S-Parameter Techniques … HP Application Note 95-1


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S-Parameter Techniques … HP Application Note 95-1

  1. 1. Test & Measurement H Application Note 95-1 S-Parameter Techniques for Faster, More Accurate Network Design http://www.hp.com/go/tmappnotes
  2. 2. H 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 © Copyright Hewlett-Packard Company, 1997. 3000 Hanover Street, Palo Alto California, USA.
  3. 3. H 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 essential part of high-frequency design, though much else Cover: A NEW MICROWAVE INSTRUMENT SWEEP MEASURES GAIN, PHASE IMPEDANCE WITH SCOPE OR METER READOUT; page 2 has changed during the past 30 years. During that time, See Also: THE MICROWAVE ANALYZER IN THE 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 original “S-Parameters the World Wide Web. We hope you find this tutorial useful. 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
  4. 4. 1 H Test & Measurement Application Note 95-1 S-Parameter Techniques Introduction Linear 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 ∇⋅D=ρ ∇×E= − 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 difficult. Efficient design you can best take advantage of their abilities, this application 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. 4
  5. 5. 2 H Test & Measurement Application Note 95-1 S-Parameter Techniques Two-Port Network Theory Although a network may have any number of ports, I1 I2 network parameters can be explained most easily by + TWO - PORT + considering a network with only two ports, an input port V1 V – 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 network of Fig. 1 is excited by voltage sources V1 and V2, the behavior of components, circuits, and systems. network currents I1 and I2 will be related by the following Lumped models are useful at equations (assuming the network behaves linearly): lower frequencies, where I1 = y 11 V1 + y 12 V2 (1) some physical effects can be ignored because they are so I 2 = y 21 V1 + y 22 V2 (2) small. Distributed models are needed at RF frequencies and higher to account for the increased behavioral impact of those physical effects. 5
  6. 6. 2 H Test & Measurement Application Note 95-1 S-Parameter Techniques Two-Port Network Theory In 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 I y 21 = 2 models use s-parameters V1 V = 0 (output short circuited) (transmission and reflection 2 (3) coefficients). 6
  7. 7. 2 H Test & Measurement Application Note 95-1 S-Parameter Techniques Two-Port Network Theory If 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). 7
  8. 8. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques 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
  9. 9. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters Another 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 transmitted. The amounts They are defined in terms of the terminal voltage Vi , the reflected and transmitted terminal current I i , and an arbitrary reference impedance Z i , are characterized by optical where the asterisk denotes the complex conjugate: reflection and transmission coefficients. Similarly, * V + Zi Ii Vi − Z i I i scattering parameters are ai = i (4) bi = (5) measures of reflection and 2 Re Z i 2 Re Z i transmission of voltage waves through a two-port 9 electrical network.
  10. 10. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques 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 positive real impedance, Z0. elements in an RF circuit 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: TWO - PORT ZL VS b1 NETWORK b2 Z0 ε = 1 377 N The s-parameters s 11 and Figure 2 s 22 are the same as optical Two-port network showing incident waves reflection coefficients; (a 1 , a 2 ) and reflected waves (b 1 , b 2 ) used in s 12 and s 21 are the same s-parameter definitions. The flow graph for as optical transmission this network appears in Figure 3. coefficients. 10
  11. 11. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters The independent variables a1 and a2 are normalized incident voltages, as follows: V1 + I1 Z 0 voltage wave incident on port 1 V i1 a1 = = = (6) 2 Z0 Z0 Z0 V2 + I 2 Z 0 voltage wave incident on port 2 V a 2= = = i2 (7) 2 Z0 Z0 Z0 Dependent variables b1, and b2, are normalized reflected voltages: V −I Z voltage wave reflected from port 1 V b1 = 1 1 0 = = r1 (8) 2 Z0 Z0 Z0 V2 − I 2 Z 0 voltage wave reflected from port 2 Vr 2 b2 = = = (9) 2 Z0 Z0 Z0 11
  12. 12. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters The linear equations describing the two-port network are Limitations of then: lumped models b 1 = s 11 a 1 + s 12 a 2 At low frequencies most (10) circuits behave in a b 2 = s 21 a 1+ s 22 a 2 (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 s 11= = Input reflection coefficient with (12) boxes. At microwave a 1 a = 0 the output port terminated by a frequencies, as circuit 2 matched load (Z =Z sets a =0) element size approaches L 0 2 the wavelengths of the b2 s 22 = = Output reflection coefficient (13) operating frequencies, a 2 a = 0 with the input terminated by a such a simplified type 1 matched load (ZS=Z0 sets Vs=0) of model becomes inaccurate. The physical b2 s 21 = = Forward transmission (insertion) (14) arrangements of the a 1 a = 0 gain with the output port circuit components can 2 terminated in a matched load. no longer be treated as b1 black boxes. We have to s 12 = = Reverse transmission (insertion) (15) use a distributed circuit a2 gain with the input port element model and a 1 =0 terminated in a matched load. s-parameters. 12
  13. 13. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters Notice that V1 − Z0 b1 I1 Z − Z0 s 11 = = = 1 (16) a1 V1 Z 1 + Z0 + Z0 S-parameters I1 S-parameters and distributed models (1 + s11 ) and Z1 = Z 0 provide a means of (1 − s11 ) (17) measuring, describing, and characterizing V1 circuit elements when where Z 1 = 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. 13
  14. 14. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters Animation 1 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 graphically Chart is used to plot reflectances, but the 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.
  15. 15. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques Using 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 to insertion gain s21 is the ‘forward of the pulse. A short trans-admittance’ y21 given by pulse with steep edges equation 3. Clearly, insertion has a signal spectrum with relatively high gain gives by far the 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
  16. 16. 3 H Test & Measurement Application Note 95-1 S-Parameter Using S-Parameters Techniques Another advantage of s-parameters springs from the simple relationship between the variables a1, a2 , b1, Radar and b2, and various power waves: The development 2 of radar, which a1 = Power incident on the input of the network. uses powerful signals = Power available from a source impedance Z 0 . at short wavelengths to detect small objects at a 2 2 = Power incident on the output of the network. long distances, provided a powerful incentive for = Power reflected from the load. improved high frequency design methods during b1 2 = Power reflected from the input port of the network. World War II. The design = Power available from a Z0 source minus the power methods employed at 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
  17. 17. 3 H Test & Measurement Application Note 95-1 S-Parameter Techniques Using S-Parameters The 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 s 22 = Power incident on the network output 2 Power delivered to a Z0 load s 21 = Power available from Z0 source = Transducer power gain with Z 0 load and source 2 s12 = Reverse transducer power gain with Z0 load and source 17
  18. 18. 4 H Test & Measurement Network Calculations with Application Note 95-1 S-Parameter Techniques Scattering 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,’ driving point parameters s11 and s22 are a measure of the input IRE Transactions on 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, No. 2, March, 1960. meaningful physical interpretation of the network performance, 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. 18
  19. 19. 4 H Network Calculations with Test & Measurement Application Note 95-1 S-Parameter Techniques Scattering Parameters In a signal flow graph, each port is represented by two nodes. Node an VS Z 0 represents the wave coming into the bS = ZS + Z 0 a1 b2 device from another device at port n, and node bn represents the wave 1 s21 leaving the device at port n. The s11 s22 ΓL = ZL - Z0 complex scattering coefficients are ZL + Z 0 then represented as multipliers on ΓS = ZS - Z0 branches connecting the nodes within ZS + Z0 b1 s12 a2 the network and in adjacent 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 a = Γ b = 0 2 L 2 19
  20. 20. 4 H Network Calculations with Test & Measurement Application Note 95-1 S-Parameter Techniques Scattering Parameters Better Smith Charts The 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 The nontouching loop rule provides a ∑ Tk ∆ k coaxial with the center of the diagram. Refinements T= k to the original form have ∆ simple method for writing the solution of 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 Tk = path gain of the kth forward path version and showed how 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. 20 ∆k = The value of ∆ not touching the kth forward path.
  21. 21. 4 H Network Calculations with Test & Measurement Application Note 95-1 ( ) S-Parameter Techniques Scattering Parameters s11 s12 A path is a continuous succession of branches, and a forward path is a path connecting the input node to the output node, where no node is encountered more than once. Path gain is s21 s22 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 bs to b2, and its gain is s21. There are two paths from bs to b1; measurement. s 11 and s 22 , the two s-parameters their path gains are s21s12ΓL and s11 respectively. There are 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 ∆ for this network respects because they are is related to the input ∆ = 1 − (s11 ΓS + s 21 s12 ΓL ΓS + s 22 ΓL ) + s11 s 22 ΓL ΓS impedance and output impedance, respectively. The Smith Chart performs The transfer function from bs to b2 is therefore a highly useful translation b2 s 21 between the distributed = and lumped models and is bs ∆ used to predict circuit and 21 system behavior.
  22. 22. 4 H Network Calculations with Test & Measurement Application Note 95-1 S-Parameter Techniques Scattering 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 ΓL and ΓS, respectively, referenced to the real characteristic impedance Z0. Transducer power gain: Power delivered to the load P GT = = L Power available from the source PavS P L = P( incident on load) − P( reflected from load) 2 2 = b 2 (1 − ΓL ) bS 2 PavS = 2 ( 1 − ΓS ) 2 b2 2 2 GT = (1 − ΓS )(1 − ΓL ) bS 22
  23. 23. 4 H Network Calculations with Test & Measurement Application Note 95-1 S-Parameter Techniques Scattering Parameters Using the non-touching loop rule, b2 s 21 = bS 1 − s 11ΓS − s 22 ΓL − s 21 s12 ΓL ΓS + s11ΓS s 22 ΓL s 21 Obtaining maximum = performance (1 − s 11ΓS )(1 − s 22 ΓL ) − s 21 s12 ΓL ΓS S-parameters are used to characterize RF and 2 2 2 microwave components s 21 (1 − ΓS )(1 − ΓL ) (18) that must operate GT = together, including 2 (1 − s11ΓS )(1 − s 22 ΓL ) − s 21 s12 ΓL ΓS 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 b1 s (1 − s 22 ΓL ) + s 21 s12 ΓL calculated, they make it s11 = ′ = 11 possible to maximize a1 1 − s 22 ΓL (19) performance in areas s s Γ such as power transfer, = s11 + 21 12 L 1 − s 22 ΓL directivity, and 23 frequency response.
  24. 24. 4 H Network Calculations with Test & Measurement Application Note 95-1 S-Parameter Techniques Scattering Parameters 2) Voltage gain with arbitrary source and load impedances V2 AV = V1 = ( a 1 + b 1 ) Z 0 = Vi 1 + Vr 1 V1 V2 = ( a 2 + b 2 ) Z 0 = Vi 2 + Vr 2 Waveguides a 2 = ΓL b 2 A radar system delivers a b 1 = s11a 1 ′ large amount of energy from a microwave (µW) (20) source to the transmitting b (1 + ΓL ) s 21(1 + ΓL ) AV = 2 = antenna. The high field a 1(1 + s11 ) (1 − s 22 ΓL )(1 + s11 ) ′ ′ 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 reflec- tions and transmission 24 quality with s-parameters.
  25. 25. 5 H Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Traveling wave amplifier Using Scattering Parameters 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. 25
  26. 26. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters Most 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 s 11 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. | s 11 | = 1. The movement of s 11 with frequency is approximately along circles of constant resistance, indicative of series capacitance and 26 inductance.
  27. 27. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters |S12| 100 MHz 10 dB/cm –30 dB S12 –110° 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 s 12. 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 s 12 increases characteristic of a shunt RC equivalent circuit. about 6dB/octave. 27
  28. 28. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters |S21| 10 dB/cm 0 dB 0 dB |S21| 10 dB/cm S21 S21 + 20° 50°/cm + 90° 50°/ cm 100 MHz 1700 MHz 100 MHz 1700 MHz Figure 4d – Magnitude and phase of s 21 . Figure 4e – Removing Linear Phase Shift. The magnitude of s 21 decays with a slope Magnitude and phase of s 21 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. 28
  29. 29. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters Figure 4f In Fig. 4f, the magnitude of s21 from Fig. 4d is Top curve: | s 21 | from Fig. 4 is replotted replotted on a logarithmic frequency scale, 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 Fig. 4d, varies linearly with |s 21 | 20 10 Magnitude (dB) frequency above about 500 MHz. Magnitude 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 – 30 u .03162 curve of Fig. 4d to that of Fig.4e 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
  30. 30. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters After removal of the constant-delay, or linear-phase, Importance of simple component, the phase angle of s21 for this transistor (Fig. 4e) approximations varies from 180° at dc to +90° at high frequencies, passing Using first-order through +135° at 125 MHz, the -3 dB point of the magnitude approximations such as curve. In other words, s21 behaves like a single pole in the equation 21 is an important frequency domain, and it is possible to write a closed step in circuit design. The intuitive sense that expression for it. This expression is designers gain from - s 210e - w j T0 developing an under- standing of these s 21 = (21) w approximations can 1+ j w 0 eliminate much frustration. The acquired insight can where save hours of time that T0 = 112 ps otherwise might be wasted w =p2 f generating designs that cannot possibly be realized w 0 = p 2 ¥ 125 MHz in the lab, while also s 210 = 11.2 = 21 dB decreasing development costs. The time delay T0 = 112 ps is due primarily to the transit time of minority carriers (electrons) across the base of this npn transistor. 30
  31. 31. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques 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 1.00 GHz. S11 Figure 5a S 22 S-parameters of 2N3478 transistor in common-emitter configuration, measured by an HP 8753 network analyzer. This plot shows s 11 and s 22 on a Smith Chart. The marker set at 47 MHz represents the -3 dB gain roll off point of s 21 . The frequency index of this point is referenced in the other plots. 31
  32. 32. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters Magnitude (dB) S12 S 21 Phase (deg) Frequency Figure 5b A plot of the magnitude and phase of s 12 . While the Figure 5c phase of s 12 depends only weakly on the frequency, A polar plot of s 21 . 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
  33. 33. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters Magnitude (dB) S 21 Phase (deg) S 21 Frequency (Hz) Frequency (GHz) Figure 5d Figure 5e The magnitude of s 21 plotted on a log The phase angle, in degrees, of s 21 . A time scale showing the 6 dB/octave roll-off delay of 167 ps was de-embedded from the above 75 MHz. measured data using the analyzer's electrical- delay feature to get a response with a single- pole transfer characteristic. Removing this time delay allows the phase distortion to be 33 viewed with much greater resolution.
  34. 34. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters In Fig. 5d, the magnitude of s21 from Fig. 5c is replotted Vector Network Analyzers 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 of 6 dB/octave. The phase angle of s21 as seen in Fig. 5e lightwave, use varies linearly with frequency above 500 MHz. To better sophisticated simulation software to model system characterize phase distortion, a compensating linear phase 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.
  35. 35. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters Since 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 T0 = 167 ps (VNA) are ideal for applica- tions requiring complete − s 210 e − jωT0 ω = 2π f network characterization. s 21 = where ω ω 0 = 2 π × 75 MHz They use narrow-band 1+ j detection to achieve wide ω0 dynamic range and provide s 210 = 8.4 = 18.5 dB noise-free data. VNAs are The time delay T0= 167 ps is due primarily to the transit time often combined with powerful 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.
  36. 36. 6 H Measurement of Test & Measurement Application Note 95-1 S-Parameter Techniques S-Parameters Explanation of Measurement Discrepancies You may have noticed a difference between the measured and caclulated data from the 1967 HP Journal article and the data obtained for this updated application note. Both sets of data are fundamentally correct. Two major sources account for these differences: Measurement techniques – Early network analyzers did not have onboard computers, an HP-IB standard, or high-resolution graphics to perform calibration, extract precision numerical data, or display electronic markers. Calibration techniques used in 1967 were procedurally and mathematically simpler than those used today. Modern network anaylzers contain sophisticated automated techniques that enhance measurement processing capabilities and reduce operator errors. Device differences – Semiconductor manufacturing processes evolve over time. Device engineers attempt to produce identical transistors with different processes. Nevertheless, successive generations of parts like the 2N3478 can exhibit unintentional, and sometimes unavoidable, performance differences, especially in characteristics not guaranteed on the datasheet. 36
  37. 37. 7 H Narrow-Band Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Suppose 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 Ω source and a 50 Ω load, 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. If the signal flow in the networks. Since reverse gain s12 for this reverse direction is much transistor is quite small—50 dB smaller smaller than the flow in than forward gain s21, according to 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 merit is a quick calculation s12 = 0, the first step is to calculate the unilateral figure of that can be used to merit u, using the formula given in Appendix B. That is, determine where this s11 s12 s 21 s 22 simplification can be u = (22) made without significantly 2 2 (1 − s11 )(1 − s 22 ) affecting the accuracy of the model. 37
  38. 38. 7 H Narrow-Band Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design A 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 High-frequency transistors introduced by using GTu instead of GT is given by the Discrete transistors were following relationship: the mainstay of high- 1 GT 1 frequency system design in < < 1967 when Dick Anderson (1 + u )2 G Tu (1 − u )2 (23) wrote the article on which this application note is From Fig. 4f, the maximum value of u is about 0.03, so the based. Thirty years later, maximum error in this case turns out to be about +0.25 dB discrete devices are still at 100 MHz. This is small enough to justify the assumption available, manufactured that s12 = 0. and selected for specific, often exceptional, Incidentally, a small reverse gain, or feedback factor, s12, is an performance characteristics. important and desirable property for a transistor to have, for Discrete transistors remain reasons other than it simplifies amplifier design. A small feedback the best choice for many factor means that the input characteristics of the completed applications, such as amplifier will be independent of the load, and the output will be sensitive first-stage independent of the source impedance. In most amplifiers, amplifiers in satellite isolation of source and load is an important consideration. TV receivers. 38
  39. 39. 7 H Narrow-Band Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Satellite Broadcast Signals Returning 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 2 2 2 s 21 (1 − ΓS )(1 − ΓL ) high power vacuum tubes, G Tu = (24) called Traveling Wave 2 2 Tubes (TWTs), which are 1 − s11ΓS 1 − s 22 ΓL 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
  40. 40. 7 H Narrow-Band Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Values 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 corresponding to s11, s*11, s22, and s*22 at 300 MHz are added to a device or plotted. Each point represents the tip of a vector leading away circuit to cancel out or compensate for from the center of the chart, its length equal to the magnitude 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 chart, representing 50 Ω, to s*11 and s*22. The circles on the network is carefully Smith Chart are constant-resistance circles; increasing series designed to transform capacitive reactance moves an impedance point counter- the 50 Ω load impedance to s*11. Another matching clockwise along these circles. 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. 40 http://www.hp.com/go/tminteractive
  41. 41. 7 H Narrow-Band Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Figure 6 In 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* 11 susceptance moves impedance points clockwise along constant- L1 conductance circles. These s* 22 circles are like the constant- resistance circles, but they are on another Smith Chart, one that is just the reverse C1 of the chart shown in Fig. 6. C2 L2 s22 The constant-conductance circles for shunt L all pass s11 through the leftmost point of 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
  42. 42. 7 H Narrow-Band Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Figure 7 Once these circles have been located, A 300-MHz amplifier with matching networks for the normalized values of L and C maximum power gain. needed for the matching networks are calculated from readings taken from the C2 reactance and susceptance scales of the C1 2N3478 Smith Charts. Each element’s reactance or 50W L1 L2 50W susceptance is the difference between the scale readings at the two end points of a circular arc. Which arc corresponds to which element is indicated in Fig. 6. X = 50 = 156 W Calculations: L 2 0.32 The final network and the element 156 values, normalized and unnormalized, L2 = = 83 nH are shown in Fig. 7. 2 p ( 0.3 x 109 ) 1 C2 = = 3 pF 2 p ( 0.3 x 10 9 )( 3.5 )( 50 ) 1 C1 = = 25 pF 2p ( 0.3 x 109 )(0.42)(50) 50 L1 = = 26 nH 2p ( 0.3 x 109 )(1.01) 42
  43. 43. 7 H Narrow-Band Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design S* 22 The animation to the right demonstrates how to use a Smith Chart to design a matching network between a transistor 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 Ω in this 300-MHz amplifier example. This matching is achieved using the LC circuit shown at the right. Starting from the 50 Ω load, the series capacitance is varied to move the impedance point along the circle of constant 50 Ω 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 Animation 2 C conductance circle as Impedance matching indicated by the admittance using the Smith Chart Smith Chart. To reach s*22, for the matching 2N3478 L 50 Ω the shunt inductance is network shown at the adjusted until the impedance left. Click over the chart point reaches s*22. to start animation. 43 http://www.hp.com/go/tminteractive
  44. 44. 8 H Broadband Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Designing a broadband amplifier, that is, one which has nearly constant gain over a GTu = G0 G1G2 prescribed frequency range, is a matter of surrounding a transistor with external 2 elements in order to compensate for the variation of forward gain, | s21| with G0 = s21 frequency. 2 This can be done in either of two ways— 1 − Γs first, negative feedback, or second, selective G1 = mismatching of the input and output 2 circuitry. We will use the second method. 1 − s11Γs When feedback is used, it is usually convenient to convert to y- or z-parameters 2 (for shunt or series feedback, respectively) 1 − ΓL using the conversion equations given in Appendix B and a digital computer. G2 = 2 Equation 24 for the unilateral transducer 1 − s22ΓL power gain can be factored into three parts, as shown to the right: 44
  45. 45. 8 H Broadband Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design When a broadband amplifier is designed by selective mismatching, the gain contributions of G1 and G2 are varied to compensate for the variations of G0= |s21|2 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 s 21 = 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. 45
  46. 46. 8 H Broadband Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design Although in the general case both a source matching network and a load matching network would be designed, G1max (i.e., G1 for Γs = s*11) for this transistor is s* 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
  47. 47. 8 H Broadband Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design 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: G2 = +4 dB 1 − g 2 s 22 at 700 MHz r2 ρ r2 = 2 2 1 − s 22 (1 − g 2 ) where r2 r2 g2 = G2 2 = G 2 (1 − s 22 ) ρ G 2 max 2 The radius of the constant-gain circle is: G2 = 0 dB ρ at 450 MHz 2 2 1 − g 2 (1 − s 22 ) ρ2 = 2 1 − s 22 (1 − g 2 ) G2 = – 3 dB at 300 MHz 47
  48. 48. 8 H Broadband Test & Measurement Application Note 95-1 S-Parameter Techniques Amplifier Design For this example, three circles will be drawn, one for G2 = -3 dB at 300 MHz, one for G2 = 0 dB at 450 MHz, and one for G2 = +4 dB at 700 MHz. Since |s22| for this transistor is constant at 0.85 over the frequency range [see Figure 4(b)], G2max for all three circles is (0.278)-1, or 5.6 dB. The three constant-gain circles are indicated in Fig. 8b. Figure 9 A combination of shunt The required matching network must transform the center of the and series inductances Smith Chart, representing 50 W , to some point on the -3 dB is a suitable matching circle at 300 MHz, to some point on the 0 dB circle at 450 MHz, network for the and to some point on the +4 dB circle at 700 MHz. There are broadband amplifier. undoubtedly many networks that will do this. One satisfactory Lseries solution is a combination of two inductors, one in shunt and one in series, as shown in Fig. 9. 50W 2N3478 Lshunt ZL = 50W 48