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The Use of Scattering Parameters in Amplifier Design

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The Use of Scattering Parameters in Amplifier Design Scholars' Mine Masters Theses Student Theses and Dissertations 1971 The use of scattering parameters in amplifier design Yousef Neman-Ebrahim Follow this and additional works at: https://scholarsmine.mst.edu/masters_theses Part of the Electrical and Computer Engineering Commons Department: Recommended Citation Neman-Ebrahim, Yousef, "The use of scattering parameters in amplifier design" (1971). Masters Theses. 5104. https://scholarsmine.mst.edu/masters_theses/5104 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. THE USE OF SCATTERING PARAMETERS IN AMPLIFIER DESIGN BY YOUSEF NEMAN-EBRAHIM, 1945- A THESIS Presented to the Faculty of the Graduate School of the UNIVERSITY OF MISSOURI-ROLLA in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING T2680 Rolla, Missouri 39 pages c.l 1971 Approved by ~(Advisor)· '-- ii ABSTRACT The scattering-parameter formulation of an active two-port device is developed from the compensation theorem. The use of scattering parameters in graphical developments of forward operating power gain and tunability is considered in detail. Finally, two design examples show the application of the graphical procedures. iii ACKNOWLEDGEMENTS My sincere thanks and gratitude to Dr. Ralph S. Carson whose advice, supervision and interpretations on the more elusive parts of this research were extremely helpful. iv TABLE OF CONTENTS Page ABSTRACT . ii ACKNOWLEDGEMENTS iii LIST OF FIGURES v I. INTRODUCTION . • . 1 A. Scattering Parameter Definitions . 2 II. DEVELOPMENT OF SCATTERING PARAMETERS • 5 A. One-Port . 5 B. Two-Port . 8 III. UNILATERAL CASE 11 A. Power Flow . 11 B. Design Relationships . 13 C. Design Example . 18 IV. NON-UNILATERAL CASE 20 A. Tunability . 20 B. Design Relationships . 23 c. Design Example . 26 V. CONCLUSIONS . 27 BIBLIOGRAPHY . 28 VITA . 29 v LIST OF FIGURES Figure Page l Voltage and current definition for the scattering parameter formulation • 30 2 Development of the scattering parameters for the one-port network • . 31 3 Development of the scattering parameters for the two-port network . 32 4 Unilateral design example . 33 5 Non-unilateral design example . 34 1 I. INTRODUCTION The difficulty of measuring the commonly accepted amplifier design parameters, such as the admittance or y parameters, increases rapidly as the frequency of interest is extended above about 100 MHz. When the desired para­ meters must be referred either to a short or an open cir­ cuit above 100 MHz, a wide-band measurement system becomes essentially impossible. A convenient way to overcome this problem is to refer the measurements to the characteristic impedance of a transmission line. A set of informative parameters which can readily be measured in terms of travelling waves on a transmission line are the scattering or "s" parameters. The ease with which scattering parameters can be measured makes them especially well suited for describing transistors and other active devices. Measuring most other parameters calls for the input and output of the device to be successively open and short circuited. This is difficult to do even at RF frequencies where inductance and capacitance make short and open circuits difficult to obtain. At higher frequencies these measurements typically require tuning stubs, separately adjusted at each measurement fre­ quency, to reflect short-or open-circuit conditions to the device terminals. Not only is this inconvenient and tedious, but a tuning stub shunting the input or output may cause a 2 transistor to oscillate, making the measurement difficult and invalid. s parameters, on the other hand, are usually measured with the device imbedded between a 50-ohm load and source, and there is very little chance for oscillations to occur. More importantly, scattering-parameter measure- ments are extremely simple. There are no adjustments to be made after the initial calibration. Because of the 50-ohm wide band terminations, it is extremely stable. A. Scattering-Parameter Definitions To apply the concept of scattering parameters to a dis- cussion of a two-port, consider the terminated two-port network N of Fig. 1. For the two-port, there are two sets of scattering variables to be considered, namely, v 1 i and v 1r at port 1 and v 2 i and v 2 r at port 2. If the incident voltages are chosen as independent variables and the reflec- ted voltages as dependent variables, then the scattering parameters s .. (i,j = 1,2) for the two-port network of Fig. 1 J..] are given by (1) where vlr vlr = sl2 = sll vli v2i v2i=o vli=o (2) v2r v2r = s22 = s21 vli v2i v .=0 v2i=() lJ.. 3 In matrix notation, (1) becomes vlr vli = [s] (3) v2r v2i where sll sl2 = (4) [ sJ s21 s22 is named the scattering matrix of the two-port N. There are special names for the scattering parameters described by (2). These are: sll = input reflection parameter s21 = forward transmission parameter sl2 = reverse transmission parameter s22 = output reflection parameter It is to be noted that there is a dual to (3) which describes the relationship between incident and reflected components of current, i.e., = (5) where the superscript I is used to denote that the scattering parameters are referred to currents. To distinguish between 4 the scattering parameters referred to currents and those referred to voltages, the s matrix for voltage relation­ ships is sometimes written with a V superscript, as = (6) However, since the scattering parameters are applied in high-frequency situations where 50-ohm transmission-line interconnections and terminations are practical, then it can be shown that = (7) Scattering parameters are being used more frequently in microwave design, because they are easier to measure at high frequencies than other kinds of parameters. They are capable of providing a surprising degree of insight into a measurement or design problem. For these reasons, manu­ facturers of high-frequency transistors and other solid­ state devices are finding it more meaningful to specify their products in terms of scattering parameters than in any other way. 5 II. DEVELOPMENT OF THE SCATTERING PARAMETERS When the scattering concept is applied to lumped cir- cuits, the actual currents and voltages can be separated into scattered components, incident and reflected, according to appropriate definitions. It is desirable, however, that such definitions should lead ultimately to relation- ships which prove useful in the analysis and design of cir- cuits and in laboratory measurements of the parameters which describe the scattering. The representation of a current or a voltage as the sum of an incident term and a reflected term becomes more meaningful through the application of the compensation theorem in the development of the scattering parameters. 1 A. One-Port Consider a one-port network with an input admittance y. If this network is connected to a current source I with a source admittance y 0 as shown in Fig. 2(a), then the voltage V is v = I (8) Redraw the circuit as shown in Fig. 2 (b) • The voltage v can now be written as v v + /J.V (9) = 0 where vo is the voltage that exists if /J.y = 0 and /J.V is the 6 change in voltage when ~y is introduced. Clearly, v = I (10) 0 2yo and ~V can be found from the compensation theorem. It should be noted that the compensation theorem relates the change in the voltage when an admittance is added in a loop, to the voltage that existed originally. As shown in Fig. 2(c) ~y v 0 ~v = (11) 2y +~y 0 where ~y = y - y (12) 0 then I(y - y ) 0 ~v = ( 13) 2Yo(y+yo) thus (9) becomes y - y V = I + I o (14) 2yo 2yo y + Yo Correspondingly, the current I 1 of the one-port is (15) where r 10 is the current that would exist for ~Y = 0, and ~I 1 is the change in current associated with ~y. Redraw Fig. 2(b) while using ~y = 0, as shown in Fig. 2(d), then I = 2 (16) 7 and from Fig. 2 (c) t.I t.v (17) 1 = -yo Substituting (13) into (17) gives I y - Yo = (18) t.Il 2 y + y 0 thus (15) becomes y - y I I 0 (19) Il = 2 2 y + y 0 If the current and voltage are normalized, (14) and (19) become y - y I I 0 - vly0 = + (20) 2/Y0 and y - y I I = -- 0 (21) 2/yo 2/Y0 Define I I. = (22) 1. = 2/Y0 I ( 2 3) I = =- -- r 2'Ya which leads to the result = I. - I ( 2 4) 1. r I = I. + I (25) n 1. r 8 By definition, the scattering parameter s is I y - y r 0 s = I. (26) ]_ The compensation theorem gives immediately the relation between ~V and V and hence the scattering parameter s, 0 so (26) describes the s parameter for the one-port network of Fig. 2. B. Two-Port The above result can be extended to a two-port as shown in the following example.Figure 3(a) shows an equivalent cir- cuit for a two-port network. Redrawing the circuit as Fig. 3(b), and following the procedures for the one-port closely, the circuits in Figs. 3(c) and 3(d) are ob- tained. In order to find the scattering parameters for the two-port network, ~v 1 and ~v 2 must be related to v 10 and v 20 • To achieve these results the following equations are obtained using the circuit in Fig.
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