Circuit technique for broadband impedancematching of passive loads

A. G. Chapman and C. S. Aitchison

Indexing terms: Frequency response, Impedance matching, Load (electric) Passive networks

Abstract: The paper examines the application of a technqiue known as reactance compensation to theproblem of broadband impedance matching of passive loads. A theory is developed which enables a Chebyshevfrequency response to be obtained for different loads in alternative compensation circuits. An additionalrelationship is developed which relates the reflection coefficient of a compensated arbitrary admittance tothe reflection coefficient of the uncompensated admittance. This relationship is confirmed experimentallyusing a microwave detector diode as the load.

1 Introduction

One of the most common problems encountered in elec-tronic circuit design is that of providing a broadbandimpedance match to a resistive source of a complex loadimpedance. Conventional impedance-matching techniquesat low frequencies involve ladder networks of lumpedinductances and capacitances, while at microwave fre-quencies, resort is frequently made to distributed elementsin the form of lengths of open-circuit or short-circuittransmission lines which approximate to lumped induct-ances and capacitances. Usually matching networks aredesigned to have a specific frequency response in thepassband1 such as the Chebyshev or Butterworth responses.Occasionally techniques are used which simply reduce themismatch without intending to achieve a particular fre-quency response ?

A technique which has been used with considerablesuccess in increasing the gain-bandwidth product ofparametric amplifiers3'4 and Impatt amplifiers5 andincreasing the tuning range of Gunn oscillators6'7 is activereactance compensation. Active reactance compensationtakes advantage of the impedance-inverting property of aquarter-wavelength transmission line. By connecting twoidentical devices together with such a line the invertedimpedance of one device compensates the impedance of theother by reducing the total circuit reactances.

So far the technique has only been applied to theproblem of broadbanding active loads, that is devices with anegative-resistance characteristic. The object of this paper isto consider its application to either increasing the band-width for a given mismatch or reducing the mismatch over agiven band for passive loads, that is loads with a positive-resistance characteristic.

2 The basic concept

Consider the circuit shown in Fig. 1. If a passive loadadmittance YL is connected to the end of a A/4 trans-mission line of characteristic admittance Yo, it is easilyshown that the input admittance of the line is

Yl " YL(1)

If yL = g + jb then

(2)

If a second load admittance, identical to the first, isconnected in parallel with the input to the transmissionline, the total input admittance Yin is given by

Yin =g2+b7 (g-jb) (3)

If the two loads are assumed to be nearly resonant then,provided g2 > b2, eqn. 3 can be rewritten as

Y2

g2(4)

By making Yo =g in eqn. 4 the total susceptance of thecircuit can be made zero and the conductance doubled.Other values of Yo will change the total circuit conductance

Paper T32 2 M, first received 16th October 1978 and in revised form16th January 1979Mr. Aitchison is, and Mr. Chapman was formerly, with theElectronics Department, Chelsea College, Pulton Place, London SW65PR, EnglandMr. Chapman is now with the Royal Signals and Radar Establish-ment, St Andrew's Road, Malvern, Worcs., England

MICROWA VES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2

Fig. 1 Basic shunt compensation circuit

43

0308-6976/79/020043 + OS $01-50/0

at the expense of not cancelling the susceptances. Thereflection coefficient of the circuit shown in Fig. 1 is givenby

2

|2 _ go - Yin

go + Yit

(5)

so, clearly, by varying the value of Yo the degree of matchof the loads to the source can be changed. The values ofg0 and Yo for minimum mismatch and maximum band-width for different types of load and circuit configurationare analysed in the next Section.

3 Theory

3.1 The lowpass model

In the following theory it is assumed that the X/4 line is aquarter wavelength at all frequencies, that is, it is an idealX/4 line.

Referring again to Fig. 1 and assuming that each load is aparallel CR lowpass, such as network shown in Fig. 2a, theneqn. 3 gives

where b = coc. From eqn. 5 we get

n2 -P

go+g

n(,,I V

Y2

+Y2

+

)

V

+ I'+ ft2 [

Yig2 + b'\

Y2

1 g2+b2

(7)

Now, let g0 = otg and Yo = (Ig, where a and |3 are constants,then

|2 —

b2 1 —

(8)

When 6 = 0 , we obtain the d.c. reflection coefficient |po|given by

IPo1

Rearranging eqn. 8 in polynomials of b

( a - I ) 2

(9)

4 [(a +/34 2/J2 (a + 1)]

(10)

We now assume the frequency response of |p | is a Chebyshevtype, and define the bandwidth as the frequency at which\p\— IPo I- Combining eqns. 9 and 10 gives

+ ft2^2 U 2 + 2 ( 1 - | 3 2 ) -

= 0 (11)

The third term of eqn. 11 is zero, hence

b2 = 8

Since b = coC, the bandwidth / c is given by

.,2 T 1 / 2

/c =g

2TTC

a - ( a 2 + 1 - 3 0 2 )

(12)

(13)

Eqn. 13 expresses the bandwidth in terms of the coef-ficients, a and (3, but a and j3 are themselves related, interms of |pol, by eqn. 9. For a given |pol there are twosolutions for a and j3 depending on whether a > (1 + /32) or

+j32)then

a =|pol)

Now

(1-IPol)

1 + lPol

(14)

l — IPOI

where So is the v.s.w.r. when b is zero, so

a = (l+(32)S0

Similarly, ii

(15)

(16)

It can be shown that a Chebyshev-type frequency responseis obtained when a < ( l 4-132) and a Butterworth responseis obtained when a > (1 + ft2). As the derivation of eqn. 13assumed a Chebyshev response, a can be eliminated bysubstituting eqn. 16 into it, which gives

2TTC |_ IS"O SQ

For maximum bandwidth dfc/d(l — 0. Hence

1 2(1+j32)- ; V _ / ^ + 3 = 0

1/2

(17)

(18)

TFig. 2 Simple lowpass loads

a Parallel CR networkb Series LR network

44 MICROWAVES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2

and1/2

From eqns. 16 and 19

3 So2 + 1

a = 2 SQ

(19)

(20)

Substituting eqn. 20 into eqn. 17, the maximum bandwidthis given by

fcmax 27ic[\ 2 So

Eqn. 21 shows that the maximum bandwidth can beexpressed in terms of only the load parameters, C and g, andthe zero frequency mismatch So.

The discussion so far has only considered the parallel CRload shown in Fig. 2a; the alternative series L-R load ofFig. 2b can be examined. In this case

YT =1

and

Y, =

r + jx

r + jx

where x = CJL and Zo = l/^o • Hence,

1 1 1 . F 1Y- = r1 in '

Now, let g0

r2 +x2

= oar and Z o = )3r, then

(22)

(23)

(24)

2 _\P\1 =

When x = 0

l . |2 _ 1

a - 1 +

2 - 20 V1 - - T

(25)

(26)

Eqns. 25 and 8 are of the same form and eqns. 26 and 9 areidentical, so the solution of eqn. 25 can be written downdirectly.

For maximum bandwidth a and |3 are as given in eqns.19 and 20 subject to their new definitions given above. Also

-\\ V'2

Eqn. 27 shows that, again, the maximum bandwidth issolely a function of the load parameters, r and L, and thezero-frequency mismatch.

At this point it might be useful to consider a designexample

Example 1: Let the load consist of a ipF capacitor inparallel with a 100 fi resistance and suppose that \p0 \ = 0-2

So =1 +0-21 - 0 - 2

= 1-5

So, from eqn. 20

3502 + 1

a =2S0

= 2-875

Hence,g0 = 0-028755 (r0 = 34-78 ft)From eqn. 19

15 =1/2

= 1-695

Hence, YQ =0-016955 ( Z o = 59-35 ft). The maximumbandwidth given by eqn. 21 is 2-6 GHz

The computed variation of |p | with frequency for thecompensation circuit with the foregoing parameters isshown in Fig. 3.

It is of interest to note that the use of a lossless L-Cmatching network requires 3 sections to produce a similarreflection coefficient behaviour.

3.2 The bandpass case

We now consider the load as a bandpass circuit. In suchcases the design can be on the basis of the lowpass modelwith the transformation achieved by replacing eachcapacitance in the lowpass model by a parallel L-C networkand each inductance by a series L-C network in theconventional manner. It is easily shown that the band edgesin the bandpass case are defined by

fc = ±/ccLOW+ / $ + [fcLOw]

\ 2 /(28)

where ^ is the resonant frequency of the load andfCLOW

the lowpass model bandwidth.If/o > ifcLOWl2) eqn. 28 reduces to

is

fc h ±fcLOW (29)

and the pass band is evenly distributed either side of theresonant frequency.Example 2: Consider the design given in example 1. Let theresonant frequency be 5 GHz, then the shunt inductance

2 2-6frequency.GHz

Fig. 3 Variation with frequency of reflection coefficient oflowpass compensation circuit in example 1

MICROWA VES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2 45

given by 1/cooC is 1 nH. Eqn. 28 gives the band edges at3-87 GHz and 647 GHz. Fig. 4 shows the variation of pwith frequency.

647

frequency, GHz

Fig. 4 Variation with frequency of reflection coefficient ofbandpass circuit in example 2

3.3 A series compensation configuration

Consider the circuit shown in Fig. 5 which shows a voltagegenerator of internal resistance r0 driving two identical loadimpedances ZL connected in series by a X/4 line. The inputreflection coefficient in this case is given by

l2 Zin ro\2 = 1

(30), l 2

\P\2 = z\Zin+r0

where Zin =ZL+ Z$ \ZL

MZL is the series L-R load shown in Fig. 2b, then

ZL =r + jx

where x = coL

AZin = r 1 +. » + J C > ' + *

(31)

Let r0 = ar and Zo = j3r, then

Z32..2

P2 =r- a —

r2 +x-

0t+ 1 +

(32)

Eqn. 32 is identical to eqn. 8 with r, Zo and A: replacing^,Yo and b, respectively. The maximum -bandwidth Chebyshevresponse solution can, therefore, be written down directlyas

A tu

6

Fig. 5 Basic series compensation circuit (dual of Fig. 1)

46

fccmaxr

2irL 2S0

- 11/2

(33)

where a and j3 are defined by eqns. 20 and 19, respectively.Similarly, if ZL is the parallel Cg network shown in

Fig. 2a then it is easily shown that by defining

ag = r0Yi

^cmax

where

and Yo = fc

- 1250

1/2

(34)

So, by appropriate definition of the a and 0 terms,compensation-type matching circuits can be easily designedfor different types of load in alternative configurations.

3.4 A special case

A further useful relationship can be derived which expressesthe reflection coefficient of a compensated load solely interms of the reflection coefficient of the uncompensatedload.

Consider an arbitrary complex load admittance YL

driven by a constant-current generator of internal admit-tance go a s shown in Fig. 6. Let YL =g + jb then

2 _ (go ~gf + b2

\Pu\l =(gi+g)2+b2 (35)

where pu is the uncompensated reflection coefficient. Now,consider the compensation circuit shown in Fig. 1. Fromeqn. 5

Y2

Yin = YL+— (36)

where YL = g + jb and

So *L rL *0a Y 4- V2 A- Y2go IL ^ rL ^ r0

Pc = (37)

where pc is the compensated reflection coefficient. Sub-stituting in eqn. 37 for YL and rationalising

Pc =b2 - Y2)+jb (g0 -2g)

(go g+g2-b2 + Y2)+jb (g0 ~2g)

Let Yo = g0 and #0 = 2g'o, then

o

(38)

Fig. 6 An uncompensated arbitrary complex load admittance

MICROWA VES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2

Pc =

Hence

-(gi-g)2+b2+j2b(gj-g)

\Pc\2 =[(gj-gf+b2]2

Kgi+g)2+b2]2

(39)

(40)

The right-hand side of eqn. 40 is the square of the right-hand side of eqn. 35, hence

IPcl = \Pu\: (41)

Eqn. 41 states that if the reflection coefficient of anuncompensated load is |p o | then the reflection coefficientof two identical loads in a shunt compensation circuit, \pc\,is | p j 2 provided the source admittance of the compen-sation circuit is twice that for the uncojnpensated load, andthe X/4 line admittance is equal to the source admittance ofthe uncompensated load. An identical relationship underequivalent conditions can be derived for the dual compen-sation circuit.

3.4 Effect of a real X/4 line

Throughout the foregoing theory it has been assumed thatthe X/4 line was ideal in that it was a quarter wavelengthlong at all frequencies. An immediate consequence of this isthat the lowpass models used in deriving the theory arestrictly unrealisable.

The effect of a real X/4 line has been studied by compu-tation. Fig. 7 shows the variation of \p\ with frequency ofthe circuit designed in example 2 with a transmission lineX/4 long at 5 GHz. The effect is simply to move the bandedges towards the centre frequency, thus reducing thebandwidth. In this instance the band edges are 413 GHzand 607 GHz, compared with 3-87 GHz and 6-47 GHz forthe ideal X/4 line.

One area in which a real X/4 line can have a beneficialeffect is in the design of compensation circuits for loadswith significantly frequency-dependent real parts to their*admittance or impedance, where it has the effect oflevelling an otherwise skewed frequency response as shownin the following example.

Example 3: Let the load be the circuit shown in Fig. 8awhich consists of a 0-92 nH inductance in series with a 1 pFcapacitance and 100H resistance in parallel. The seriesresonant frequency of this circuit is again 5 GHz, and the

02

01

3 U 5 6 7frequency, GHz

Fig. 7 Variation with frequency of reflection coefficient ofexample 2 showing the effect of a real \/4 line

• ideal \/4 line

variation with frequency of the real and imaginary parts,R (co) and X(cS), respectively, are shown in Fig. 9.

The design of a compensating matching circuit is bestdone by considering the load as the simple series resonantcircuit shown in Fig. 8b where L = 0-91 nH, C— 1 1 pF,R = 9 -2 f2 which are the equivalent series values of theoriginal circuit at 5 GHz.

LO-

0-91nH \ \MpF

b

Fig. 8 Circuits for example 3

a Load with a frequency-dependent real part to its admittance aridimpedanceb Equivalent circuit of load at 5 GHz

real \/4 line

MICROWAVES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2

-20 -

-251

Fig. 9 Variation with frequency of real and imaginary parts ofimpedance of circuit shown in Fig. 8a

• • reactance (X)x x resistance (R)

47

Assuming a shunt compensation circuit of the formshown in Fig. 1 then if |po | = 0-2,eqns. 19 and 20 give

go = 0 09

and

Yo = 0 064

From eqns. 27 and 28 the band edges are at 3-86 GHz and6-49 GHz.

The above values of g0 and Yo will yield the correctmatch at 5 GHz but, because the design assumed the load tobe the equivalent circuit shown in Fig. 8b, the band edgeswill not be those calculated above.

Fig. 10 shows the variation with frequency of thecompensated circuit with an ideal A/4 line and a real A/4"line. With the ideal A/4 line the curve is very asymmetricaland the reflection coefficient continues to increase about0-2 immediately beyond 5 GHz. The measured and theor-etical bandwidths are 1-5 GHz and 1-54 GHz, respectively,so there is close agreement in bandwidth terms but not inthe position of the band edges since the simple series circuitwould be almost symmetrically positioned about 5 GHz.The effect of the real A/4 line, on the other hand, is toremove some of the asymmetry from the curve and conse-quently increase the bandwidth at the higher frequencies atthe expense of a reduction at the lower frequencies. In thiscase the bandwidth extends from 3-94 GHz to 6-3 GHzwhich is much better than that obtained with the ideal A/4line. In both cases |p| rises slightly about 0-2 beyond 5 GHzbut not significantly in the case of the real A/4 line. It isalso possible that the response could be improved by usinga computer optimisation routine to optimise the A/4 linelength and characteristic impedance.

4 Experimental work

To confirm the foregoing theory experimentally a detectordiode was selected as a suitable complex load, with theexperimental circuits fabricated in stripline. The exper-imental work falls into two separate parts. The first part isconcerned with the characterisation of the detector diodeand the second with confirming experimentally the specialcase relationship discussed in Section 3.4. In all the exper-imental work the circuits were made from aluminium-

0-3

0 2

0-1

0

\\\

V \\ \_ \

\\\\

\

\\

\\\\\\\\\

\ \

\\\ )

/

/

//

y1\

/ // // /

\ i\ i\ /\ i\ i\ !

\\\ /\ /

frequency.GHz

Fig. 10 Variation with frequency of reflection coefficient of loadshown in Fig. 8a showing effect of real and ideal \/4 lines

ideal \/4 linereal \/4 line

backed Polyguide manufactured by Electronized ChemicalsCorporation. The material consists of a 0-159 cm thicksheet of Polyguide, which has a relative dielectric constantof 2-32, clad on one side with the same thickness ofaluminium and on the other side with a 35-6jxm thick layerof copper onto which the circuit pattern is etched. Atriplate structure was used so the copper was completelyremoved from one of the pieces of Polyguide in eachcircuit.

The type of diode used was the AEI Semiconductors LtdDC1303 Schottky barrier detector.

4.1 Characterisation of diode

A diode was characterised by mounting it on the end of a50 £1 stripline and terminating it in a short circuit consistingof two OBA screws, one through each ground plane, whichgently clamp the diode tab in the plane of the centreconductor. The position of the diode relative to the 50 £2line and short circuit is shown in Fig. 11.

The diode was forward biased by 150jLtA and the reflec-tion coefficient of the whole stripline circuit measured on aHewlett-Packard network analyser over the band 2—12 GHzin 200 MHz steps, at an incident power level of — 20 dBm ±1 dB. The diode impedance characteristic was obtained byreferring the measured reflection coefficient to the end ofthe 50 £1 stripline (point T in Fig. 1) making due correctionfor the coaxial to stripline transition of the SMA launcher.8

A computer optimisation program was used to derive valuesfor the components of the diode equivalent circuit inFig. 12 using the previously obtained diode impedancedata. Fig. 13 shows the measured and computed impedancecharacteristics over the band 2-12 GHz. The agreementbetween the curves is good over the entire band. Theremaining disagreement is probably due to the uncorrectednetwork analyser errors and the idealised nature of theequivalent circuit. It was considered that the model shownin Fig. 12 adequately represented the diode under the givenmeasurement conditions. In addition, the impedance wasmeasured over X-band (8-12 GHz) at incident power levelsof —15, —10, —5 and OdBm in order to quantify thepower dependence of the diode junction parameters (C2

and R in Fig. 12). Fig. 14 shows how the parameters vary

Fig. 11 Plan view of position of diode in stripline characterisationcircuit

Fig. 12 Optimised equivalent circuit of the detector diode

48 MICROWA VES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2

with incident power, and it is clear that the video resistanceis the dominant parameter. This information is particularlyimportant when the diodes are used in a compensationmatching circuit, as the power distribution between thediodes is frequency dependent and consequently affects thediode impedance in addition to its normal frequencydependence.

Fig. 13 Measured and computed impedance characteristics of thedetector diodem——• measured• • computed

0-2

015

0-1

700

600

d

400

300

- 200-20 -15 -10

incident power,dBm-5

Fig. 14 Variation with incident power of diode junctionparameters

4.2 The compensation circuit

The object of this experiment was to confirm that compen-sation occurred experimentally and that the matchimprovement was in accordance with eqn. 41. A compen-sation circuit was designed on the basis of a 50 ft sourceimpedance since this was the characteristic impedance ofthe measurement system. The X/4 line impedance wasthereby fixed at 100 ft, which was also the sourceimpedance for the uncompensated diode. The reflectioncoefficient of the compensation circuit with a 50 ft sourceand 100ft X/4 line should, theoretically, be the square ofthe reflection coefficient of a single diode driven from a100ft source. To confirm the theory in its most generalform a centre frequency of 6 GHz was chosen as this waswell removed from the series resonance of the diode at10-4 GHz. The stripline conductor pattern is shown inFig. 15 in which the 100ft line length is X/4 at 6 GHz. Thediode impedance, for the purposes of the compensationcircuit, therefore, includes the 0-2cm line preceding theactual diode reference plane which means the characteristicis not the same as that shown in Fig. 13.

The impedance of the compensated diodes was measuredat an incident power level of — 20dBm and total forwardbias current of 300 juA which was assumed to divide equallybetween the two diodes. The measurements were made overthe band 2—12 GHz, but the band of most interest is that inthe vicinity of 6 GHz, as this was the design frequency.

Before examining the results, consideration must begiven to the power dependence of the diodes. It can beshown that the compensation circuit does not divide theincident power equally between the two loads. This isunimportant for power-independent loads but can besignificant in the case of dectector diodes. In computing thevariation of reflection coefficient with frequency of thecircuit the model of each diode was modified according tothe level of the incident power using the data given inFig. 14. Fig. 16 shows the measured impedance of thecompensation circuit over the band 2—12 GHz thecomputed impedance with power correction and a real X/4line, and the theoretical impedance assuming powerindependent loads and an ideal X/4 line. The agreementbetween the computed curve and the measured character-istic is good with the measured curve exhibiting a slightlylower v.s.w.r. than expected over the whole octave band-width. The theoretical curve is of the same general shape asthe other curves but exhibits a larger v.s.w.r. than predictedand measured. This is probably due to the unequal power

0 02cm

5011 line

Po .

diodeposition

Fig. 15 Stripline conductor pattern of experimental compensationcircuit

MICROWA VES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2 49

distribution between the diodes. The disagreement outsidethis region is due to the 100ft line not being X/4 longexcept at 6 GHz. At low frequencies there is less than 90°phase difference between the two impedances, thus tendingto make the diodes appear in parallel. At high frequenciesthe phase difference increases (as the length tends to X/2)but approaching from the opposite direction around theSmith chart. Fig. 17 shows the v.s.w.r. taken from Fig. 16.The experimental v.s.w.r. at 4 GHz is 1-4 : 1 compared tothe theoretical figure of 2-6 : 1, which is significantly better.Over the rest of the band the experimental and predictedv.s.w.r.s are comparable and marginally lower than thetheoretical v.s.w.r. The technique has changed the v.s.w.r.of the diode, referred to a 100ft source impedance, from4-3:1 to 1-4:1 at 4 GHz, and from 5 : 1 to 2-2 : 1 at8 GHz which is useful improvement. Improvement can alsobe obtained over a somewhat wider bandwidth.

Fig. 16 Impedance characteristics of compensated diodes

x—x measured• • predicted• • theoretical

For the greatest improvement in match it is necessary toestablish the source impedance which minimises themismatch of the uncompensated diode over the band ofinterest and design the compensation circuit accordingly.For this particular diode over the 4—8 GHz band thecompensation circuit requires a source impedance of36-75 ft and a X/4 line impedance of 72-5 ft.

It is concluded from this experimental work that thetheory is valid and a useful improvement in impedancematch can be obtained with this compensation techniquewith appropriate complex loads.

5 Conclusions

An exact theory has been developed which enablescompensation-type impedance matching circuits to bedesigned with a Chebyshev like frequency response. Aparticular case has also been determined where the compen-sated reflection coefficient is the square of the uncompen-sated reflection coefficient. This relationship is particularly

useful because it does not assume the load to be resonantand, furthermore, the design of such a matching cirucitdoes not necessarily require an intimate knowledge of theload topology since measured impedance or admittancedata are sufficient.

The special case relationship has been confirmedexperimentally using detector diodes for the loads and asimple X/4 transmission line as the impedance invertingelement. The theoretical and experimental impedancecharacteristics exhibited satisfactory agreeement over anoctave bandwidth.

6 7 8 9frequency, GHz

10 11

Fig. 17 Variation with frequency of v.s.w.r. of compensateddiodes

x x single diode 100 SI system• • predictedo o measured \ compenSated• •theoretical J

The compensation technique provides a simple methodof significantly improving the impedance match of a passivecomplex load.

6 Acknowledgements

The support for this work of the MEL Equipment Co. Ltd,a division of Philips Electronic and Associated Industries;and the UK Science Research Council is gratefullyacknowledged.

7 References

1 MATTHAEI, G. L. et. al: 'Microwave filters, impedancematching networks, and coupling structures' (McGraw Hill, NewYork 1964)

2 'Impedance matching techniques for mixers and detectors',Hewlett-Packard Application Note 963

3 AITCHISON, C. S., and WILLIAMS, J. C: 'Active reactancecompensation of parametric amplifiers', Electron. Lett., 1969,5,pp. 139-140

4 AITCHISON, C. S., and WILLIAMS, J. C: 'Actively compen-sated parametric amplifiers - some further results', ibid., 1972,8, pp. 567-568

5 BAINS, A. S., and AITCHISON, C. S.: 'Active broadbanding ofan X-band Impatt diode amplifier', ibid., 1977, 13, pp. 289-291

6 AITCHISON, C. S.: 'Gunn oscillator electronic tuning range andcompensation: an experimental result at X-band', ibid., 1974,10, pp. 488-489

7 AITCHISON, C. S., and GELSTHORPE, R. V.: 'A circuittechnique for broadbanding the tuning range of Gunn oscillators',IEEEJ. Solid-State Circuits, 1977, SC-12, pp. 21-28

8 CHAPMAN, A. G., and AITCHISON, C. S.: 'A frequencyindependent broad band model of a coaxial to striplinetransition'. Submitted to IEEE Trans., MTT

50 MICROWA VES, OPTICS AND ACOUSTICS, MARCH 1979, Vol. 3, No. 2