Gain-Sensitivity Analysis for Cascaded Two-Ports and Application to Distributed-Parameter Amplifiers

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1 Gain-Sensitivity Analysis for Cascaded Two-Ports and Application to Distributed-Parameter Amplifiers Filiz Güneş, Serhat Altunç Yildiz Technical University, Electronic and Communication Engineering, Besiktas, Istanbul, Turkey Received 9 September 2003; accepted 9 March 2004 ABSTRACT: Enhancement of a gain-sensitivity analysis of electrical networks is presented by computing gain sensitivities with respect to network parameters. A simple and versatile method. The so-called chain-sensitivity matrix is presented and compared with the current method in the literature, gain factorization, for the gain sensitivities of the cascaded networks. Analytical formulas are derived to calculate gain sensitivities of the T and types of distributed-parameter amplifiers with respect to the physical length l and characteristic impedance Z 0, rather than using a time-intensive computer-based perturbation method. The numerical results of the T- and -type amplifiers for the design targets of noise figure F req 0,46 db (N 1, 12) input VSWR V ireq 1, power gain G Treq 12 db (N 15, 86) and the bandwidth B 2 GHz 11 GHz are given in comparison to each other Wiley Periodicals, Inc. Int J RF and Microwave CAE 14: , Keywords: sensitivity; transducer gain; chain parameters; matching circuit; characteristic impedance; phase constant I. INTRODUCTION Sensitivities, or partial derivatives of certain network functions with respect to circuit parameters, are very important in the design of microwave circuits. An aspect of the circuit design is the optimization of the electrical and noise performances. Efficient gradientbased optimization requires calculation of the circuit sensitivities. The study of the network sensitivity and its automated calculation was developed at least as early 1972 by Calahan [1]. Since then, sensitivity calculation by the adjoin method and its variants, as well as direct computation of network partial derivatives, has become textbook material [2, 3]. Nevertheless, until now, there have been a very limited number of considerable works on the analytical expressions for the sensitivities of the performance-measure functions of microwave networks in the literature [4 7]. Correspondence to: F. Güneş; gunes@yildiz.edu.tr. Published online in Wiley InterScience ( wiley.com). DOI /mmce In [4], analytical formulas of the gradients were supplied to the optimization of a distributed amplifier with T-type matching circuits. In [4], the analysis the gradient of the gain was worked out by the so-called gain factorization method (utilized in our work as well), in which gradients of the other performancemeasure functions, such as noise, input VSWR, and output VSWR, are expressed in terms of gain gradients with the proper terminations. In [5], noise figure and spot-noise parameters sensitivities were derived analytically by using a nodal approach for both single and cascaded two-port networks, and an application was derived for a two-stage amplifier with commongate active matching. In addition, all the results were verified by the numerical-perturbation method. In [6], excellent information of the sensitivity analysis was gathered for the microwave circuits. In our work, an enhancement of the gain-sensitivity analysis for the cascaded two-ports is presented by using the two different abovementioned approaches: gain factorization and chain-sensitivity matrix. These 2004 Wiley Periodicals, Inc. 462

2 Gain-Sensitivity Analysis for Cascaded Two-Ports 463 sensitivities can be related to any passive or active parameter of the element in any position of the network. Rather than employing the adjoin method, which is one of the major techniques with regard to sensitivity in linear-circuit theory and can be applied to any type of circuit configuration, a simple and easy method known as chain sensitivity is established for formulation of sensitivity, which straightforwardly uses the configuration knowledge of the original network and results in weighting coefficients that are dependent upon the position of the two-port of the element. This type of sensitivity formulation was applied to many microwave circuits with distributed parameters in [7], among which the microwave amplifiers, considered as cascaded networks with T- or -type matching circuits, were given the most attention. Cascaded two-ports are very commonly used in the microwave network theory. One considerable application for cascaded two-ports is the synthesis of microwave amplifiers using lumped and/or distributed elements. In particular, after solving the problem of definite target space [8], a cascaded configuration is very easy to use in the design with together the Darlington representation of the source and load impedances [9]. Designing a microwave amplifier using the potential-noise, input-vswr, and gain performances, with predetermined bandwidth for the employed active device, can be considered as a significant development in current state-of-the-art microwave technology. The motivation underlying this work is essentially to investigate how and in what specific ways the sensitivities of the distributed parameters effect the circuit performance. This can provide the answer to the question of the tolerance limitations of the technology utilized in the realization process. At the same time, an equally important application area is the fact that analytical gradient expressions obtained by an easy and simple method can be used in the optimization procedure of microwave amplifiers. In the CAD of microwave circuits, direct expressions for the gain sensitivities can improve the speed of computations, and also safeguard against possible numerical difficulties associated with the perturbation method. In the realization and production stages, the network and system parameters usually change; the effect of these changes upon the system performance can be analyzed by using the system sensitivities. Real networks and systems must usually operate under changing environmental conditions (for example, at changing temperature, humidity, atmospheric pressure, electromagnetic radiation, and so forth). Circuit elements change their values due to natural aging. By taking into account these dependencies, we can determine network or system sensitivities with respect to environmental-condition parameters. Information about network sensitivities may be used extensively in computer-aided optimal-tolerance assignment, optimal centering, and postproduction tuning. II. SENSITIVITY In this work, sensitivity is defined as the partial derivative of a derivable function F with respect to the x parameter: D x F F x. (1) The common normalized sensitivity can be defined in the form of either or S x F S x F ln F ln x x F D xf F F x x F D xf. x (2a) (2b) From the definition in eq. (2b), the relative change in F can be related to its cause x/x as follows: F F S xf x x. (2c) So in order that the F function will not be sensitive to x, the limits of the S x (F) function should be 1 S x F 1. (2d) For a general case, F and x may be vectors consisting of m real-circuit functions and k real parameters, respectively: F x F 1 x, F 2 x,...f m x T, (3) x x 1, x 2,...,x k T. (4) Here, the x-circuit parameter vector can be combined as follows:

3 464 Güneş and Altunç Figure 1. Block diagram of the m cascaded two-ports. x x x, (5a) Where x o,1 k is a dimensioned nominal parameter vector and x is a differential vector with the same dimensions: x x 1, x 2...x k T, x x 1, x 2...x k T. (5b) (5c) Provided that the parameter changes are infinitesimal and the circuit function F is analytical at x x. We have a linear relation between the parameter changes and the resultant change in the single-circuit function F. Thus, for the changes in the k parameters, we obtain the resultant change in F as follows: or, in matrix notation: k F F x x i i i1 F Fx, (6a) (6b) III. SENSITIVITY ANALYSIS BY GAIN FACTORIZATION A. Formulation In the analysis of larger networks, it is often convenient to treat the whole network as a connection of smaller subnetworks and obtain an overall solution by combining individual solutions. One particular case, often occurring, is that of a cascaded network, as shown in Figure 1. The network consists of the inner (index I) two-port embedded between left (index L) and right (index R) two-ports. The two-port of index n is described by its chain matrix T n. Let us assume that our objective is to optimize gain performance G T of the whole network in terms of some parameter x n from the inner two-port. So, the overall transducer power gain G T can be factorized into three terms as follows: G T P L P AVs P AVL P AVs P LI P AVL1 P L P LI, (7a) where all the power components have common definitions and are given in Figure 1. So, where the gradient operator is then given as G T G AL G TI G PR, (7b),,..., x 1 x 2 x k. (6c) If the parameter changes x i are incremental, then the resultant change in a single-circuit function should be expressed as the Taylor series expansion including the 2 nd -order derivatives. Similar reasoning can be applied to the vector cases in eqs. (3), (4), and (5). where G TI is the transducer power gain of the inner two-port, which is the function of the inner two-port parameters x n, Z SI, and Z LI terminations; G AVL is the available power gain of the left-hand subnetwork, which does not depend upon any inner two-port s parameter x n and can be expressed as the chain parameters and source impedance of the left-hand subnetwork:

4 Gain-Sensitivity Analysis for Cascaded Two-Ports 465 Figure 2. T-type amplifier. n1 G AL G Ai i1 where T L A L C L R S A L C L Z S Real B 2 L D L Z S A L C L Z S, (7c) B L D L A 1 B 1 C 1 D 1 A 2 B 2 C 2 D 2 A n1 C n1 B n1 D n1. (7d) Similarly, G PR in eq. (7b) is the operating power gain of the right-hand subnetwork, which consists of the final mn cascaded two-ports and is independent from the inner-network parameters. It can be expressed in terms of the chain parameters and load of the right-hand subnetwork: m G pr G pi in1 where T R A R C R A n1 B R D R B n1 R L D R C R Z L Real A RZ 2 L B R C R Z L D R, (8a) C n1 D n1 A n2 B C n2 D n2 A m B m C m D (8b) m. Thus, using the equality in eq. (7b), the derivation of the overall transducer power gain with respect to the x n parameter of the n th two-port of m cascaded twoports can be given as x n G AL n x n G PR (9) and gain sensitivity matrix of the m-cascaded network can be expressed as D x x1, xn, xm, (10a) where the gradient of the n th two-port xn (G T ) can be given by xn T p 1 p. (10b) Here, the n th port is assumed to have sensitivity parameters. B. Formation of the Gain-Sensitivity Matrix for the IMC (T-Type) Transistor/ OMC (T-Type) Circuit A microwave amplifier with a T-type distributed matching circuit can be considered as the cascaded connection of seven two-ports between the source (V S, Z S ) and the load (Z L ), as shown in Figure 2. There are two typical two-ports used in the matching circuits whose sensitivities should be taken into account in the overall sensitivity matrix: (i) the series-line and (ii) the parallel-line two-ports, whose respective chain matrices can be given by T S cos jzo sin j sin cos, (11a) Z 0

5 466 Güneş and Altunç T P 1 0 j Z 0 tan 1, (11b) where and Z 0 are the physical length and the characteristic impedance for a line, respectively. The transducer power gain of a two-port characterized by the chain parameters can be given by G T 4R S R L AZ L B Z S CZ L D 2, (12) where Z S R S jx S and Z L R L jx L are the source and load impedances of the two-port, respectively. Therefore, the gradient of a typical series-line twoport can be written as: where 4R SR L S P 2 S S T Z 0S, (13a) S 2 2 Z 2 Z 0S S 3sin 2 S 0S 2Z 0S S 4 S 5 Z 0Scos 2 S, 4R SR L Z 0S P 2 2Z 0S S 2 3 sin 2 S Z 0S P S 1 S 2 Z 2 Z 0S S 3sin 2 S 0S (13b) S 4 S 5 2 Z sin 2 S, (13c) 0S Z 0S S 4 S 5 Z 0S sin 2 S, (13d) S 1 Z L 2 Z S 2 2R S R L X S X L, (13e) S 2 Z L 2 Z S 2, (13f) S 3 4X S X L Z L 2 Z S 2, (13g) S 4 X L X S, (13h) S 5 X S Z L 2 X L Z S 2. (13i) Similarly, the gradient vector for a typical parallelline two-port can be expressed as P T P Z 0P, (14a) where 4R SR L 2 T P P 2 Z 0P sin 2 2 T Z 0P tg 3, P 4R SR L 2 Z 0P P 2 2 tg P P Z 0P 1 T 1 Z 0P tg P T 2 Z 0 tg P T 3, T 2 Z 0P tg P 2T 3, (14b) (14c) (14d) T 1 Z L 2 Z S 2 2R S R L X S X L, (14e) T 2 Z L 2 Z S 2, (14f) T 3 X S Z L 2 X L Z S 2. (14g) Using these typical gradient vectors in the formation of the overall sensitivity matrix, the following points should be taken in to consideration: (i) Z S and Z L are the input and output impedances of the left- and right-hand subnetworks, respectively, and (ii) G AVL and G PR should be placed as previously defined with respect to the inner two-port. IV SENSITIVITY ANALYSIS USING THE CHAIN-SENSITIVITY MATRIX A. Formulation This method begins by obtaining the chain-sensitivity matrix of the overall network. The overall chain matrix is given by T T L T I T R, (15) where the sensitivity of the T matrix with respect to the x n parameter of the n th port is given by T x n T L T I x n T R, (16) where T I / x n is the chain-sensitivity matrix of the inner subnetwork with the sensitivity parameters.

6 Gain-Sensitivity Analysis for Cascaded Two-Ports 467 Figure 3. -type amplifier. The chain-sensitivity matrix is employed in the gain-sensitivity formula. For this purpose, the transducer gain can be expressed as where P G T 4R SR L P, (17a) AZ L B Z S CZ L D 2 or P MM* M AZ L B Z S CZ L D (17b) The sensitivity of G T with respect to parameter x n can be written as 4 G x n T T İ i1 4 T İ x n İ1 T* İ T* İ x n, (18a) where T İ stands for one of the overall chain parameters. The gain-sensitivity formula in eq. (18a) can be rewritten in matrix form as x n T İ T İ x n T* İ T * İ x n, (18b) where T İ 4R SR L P 2 Z L M* M* Z S Z L M* Z S M*, T* İ 4R SR L P 2 Z* L M M Z* S Z* L M Z* S M, and because 4R S R L /P 2 is real, we obtain (19a) (19b) T* İ T İ*, and the chain-sensitivity matrix can be defined as (19c) T İ x n A T B C D x n x n x n x n, (20) where we also obtain the following relation: T * İ x n T İ x n*. (21) B. Formation of Gain-Sensitivity Matrix for the IMC (-Type) Transistor/OMC (-Type) Circuit Again we have seven cascaded two-ports between the source and the load in the -type amplifier and the two typical matching circuits: (i) series-line and (ii) parallelline two-ports (Fig. 3). Thus, the two typical chainsensitivity matrices can be expressed as follows: T s sin s j cos s Zos jzo cos s sin, s (22a) Zos T 0 j sin s j sin s 0, (22b) Zos 2 T p j, (22c) Zop sin 2 p Zop T j. (22d) Zop 2 tan p

7 468 Güneş and Altunç While using the above formulas, the order of the two-port in the chain should be taken into account by the relation given in eq. (16). As an example, let us formulate the gain gradient ƒ x5 (G T ) of the fifth port in Figure 3: x5 5 Z 05 (23) for the partial derivative with respect to 5 : 5 T İ T İ 5 T* İ T * İ 5, T 5 T 1 T 2 T 3 T TR T 4 T 5 5 T 6 T İ 5 (24a) 4 sin 5 jz 05 cos 5 T 1 T 2 T 3 T TR T j cos 5 sin T 5 6, Z 05 T 5 T 5, (24b) (24c) T 5 1, 1 T 5 1, 2 T 5 2, 1 T 5 2, 2 T, (24d) 5 4R SR L P 2 Z L M* M* Z S Z L M* Z S M*, (24e) T 5 1, 1 T 5 1, 2 T 5 2, 1 T 5 2, 2 T, (24f) Z* L M M Z* S Z* L M Z* S M, (24g) T 5 1, 1* T 5 1, 2* T 5 2, 1* T 5 2, 2* T }, and for the partial derivative with respect to Z 05 : Z 05 T İ T İ Z 05 T* İ T * İ Z 05, (24h) (25a) T T 5 Z 05 T 1 T 2 T 3 T TR T 4 T 4 Z j sin 5 T 1 T 2 T 3 T TR T j sin T 6, (25b) Z 05 T Z05 T Z 05, T İ Z 05 T Z05 1, 1 T Z05 1, 2 T Z05 2, 1 T Z05 2, 2 T, (25c) (25d) Z 05 4R SR L P 2 Z L M* M* Z S Z L M* Z S M*, (25e) T Z05 1, 1 T Z05 1, 2 T Z05 2, 1 T Z05 2, 2 T, Z* L M M Z* S Z* L M Z* S M, (25f) T Z05 1, 1* T Z05 1, 2* T Z05 2, 1* T Z05 2, 2* T }. V. COMPUTED RESULTS (25h) A general computer program for the sensitivity analysis of the T and types of the distributed-parameter microwave amplifier iss developed, based on the theory presented above. In addition, all the results of the analytical formulas were verified by the numerical perturbation method. The classical definition [eq. (2b)] of the sensitivity of the function with respect to parameter x was used, which led to the following expressions in the perturbation-method case: S x x F F x x F x x, (26) it was assumed that x 10 4 cm and 10 4 The validity of the techniques was tested on the T and types of the distributed-parameter circuits. Firstly, gain-sensitivity analyses of the IMC and OMC circuits of the T and distributed-parameter amplifiers are made and computed separately. In the analysis, it is assumed that the IMC circuits to be loaded by the complex conjugate of the transistor s required source impedance, Z* STR ( i ), and the OMC

8 Gain-Sensitivity Analysis for Cascaded Two-Ports 469 TABLE I. Physical Lengths and Characteristic Impedances for Matching Circuits IMC (T-Type) cm cm cm Z Z Z OMC (T-Type) cm cm cm Z Z Z IMC (-Type) cm cm cm Z Z Z OMC (-Type) cm cm cm Z Z Z circuits to be driven by the complex conjugate of the transistor s required load impedance, Z* LTR ( i ). In all cases, the three methods gave almost the same results, thus confirming validity of the derived analytical formulas; their plots are not included for the sake of brevity. Here, only the plots of the sensitivities for the T and types of amplifiers are presented, in comparison to each other. The design of these amplifiers was originally presented by Güneş and Cengiz in [10]. The design targets of the noise, input VSWR, and gain for both types of amplifiers were F req 0,46 db (N 1, 12) and V ireq 1, G Treq 12 db (N 15, 86), respectively, in the frequency range 2 11 GHz. NE329S01 was used as the active element. The nominal values of the transmission-line segments used in the matching circuits are given in Table I and the corresponding gain-frequency characteristics of the T and types of circuits are given in Figures 4(a) and 4(b), and Tables II and III, respectively. The desired behavior of the sensitivity of an element can be defined as the magnitude of its normalized sensitivity to take values between zero and unity, as in eq. (2d): 0 S x F 1. (27) The sensitivity-frequency curves of the T and types of amplifiers with respect to the parameters of the matching circuits are given in Figures 5 16, in comparison to each other. As seen from the figures, gain can be considered not sensitive to the physical lengths and characteristic impedances of the lines in both configurations along the operation bandwidth, except the sensitivity with respect to the physical lengths and characteristic impedances s, Z os of the fifth element Figure 4. Gain-frequency characteristic of the (a) T-type amplifier and (b) -type amplifiers. in the configuration. Thus, the T-type amplifier is preferable to the -type amplifier. However, as it is relatively very simple to form the derivative vectors of [ /T i ] and [T i / x n ], the chain-sensitivity matrix method is preferable to the gain-factorization method. A clear advantage of the analytical methods can be found in [5], where a comparison of the CPU time with respect to the number of variables for the two methods is graphically presented. VI. CONCLUSION The significance of this work can be discussed in two aspects: (i) the contributions of the sensitivity analysis developed for the cascaded two-ports to the circuit theory and (ii) the importance of the problem focused upon in this work, with regard to applications. As far as the developed theory is concerned, a simple method called the chain-sensitivity matrix has been suggested to formulate the gain sensitivities with respect to the circuit parameters of the cascaded twoports due to the following reasons: (i) this method is relatively simple, as compared to the gain factorization method in the literature [4]; (ii) it is easily extendable to the sensitivities of the other performance measure functions, such as noise figure, input VSWR, and output VSWR; (iii) its fundamentals are also to extendable to other types of configurations. As far as applications are concerned, the significance of the work can be summarized briefly as follows. Efficient, gradient-based optimization techniques require calculation of the circuit-performance sensitivities. In this work, analytical formulas for direct computation are presented in a form of compatible with the analysis of the given circuit. These

9 470 Güneş and Altunç TABLE II. Gain-frequency Characteristics of the T-Type Amplifier Frequency (GHz) G T (ratio) TABLE III. Gain-frequency Characteristics of the -Type Amplifier Frequency (GHz) G T (ratio) Figure 5. Sensitivity-frequency curves of the T-type amplifier with (a) 1 variation and (b) Z o1 Figure 6. Sensitivity-frequency curves of the -type amplifier with (a) 1 variation and (b) Z o1 Figure 7. Sensitivity-frequency curves of the T-type amplifier with (a) 2 variation and (b) Z o2

10 Gain-Sensitivity Analysis for Cascaded Two-Ports 471 Figure 8. Sensitivity-frequency curves of the -type amplifier with (a) 2 variation and (b) Z o2 Figure 9. (a) Sensitivity-frequency curve of the T-type amplifier with 3 variation and (b) Z o3 Figure 10. Sensitivity-frequency curves of the -type amplifier with (a) 3 variation and (b) Z o3 Figure 11. Sensitivity-frequency curve of the T-type amplifier with (a) 4 variation and (b) Z o4

11 472 Güneş and Altunç Figure 12. Sensitivity-frequency curves of the -type amplifier with (a) 4 variation and (b) Z o4 Figure 13. Sensitivity-frequency curve of the T-type amplifier with (a) 5 variation and (b) Z o5 Figure 14. Sensitivity-frequency curves of the -type amplifier with (a) 5 variation and (b) Z o5 variation [Color figure can be viewed in the online issue, which is available at Figure 15. Sensitivity-frequency curve of the T-type amplifier with (a) 6 variation and (b) Z o6 wiley.com]

12 Gain-Sensitivity Analysis for Cascaded Two-Ports 473 Figure 16. Sensitivity-frequency curves of the -type amplifier with (a) 6 variation and (b) Z o6 variation, [Color figure can be viewed in the online issue, which is available at sensitivities can be related to any passive or active parameter of the network. In the CAD of amplifiers with high gain, the direct expressions for gain sensitivities improve the speed of computations, and also safeguard against possible numerical difficulties associated with the perturbation method. The other important application area with regard to sensitivity analysis, without a doubt, is tolerance analysis. Since sensitivity can be used to predict the response variations due to changes in the values of the circuit elements for various reasons, this analytical work is expected to be a good reference for the computer-aided optimal-tolerance assignment, optimal centering, and postproduction tuning. REFERENCES 1. D.A. Calahan, Computer-aided circuit design, McGraw-Hill, New York, L.O. Chua and P.M. Lin, Computer-aided analysis of electronic crcuits, algorithms, and computational techniques, Prentice Hall, New York, J. Vlach and K. Singhal, Computer methods for circuit analysis and design, Van Nostrand Reinhold, New York, M. Güneş, The design of low-noise microwave bipolar transistor amplifiers, Ph.D. thesis, submitted to University of Bradford, UK, O.P. Paixao and A.K. Jastrzebski, Analysis and sensitivities of noisy network using nodal approach, IEEE Trans Microwave Theory Tech 42 (1994), A. Dobrowolski, Introduction to computer methods for microwave circuit analysis and design, Artech House, Norwood, MA, 1991, pp S. Altunç, Gain sensitivity analysis for the cascaded two-ports and application to the distributed amplifiers, M.Sc. thesis, submitted to Yildiz Technical University, 2003 (in Turkish). 8. F. Güneş and C. Tepe, Gain-bandwidth limitations of microwave transistor, Int J RF and Microwave CAE 12 (2002), B.S. Yarman, A. Aksen, and A. Kilincç, An immittance-based tool for modelling passive one-port devices by means of Darlington equivalents, Int J Electron and Commun AEÜ, 55 (2001), F. Güneş and Y. Cengiz, Optimization of a microwave amplifier using neural performance data sheets with genetic algorithms, Lecture notes in computer science, Springer-Verlag, New York, 2003, pp

13 474 Güneş and Altunç BIOGRAPHIES Filiz Güneş received her M.S. degree in electronic and communication engineering from the Istanbul Technical Univrsity in She attained her PhD. degree in communication engineering from Bradford University, London, in She worked as a research fellow at the same university between 1979 and 1983 with contracts from the European Space Agency and British Defence Minister in the areas of the propagation and electromagnetic compatibility. Since 1983, she has been with the Yildiz Technical University, where she is currently a Full Professor and head of the Electronic and Communication Engineering Department. Her current research interests are in the areas of multivariable network theory, device modeling, computer-aided circuit design, microwave amplifiers, microwave filters, broadband matching circuits, monolithic microwave integrated circuits, and electromagnetic compatibility. Serhat Altunç was born in Turkey. He earned an undergraduate degree in Electronics and Communications Engineering. After receiving his M. Engg. Degree from Istanbul Technical University, he continued his graduate study at Yildiz Technical University receiving a MS (Communication Engineering, 2003). He worked as a Research Assistant at Yildiz Technical University from 2001 to He has published two papers.