When employing conditionally stable transistors in microwave amplifier design, it is impossible to
have both ports perfectly matched. One can easily design for a perfect match at one of the ports, but the
resulting mismatch at the remaining port may be unacceptable. Through trial and error using graphical
methods, the designer may trade-off the perfect port match for improved match at the other port. In this
paper, we propose a systematic approach whereby the maximisation of an objective function results in
maximum gain and port match, for given stability safety margins

Eccleston, K.W.

UC Research Repository (University of Canterbury)

OPTIMUM DESIGN OF SMALL-SIGNAL MICROWAVE AMPLIFIERS WITHSPECIFIED STABILITY SAFETY MARGINK. W. ECCLESTONDept of Electrical EngineeringNational University of SingaporeSINGAPORE  119260E-mail: e ekwe@nus.edu.sgWhen employing conditionally stable transistors in microwave amplifier design, it s impossible tohave both ports perfectly matched.  One can easily design for a perfect match at one of the ports, but theresulting mismatch at the remaining port may be unacceptable.  Through trial and error using graphicalmethods, the designer may trade-off the perfect port match for improved match at the other port.  In thispaper, we propose a systematic approach whereby the maximisation of an objective function results inmaximum gain and port match, for given stability safety margins.1 IntroductionWhen designing microwave amplifiers, it is the task to select suitable values for the source (inputgenerator) and load reflection coefficients for the transistor.  The source and load reflectioncoefficients are chosen to achieve minimum gain and port matching, as well as ensure stability.  Whenusing conditionally stable transistors, it is impossible to simultaneously achieve perfect match at bothports.  However, one can easily design an amplifier with one of the ports perfectly matched, but theresulting mismatch at the other port is often unacceptable.  Therefore it is often necessary to trade-offthe perfect port match for improved match at the other port.Amplifier design typically begins with the selection of either the source or load reflectioncoefficient so that a certain gain can be achieved as well as allowing conjugate matching at the otherport [1].  The appropriate gain circles, along with the appropriate stability circle, are plotted on eitherthe source or load reflection coefficient plane to aid selection of the first reflection coefficient [2].  Anappropriate safety margin between the chosen value of the first reflection coefficient and stabilitycircle is necessary to ensure conjugate matching can be used at the other port [1], but also allows formanufacturing variations [3].  One can further ensure adequate stability safety margin at the remainingport [3].  Although the resulting amplifier is matched at one of its ports, the mismatch at the other portis typically severe and necessitates the trade-off perfect match at one port for improved match at theother.  Circles of constant mismatch or VSWR [2] can be described on the reflection coefficient planesto aid this trade-off.The attractiveness of these graphical design methods is that the functional behaviour of gain,and mismatch with respect to the source and load reflection coefficients is obtained by simply plottingcircles whose centres and radii need only be calculated using well known formulae [2].  Adisadvantage of this approach is that the selection of the source and load reflection coefficients issubjective, particularly when the contours of more than two performance parameters are plotted on thesame reflection coefficient plane.  "Trial and error" is often required to obtain acceptable performance,and inevitably, circuit optimisers are required.Recent research activities in microwave amplifier design have focussed on stability [1][3][4]or noise performance [5] – [7]; but apart from Albinsson [6], do not addressed mismatch.  In previouswork by the author [8], the design of a low noise microwave amplifier was considered.  To optimisethe LNA, two objective functions were considered: one being the noise measure, and the other beingthe product of the two port mismatches.  The former was first minimised with respect to the sourcestability safety margin and intermediate variable, whilst the later was maximised with respect to theload stability safety margin and another intermediate variable.  By adopting this approach, the stabilityAuthorized licensed use limited to: University of Canterbury. Downloaded on November 29, 2009 at 22:13 from IEEE Xplore.  Restrictions apply. safety margin is inherently incorporated into the design since it is a design variable [8].  However, aproblem with this approach is that being an LNA design, it was necessary to consider two objectivefunctions and hence optimisation in two steps.In this paper, we propose a design approach for a microwave amplifier gain stage whereby itis possible to incorporate gain and port mismatch into one objective function meaning that theamplifier can be optimised in one step.  In a similar approach to the previous work [8], this objectivefunction can be related to the source and load stability safety margins, as well as intermediatevariables, rather than directly to the source and load reflection coefficients.  For given stability safetymargins, the objective function is easily maximised with respect to two intermediate variables using asimple direct search method.2 DefinitionsIn the work that follows, the definitions of the various reflection coefficients refer to the single-stageamplifier depicted in Fig. 1.  The transistor is described by its S-parameters, the effective source andload reflection coefficients presented to the input and output ports of the transistor are GS andGLrespectively.  In this work, we will assume that the magnitudes of the transistor S-parameters S11 andS22 are both less than unity and that the stability circles do not enclose the origin.  These assumptionsare valid for many microwave transistors used in amplifier circuits.  G1 i th  reflection coeffi ientlooking into the input port of the transistor and is a function of GL via the well known bilineartransformation [2].  Similarly G2 is the ref ection coefficient looking into the output port of thetransistor and is a function of GS via the well known bilinear transformation [2].Figure 1. Microwave amplifier circuit.In general, GS ndG1 are mismatched with mismatch factor M1:( ) ( )212121111GG-G-G-=SSM ...(1)similarly, GL and G2 are mismatched with mismatch factor M2:( ) ( )222222111GG-G-G-=LLM ...(2)In the presence of statistical variation of both transistor and circuit parameters, it is important whenselecting GS and GL that an adequate safety margin with respect to their respective stability circles bemaintained.  Eccleston [3] has previously developed equations for maximum gain design usingconditionally stable transistors and defined a new performance parameter, the stability safety margin.Let CS and rS be the centre and radius of the source stability circle, and CL and rL be the centre andradius of the load stability circle.  The source safety margin (SSM) is the distance between thenominal (and allowable) value f  GS and the source stability circle.  Likewise, the load safety margin(LSM) is the distance between the nominal (and allowable) value of GL and the load stability circle.Authorized licensed use limited to: University of Canterbury. Downloaded on November 29, 2009 at 22:13 from IEEE Xplore.  Restrictions apply. 3 Objective Function FormulationIn the approach that we propose, the functional behaviour of noise measure and port mismatch aredescribed as functions of the stability safety margins rather than direct functions of GS and GL. Sincewe assume that the transistor S-parameters S11 and S22are both l s than unity and that the stabilitycircles do not enclose the origin, we can write GS and GLin terms of stability circle centres and radii,SSM and LSM :GS   =   CS +  (rS + SSM) exp(jq ) ...(3)GL   =   CL +  (rL + LSM) exp(jf ) ...(4)where q and f are intermediate design parameters.  With GS and GL described in this manner, both gainand mismatch factor can be described as functions of SSM, LSM, q and f.  Thus SSM and LSM aredesign parameters as well as performance parameters.The objective functions we wish to develop consider both gain and mismatch at both ports.Therefore it is possible to define two such objective functions:FA (q, f, SSM, LSM)  =  M1 GA M2 ...(5)and FP (q, f, SSM, LSM)   =  M1GP M2 ...(6)where GA and GP are the available gain and power gain respectively. Because the products GA M2 andM1 GP give the transducer gain, it is not necessary to consider another objective function involving thetransducer gain.  We see that maximising either objective function with respect to q and f f r givenstability safety margins (SSM and LSM) will simultaneously yield high gain and good port match.The choices available is between application of either (5) or (6) and the appropriate values of SSM andLSM.  The appropriate choice of SSM and LSM is related to expected manufacturing processvariations.   The search domain for q nd f may be determined from (3) and (4) by noting that theresulting values of GS and GL must have magnitudes less than unity.  With optimum values of q andfdetermined, the corresponding optimum values of GS and GL may be obtained using (3) and (4).4. Numerical ExamplesTo illustrate the application of the above objective functions (5) and (6), we can consider a transistorwhose S-parameters are:S11 =  0.809 /-148° S12 =  0.111  / -4°S21 =  1.940 /  57° S22 =  0.634 / - 8°Figure 2 shows the contour plot of FA and FP as func i ns of q and f ( or complete search domain) forSSM and LSM both equal to 0.1.  We see from Figure 2 that the local maximum is well defined forboth objective functions and can be easily located by direct search of the d scretised qf plane.  Forexample, FA is maximised at q = -0.3719, f = 4.81, and hence application of (3) and (4) yields thecorresponding optimum values for reflection coefficients:  GS = 0.804 / 171°  and GL = 0.812 / 62°.Table I shows the maximum objective function value, and corresponding gain and mismatch, forvarious values of SSM and LSM.  The maximum of FA or FP was located by sampling the searchdomain at the grid points of a 100 by 100 grid discretisation of the qf plane.  We see from Table I, thatthe outcome when either using FA or FP are  both cases good but using FP results in about 0.7 dBimprovement over using FA in this case.5. ConclusionIn this paper we have shown a method for selecting the values of the source and load reflectioncoefficients to achieve maximum gain and match at both ports, for given stability safety margins.Authorized licensed use limited to: University of Canterbury. Downloaded on November 29, 2009 at 22:13 from IEEE Xplore.  Restrictions apply. This method is based on an objective function that is described in terms of stability safety margins,thereby incorporating stability into the design parameters rather than consider them separately.  Theobjective functions are straightforward to maximise without sophisticated optimisers.References[1] M. L. Edwards, S. Cheng & J. H Sinsky, "A Deterministic Approach for Designing ConditionallyStable Amplifiers", IEEE T. on MTT, 41(2), July 1995, pp 1567 - 1575.[2] G. Gonzalez, Microwave Transistor Amplifiers, Analysis and Design, Prentice-Hall 1997.[3] K. W. Eccleston, "Design Formulae for Microwave Amplifiers Employing Conditionally-StableFETs", IEICE Trans. on Electronics, E82-C(7), July 1999, pp 1054 – 1060.[4] M. L. Edwards & S. Cheng, "Conditionally Stable Amplifier Design Using Constant m- Contours",IEEE T. on MTT, 44(12), Dec. 1996, pp 2634 - 2640.[5] C. R. Poole & D. K. Paul, "Optimum Noise Measure Terminations for Microwave TransistorAmplifiers", IEEE T. on MTT, 33(11), November 1985, pp 1254 -1257.[6] B. M. Albinsson, "A Graphic Design Method for Matched Low-Noise Amplifiers, IEEE T. onMTT, 38(2), February 1990, pp 118-122.[7] G. N. Link & V. S. Gudimetla, "Analytical Expressions for Simplifying the Design of BroadbandLow Noise Microwave Amplifiers", IEEE T. on MTT, 43(10), October 1995, pp 2498 - 2501.[8] KW Eccleston, “Stability Safety Margin Based Design of Low Noise Microwave Amplifiers”,APMC '99, 30 Nov – 3 Dec 1999, Singapore, pp 1 – 4.Table I Optimum Gain and Mismatch FactorsSSM /LSMMaximising objective function FA Maximising objective function FPMax FA GT(dB)M1 M2 Max FP GT(dB)M1 M20.1 19.9 13.9 0.818 0.835 24.6 14.2 0.587 0.9330.2 15.4 12.5 0.859 0.898 18.4 12.8 0.708 0.9720.3 12.4 11.6 0.864 0.903 14.6 11.8 0.734 0.9740.4 10.0 10.7 0.851 0.889 11.8 10.9 0.730 0.964                  Figure 2.  Contour plots of objective functions FA (left) and FP (right) for SSM and LSM equal to 0.1.Authorized licensed use limited to: University of Canterbury. Downloaded on November 29, 2009 at 22:13 from IEEE Xplore.  Restrictions apply. 

Optimum design of small-signal microwave amplifiers with specified stability safety margin

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Optimum design of small-signal microwave amplifiers with specified stability safety margin

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