S-Parameter Techniques – HP Application Note 95-1

Home , Admittance parameters, Electrical network, Gain (electronics), Scattering parameters, Smith Chart

Test & Measurement Application Note 95-1 S-Parameter Techniques Contents 1. Foreword and Introduction 2. Two-Port Network Theory 3. Using S-Parameters 4. Network Calculations with Scattering Parameters 5. Amplifier Design using Scattering Parameters 6. Measurement of S-Parameters 7. Narrow-Band Amplifier Design 8. Broadband Amplifier Design 9. Stability Considerations and the Design of Reflection Amplifiers and Oscillators Appendix A. Additional Reading on S-Parameters Appendix B. Scattering Parameter Relationships Appendix C. The Software Revolution Relevant Products, Education and Information Contacting Hewlett-Packard

Test & Measurement Application Note 95-1 S-Parameter Techniques Foreword HEWLETT-PACKARD JOURNAL This application note is based on an article written for the February 1967 issue of the Hewlett-Packard Journal, yet its content remains important today. S-parameters are an Cover: A NEW MICROWAVE INSTRUMENT SWEEP essential part of high-frequency design, though much else MEASURES GAIN, PHASE IMPEDANCE WITH SCOPE OR METER READOUT; page 2 See Also:THE MICROWAVE ANALYZER IN THE has changed during the past 30 years. During that time, FUTURE; page 11 S-PARAMETERS THEORY AND HP has continuously forged ahead to help create today's APPLICATIONS; page 13 leading test and measurement environment. We continuously apply our capabilities in measurement, communication, and computation to produce innovations that help you to improve your business results. In wireless communications, for example, we estimate that 85 percent of the world’s GSM (Groupe Speciale Mobile) telephones are tested with HP instruments. Our accomplishments 30 years hence may exceed our boldest conjectures. This interactive application note revises and updates the1967 article for online electronic media. It reflects FEBRUARY 1967 the changes in our industry, while reminding us of the February 1967 HP Journal underlying scientific basis for the technology, and takes Cover of issue in which advantage of a potent new information dissemination capability, the World Wide Web. We hope you find this tutorial useful. the original “S-Parameters Theory and Application,” written during Christmas Richard Anderson, holiday 1966, first appeared. HP Vice President and General Manager, HP Journal is now online at: 3 Microwave and Communications Group www.hp.com/go/journal H

Test & Measurement Application Note 95-1 S-Parameter Techniques Introduction 1Linear networks, or nonlinear networks operating with signals Maxwell’s equations sufficiently small to cause the networks to respond in a linear All electromagnetic manner, can be completely characterized by parameters behaviors can ultimately be measured at the network terminals (ports) without regard to explained by Maxwell’s four the contents of the networks. Once the parameters of a basic equations: network have been determined, its behavior in any external ∂B environment can be predicted, again without regard to the ∇⋅ DE= ρ∇×=− ∂ contents of the network. t ∂D S-parameters are important in microwave design because they ∇⋅B= 0∇× H=j+ ∂t are easier to measure and work with at high frequencies than other kinds of parameters. They are conceptually simple, However, it isn’t always analytically convenient, and capable of providing a great possible or convenient to insight into a measurement or design problem. use these equations directly. To show how s-parameters ease microwave design, and how Solving them can be quite you can best take advantage of their abilities, this application difficult. Efficient design requires the use of note describes s-parameters and flow graphs, and relates them approximations such to more familiar concepts such as transducer power gain and as lumped and distributed voltage gain. Data obtained with a network analyzer is used to models. illustrate amplifier design.

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Test & Measurement Application Note 95-1 S-Parameter Techniques Two-Port Network Theory

I 2Although a network may have any number of ports, 1 I2 network parameters can be explained most easily by ++TWO - PORT considering a network with only two ports, an input port V V 1 –NETWORK –2 and an output port, like the network shown in Figure 1. To characterize the performance of such a network, any Port 1 Port 2 of several parameter sets can be used, each of which has certain advantages. Each parameter set is related to a set of Figure 1 four variables associated with the two-port model. Two of these General two-port network. variables represent the excitation of the network (independent variables), and the remaining two represent the response of Why are models needed? the network to the excitation (dependent variables). If the Models help us predict the behavior of components, network of Fig. 1 is excited by voltage sources V1 and V2, the network currents I and I will be related by the following circuits, and systems. 1 2 Lumped models are useful at equations (assuming the network behaves linearly): lower frequencies, where =+ IyVyV1 11 1 12 2 (1) some physical effects can be ignored because they are so =+ small. Distributed models IyVyV2 21 1 22 2 (2) are needed at RF frequencies and higher to account for the increased behavioral impact of those physical effects. 5 H

Test & Measurement Application Note 95-1 S-Parameter Techniques Two-Port Network Theory

2In this case, with port voltages selected as independent variables and port currents taken as dependent variables, the relating parameters are called Two-port models short-circuit admittance parameters, Two-port, three-port, and or y-parameters. In the absence of n-port models simplify the additional information, four input / output response of measurements are required to determine the four parameters active and passive devices y11, y12, y21, y22. Each measurement is made with one port and circuits into “black of the network excited by a voltage source while the other boxes” described by a set port is short circuited. For example, y21, the forward of four linear parameters. transadmittance, is the ratio of the current at port 2 to the Lumped models use voltage at port 1 with port 2 short circuited, as shown in representations such as equation 3. Y (conductances), Z (resistances), and h (a mixture of conductances and resistances). Distributed I2 models use s-parameters y 21 = (transmission and reflection V1 V =0 (output short circuited) 2 (3) coefficients).

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Test & Measurement Application Note 95-1 S-Parameter Techniques Two-Port Network Theory

2If other independent and dependent variables had been chosen, the network would have been described, as before, by two linear equations similar to equations 1 and 2, except that the variables and the parameters describing their relationships would be different. However, all parameter sets contain the same information about a network, and it is always possible to calculate any set in terms of any other set. “Scattering parameters,” which are commonly referred to as s-parameters, are a parameter set that relates to the traveling waves that are scattered or reflected when an n-port network is inserted into a transmission line. Appendix B “Scattering Parameter Relationships” contains tables converting scattering parameters to and from conductance parameters (y), resistance parameters (z), and a mixture of conductances and resistances parameters (h).

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Test & Measurement Application Note 95-1 S-Parameter Techniques 3 Using 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 opened and short circuited. This can be hard to do, especially at RF frequencies where lead 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 frequency, 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 transistor to oscillate, making the measurement invalid.

S-parameters, on the other hand, are usually measured with the device imbedded between a 50 Ω load and source, and there is very little chance for oscillations to occur. 8

Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters

3Another important advantage of s-parameters stems from the fact that traveling waves, unlike terminal voltages and currents, do not vary in magnitude at points along a lossless transmission line. This means that scattering parameters can be measured on a device located at some distance from the measurement transducers, provided that the measuring device and the transducers are connected by low-loss transmission lines. Derivation Transmission and Reflection Generalized scattering parameters have been defined by When light interacts with a K. Kurokawa [Appendix A]. These parameters describe lens, as in this photograph, the interrelationships of a new set of variables (ai , bi). part of the light incident on The variables ai and bi are normalized complex voltage waves the woman's eyeglasses is incident on and reflected from the ith port of the network. reflected while the rest is They are defined in terms of the terminal voltage V , the transmitted. The amounts i reflected and transmitted terminal current Ii , and an arbitrary reference impedance Zi , are characterized by optical where the asterisk denotes the complex conjugate: reflection and transmission coefficients. Similarly, * VZIiii+ VZIiii- scattering parameters are a i= (4)bi= (5) measures of reflection and 2ReZ 2 Re Z i i transmission of voltage waves through a two-port 9 electrical network.

Test & Measurement Application Note 95-1 S-Parameter Techniques 3 Using S-Parameters For most measurements and calculations it is convenient Scattering parameters to assume that the reference impedance Zi is positive relationship to optics and real. For the remainder of this article, then, all Impedance mismatches variables and parameters will be referenced to a single between successive elements in an RF circuit positive real impedance, Z0. relate closely to optics, The wave functions used to define s-parameters for a where there are successive two-port network are shown in Fig. 2. differences in the index of refraction. A material’s ZS characteristic impedance, Z0, is inversely related to a1 a2 the index of refraction, N: V TWO - PORT Z S NETWORK L e b1 b2 Z = 1 0 377 N

The s-parameters s11 and Figure 2 s22 are the same as optical Two-port network showing incident waves reflection coefficients; (a1, a2) and reflected waves (b1, b2) used in s12 and s21 are the same s-parameter definitions. The flow graph for as optical transmission this network appears in Figure 3. coefficients. 10 H

Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters

3The independent variables a and a are normalized 1 2 incident voltages, as follows: V VIZ110+voltage wave incident on port 1 i1 a 1=== (6) 2ZZ00Z0 + VIZ220voltage wave incident on port 2 Vi2(7) a 2=== 2ZZ00Z0

Dependent variables b1, and b2, are normalized reflected voltages:

VIZ110−voltage wave reflected from port 1 Vr1 b1===(8) 2ZZ00Z0 V VIZ220−voltage wave reflected from port 2 r2 b2===(9) 2ZZ00Z0

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Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters

3The linear equations describing the two-port network are Limitations of then: lumped models b=+sasa At low frequencies most 1 111 12 2 (10) =+ circuits behave in a b2 s21a1 s22 a2 (11) predictable manner and can be described by a The s-parameters s11, s22, s21, and s12 are: group of replaceable, lumped-equivalent black b1 (12) boxes. At microwave s11= = Input reflection coefficient with a1 the output port terminated by a frequencies, as circuit a 2=0 matched load (ZL=Z0 sets a2=0) element size approaches the wavelengths of the b 2 = Output reflection coefficient (13) operating frequencies, s22= with the input terminated by a such a simplified type a 2 a =0 1 matched load (ZS=Z0 sets Vs=0) of model becomes inaccurate. The physical b 2 = Forward transmission (insertion) (14) arrangements of the s 21 = gain with the output port circuit components can a1 a = 0 2 terminated in a matched load. no longer be treated as b black boxes. We have to 1 (15) s12 = = Reverse transmission (insertion) use a distributed circuit a 2 gain with the input port element model and a1=0 terminated in a matched load. 12 s-parameters. H

Test & Measurement Application Note 95-1 S-Parameter Techniques 3Using S-Parameters Notice that

V1 −Z0 b1 I1 ZZ10− s11 == = (16) a1 V1 ZZ10+ +Z S-parameters I0 1 S-parameters and distributed models ()1 + s ZZ= 11 provide a means of and 10− (17) ()1 s11 measuring, describing, and characterizing circuit elements when V1 where Z1 = is the input impedance at port 1. traditional lumped- I1 equivalent circuit models cannot predict This relationship between reflection coefficient and impedance circuit behavior to the is the basis of the Smith Chart transmission-line calculator. desired level of Consequently, the reflection coefficients s11 and s22 can be accuracy. They are used plotted on Smith charts, converted directly to impedance, and for the design of many easily manipulated to determine matching networks for products, such as optimizing a circuit design. cellular telephones.

Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters Animation 1 3 Transformation between Smith Chart Transformation the impedance plane and the Smith Chart. Click The movie at the right animates the over image to animate. mapping between the complex impedance plane and the Smith Chart. The Smith Showing transformations Chart is used to plot reflectances, but the graphically To ease his RF design circular grid lines allow easy reading of work, Bell Lab’s Phillip H. the corresponding impedance. As the Smith developed increas- animation shows, the rectangular grid ingly accurate and power- lines of the impedance plane are ful graphical design aids. One version, a polar co- transformed to circles and arcs on the ordinate form, worked for Smith Chart. all values of impedance components, but Smith Vertical lines of constant resistance on the impedance plane suspected that a grid with are transformed into circles on the Smith Chart. Horizontal orthogonal circles might lines of constant reactance on the impedance plane are be more practical. In 1937 transformed into arcs on the Smith Chart. The transformation he constructed the basic between the impedance plane and the Smith Chart is nonlinear, Smith Chart still used today, using a transforma- causing normalized resistance and reactance values greater tion developed by co- than unity to become compressed towards the right side of the workers E.B. Ferrell and Smith Chart. J.W. McRae that accom- modates all data values 14 from zero to infinity.

Test & Measurement Application Note 95-1 S-Parameter Techniques 3Using S-Parameters Advantages of S-Parameters The previous equations show one of the important advantages of s-parameters, namely that they are simply gains and reflection coefficients, both Digital pulses familiar quantities to engineers. Digital pulses are By comparison, some of the y-parameters comprised of high-order described earlier in this article are not so familiar. harmonic frequencies For example, the y-parameter corresponding that determine the shape of the pulse. A short to insertion gain s21 is the ‘forward pulse with steep edges trans-admittance’ y21 given by equation 3. Clearly, insertion has a signal spectrum gain gives by far the with relatively high power levels at very high greater insight into frequencies. As a result, the operation of some elements in the network. modern high-speed digital circuits require characterization with distributed models and s-parameters for accurate performance prediction. 15 H

Test & Measurement Application Note 95-1 S-Parameter Using S-Parameters Techniques

3Another advantage of s-parameters springs from the simple relationship between the variables a1, a2 , b1, and b , and various power waves: Radar 2 The development 2 of radar, which a1 = Power incident on the input of the network. uses powerful signals = Power available from a source impedance Z0. at short wavelengths to detect small objects at 2 long distances, provided a2 = Power incident on the output of the network. = a powerful incentive for Power reflected from the load. improved high frequency design methods during 2 World War II. The design b1 = Power reflected from the input port of the network. methods employed at = Power available from a Z0 source minus the power delivered to the input of the network. that time combined distributed measurements 2 and lumped circuit b2 = Power reflected from the output port of the network. design. There was an = Power incident on the load. urgent need for an = Power that would be delivered to a Z0 load. efficient tool that could integrate measurement and design. The Smith Chart met that need. 16 H

Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters

3The previous four equations show that s-parameters are simply related to power gain and mismatch loss, quantities which are often of more interest than the corresponding voltage functions: 2 Power reflected from the network input s11 = Power incident on the network input

2 Power reflected from the network output s22 = Power incident on the network output

2 Power delivered to a Z0 load s21 = Power available from Z0 source = Transducer power gain with Z0 load and source

2 = s12 Reverse transducer power gain with Z0 load and source

17 H Test & Measurement Application Note 95-1 Network Calculations with S-Parameter Techniques 4Scattering Parameters Signal Flow Graphs References Scattering parameters turn out to be particularly convenient J. K. Hunton, ‘Analysis of in many network calculations. This is especially true for power Microwave Measurement and power gain calculations. The transfer parameters s12 and Techniques by Means of s21 are a measure of the complex insertion gain, and the Signal Flow Graphs,’ IRE Transactions on driving point parameters s11 and s22 are a measure of the input and output mismatch loss. As dimensionless expressions of Microwave Theory and gain and reflection, the s-parameters not only give a clear and Techniques, Vol. MTT-8, meaningful physical interpretation of the network performance, No. 2, March, 1960. but also form a natural set of parameters for use with signal N. Kuhn, ‘Simplified Signal flow graphs [See references here and also in Appendix A ]. Flow Graph Analysis,’ Of course, it is not necessary to use signal flow graphs in order Microwave Journal, Vol. 6, to use s-parameters, but flow graphs make s-parameter No. 11, November, 1963. calculations extremely simple. Therefore, they are strongly recommended. Flow graphs will be used in the examples that follow.

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Test & Measurement Application Note 95-1 Network Calculations with S-Parameter Techniques Scattering Parameters 4In a signal flow graph, each port is represented by two nodes. Node an VZS 0 represents the wave coming into the bS= ZZS +0 device from another device at port n, a1 b2 and node b represents the wave s n 1 21 ZZ- leaving the device at port n. The Γ=L 0 s11 s22 L complex scattering coefficients are ZZL+0 ZZ- then represented as multipliers on Γ=S 0 S branches connecting the nodes within ZZS+0 b s a the network and in adjacent 1 12 2 networks. Fig. 3, right, is the flow graph representation of the system of Fig. 2. Figure 3 Figure 3 shows that if the load reflection coefficient ΓL is Flow graph for zero (ZL = Z0) there is only one path connecting b1 to a1 two-port network (flow graph rules prohibit signal flow against the forward appearing in Figure 2. direction of a branch arrow). This confirms the definition of s11:

b1 s11 = a1 ab22==ΓL0 19 H

Test & Measurement Application Note 95-1 Network Calculations with S-Parameter Techniques Scattering Parameters

Better Smith Charts 4The simplification of network analysis by flow graphs results On the copyrighted Smith from the application of the “non-touching loop rule.” This rule Chart, curves of constant applies a generalized formula to determine the transfer standing wave ratio, function between any two nodes within a complex system. The constant attenuation, and non-touching loop rule is explained below. constant reflection coefficient are all circles The Nontouching Loop Rule ∑Tkk∆ coaxial with the center of The nontouching loop rule provides a the diagram. Refinements k to the original form have simple method for writing the solution of T = enhanced its usefulness. any flow graph by inspection. The solution ∆ In an article published in T (the ratio of the output variable to the 1944, for example, Smith input variable) is defined, where: described an improved th version and showed how Tk = path gain of the k forward path to use it with either ∆ = 1 – Σ(all individual loop gains) impedance or admittance + Σ(loop gain products of all possible coordinates. More recent combinations of 2 nontouching loops) improvements include – Σ(loop gain products of all possible double Smith Charts for combinations of 3 nontouching loops) impedance matching and + . . . a scale for calculating phase distance. th 20 ∆k = The value of ∆ not touching the k forward path.

Test & Measurement Application Note 95-1 Network Calculations with S-Parameter Techniques Scattering Parameters ss11 12 4A path is a continuous succession of branches, and a forward path is a path connecting the input node to the output node, ss21 22 where no node is encountered more than once. Path gain is () the product of all the branch multipliers along the path. A loop S-parameters & Smith Charts is a path that originates and terminates on the same node, no Invented in the 1960’s, node being encountered more than once. Loop gain is the S-parameters are a way product of the branch multipliers around the loop. to combine distributed For example, in Figure 3 there is only one forward path from design and distributed measurement. s and s , bs to b2, and its gain is s21. There are two paths from bs to b1; 11 22 their path gains are s s G and s respectively. There are the two s-parameters 21 12 L 11 typically represented using three individual loops, only one combination of two Smith Charts, are similar to nontouching loops, and no combinations of three or more lumped models in many nontouching loops. Therefore, the value of D for this network respects because they are is related to the input DGGGGGG=-1(s + s s + s) + s s impedance and output 11SLSLLS 21 12 22 11 22 impedance, respectively. The Smith Chart performs The transfer function from b to b is therefore s 2 a highly useful translation b s21 between the distributed 2 = and lumped models and is bs D used to predict circuit and 21 system behavior.

Test & Measurement Application Note 95-1 Network Calculations with S-Parameter Techniques 4Scattering Parameters Transducer Power Gain Using scattering parameter flow-graphs and the non-touching loop rule, it is easy to calculate the transducer power gain with an arbitrary load and source. In the following equations, the load and source are described by their reflection coefficients GL and GS, respectively, referenced to the real characteristic impedance Z0. Transducer power gain:

Power delivered to the load PL GT== Power available from the source PavS PPL =-()()incident on load P reflected from load 22 =-b2()1GL 2 bS PavS = 2 ()1-GS 2 b2 22 GT =-()()11GGSL - bS 22 H

Test & Measurement Application Note 95-1 Network Calculations with S-Parameter Techniques Scattering Parameters

4Using the non-touching loop rule, b 2= s21 bSSLLSSL1 −−−ssssss11ΓΓ 22 21 12 ΓΓΓΓ + 11 22 Obtaining maximum = s21 −−−ΓΓ ΓΓ performance ()()11ssss11SL 22 21 12 LS S-parameters are used to characterize RF and 22 2 microwave components s21 ()()11−−ΓΓSL (18) that must operate G= together, including T 2 ()()11−−−ssss11ΓΓSL 22 21 12 ΓΓ LS amplifiers, transmission lines, and antennas (and Two other parameters of interest are: free space). Because s-parameters allow the 1) Input reflection coefficient with the output interactions between termination arbitrary and Zs = Z0. such components to be simply predicted and bss()1 −+ΓΓ ss calculated, they make it s′ ==1 11 22LL 21 12 11 a 1 −sΓ possible to maximize 1 22 L (19) performance in areas ssΓ =+s21 12 L such as power transfer, 11 −Γ directivity, and 1 s22 L 23 frequency response. H

Test & Measurement Application Note 95-1 Network Calculations with S-Parameter Techniques Scattering Parameters 42) Voltage gain with arbitrary source and load impedances V2 AV=V1=()a1+b1Z0=Vi1+Vr1 V1 V=+()ab Z=V+V 2220ir22 Waveguides ab=Γ 22L A radar system delivers a b1=s1′1a1 large amount of energy from a microwave (µW) (20) source to the transmitting b2()1 +ΓΓLLs21()1 + antenna. The high field AV= = as1()1 +1′1(11−ss22 ΓL)(+11′) strengths cause short circuits in standard wires, Appendix B contains formulas for calculating many often-used cabling, and coax, so waveguides are used. network functions (power gains, driving point characteristics, These hollow metal tube etc.) in terms of scattering parameters. Also included are constructions conduct µW conversion formulas between s-parameters and h-, y-, and energy much like a z-parameters, which are other parameter sets used very often plumbing system. In the for specifying transistors at lower frequencies. design of waveguides, we can test for signal reflections and transmission 24 quality with s-parameters.

Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Using Scattering Parameters Traveling wave amplifier 5 S-parameters are extensively The remainder of this application note will show with several used for designing RF/µW examples how s-parameters are used in the design of transistor circuits such as the HP TC702 amplifiers and oscillators. To keep the discussion from becoming distributed traveling wave bogged down in extraneous details, the emphasis in these amplifier (TWA) enlarged examples will be on s-parameter design methods, in the photograph below. and mathematical manipulations will The frequency-dependent be omitted wherever possible. impedances (or dispersion) in this integrated circuit can The HP 83017A microwave not be modeled by lumped- system amplifier achieves equivalent circuit elements, 0.5–26.5 GHz bandwidth by but s-parameters can incorporating the HP TC702 accurately characterize the GaAs MESFET TWA IC. amplifier's response.

Test & Measurement Application Note 95-1 Measurement of S-Parameter Techniques S-Parameters 6Most design problems will begin with a 1700 MHz tentative selection of a device and the measurement of its s-parameters. Figures 4a – 4e, which appear to the right and on the next two pages, are a set of oscillograms showing complete s-parameter data btween 100 MHz and 1.7 GHz for a 2N3478 transistor in the common-emitter configuration. These graphs are the results of swept- S11 frequency measurements made with the classic HP 8410A microwave network analyzer. They were originally published as part of the 1967 HP Journal article. 100 MHz Measurements made with a modern Figure 4a network analyzer are presented at the end of this section. While the s11 of a 2N3478 transistor measured with the measurement tools have changed over classic HP 8410A network analyzer. Outermost the past 30 years, the basic circle on Smith Chart overlay corresponds to measurement techniques have not. |s11| = 1. The movement of s11 with frequency is approximately along circles of constant resistance, indicative of series capacitance and 26 inductance.

Test & Measurement Application Note 95-1 Measurement of S-Parameter Techniques 6 S-Parameters |S12| 100 MHz 10 dB/cm –30 dB

–110° S12 S22 50°/cm

1700 MHz 100 MHz 1700 MHz Figure 4b Figure 4c

Displayed on the same scale as Figure 4a, Magnitude and phase of s12. While the s22 moves between the indicated frequencies phase of s12 is relatively insensitive to the roughly along circles of constant conductance, frequency, the magnitude of s12 increases characteristic of a shunt RC equivalent circuit. about 6dB/octave.

Test & Measurement Application Note 95-1 Measurement of S-Parameter Techniques 6S-Parameters |S21| 10 dB/cm 0 dB 0 dB |S21| 10 dB/cm S S 21 21 +20° 50°/cm +90° 50°/cm 100 MHz 1700 MHz 100 MHz 1700 MHz Figure 4d – Magnitude and phase of s21. Figure 4e – Removing Linear Phase Shift. The magnitude of s21 decays with a slope Magnitude and phase of s21 measured of about 6 dB/octave, while the phase with a line stretcher adjusted to remove decreases linearly above 500 MHz. the linear phase shift above 500 MHz.

Test & Measurement Application Note 95-1 Measurement of S-Parameter Techniques S-Parameters Figure 4f 6In Fig. 4f, the magnitude of s21 from Fig. 4d is Top curve: |s | from Fig. 4 is replotted replotted on a logarithmic frequency scale, 21 on a logarithmic frequency scale. Data along with additional data on s21 below below 100 MHz was measured with an 100 MHz, measured with a vector voltmeter. HP 8405A vector voltmeter. The bottom The magnitude of s21 is essentially constant curve u, the unilateral figure of merit, to 125 MHz, and then it rolls off at a slope of calculated from s-parameters. 6 dB/octave. The phase of s21, as seen in |s21| Fig. 4d, varies linearly with 20 10 frequency above about 500 MHz. By adjusting a calibrated line 10 3.162 stretcher in the network 0 1 analyzer, a compensating linear phase shift was introduced, and –10 .3162 the phase curve of Fig. 4e –20 .1 resulted. To go from the phase u Magnitude curve of Fig. 4d to that of Fig.4e Magnitude (dB) –30 .03162 required 3.35 cm of line, that is equivalent to a pure time 10 MHz 100 MHz 1 GHz 10 GHz delay of 112 picoseconds. Frequency 29

Test & Measurement Application Note 95-1 Measurement of S-Parameter Techniques 6 S-Parameters The s-parameters of an 2N3478 transistor shown in Figures 4a through 4f were measured with the classic HP 8410A network analyzer. In Figures 5a through 5e, the s-parameters of an 2N3478 transistor are shown re-measured with a modern HP 8753 network analyzer. Figures 5a through 5e represent the actual s-parameters of this transistor between 0.300 MHz and 1.00GHz. S11

Figure 5a S22 S-parameters of 2N3478 transistor in common-emitter configuration, measured by an HP 8753 network analyzer. This plot shows s11 and s22 on a Smith Chart. The markerset at 47 MHz represents the -3 dB gain roll off point of s21. The frequency index of this point is referenced in the other plots. 31 H

Test & Measurement Application Note 95-1 Measurement of S-Parameter Techniques 6 S-Parameters

S S12 21 Magnitude (dB)

Frequency Phase (deg)

Figure 5b A plot of the magnitude and phase of s12. While the Figure 5c phase of s12 depends only weakly on the frequency, A polar plot of s21. The frequency the magnitude increases rapidly at low frequencies. marker shown is at the -3 dB point. Both the phase angle and magnitude decrease dramatically as the frequency is increased. 32

Test & Measurement Application Note 95-1 Measurement of S-Parameter Techniques S-Parameters

6 Vector Network Analyzers In Fig. 5d, the magnitude of s21 from Fig. 5c is replotted on a logarithmic frequency scale. The magnitude of s21 is Most modern design essentially constant to 75 MHz, and then rolls off at a slope projects, RF through lightwave, use of 6 dB/octave. The phase angle of s21 as seen in Fig. 5e varies linearly with frequency above 500 MHz. To better sophisticated simulation characterize phase distortion, a compensating linear phase software to model system performance from shift was introduced electronically in the network analyzer. components through This established an accurate calibration for measuring the subsystems. These device, resulting in the second phase curve of Figure 5e. programs require complete s-parameter data on each To go from the first phase curve of Fig. 5e to the second phase component. Measurements curve required removing a pure time delay of 167 picoseconds. are made with a VNA, Thirty years ago this operation was accomplished by Vector Network Analyzer, de-embedding 5.0 cm of line using a calibrated line stretcher. an instrument that Today it’s performed by software in the network analyzer. accurately measures the s-parameters, transfer After removal of the constant-delay, or linear-phase, function, or impedance component, the phase angle of s21 for this transistor characteristic of linear (Fig. 5e) varies from 180° at dc to +90° at high frequencies, networks across a broad passing through +135° at 75 MHz, the -3 dB point of the range of frequencies. magnitude curve. In other words, s21 behaves like a single 34 pole in the frequency domain.

Test & Measurement Application Note 95-1 Measurement of S-Parameter Techniques S-Parameters

6Since s21 behaves like a single pole in the frequency domain, Complete network it is possible to write a closed expression for it. This expression characterization is the same as equation 21, repeated here. Vector Network Analyzers Tps0= 167 (VNA) are ideal for applications requiring complete se-wjT0 - 210 wp= 2f network characterization. s21 = where w They use narrow-band 1 + j wp0=´275MHz detection to achieve wide w0 s==8.. 4 18 5 dB dynamic range and provide 210 noise-free data. VNAs are often combined with powerful The time delay T0= 167 ps is due primarily to the transit time of minority carriers (electrons) across the base of this npn computer-based electronic transistor. Removing this time delay using the electrical-delay design automation (EDA) feature of the vector network analyzer allows the phase systems to both measure distortion to be viewed with much greater resolution. data, and simulate and optimize the performance Using the first-order, single-pole approximation for s21 is an of the complete system important step in circuit design. Today, however, we have design implementation being technology undreamed of in 1967. Subsequently, through the developed. process of electronic design automation (EDA), computer-aided engineering (CAE) tools now can be used iteratively to simulate and refine the design. These tools combine accurate models 35 with performance-optimization and yield-analysis capabilities.

Test & Measurement Application Note 95-1 Narrow-Band S-Parameter Techniques Amplifier Design

7Suppose now that this 2N3478 transistor is to be Unilateral figure of merit used in a simple amplifier, operating between a All two-port models are 50 Ωsource and a 50 Ωload, and optimized bilateral, so both the for power gain at 300 MHz by means of forward and reverse signal lossless input and output matching flow must be considered. networks. Since reverse gain s for this If the signal flow in the 12 reverse direction is much transistor is quite small—50 dB smaller smaller than the flow in than forward gain s , according to 21 the forward direction, Fig. 4—there is a possibility that it can be it is possible to make the neglected. If this is so, the design problem will simplification that the be much simpler, because setting s12 equal to zero will make reverse flow is zero. the design equations much less complicated. In determining how much error will be introduced by assuming The unilateral figure of s = 0, the first step is to calculate the unilateral figure of merit is a quick calculation 12 that can be used to merit u, using the formula given in Appendix B. That is, determine where this simplification can be s11s12s21s22 u = (22) made without significantly 2 2 affecting the accuracy of (1−−ss11 )(1 22 ) the model.

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Test & Measurement Application Note 95-1 Narrow-Band S-Parameter Techniques Amplifier Design

7A plot of u as a function of frequency, calculated from the measured parameters, appears in Fig. 4f. Now if GTu is the transducer power gain with s12 = 0 and GT is the actual transducer power gain, the maximum error introduced by using G instead of G is given by the High-frequency transistors Tu T Discrete transistors were following relationship: the mainstay of high- frequency system design in 1 <

Test & Measurement Application Note 95-1 Narrow-Band S-Parameter Techniques Amplifier Design Satellite Broadcast Signals 7Returning now to the 300-MHz amplifier design, the unilateral Satellites provide broad expression for transducer power gain, obtained either by geographical signal setting s12 = 0 in equation 18 or looking in Appendix B, is coverage over a wide band of frequencies by using 22 2 high power vacuum tubes, s21 ()()11−−ΓΓSL G = called Traveling Wave Tu 2 2 (24) 11−−ssΓΓ Tubes (TWTs), which are 11 SL22 best characterized by s-parameters. When |s11| and |s22| are both less than one, as they are in this case, maximum GTu occurs for ΓS = s*11 and ΓL = s*22 (Appendix B). The next step in the design is to synthesize matching networks that will transform the 50 Ω load and source impedances to the impedances corresponding to reflection coefficients of s*11 and s*22, respectively. Since this is to be a single-frequency amplifier, the matching networks need not be complicated. Simple series-capacitor, shunt-inductor networks will not only do the job, but will also provide a handy means of biasing the transistor—via the inductor—and of isolating the dc bias from the load and the source. 39 H

Test & Measurement Application Note 95-1 Narrow-Band S-Parameter Techniques Amplifier Design

7Values of L and C to be used in the matching networks for the Matching networks 300-MHz amplifier are determined using the Smith Chart of Matching networks are Fig. 6, which is shown on the next page. First, points extra circuit elements added to a device or corresponding to s11, s*11, s22, and s*22 at 300 MHz are plotted. Each point represents the tip of a vector leading away circuit to cancel out from the center of the chart, its length equal to the magnitude or compensate for undesired characteristics of the reflection coefficient being plotted, and its angle equal or performance variations to the phase of the coefficient. at specified frequencies. Next, a combination of constant-resistance and constant- To eliminate reflections in conductance circles is found, leading from the center of the an amplifier, one matching network is carefully chart, representing 50 Ω, to s*11 and s*22. The circles on the Smith Chart are constant-resistance circles; increasing series designed to transform Ω capacitive reactance moves an impedance point counter- the 50 load impedance clockwise along these circles. to s*11. Another matching network transforms the You will find an interactive Impedance Matching Model at the 50 Ω source impedance to HP Test & Measurement website listed below. Challenge s*22. yourself to impedance matching games based on principles and examples discussed in this application note! Click on the URL below or type the address in your browser.

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Test & Measurement Application Note 95-1 Narrow-Band S-Parameter Techniques Amplifier Design Figure 6 7In this case, the circle to be used for Smith Chart for 300-MHz finding series C is the one passing amplifier design example. through the center of the chart, as shown by the solid line in Fig. 6. Increasing shunt inductive s* susceptance moves impedance 11 points clockwise along constant- L1 conductance circles. These s* circles are like the constant- 22 resistance circles, but they are on another Smith Chart, one that is just the reverse C1 of the chart shown in Fig. 6. s C2 L 22 The constant-conductance 2 circles for shunt L all pass s through the leftmost point of 11 the chart rather than the rightmost point. The circles to be used are those passing through s*11 and s*22, as shown by the dashed lines in Fig. 6. 41

Test & Measurement Application Note 95-1 Narrow-Band S-Parameter Techniques Amplifier Design

7The animation to the right demonstrates how to use a * S Smith Chart to design a matching network between a transistor 22 output and a resistive load. As previously described, to maximize the power delivered to the load, the s*22 parameter of the transistor must be matched to the load impedance, 50 W in this 300-MHz amplifier example. This matching is achieved using the LC circuit shown at the right. Starting from the 50 W load, the series capacitance is varied to move the impedance point along the circle of constant 50 W C = 1000 pF resistance on the Smith Chart. The capacitance is adjusted until L = 10000 nH it intersects the constant conductance circle on which s*22 is sitting. Varying the shunt inductance then moves the impedance point along this constant C Animation 2 conductance circle as Impedance matching indicated by the admittance using the Smith Chart Smith Chart. To reach s* , for the matching 22 2N3478 L 50 W the shunt inductance is network shown at the adjusted until the impedance left. Click over the chart to start animation. point reaches s*22.

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Test & Measurement Application Note 95-1 Broadband S-Parameter Techniques 8Amplifier Design Designing a broadband amplifier, that is, one which has nearly constant gain over a G =G G G prescribed frequency range, is a matter of Tu 0 1 2 surrounding a transistor with external 2 elements in order to compensate for the G = s variation of forward gain, |s21| with 0 21 frequency. 2 This can be done in either of two ways— 1− Γs first, negative feedback, or second, selective G = mismatching of the input and output 1 2 circuitry. We will use the second method. 1s− Γs When feedback is used, it is usually 11 convenient to convert to y- or z-parameters 2 (for shunt or series feedback, respectively) 1− Γ using the conversion equations given in G = L Appendix B and a digital computer. 2 2 Equation 24 for the unilateral transducer 1s− Γ 22 L power gain can be factored into three parts, as shown to the right:

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Test & Measurement Application Note 95-1 Broadband S-Parameter Techniques Amplifier Design

8When a broadband amplifier is designed by selective mismatching, the gain contributions of G1 and G2 are varied to 2 compensate for the variations of G0= |s21| with frequency. Suppose that the 2N3478 transistor whose s-parameters are given in Fig. 4 is to be used in a broadband amplifier that will operate from 300 MHz to 700 MHz. The amplifier is to be driven from a 50 Ω source and is to drive a 50 Ω load. According to Figure 4f, 2 s21 = 13 dB at 300 MHz = 10 dB at 450 MHz = 6 dB at 700 MHz To realize an amplifier with a constant gain of 10 dB, source and load matching networks must be found that will decrease the gain by 3 dB at 300 MHz, leave the gain the same at 450 MHz, and increase the gain by 4 dB at 700 MHz.

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Test & Measurement Application Note 95-1 Broadband S-Parameter Techniques 8 Amplifier Design Although in the general case both a source matching network and a load matching * network would be designed, G1max (i.e., s G1 for Γs = s*11) for this transistor is 22 less than 1 dB over the frequencies of interest, which means there is little to be gained by matching the source. 700 Consequently, for this example, only a load-matching network will be 450 designed. Procedures for designing 300 source-matching networks are identical to those used for designing load-matching networks. The first step in the design of the load-matching network is to plot s*22 over the required frequency range on the Smith Chart, Fig. 8a. Figure 8a A plot of s*22 over the frequency range from 300 MHz to 700 MHz. 46 H

Test & Measurement Application Note 95-1 Broadband S-Parameter Techniques Amplifier Design 8 Figure 8 Next, a set of constant-gain circles is drawn. As shown in Fig. 8b, Constant-gain circles. each circle is drawn for a single frequency; its center is on a line between the center of the Smith Chart and the point representing s*22 at that frequency. The distance from the center of the Smith Chart to the center of the constant gain circle is given by the following equations, which also appear in Appendix B: − GdB2 = +4 ρ 1 gs222 at700 MHz r r2= 22 −−2 11sg22 ()2 where r r2 G2 2 2 ρ g2==−Gs222()1 G2max 2 GdB= 0 ρ The radius of the constant-gain circle is: 2 at450 MHz 2 2 11−−gs222() ρ2 = 2 GdB=–3 11−−sg22 ()2 2 at300 MHz 47

Test & Measurement Application Note 95-1 Broadband S-Parameter Techniques Amplifier Design Figure 10a 8Shunt and series elements move impedance points Constant resistance Matching paths on the Smith Chart along constant-conductance circles-Transformationfor shunt and and constant-resistance circles, as explained in due to Lseries series L. the narrow-band design example. As shown in Γ Fig. 10a, the shunt inductance transforms the L locus 50 Ω load alng a circle of constant conductance and varying (with frequency) inductive Hz M 0 z susceptance. The series inductor transforms 0 H 3 M the combination of the 50 Ω load and the 0 z 5 H 4 M shunt inductance along circles of constant 0 0 resistance and varying inductive reactance. 7 Optimizing the values of shunt and series L is an iterative process with two goals: – the transformed load reflection terminates on the right gain circle at each frequency, and – the susceptance component decreases Constant with frequency and the reactance component conductance increases with frequency. (This rule applies to circle-Transformation inductors; capacitors would behave in the due to Lshunt 49 opposite way.) H

Test & Measurement Application Note 95-1 Broadband S-Parameter Techniques Amplifier Design 8 36.4 nH Once appropriate constant- conductance and constant- Ω resistance circles have been found, ZS = 50 2N3478 the reactances and susceptances Ω of the elements can be read 20.4 nH 50 directly from the Smith Chart. Then the element values are calculated, the same as they were for the narrow-band design. Inductance Calculations: Figure 10b is a schematic diagram jLω of the completed broadband From 700 MHz data, series =-=jj(.364 044 . ) 32 . Z amplifier, with unnormalized 0 (.)32 () 50 element values. Lseries = nH= 36. 4 nH 207π (.) Z From 300 MHz data, 0 =-j13. jLω Figure 10b shunt Broadband amplifier with constant gain 50 Lshunt ==20. 4 nH of 10 dB from 300 MHz to 700 MHz. (.13 ) () 2π (.) 03

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Test & Measurement Application Note 95-1 S-Parameter Techniques Stability Considerations

9 Two port with Design of Reflection Amplifiers s'11 > 1 and Oscillators (Real part of input When the real part of the input impedance of a impedance is network is negative, the corresponding input negative) reflection coefficient (Equation 17) is greater than one, and the network can be used as the basis for two important types of circuits, reflection amplifiers and oscillators. A reflection amplifier (Fig. 11) can be realized with a circulator—a nonreciprocal three-port device— Circulator and a negative-resistance device. The circulator is used to separate the incident (input) wave from the larger wave reflected by Input Output the negative-resistance device. Theoretically, if the circulator is perfect and has a positive real characteristic impedance Z0, an amplifier with infinite gain can be built by selecting a negative- Figure 11 resistance device whose input impedance has a Reflection amplifier real part equal to -Z0 and an imaginary part consists of circulator and equal to zero (the imaginary part can be set transistor with negative equal to zero by tuning if necessary). input resistance. 51

Test & Measurement Application Note 95-1 S-Parameter Techniques Stability Considerations Computer Aided 9 Engineering tools (CAE) Amplifiers, of course, are not supposed to oscillate, whether they CAE software tools are are reflection amplifiers or some other kind. There is a convenient used in the design process criterion based upon scattering parameters for determining whether to simulate actual device a device is stable or potentially unstable with given source and load and circuit behavior so designs can be evaluated impedances. Referring again to the flow graph of Figure 3, the ratio before they're built. The of the reflected voltage wave b1 to the input voltage wave bs is CAE approach is faster, produces accurate results, b s¢11 1= and is easier to follow b 1 - G s¢ than manual methods ss11 using graphical design aids. where s¢11 is the input reflection coefficient with Gs= 0 ( Z2= Z0 ) CAE tools are part of the and an arbitrary load impedance Z , as defined in Equation 19. total engineering solution. L If at some frequency

Gs s¢11 = 1 (25) the circuit is unstable and it will oscillate at that frequency. On the other hand, if 1 s¢11 < Gs the device is unconditionally stable and will not oscillate, whatever 52 the phase angle of Gs might be. H

Test & Measurement Application Note 95-1 S-Parameter Techniques Stability Considerations Figure 12 The transistor 9 oscillator is designed by To see how these principles of stability are applied in design choosing a tank circuit Γ ′ = problems, consider the such that s s11 1 transistor oscillator design ω illustrated in Fig. 12. In this 0, Q case the input reflection coefficient s′11 is the reflection Parallel coefficient looking into the Γ 150 Ω T Tank collector circuit, and the 1 Circuit ‘source’ reflection coefficient Γs is one of the two tank-circuit refection coefficients, ΓT1 or Gain Ω Γ ΓT2. From equation 19, s' 200 Z , Element 11 L L ssΓ ss′ =+12 21 L 11 11 −Γ 1 s22 L 15 Ω Series Γ Tank ω , Q T2 0 Circuit

Test & Measurement Application Note 95-1 S-Parameter Techniques Stability Considerations

9To make the transistor oscillate, s¢ and G must be adjusted so 11 s that they satisfy equation 25. There are four steps in the design procedure: – Measure the four scattering parameters of the transistor as functions of frequency.

– Choose a load reflection coefficient GL that makes s¢11 greater than unity. In general, it may also take an external feedback element that increases s12 s21 to make s¢11 greater than one.

– Plot 1/s¢11 on a Smith Chart. (If the network analyzer is being used to measured the s-parameters of the transistor, 1/s¢11 can be measured directly by reversing the reference and test channel connections between the reflection test unit and the harmonic frequency converter. The polar display with a Smith Chart overlay then gives the desired plot immediately.) – Connect either the series or the parallel tank circuit to the collector circuit and tune it so that GT1 or GT2 is large enough to satisfy equation 25. (The tank circuit reflection coefficient plays the role of Gs in this equation.) 54

H Test & Measurement Application Note 95-1 Additional Reading S-Parameter Techniques on S-Parameters

AIn addition to previous references listed earlier and repeated again here, the following papers and books were listed in the 1967 HP Journal article as sources for information on s-parameter design procedures and flow graphs. Current references are also mentioned. – J. K. Hunton, ‘Analysis of Microwave Measurement Techniques by Means of Signal Flow Graphs,’ IRE Transactions on Microwave Theory and Techniques, Vol. MTT- 8, No. 2, March, 1960. – D.C Youla, ‘On Scattering Matrices Normalized to Complex Port Numbers,’ Proc. IRE, Vol. 49, No. 7, July, 1961. – J.G. Linvill and J.F. Gibbons, ‘Transistors and Active Circuits,’ McGraw-Hill, 1961. (No s-parameters, but good treatment of Smith Chart design methods.) – N. Kuhn, ‘Simplified Signal Flow Graph Analysis,’ Microwave Journal, Vol. 6, No, 11, November, 1963. – K. Kurokawa, ‘Power Waves and the Scattering Matrix,’ IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-13, No. 2, March, 1965. 56

Test & Measurement Application Note 95-1 Additional Reading S-Parameter Techniques on S-Parameters

A– F. Weinert, ‘Scattering Parameters Speed Design of Keep up to date High-Frequency Transistor Circuits,’ This book by P. H. Smith Electronics, Vol. 39, No. 18, Sept. 5, 1966. referenced here must be considered the ultimate – G. Fredricks, ‘How to Use S-Parameters for Transistor source on Smith Charts. Circuit Design,’ EEE Vol. 14, No. 12, Dec., 1966. Many excellent articles on Among many modern reference sources on the subject, the the use of Smith charts following book, first published in 1969, is definitely a classic: have appeared in trade publications such as the – Smith, Phillip H., ‘Electronic Applications of the Smith Microwave Journal, and Chart in Waveguide, Circuit and Component Analysis,’ educational Smith Chart Noble Publishing Classic Series, Tucker, Georgia, 1995, software is sold over the ISBN-1-884932-39-8, 237 pp. internet. We also mention a useful textbook containing 5 chapters, 2 appendices, and problem sets. This text presents a unified treatment of the analysis and design of microwave transistor amplifiers using scattering parameters techniques: – G. Gonzalez, ‘Microwave Transistor Amplifiers: Analysis and Design,’ Prentice Hall, 1984, ISBN 0-13-581646-7, 240 pp. 57 H

Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques BRelationships + + a1 a2 b1 =+s11a1 s12a2 V TWO - PORT 1 V2 =+ b NETWORK b b2 s21a1 s22a2 – 1 2–

Input reflection coefficient with arbitrary ZL ssΓ ss′=+1221 L 11 11 −Γ 1 s22 L

Output reflection coefficient with arbitrary Zs ssΓ ss′=+1221 S 22 22 −Γ 1 s11 S Voltage gain with arbitrary ZL and ZS V s ()1 +Γ A ==2 21 L V −Γ+′ V1(11ss22 L)(11) 58

Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques BRelationships Power Gain 22æö s21 ç1-GL÷ Power delivered to load èø G == Power input to network æ2ö2æ22ö ç1-ssD11 ÷+-GLç22 ÷-2Re(GL N) èøèø

Available Power Gain 22æ ö s21 ç1- GS÷ Power available from network è ø GA = = Power available from source æ 22ö æ22ö ç1- ssD22 ÷ +-GSç11 ÷-2Re(GS M) è ø èø

Transducer Power Gain 22æ öæ 2ö s 21 11- GGSLç - ÷ Power delivered to load è øè ø GT = = Power available from source 11- ssssGG- - GG)2 ( 11SL)( 22) 12 21 LS 59 H

Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques BRelationships Unilateral Transducer Power Gain (s12 =0) 2 Gs= 222 021 s 21 11−ΓΓSL−  2 GTu = =G0G1G2 −Γ 2 2 1 S 11−−ssΓΓ G1= 11 SL22 2 1 −s11ΓS

2 1 −ΓL G= < 2 Maximum Unilateral Transducer Power Gain when s 11 1 2 1 −s22 ΓL <Γ=**Γ = and s 22 1. Maximum obtained for SLss11 and 22 1 Gimax = 2 2 1 −s s21 ii Gu== GG01max G2 max 22 i=12, 11−ss− 11 22 

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Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques Relationships BConstant Gain Circles (Unilateral case: s = 0) 12 • center of constant gain circle is on line between center of Smith Chart and point representing s*ii

• distance of center of circle from center of Smith Chart: Constant Gain Circle gs r= iii i 2 11−sgii ()− i * • radius of circle: S 2 ii 11−−gs() r ρ= iii i i 2 11−−sg() ii i ρ

where i = 1, 2, and i Gi 2 gi==Gsi(1−ii) Gimax 61 H

Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques Relationships

B Unilateral figure of merit Unilateral Figure of Merit All two-port models are bilateral, so both the s11s22s12s21 u = forward and reverse signal 2 2 flow must be considered. (11−−ss11 )(22 ) If the signal flow in the reverse direction is much smaller than the flow in the forward direction, it’s possible to make the simplification that the reverse flow is zero. Error Limits on Unilateral Gain Calculations

Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques Relationships

B as.,<<11 s Conditions for Absolute Stability : 11 22

No passive source or load will cause a network to ss- M* oscillate, if conditions a, b, and c are all satisfied. 12 21 b. >1 22 sD11 - Condition that a two- port network can be simultaneously * matched with a real source and load : ss12 21 - N c. >1 KC>11 or < where C = Linvill C Factor and 22 sD22 -

CK= -1 Dss=-11 22 ss 12 21 2 2 2 1+-Ds11 - s22 M s Ds* K= =-11 22 2ss * 12 21 Ns=-22 Ds 11

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Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques Relationships BSource and load for Simultaneous Match

 2  B±−B4 M2 *11  ΓmS =M    2 (Use minus sign 2 M 222   B=+1s −s −D when B1 is positive, 11122  2  B±−B4 N2 plus sign when =+ 2− 22− *22  B21s22s11 D ΓmL =N  B is negative.)  2  1 2 N 

Maximum Available Power Gain s KG>11=21 KK±2 − If , A max  s12 − where KC=1

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Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques BRelationships s-parameters z-parameters in terms of z-parameters in terms of s-parameters

−+− (z11 11)(z22 )z12z21 (11+s11 )(−s22 )+s12s21 s11 = z11 = (z11 +11)(z22 +)−z12z21 (11−s11 )(−s22 )−s12s21

2z12 2s12 s12 = z12 = (z11 +11)(z22 +)−z12z21 (11−s11 )(−s22 )−s12s21

2z21 2s21 s21 = z21 = (z11 +11)(z22 +)−z12z21 (11−s11 )(−s22 )−s12s21

(z+11)(z−)−zz (11+s)(−s)+ss s=11 22 1221 z=22 11 1221 22 ++− 22 −−− (z11 11)(z22 )z12z21 (11s11 )(s22 )s12s21

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Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques BRelationships s-parameters y-parameters in terms of y-parameters in terms of s-parameters

(11−y)(+y)+yy (11+s)(−s)+ss s =11 22 1221 y =22 11 1221 11 ++− 11 (11y11 )(y22 )y12y21 (11+s11 )(+s22 )−s12s21

−2y −2s s= 12 y= 12 12 ++− 12 (11y11 )(y22 )y12y21 (11+s11 )(+s22 )−s12s21

−2y −2s s= 21 y= 21 21 ++− 21 (11y11 )(y22 )y12y21 (11+s11 )(+s22 )−s12s21

(11+y)(−y)+yy (11+s)(−s)+ss s=11 22 1221 y=11 22 1221 22 ++− 22 ++− (11y11 )(y22 )y12y2211 (11s11 )(s22 )s12s2211

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Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques BRelationships s-parameters h-parameters in terms of h-parameters in terms of s-parameters

(h−11)(h+)−hh (11+s)(+s)−ss s =11 22 1221 h=11 22 1221 11 ++− 11 (h11 11)(h22 )h12h21 (11−s11 )(+s22 )+s12s21

2h 2s s= 12 h= 12 12 ++− 12 (h11 11)(h22 )h12h21 (11−s11 )(+s22 )+s12s21

−2h −2s s= 21 h= 21 21 ++− 21 (h11 11)(h22 )h12h21 (11−s11 )(+s22 )+s12s21

(11+h)(−h)+hh (11−s)(−s)−ss s=11 22 1221 h=22 11 1221 22 ++− 22 −++ (h11 11)(h22 )h12h21 (11s11 )(s22 )s12s21

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Test & Measurement Application Note 95-1 Scattering Parameter S-Parameter Techniques Relationships B Parameter Normalization The h–, y–, and z–parameters listed in previous tables are all normalized to Z . The various scattering 0 parameters are all If h′, y′, z′are the actual parameters, then: normalized by the reference impedance, Z0. This impedance is usually y′= y/ Z ′= z11′ = z11Z0 11 11 0 h11 h11Z0 the characteristic impedance of the y′= y/ Z ′= transmission line in which z12′ = z12Z0 12 12 0 hh12 12 the network of interest is ′= y′= y/ Z hh′= embedded. Normalizing z21 z21Z0 21 21 0 21 21 the scattering parameters makes the Smith Chart ′= y′= y/ Z h′= h/ Z z22 z22Z0 22 22 0 22 22 0 readily applicable to transmission lines of any impedance. In addition, impedance and admittance values can be plotted on the same chart.

Test & Measurement Application Note 95-1 CAE tools for High- S-Parameter Techniques Frequency Design Circuit Envelope Simulation CThe CAE tools of interest to RF and The waveform above microwave designers include those typifies the modulated summarized below: and transient signals that Small-signal (s-parameter) simulation — can be efficiently analyzed Small-signal analysis CAE tools simulate using circuit envelope response over a range of frequencies, so actual implementations simulator software. Circuit perform more closely to design parameters. Matching circuits are envelope simulation is easily determined, and can be readily optimized, saving time. orders of magnitude faster A yield-analysis feature allows the selection of components in than traditional SPICE matching networks for the best production yield, saving costs. simulation software if the Large-signal simulation — This powerful analysis tool envelope bandwidth of the includes the harmonic balance implementation, useful for RF carrier frequency is oscillator design and many other problems. much smaller than the carrier frequency itself. Circuit Envelope simulation — Efficiently analyzes circuits This is the case in many and feedback loops in the presence of modulated or transient communications, and radar high-frequency signals. circuits and subsystems.

Test & Measurement Application Note 95-1 CAE tools for High- S-Parameter Techniques Frequency Design CTime-domain analysis — A CAE tool that is especially useful for simulating the response of digital systems at high clock rates. Planar electromagnetic analysis — This simulator accurately computes the s-, y-, or z-parameters of arbitrarily shaped, multilayer planar structures such as striplines. 3D electromagnetic analysis — This CAE tool accurately High Flying Software computes the s-parameters for passive, three-dimensional, The pattern of a horn multiport structures. antenna, displayed using HP High-Frequency System analysis — System and board-level simulators offer Structure Simulator discrete-time and frequency-domain capabilities; can analyze 3D electromagnetic and optimize complicated system topologies; handle complex visualization software, waveforms; and perform physical layout design. shows beam shapes Modeling systems — Hardware and software are combined to in both azimuth and elevation in a single plot. extract parameters needed for accurate active device modeling.

Visit the HP EEsof website for the latest in CAE news, products, software, and solutions.

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Test & Measurement Application Note 95-1 Relevant Products S-Parameter Techniques HP 8720D Series Vector Network Analyzers Vector network analyzers with built-in synthesized sources cover 50 MHz to up to 40 GHz The HP 8720D vector network analyzer family characterizes RF and microwave components from 50 MHz up to 40 GHz. They combine a fast synthesized source, tuned receiver, and S-parameter test set in a single instrument. The devices have the performance and flexibility to solve difficult measurement problems and cut test times, all at an attractive cost. Use them to quickly and accurately measure magnitude and phase of all four s-parameters, as well as group delay, plus the absolute output power of microwave components. Features of HP8720D VNA’s • Built-in synthesized source Productivity is enhanced with pass/fail testing, direct printer/ with 1 Hz resolution plotter output of results, advanced marker functions, save/ • Allows measuring all four recall of test configurations to internal memory or a built-in s-parameters with a floppy disk drive, and test sequencing for automation. Options single connection allow high-power tests, frequency offset mixer testing, and • Continuous updates for high-accuracy noncoaxial and on-wafer measurements. two-port error correction http://www.hp.com/go/tmdatasheets H

Test & Measurement Application Note 95-1 Relevant Products S-Parameter Techniques HP 8510C Microwave Network Analyzers Microwave network analyzers with coverage up to 110 GHz The HP 8510C microwave vector network analyzer family provide complete solutions for characterizing the linear behavior of either active or passive networks over the 45 MHz to 110 GHz range. A complete system consists of the HP 8510C network analyzer, an s-parameter test set, and a compatible RF source. Also available are fully integrated systems, tested and verified prior to shipment. Features of the HP8510C The HP 8510C displays measurement results in log/linear • 45 MHz to 110 GHz magnitude, phase, or group delay format on a large, color CRT frequency range with two independent, yet identical, channels. The impact of • Real-time error-corrected systematic errors is removed by virtually “real-time” error measurements correction, so a test device can be adjusted while it’s being • 60 dB effective directivity measured. Effective directivity and source match can be improved and source match to as much as 60 dB. The HP 85161B software leads the operator • Up to 100 dB dynamic one step at a time, from setup and calibration to hardcopy results. range • 0.001 dB, 0.01 degree, Visit the HP Test & Measurement website and find more than 0.01 ns resolution 1,000 up to date product datasheets. (Adobe Acrobat users with • Optional time domain the Weblink plug-in may click directly on the URLbelow.) and pulsed RF measurement capability http://www.hp.com/go/tmdatasheets

Test & Measurement Application Note 95-1 Relevant Products S-Parameter Techniques HP 8753D RF Network Analyzer RF network analyzer with integrated s-parameter test set performs characterizations from 30 kHz to 6 GHz The HP 8753D RF vector network analyzer will simplify and speed your device, component, or network measurements in the 30 kHz to 6 GHz range. A 1-Hz resolution swept synthesized source, s-parameter test set, and sensitive receiver are integrated into this compact instrument, which is simple to set up and use in the lab or on the production line. The HP 8573D provides magnitude and phase information, offers up to 110 dB dynamic range, makes group delay and time domain measurements, and uses vector accuracy Features of the HP 8753D enhancement to minimize measurement uncertainty. • Built-in s-parameter test To increase your throughput in production, the HP 8753D offers set, synthesized source features such as the test sequence function, which allows you to • Optional time-domain and swept-harmonic make a measurement once from the front panel and automatically measurements save the keystrokes without an external computer. The analyzer’s • Up to 110 dB dynamic range fast CPU clock rate, LIF and DOS formats for output to the • Superb accuracy, with built-in disk drive or an external disk drive, and a 512 KB comprehensive calibration nonvolatile memory also help improve your productivity. • Save/recall to built-in disk drive http://www.hp.com/go/tmdatasheets

Test & Measurement Application Note 95-1 Relevant Products S-Parameter Techniques HP 85180A High Frequency Structure Simulator Fast, accurate electromagnetic simulation of 3D high-frequency structures saves design time With today’s computer technology, the best method for designing high frequency circuits and machined structures is to compute their behavior with 3D electromagnetic (EM) simulator software, making improvements through successive simulations, then building designs to the refined specifications. This approach is more cost effective and takes less time than building and testing prototype after prototype in the lab. The HP High Frequency Structure Simulator (HP HFSS), Release 5.0, models arbitrarily-shaped, passive 3D structures Features of the HP 85180A such as antennas, machined components, and RF and digital • Reduces "cut and try" circuits. Using accuracy-driven adaptive solution refinement, prototyping, for lower the software produces accurate results ten times faster, using development costs and a half the memory of previous releases. Its drawing environment quicker time to market and parts library simplify specifying complex structures. • Yields accurate results ten For comprehensive evaluations, animated EM fields, surface times faster, using half the currents, vector plots, antenna polar patterns and tabular, memory of previous releases generalized s-parameters and Smith charts can be viewed. • Requires minimal user knowledge of EM field theory • For PC and UNIX platforms http://www.hp.com/go/hpeesof

Test & Measurement Application Note 95-1 Relevant Services S-Parameter Techniques RF and microwave education and training

HP offers an extensive curriculum of education services at locations worldwide. To help high-frequency design engineers learn how to use CAE tools quickly, we provide training courses taught by the experts from HP’s EEsof Division. Advanced RF/microwave CAE courses are also conducted to help extend the capabilities and hone the skills of experienced CAE users. HP education classes are scheduled regularly. The curriculum can be tailored to a company’s special needs, and training sessions can be conducted at its site.

Visit HP’s World Wide Website for the complete and continuously updated list of Test & Measurement class schedules and locations around the world. Click on the URL below or type the address in your browser. Register directly online!

http://www.hp.com/go/tmeducation H

Test & Measurement Application Note 95-1 Worldwide Contacts S-Parameter Techniques Hewlett-Packard Test & Measurement offices

Asia-Pacific Canada Latin America Hewlett-Packard Asia Pacific Ltd. Hewlett-Packard Ltd. Latin American Region Hewlett-Packard Hong Kong Ltd. 5150 Spectrum Way Headquarters 17-21/F Shell Tower,Times Square Mississauga, Ontario L4W 5G1 Hewlett-Packard Company 1 Matheson Street, Causeway Bay 905 206 4725 Waterford Building, Suite 950 Hong Kong 5200 Blue Lagoon Europe/Middle East/Africa 85 2 599 7777 Miami, FL 33126 Hewlett-Packard S.A. 305-267-4245 For Japan: 0120-421-345 150, Route du Nant-d’Avril 1217 Meyrin 1 United States of America Australia/New Zealand Geneva, Switzerland Hewlett-Packard Company Hewlett-Packard Australia Ltd. For Europe: 800-452-4844 31-41 Joseph Street 41 22 780 81 11 Blackburn, Victoria 3130 Australia For Middle East/Africa: 1 800 627 485 41 22 780 41 11

Visit HP’s World Wide Website for the complete and up to date listing of Test & Measurement sales office addresses and call center phone numbers. Click on the URLbelow or type the address in your browser. http://www.hp.com/go/tmdir H

Test & Measurement Application Note 95-1 Copyrights and Credits S-Parameter Techniques Authors, contributors and producers Authors Graphic & Dick Anderson– original author Interaction Design of this application note. Ev Shafrir– original concept, Lee Smith– S-parameter guru, art direction, interaction design, mathematical programming, digital publishing & production, technical illustrations, updated & content supervision, overall revised content, and animations. direction, & project management. Jeff Gruszynski– expert domain Christina Bangle– free-style consultant, content perspectives, illustrations.and graphic reviews. side-bars text, and source images. Kathy Cunningham – page layout design and Quark guru. Contributors Leann Scully – source images. Richard W. Anderson Walt Patstone– technical Holds a 1959 BSEE degree content and side-bar editing. Business Manager from Utah State University Chuck McGuire, Mike Nyna Casey– funding, support, and a 1963 MSEE degree C’debaca – new s-parameter excitement, and encouragement. from Stanford University. During his 37 years with HP measurements & device output. Dick has contributed to the development of numerous © Copyright Hewlett-Packard Company 1996-1997 All Rights Reserved. microwave and other T&M Adaptation, reproduction, translation, extraction, dissemination or dis- instruments. Currently, he assembly without prior written permission is prohibited except as allowed is HP Vice President, and under the copyright laws.S-Parameter Techniques for Faster, More General Manager of the Accurate Network Design is electronically published as part of the HP T&M Microwave and Test & Measurement Digital Application Note Library for the World Wide Communications Group. Web, November© 1996. Original printed publication Number 5952-1130.