6)(S + .53)(B 2 .00023 8 + .0671)12 (B + 2)(S + 4)(B + 0)(B 8)

1 .341 306.5 8 3500 + 1313.6 SY22 = 22i . 2 F + 4 + 8 + 8

and

1 f 3.181 5 736.5 5 4120 S 223.55 SYL " 225 + 2.48 + 8 + 4.467 + 8 + 6522 + +10

We synthesize the four terminal network with the above Y-parameters by

Guillemin's method . The two ladders have , Y and Y12 parameters as given

below, blt for a scaling factor.

Y lla M Yl ii Yl ip ',

a 4 + .5964 8 3 .1022 S2 + .04 S + .00237"Y 12a (S + 2)(8 + 4)(S + 6)(S + 8)' & "12& -( + 2)(6 + 4)"' + 6)(S + 8)

The two ladders ( and P) synthesized from the above Y and Y12 par&-

meters, which give exact Y but Y12 within a constant multiplier, after un-

scaling in frequency and magnitude are shown in Figure 10.

The a and P ladders when connected in parallel has the Y-parameters asgiven in Equation (8).

, 61.2375 (S + 1)(S + 3)(8 + 5)(S + 7)11 (S + 2)(S + 4)(S + 6)(S + 8)

, (S + .066)(8 + .53)(S 2 - .00023 S + .067) (8)12 300.6 (S + 2)(S + 4)(S 4 6)(S + 8)

Y, .2464 (S4 + 13.93 53 4. 57.45 S2 + 86.1 5 + 33.48)22 72 (S + 2)(S + 4)(S + I)(S + 8)

2.835 K10.64 2.71 .01735 300.6

.0597 .038 091 126

a LADDER29.7 K 238.5 K

F89 1251 17.9 5 K2

316.9~ 492,u -i 859II 2 '9s2

1 LADDER

Figure 10. a and B Ladders of the Four Terminal Network

:L 4

Since we need the Y-purametoru of this four terminal network as in

Equation (6), we use a method of scaling analogoulto that of impedance scaling

Darlington network; as shown below.

We have from Equation (5)

y (S) y (Y11 11 Y21,1 - L

(-Y 1 2)2

11 (Y22 YL-,-(- 121

2

(y -

ll (22'"Y

where

Y11' " p Y1 2 k Y12P Y2 2 Y, L k

Using the ebove in Equations (7) and (8) we have k 1/300.6; and

(V y 1.675 8 10-8 +.1504 10-4 1.72 10 -4 + .645 10-422 L 0 + 2 5 + 4 0 + 6 8 +

1.56 8 10- + .362 10-4 + 2.025 104 + 10+ 2.48 8 + 4.467 1 + 6.522 10 +U '10

(9)

But we have Y2 2 ' as in Equation (8), hencr we obtain using Equation (9);

1,726 1 10-3 + .37 10- 3 +.220S0o3 + 22 10-3 6 10-7L { + 2 8 + 4 8+ 6 8 +8 5 + 2.48

.362 0 2.025 .11 6 10- 415 + 4.467 0 + 0.522 S 10 /I

15

The two terminal network with input admittance Y L' is shown in Figure 11.

0L

6.41M 0 4S4K :15,795K 14.43K

1 427K 91K 2.66K 4.41K

L it 8.1 I .1P 94p 28.3 V62.8 ~ 31.1 p 8 6.3 p3771

0- -

Figure 11. Two Terminal Natworkl with Input Imped.inco Y

Unscale the two networks of Figures 10 and 11 by 108 in frequency and 1/120

in magnitude to obtain the complete network as shown in Figure 12 to give the

required YI(S) = 2X(s). This network must be used in turn in complete matching

system as shown in Figure 8.

The scaling of each element is achieved using the tollowing relations, where

* denotes the scaled elements.

R, 2 10- 8 CR. 120R ; C ="- C

120

4.3 Synthesis of Z1 (S) using LC elements

We now synthesize the impedance ZI (B) as obtained in Section 4.1, by1 6

means of LC elements based on the methoG used by Rohrer

We have

Z () -14.4 - s225 S4 + 29.94 2 + 1

~~11

2

it A

0 vb

In CLi)ANa

CLIa.k

00

In 06

C-I-

a.6

c"04

IrCI21

NYNJ A

Figue 1, Coplee RC.machig newor fortheZif)

17

which is twice the odd part of impedance Z(S), utter frequency scaling by8 1

10 and magnitude scaling by -L where,

202

Z(S) - o125852 + .58

1582 + .25 + I

Determine the nositive real function Z'(8) assooiated with Z1(a) related

to it by Z1(8) a odd Z'(8), to obtain

ZI(S) x 61.05282 + ,00148 + 4.0865 m + n1

1552 + .2458 + 1 m2 + n2

using the condition that (m1m2 - n n 2) is a perfect square, whore m is

an even functton and n an odd function.

Then we have

n1 n2

Z (S) = 1 2n2 m2 n2n2 m2

1 -. ZL

Y221 Z 22 - L

where we realize Z (8) as a four-terminal network terminated by a negative

impedance ZL as in RC elements case, but now using LC elements.

Thus we identify

"'1 "'2'Zl n' 2 " 22 • n2

m2m - nIn 2 n2

z12 • n2 ; L m2

7 3 , 0 I C07)L .0 2 0 8 ,u ( "

Figure 13. Complete LC Mntchingt Nrotwork for the Zi (S)

19

Hence we have the Z Amrameters of the casoado structure as follows:

2]61.0528 + 4.086511 = 245S

22.0215 (14.94S + 1)

12 .245S(10)

15S + 1Z222 .245S

Z u .245S

158 + 1

Unscaled cagcade network realizing ZI(S) is shown in Figure 13, where the

scaled networ', has Z parameters of Equation (10).

This network is In turn used in system of Figure 8 to obtain the complete

system used for matching the impedance Z(S).

4.4 Simplification of Network by means of Approximation

We find in Sections 4.2 and 4.3 that the networks realizing Z 1 vb), which

is twice the negative of odd part of Z(S), are very complicated and sometimes

lead to impractical element values. We find it possible to simplify this

network considerably if we approxinmate the Z1 (s), as shown below in case of a

Z(S) different from that used in Sections 4.1, 4.2, and 4.3.

Consider the antenna equivalent impedance Z(s) as shown in Figure 14.

IMPENOANCE AT 10 Mc/$

C R:I. R :1..

Z(S) L C 250 pF XL: 628 ILL 1OWJH XC: 65 .

Figure 14. A Second.,,Antenna.Equivalent Impedance

20

Hence we have

.0+400S

After frequency scaling of 10 7.and

X= 399*99S 3+ 1600S 11

SS~ + 7-9999S 1

and

Z (s) =-2X (S) - 9.8~ 3200S

1eq S 4 79999S52 + 164

If the matching network for this Z (b) is synthesized on the line of1I

Section 4.3, the network has quite impractical 'element values. But if we

approximate X eqby X1 ' as in Equation (12), we find the network a very

simple one.

3X? 400S + 16008 400S

2 -2(12)eq + +8 + 16 S + 4

and

ZI(S) ~2Xt - 800S1j8 -Xeq- 2 4

Figure 15. A Simplified LC Matching Network

21

Figure 15 shows the network realizing Z I(b). We now check the error

involved in this approximation and the limitation on the frequency band over

which it is valid. To determine this one muxt consider the ratio of difference

in XeIS to the Req of the known impedance Z(B) since this affects the pror in

the impedance faced by the source. Tabulated in Table I is the factor -eg eg

Reqat different frequencies over thc range of 10 to 1. It is clear that this

ratio is less than 1 over this frequency, and is very small for most of the

frequencies. Thus we can conclude that over the frequency range f.Oriwhich

the , antenna - equivalent impedance of the form shown in Figure 14, the

matching network is exceedingly simple if an approximation of neglecting R is

used.

22

Table I

Frequency ex eq

in X x Req Re0NC/S OR._e

1 69.?70 69.711 0.014; .1814

2 207.63 207.635 0.426 -0.001

3 1679.709 1686.7651 62.915 +0.111

4 -433.td81 -433.963 7.434 -0.0115

5 -214.u76 -214.091 2.827 -0.005

6 -147.654 -147.662 1.937 -0.004

7 -114.648 -114.653 1.589 -0.003

8 - 94.542 - 94.545 1.412 -0.002

9 - 80.843 - 80.846 1.306 -0.005

10 - 70.837 70.839 1.238 +1.688

20 - 32.658 - 32.6582 1.05 -0.0002

23

OUTPUT

R, X, R

INPUT, X1

(a) NEGATIVE IMPEDANCE INVERTER MATCHING NETWORK.

R1 :1R -R OUTPUT.

1 I - 0 2

(b) IMPEDANCE MATCHING BY MEANS OF PADDING RESISTER.

Figure 16. (a) Negative Impedance Inverter Matching Network(b) Network Matching Impedance by Means of Padding Resistor

k4

5. NOISE AND POWER CONSIDERATIONS

5.1 General Considerations

In receiving antennas the noise performance of the receiving system Ls a

very important factor, while in a transmitting antenna the efficiency of the

matching system is of primary importance. Thus any matching system should also be

analyzed from point of view in order to ascertain its usefulness. It is often

difficult to make any general statemonts about various systems, and a specific

analysis has to be mada before coming to a definite conclusion.

Let us consider a simple negative impedance inverter type of active matching

network shown in Figure 16. Its performance will be compared with that of the

simple active matching circuit which uses resistance amplification phenouenon.

These two networkp are shown in Figure 16(a) and (b).

5.2 Noise Considerations

5.2.1 Noise Figure of Network of Figure 16(a)

Figure 16(a) shows the negative impedance inverter type of network. R1 and

X1 are the real and imaginary parts of the impedance Z(S) of an electrically

short antenna. (-R) is a negative resistance and X2 an inductance of the value

shown. The noise sources in the system are the thermal noise generators of the

resistors R., and R's. The negative resistance is considered to be a tunnel diode

and its noise current generator ie of magnitude V2el'dAf. It being the d.c.dc dcbias current, e the electron charg-, and Af is the bandwidth. The network of

Figure 16(a) with all its noise sources present is shown in Figure 17(a).

I 2 se

e., R1 eg vR eN eN

Figuire 17(a). Negative Impedance 'Inv-erter'Network with all Noise Sources

25

where

Ni a Thermal noise of resistance R1 " /4k_Ill

•N = Thermal noise of resistance R = 4 (13)

in = noise current of tunnel diode = 2e I'dc

k is Boltzmann's constant.

The noise figure defined on the basis of short circuit current at the

output terminals is

Noise Figure = N.F.

Sum of the square of short circuit noise current at the output terminals

due to all the noise sourcesS'm of the square of short circuit noise current at the output terminals

due to external noise sources alone

We have in this case a negative impedance inverter and associated matching net-

work as the system under consideration for the noise figure analysis. Hence the

noise figure of network of Figure 16(a) is given by

2 e 2 I 12 2 e

N.F. 1 +X+ J X ) - JXI+ J j (R.F : 1 X 12 + j X 2(R- R 1) ' i( + R

2

+ R(R + JX (R JX J 2 2 R)

+ It e2x 2 (R 1 - jX 1 + j - - R alX2

Algebraic simplification using Equation (13) gives

112 R2 + 12 e I(N.F. = 1 + - + - + _R1 - I (14)

26

5.2.2 Noise Figure of Network of Figure 16(b)

Figure 16(b) shows a matching network using a padding resistor formed

by the parallel combination of R2 and (-R). (-R) is a negative resistance

amplifier, and again we consider it to be a tunnel diode.

Shown :n Figure 17(b) is the network of Figure 16(b) with all its noise

sourc3s present.

I RI I T 2R2 -R I

NIlyeN 2

Figure 17(b). Network Matching'Impedance" 6y Means of Padding ResistorWith All Noise Sources

Where we have

eN1 = ,V4kTRl '

aN2 - V(15)

in /2Idc

Analyzing along the lines of Section 5.2.1 for noise figure on the basis

of output short circuit current we obtain, after algebraic simplification

using Equation (15),R2 +X2 12 +12 e

N.F.=1+ R1 + R1 +11 Itdc (16)R 1R 2 R 1 (kT d

From Equation's (14) and (16), one can readily verify that under conditionsX >>R; and X2 Z R (necessary for large bandwidth), these two expressions are

approximately identical. Thus for all practical purposes the noise perfor-

27

mance can be considerud about equal for the two circuits,

5.3 Power Consideration,

To compare Lhe performance of the two systems of Figure 16 for power

requirementmwe determine the power input: to the systems for the same power

dissipated in the antenna equivalent radiation resistance R 1 . The main power

source shown here as in Figure 18 is assumed to hove an internal impedance R ggLet I be the current flowing through the resistance R 1 . A simple analysis

leads to the following results in the two cases under the assumption that the

current delivered to the load is the same in each case.

Case I. Negative impedance inverter matching network of Figure 18(a).

(a) Power output to RIL = Pt = 12R1

(b) Power input by the generator P' ml

- I(R 4 R 1)

(c) Power input by the negative resistor (-R) = PI

= 12 (R + X)

(d) Total power input Pin = Pinl + P'in

= 12j[Rg + R1 + R + X12/RJ (17)

Case I. Padding resistor matching network of Figure 18(b)

(a) Power output to RI = Pout =I2R1

(b) Power input by the generator = P inl

= 12 tRg + R, + Re. "jI 2 XI

(c) Power input by the negative resistor (-R) = Pi n2

(d) Total power input = Pn = P nl + P n2

I2 {Rg 4 R, + Re (T -) -j 12X1 (18)

28

R I jXI R R R I jX1

V-j ____ Xt l R -RV.' V.

(a) (b)

Figure 18. (a) Negative Impedance Invertor Matching Network for Power Considerations(b) Padding Resistor Matching Network for Power Considerations

29RR9

where R R2* R-R2

Comparing the two expressions for the total power input i, two cases given by

Equai'U on (17) and (18); we readily notice that whereas we need a large reactive

power input for Case 11, the input power required in Case I is only roal power,

In Case II Ra is of the order of X1 at the lowest frequency of matching. Hence

it can be easily verified that the input power is less- in Case I. Thus the

system of Figure 16(a) offers a definite saving in power requirement.

3)

6. EXPBRIMRNTAL WORK

The experimental work conducted during this investigation has heon con-

cerned primarily with the matching networks for simple antenna equivalent circuits,

and using negative impedance converters as active elements. The N.I.C.'s ais a

positive fecdback network has, as a rule, a small useful frequency band. Thus

a number of NIC's were built to function in different frequency bands from

1 XC/S to about 10Mc/S. NIC's using tubes were built to function in the

lower frequency range of up to 10 IC/S and successfully tested. Later on,

transistors have been used all throughout in this investigation.

The basic design of these NIC's was brsed on the circuits given by Linvill9

and Bonner 10 in their papers. Figures 19, 20, 21 show three different types of NIC's

using transistors. These NIC's were used in matching simple equivalent impedances

of antenna. The results of these are shown in Figures 22, 23, and 24. In each

figtire is shown the signal across the antenna equivalent resistance with and

without the matching networks compensation.

It is observed that there is a limitation set up at high frequencies, and

this is the limitation in the performance of NIC at these frequencies due to

the appreciable phase shift introduced by the transistors used. The low fre-

quency limitation observed is due to the large ratio of reactive to real

impedance of the antenna equivalent impedance. This ratio is about 25 to 35

at the lowest frequency.

Figures 25, 26 show. the results obtained using twc NIC compensations.

4K IO;0 , K OOK

2N270 2N270INPUT 10~--l014F

tpu F

10K 270XL I1K I0K

LOAD

Figure 19. Negative Impedance Converter Number 1.

32

~-L2.2 K 50K 2.2K

2N270-2N270

4.7K 4.7 K

a+

YL

Figure 20. Negative Impedance Cinverter Numbe~r 2.

2N711 DARLINGTON PAIR

INPUT LOAD

IOK 10K10K

2N711 DARLINGTON PAIR

Figure 21.. Negative Impcdancc Converter Number 3.

0

z J30 0P -4

zw 00 0

.0

8/00

$1 I

0-U.

j*a

C4)4

00

0.4 0

d4.)

z

zz

t 4 J4

8. caUw

00

001-4d..

360

00z0

x

41 wEn a

0.

2

0

00

0

.41

z

U. 4J

290 0

0~ U)0~

.Y.

0 00.-4U

&L. 4J

33

0

I- -

0 0

0 ~x x

0h 0

2. 2

co r0m

z 9f.00

Q U1

C1 ddAi

38

00

0

u0 0

E[

I-00

~ U0zI- I-

0cv.

d4.)

39

7. CONCLUSIONS

(1) Active networks consisting of negative impedance converters or negative

resistances may be successfully used to design wideband matching circuits

for arbitrary load impedances. The design procedure using two active

ele&lents is very straightforward. An R-C matching circuit using one

active element only may be designed using a procedure outlined by Kinariwala,

but the method does not always lead to practical values of circuit components.

L-C circuits using one active element may also be designed to approximately

match a given load. One usually pays the price for reduced number of active

elements in terms of increased complexity of design.

(2) It is difficult to form a general conclusion in regard to tbh comparative

advantages of Pitive matching networks, over a simple active padding network

from the point of view of the noise performances because there are various factors

including the nature of the load ilpedance which determines this behavior. For

a particular case it was found that there is little relative advantage of the

active matching circuit over the padding network, inzofar as the noise

performance is concerned. However, for the same case it was found that from the

point power performance the active circuit had a definite advantage.

(3) Experimental studies demonstrate the feasibility of building the

theoretically designed circuits and show that active matching circuits

perform satisfactorily in the design frequency band limited only by the

bandw'Ath of the active elements and their signal handling capacity.

The general conclusion is that wideband matching may be accomplished

through the use of active elements. However, the complexity of design, the

noise and power performance etc., depend strongly on the nature of the load

impedance to ba matched and the relative bandwidth desired. A study of some

of these aspects has not yet been completed and a continuation of the study

along this line is recommended.

40

REFERENCES

1. H. J. Carlin & R. LaRosa, "Broadband Reflectionless Matching withMinimum Insertion Loss", Proo. of Symposium on Modern Network Synthesis,Polytech. Inst. of Brooklyn, April 1952.

2. N. Yaru, "A Synthesis Method for Broadband Antena Impedance MatchingNetworks", P!..D. Thesms, University of Illinois, 1955.

3. R. M. Fano, "Theoretical Limitations on the Broadband Matching of ArbitraryImpedances," J. Franklin Inst. V 249, p. 57, Jan. 1950 and p. 139 Feb. 1950.

4. B. K. Kinariwala, "Synthesis of Active RC Networks", B.S.T J., V. 38, pp.1269-1316, September 1959.

5. H. J. Carlin & D. C. Youla, 'Network Synthesis with Negative Resistors,"

Proc. IRE, V. 49, No. 5, pp. 907-920, May 1961.

6. R. A. Rohrir, "Active Network Matching of Arbitrery Loads", Electronics Ras.Lab., University of California, Berkeley, Report No. 367, Series 60.June 8, 1961.

7. A. Fialkow & I. Gerst, "The Transfer Function of General Two TerminalPair RC Networks", Quart. Appl. Math., Vol. 10, July 1952.

8. M. E. Van Valkenburg, "Introduction to Modern Network Synthesis", JohnWiley, New York, 1960, Sec. 11.5.

9. J. G. Linvill, "Transistor Negative Impedance Converters", Proc. IRE,V. 41, 7une 1953, pp. 725.

10. A. L. Bonner & J. L. Garriser et. al., "The E 6 Negative ImpedanceRepeater", B.S.T.J., V. 39, n. 6, Nov. 1960, p. 1455.

ANT2NVA LABORATORY

TECHNiCAL REPORTS AND MEMORANDA ISSUED

Contract AF33(616-310

"Synthesis of Aperture Antennas," Technical Report No. 1, C.T.A. Johnk,October, 1954.*

"A Synthesis Uetbrod for Broad-band Antenna Impedance Matching Networks,"Technical Report No. 2, Nichol&@ Yaru, 1 February 1955.* AD 61049.

"The Assymmetrically Excited Spherical Antenna," Technical Report No. 3,Robert C. Hansen, 30 April 1955.*

"fAnalysis of an Airborne Homing System," Technical Report No. 4, Paul H.Mayes, 1 June 1955 (CONFIDENITIAL).

"Coupling of Antenna Elements to a Circular Surface Waveguide," TechnfealReport No. _5, H. S. King and R. H. DuHamel, 30 June 1055.*

"Axially Excited Surface Wave Antennas , Technical Report No. 7, D. E. Royal,10 October 1955.*

"Homing Antennas for the F-66F Aircraft (450-2500 mc)," Technical ReportNo. 8, P. E. Mayes, R. F. Hyneman, and R. C. Backer, 20 February 1957,

(CNIDENTIAL).

"Ground Screen Pattern Fange," Technical Memorandum No. 1, Roger R. Trapp,10 July 1955.*

Contract AF33(610'-3220

"Effective Permeability of Spheroidal Shells," Technical Report No. 9,E. J. Scott and R. H. Duliamel, 16 April 19.56.

"An Analytical Study of Spaced loop AD' Antenna S~'stems," Technical Report,No. 10, D. 0. Berry and J. B. Ireer, 10 May 1956. AD 98615.

"A Tochnique for Controlling thie Radiation from Dielectric Rod Wavcgutdes,"Technical Report No. 11, J7. W. D.mcan anid F. H. DuHazael, 15 July 1956.*

"Direction Characterigtics of a 17-Shapec 5'ot Artenna,' Technical Reportlo. 12, Richard C. Becker, 30 September 1956.**'

"Impedance of Ferrite Loop Ante~inas," Technical Report No. 13, V. H. Rumseytind W. L. Week.,p 15 October 3956. AD 1!9780,

'Closely Spaced Tras,sverse Slots in Pec'angulkr Waveguide," Technical ReportNo 14, Richard F. Pyreman, 20 December 1956.

37Distributed Coupling to Surface Wave Anternas," Technical Report No. 15,Ralph Richard Hodges, Jr., 5 January 1957.

It The Characteristic Impedance of the Fin Antenna of Infinite Length,"I Tech-nicalPReport No. 16, Robert L. Carrel, 15 January 1957.*

"On the Ist.mAtion of Ferrito Loop Antenna Impedance," Technical Report No. 17,Walter L. Weeks, 10 April 1957.* AD 143969.

"A Note Concer"ing a Mechanical Scanning System for a Flush J'ounted LineSource Antenna,' Technical Report No. 16. Walter L. Weeks, 20 April 1957.

"Broadband Logarithmically Periodic Antenna Structures," Technical Report No.19, R.. H. DuHamel and D. E. Isbell, 1 May 1957. AD 140734

"Frequency Independent Antennas," Technical Report K2ro. 20' V. ff. Rumsey,25 October 1957.

"The Equiangular Spiral Antenna," Technical Report No. 21, J. D. Dyson,15 September 10,67. AD' 145019.

"Experimental 1nvestigion of the Conical Spiral Antenna," Technical ReportNo. 22, R. L. Carrel, 25 May 1957.** AD 144021

"Coupling between a Parallel Plate Waveguide, and a Surface Waveguide,"Technical Rp~er1,No. 23, E. J. Scott, 10 August 1957.

"Launching Efficiency of Wires and Slots for a Dielectric Rod Vaveguide,"Tehia epr o 4 J. W. Duncan and R. H. Duffemel, August 1957.

'The Characteristic Impedance of an Infinite Biconical Antenna of ArbitraryCross Section," Technical Report No._25, Robert L. Carrel, August 1957.

"Cavity-Baclred Slot Antennas," Technical Report No. 26, R. J. Tector, 30October 1957.

"Coupled Waveguide Excitation of Traveling 'Vive Slot Antennas," TechnicalRoport No. 27, W, L. Weeks, 1 December 1951,

'Pha'ie Velocities in Rectangular Vaveguide Partially Y'illed with Dielectric,"recncal Rert No. 28 W. L. Weeks, 20 December 1957.

'M3asuring the Capaci±tan'ce per 'Unit Length of 91 conical Structures of Arbi-trary Cross Section," Technical epo2rt No. 29, J. D. Dyson, 10 January 1958.

'Non-Planar Logarithmically Periodic Antenia Structure.," Technical ReportNo. 30, D. P. Isbell, 20 February 1958. AD 156203.

Electromagnetic Fields in Rectangular Slots," Technical Report No. 31, X. J.Kuhn and P. E. Mast, 10 March 1958.

"oThe Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder,"Technical Report No. 32, J. W. Duncan, 25 May 1958.

" A Unidirectionul, Equiangular Spiral Antenna," Technical Report N~o. 33,J. D. Dyson, 10 July 1958. AD 201135

"Dielectric Coated Spheroidal Radiators," Technical "t~port No. 34, W. L.Weeks. 12 September 1958. AD 204547

"A Theoretical Study of the Equiangular Spiral Antenna," Technical ReportNo. 35, P. E. Vast, 12 September 1958. AD 20454R,

Contract AF33(616)-6079

"tUse of Coupled Waveguides In a Traveling Wave Scanning Antenna," TechnicalReport No 36, . H. MacPhis, 30 .April 1959. AD 215558

" On the Solution oL a Class of Wiener-Hopf Integral Equations in Finite andlnfinite Ranges," Technical Report No. 37, Raj hittra, 15 Kay 1959.

"Prolate Spheroidal Wave Functions for Electromagnetic Theory," TechnicalReport No. 38, W. L. Weeks, 5 June 1959.

"Log Periodic Dipole Arrays," Technical Report No. 39, D. Z. Isbell, 1 June1959. AD 220651

"tA Study of the Com-Corrected Toned Mirror by Diffraction Theory," TechnicalReport No. 40,, S. Daegupta and Y. T. Lo, 17 July 1959.

"The Radiationt Pattern of a Dipole on a Finite Dielectric Sheet," TechnicalReport No. 411, K. G. Ealmain, 1 August 1959.

"tThe Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problemin a Waveguido," Technical Report No. 42, Raj Mittra, 1 October 1959.

"1Impedance Properties of Complementary Multiteruival Planar Structures,"iTechnical Report No. 43, 0. A. Deschamps, 11 November 1959.

"On the Synthesis of Strip Sources," Technical Report No._44. Raj Kittra,4 December 1959.

"Numerical Analysis of the Eigenvalie Problem of Waves In Cylindrical Wave-guides," Tee-tia lRepor~t No. 45, C. If. Tang atd Y. T; Lo, 11 March 1960.

"New Circularly Polarized Frequency Independent Antennas with Conical Deanor Omnidirectional Patterns,"I Tech"ical Report No. 46, J. D. Dyson and P. Z.Mayes, 20 June 1960. AD 241321

"Logarithmically Periodic Resonant-V Arrays,' Technical Report No. 47, P. Z.Mayes. and R. L. Carrel; 15 July 1960. AD 240302

'A Study of Chromatic Aberration of a Coma-Corrected Zoned Mirror," Tech-nic.al Report No. 48, Y. 1'. Lo, June 1960.

"Evaluation of Cross-Correlation Methods in the Utilization of Antenna Systems,"Technical Report No. 49, R. H. acPhie, 25 January 1961.

"Syntheel. of Antenr.s Products Patterns Obtained from a Single Array," TechnicalReport No. 50, R. H. MacPhle, 25 January 1961.

"On the Solution of a Class of Dual Integral iquacions," Technical Report No. 51,

F., Mittra, I C-tober 1961. AD 264557

"Analysis and Design of the Log-Periodic Dipole Antenna," Technical Report No.52, Robert L. Carrel, 1 October 1961.* AD 264556

'A Study of the Non-Uniform Convergence of the Inverse of a Doubly-InfiniteMatrix Associated with * Boundary Value Problem in a Waveguide," Technical!t2pr~to , P,, Mttrap I October 1961. AD 264556

Copies available for a threti-week loan period.

Copies no longer availAblo.

I1

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San A!ntonio, rexas West~ood, Massachusetts

New Mexico Staf- Unv crsity Aer onaut Ic DiviS ion

Head Antenna Department Ford Motor Company

Physical Scinc(. Laboratory Ford RoadLniv.rsity Park. New Mexico Attn Mr. ) M Black

Bell Telephonc Laboratories, Inc Uni'ersity of DaytonWhippany, New Jersey Research Institute

Att,. Te.ihhical Reports Libiariar. Attn Professor Dotrilas HannemanRoom 2A--ICZ5. 300 College Parc

Robert C. ilansen Techni .che hochs-:hule

Aerospace Corooration Attn H 1i M.inke

Cox 95085 Mcn-c'-, Gorma:y

Los Angeles 45, CaliforniinNASA Guddard Space Flight Ccnter

Dr R.chard C. Bch:r AtT Auim.m Frt'u h,

10829 Bcrkshire Mr L.atz

Westchester, Illinois Grcenbelt Maryland

Dr W. M Hall Professor A. A Abner

Raytheon Company Polytcchnic Institute of Brooklyn

Surface Radar Arn~i Navagation Microwave Research Ilnstitu'e

Operatior 35 Johnson Street

Boston Post :Noad Brooklyn 1, New York

Wayland, Massachusetts

U. S Naval Drdnan.q. Laboratory

Dr Robert L Carrel Attn. Te.chnical Libruty

Collins Radio Corporation Corona. Ca, fo-.nia

Antenna Section

D llas, Texas Avco Ccrporation

Research and Adianced Detiopment Dv1isLonDr. A. K Chatter.ee At In Re-ei.-rc.i Library T A RuDarepht

lice Principai & Head of the Depart-nent 201 Lowell Strcei

of Research Wilm'ngton. Mass

Birla Institute of TechnologyP 0 Mesra Raytheon Company

District-Ranchi (Bihar) India ?:hssle and Spase Di'.iuion

Attn R search LnritryAeronautical Svster .. Division Bedford. ,;s

Attn ASAD - Library

Wright-Patterson Air Force Base Ariirt¢ n 6%5t,exs ,ncorgcrated

Ohio Att, Tscho cI t . ibrarv An'enna Section

National Bureau of Standards

Deprritmen' of ComimerceAt t n Dr A G McNitsh

Washxngtoz, 25. D. C.

IIE

I

kivional Research CounciIAttn. Micro*avc Section

Oitawa 2. Canada

Sichak Associates

Attn- W Sichak518 Franklin Al.enue

Nutlev. New Jersey

W. T. PattonZ203 New Abany load

Cznt,. Township

Rtterton Post Of ice

New Jersey

Radio Corporitio.: of America

Mist"Ile and Ser'icc Radar Division

Attn- ManaLEr

Ail(. ,na Engineering S ill Center

Mores'own, Ne% J(cSe)

Commander

Air Force Systems Command

Aeronautical Systems Di%-sion

Wright-Pattverson Air Force Base

Ohio

Attn- ASNCSO

Commander

Air Force Systems Command

Aeronautical Systems Di-ision

Wright-Pattercon Air force Base

Attn ASNPOT, Mr. Finochoro