X-Parameters Characterization Modeling and Design of Nonlinear RF and Microwave Components_Root.pdf

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X-Parameters

This is the de nitive guide to X-parameters, written by the original inventors and developers of this powerful new paradigm for nonlinear RF and microwave components and systems.

Learn how to use X-parameters to overcome intricate problems in nonlinear RF and microwave engineering, as the general theory behind X-parameters is carefully and intuitively introduced, and then simpli ed down to speci c, practical cases, providing you with useful approximations that will greatly reduce the complexity of measuring, modeling, and designing for nonlinear regimes of operation.

Containing real-world case studies, de nitions of standard symbols and notation, detailed derivations within the appendices, and exercises with solutions, this is the de nitive stand-alone reference for researchers, engineers, scientists, and students looking to remain on the cutting edge of RF and microwave engineering.

David E. Root

David E. Root is an Agilent Research Fellow at Agilent Technologies. He co-led the Agilent research and technical development of X-parameters through its commercial-ization. He is a Fellow of the IEEE, and co-editor of Nonlinear Transistor Model Parameter Extraction Techniques (2011).

Jan Verspecht

Jan Verspecht is a Master Research Engineer at Agilent Technologies. He invented X-parameters in 1996, and is a Fellow of the IEEE.

Jas

Jason on HorHornn is an Expert Design Engineer at Agilent Technologies, who has been heavily involved in the development of X-parameter measurements.

Mihai Marcu

Mihai Marcu is a Senior Consultant at Agilent Technologies, who is deeply involved in the development and application of X-parameters for nonlinear modeling.

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The Cambridge RF and Microwave Engineering Series The Cambridge RF and Microwave Engineering Series

Series Editor

Steve C. Cripps, Distinguished Research Professor, Cardiff University

Peter Aaen, Jaime Plá and John Wood, Modeling and Characterization of RF and Microwave Power FETs

Dominique Schreurs, Máirtín O’Droma, Anthony A. Goacher and Michael Gadringer, RF Ampli er Behavioral Modeling

Fan Yang and Yahya Rahmat-Samii, Electromagnetic Band Gap Structures in Antenna Engineering

Enrico Rubiola, Phase Noise and Frequency Stability in Oscillators Earl McCune, Practical Digital Wireless Signals

Stepan Lucyszyn, Advanced RF MEMS

Patrick Roblin, Nonlinear RF Circuits and the Large-Signal Network Analyzer Matthias Rudolph, Christian Fager and David E. Root, Nonlinear Transistor Model

Parameter Extraction Techniques

John L. B. Walker, Handbook of RF and Microwave Solid-State Power Ampli ers Anh-Vu H. Pham, Morgan J. Chen and Kunia Aihara, LCP for Microwave Packages

and Modules

Sorin Voinigescu, High-Frequency Integrated Circuits Richard Collier, Transmission Lines

Valeria Teppati, Andrea Ferrero and Mohamed Sayed, Modern RF and Microwave Measurement Techniques

Nuno Borges Carvalho and Dominique Schreurs, Microwave and Wireless Measurement Techniques

David E. Root, Jan Verspecht, Jason Horn and Mihai Marcu, X-Parameters Forthcoming

Richard Carter, Theory and Design of Microwave Tubes

Hossein Hashemi and Sanjay Raman, Silicon mm-Wave Power Ampli ers and Transmitters

Earl McCune, Dynamic Power Supply Transmitters

Isar Mostafanezad, Olga Boric-Lube cke and Jenshan Lin, Medical and Biological Microwave Sensors

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X-Parameters

Characterization, Modeling, and Design of

Nonlinear RF and Microwave Components

DAVID E. ROOT

Agilent

Agilent Technologies, Inc.Technologies, Inc.

JAN VERSPECHT

Agilent

JASON HORN

Agilent

MIHAI MARCU

Agilent

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University Printing House, Cambridge CB2 8BS, United Kingdom

Published in the United States of America by Cambridge University Press, New York

Cambridge University Press is part of the University of Cambridge.

It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence.

www.cambridge.org

Information on this title: www.cambridge.org/9780521193238

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2013

Printing in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data

Root, David E.

X-parameters : characterization, modeling, and design of nonlinear RF and microwave components / David E. Root, Agilent Technologies Inc., Jan Verspecht, Agilent Technologies Inc., Jason Horn, Agilent Technologies Inc., Mihai Marcu, Agilent Technologies Inc.

pages cm – (The Cambridge RF and microwave engineering series) Includes bibliographical references.

ISBN 978-0-521-19323-8 (Hardback)

1. Microwave circuits – Design and construction – Mathematics. 2. Electric circuits, Nonlinear – Design and

construction – Mathematics. 3. Parametric devices – Design and construction – Mathematics. 4. Differential equations. I. Verspecht, Jan. II. Horn, Jason. III. Marcu, Mihai. IV. Title. TK7876.R66 2013

621.38410

2 – dc23 2013013915

“X-parameters” is a trademark of Agilent Technologies, Inc. ISBN 978-0-521-19323-8 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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For Marilyn, with thanks for her patience, support, and, most of all, her love.

David

In memory of Petrus Verspecht.

Jan

To Jessica, Jonathan, and Elise, my inspiration.

Jason

To Domnica, for the patience shown in the many evenings and weekends

that I have spent away from her.

Mihai

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“Just as the S-parameters revolutionized linear microwave circuit engineering nearly 60 years ago, the relatively new development of the X-parameters and the mixer-based VNA provides a truly scienti c approach to nonlinear RF and microwave circuit design. This book, written by experts, contains a wealth of information about the characterization and modeling of

nonlinear components as well as their applications to various types of designs. I can only wish that such capability and textbook had been available when I was a design engineer. ”

Les Besser

Founder of Compact Software and Besser Associates

“S-parameters revolutionized linear RF and Microwave design in the 1970s and X-parameters are

doing the same for non-linear design today. Starting with the familiar foundation of S-parameters, the text guides the reader through the additional non-linear terminology needed to provide a clear and practical view of X-parameters. Many practical examples show how to apply them in real world designs and answers are provided to some of the more subtle concepts of cross-frequency phase and memory effects. In a world where wireless is proliferating, this book will be an invaluable reference for any RF designer to reduce design turns and improve

their rst-pass designs.”

Mark Pierpont

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Preface

The need for a rigorous, yet practical, framework for characterization, modeling, and design of nonlinear electronic components at high frequencies has never been more urgent. The communications revolution is inexorably forcing active devices into more and more strongly nonlinear regimes of operation. This is a consequence of the relentless drive for more ef ciency in order to save power, extend battery life, and minimize cooling. The price for ef ciency is nonlinearity. Dealing with nonlinearity means that new measurement instrumentation and new modeling and design method-ologies are required that go far beyond linear S-parameters. Fortunately, there is an overarching, interoperable paradigm combining all these pieces of the nonlinear puzzle together, seamlessly. The new paradigm is called X-parameters, 1and that is what this book is about.

The book is intended as a comprehensive introduction to X-parameters. It is aimed at a diverse audi ence with a wide range of backgrounds. This is quite a challenging undertaking! We are targeting professional microwave engineers, device modeling engineers and scientists, RF and microwave circuit designers, electronic and communi-cations engineers, CAE professionals developing simulator algorithms, and microwave and RF professionals developing new high-speed instrumentation for a wide range of nonlinear characterization applications. The inherent interdisciplinary nature of X-parameters is the prime reason we seek to appeal to this broad audience. The practical solutions based on X-parameters deployed by industry over the past several years depend on contributions in all of these areas.

With this diverse audience in mind, we have chosen a particular sequence with which to introduce the subject. We start with a concise summary of the well-known time-invariant linear theory, namely S-parameters. We choose this context, familiar to many readers, to introduce more advanced concepts that will be needed for the remainder of the book. Chapter 2 introduces X-parameters, based on multi-tone nonlinear spectral maps de ned on a harmonic grid, and goes into signi cant detail about the application and implications of the constraint of time invariance. Chapter 3 simpli es the general discussion to simple practical cases, based on the application of spectral linearization, a useful approximation that reduces complexity, enabling practical applications. Several examples are presented demonstrating the power, utility, and relative simplicity of these

1

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simplest X-parameters. The srcins of “conjugate” terms in the spectral linearization are discussed. Chapter 4 is devoted to how X-parameters are measured, and also to how they are computed (generated) from within a circuit simulator. The functional block diagram of the main instrument (the nonlinear vector network analyzer – NVNA) is

discussed, and the application of measurements using a pulse generator phase reference to obtain the key X-parameter quantities is reviewed. Chapter 5 extends the treatment of X-parameters to multiple large signals and multiple ports, as is necessary in the

treatment of many mixers, the treatment of intermodulation with phase, and the large-signal response of power ampli ers as nonlinear functions of both input power and re ections of electrical signals back into the device due to large mismatch, going beyond the rst spectral linearization approximation of Chapter 3. Finally, Chapter 6

extends the treatment of X-parameters to dynamic “memory effects,” important

phe-nomena exhibited by practical modern high-speed devices in response to wide-band communication signals, for example. Several appendices are provided for detailed derivations, standard symbol and notational de nitions, and further elaboration of some parts of the main text to help serve as a reference for workers in the eld.

The book is appropriate as a text for an advanced undergraduate or graduate course in electrical engineering. In fact, we perceive an acute need to make X-parameters a standard part of the electrical engineering curriculum. The book may also be appropriate for applied mathematicians and scientists with an interest in rigorous and practical foundations for applications to a wide range of nonlinear systems well beyond electronics.

The background needed by readers of this book is not much more than rst-year calculus, basic circuit theory, and simple Fourier analysis. Rudimentary knowledge of electronic power ampli ers and transistors, S-parameter fundamentals, differential equations, circuit design, and circuit simulation would certainly be helpful.

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Acknowledgments

The authors are profoundly grateful to our many dedicated and talented colleagues who collaborated with us to develop and deploy X-parameter technology, products, support, and services. We are grateful to our many customers, academic researchers, and practicing professionals for their thoughtful feedback, stimulating discussions, and

creative applications of this technology. We thank Agilent management for their vital support. Finally, we thank the staff at Cambridge University Press for their commitment to this project, their cheerful professionalism, and their patience.

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1 1 S-p

S-para

aramet

meters

ers –

– a

a con

concis

cise

e rev

review

iew

1 1..1

1 IIn

nttrro

od

du

uc

cttiio

on

n

This chapter presents a concise treatment of S-parameters, meant primarily as an introduction to the more general formalism of X-parameters. The concepts of time invariance and spectral maps are introduced at this stage to enable an easier general-ization to X-parameters in the ensuing chapters. The interpretations of S-parameters as calibrated measuremen ts, intrinsic properti es of the device under test (DUT), IP-secure component behavioral models, and composition rules for linear system design are presented. The cascade of two linear S-parameter components is considered as an

example to be generalized to the nonlinear case later. The calculation of S-parameters for a transistor from a simple nonlinear device model is used as an example to introduce the concepts of (static) operating point and small-signal conditions, both of which must be generalized for the treatment of X-parameters.

1 1..2

2 S

S--p

pa

arra

am

me

ette

errss

Since the 1950s, S-parameters, or scattering parameters, have been among the most important of all the foundations of microwave theory and techniques.

S-parameters are easy to measure at high frequencies with a vector network analyzer (VNA). Well-calibrated S-parameter measurements represent intrinsic properties of the DUT, independent of the VNA system used to characterize it. Calibration procedures [1] remove systematic measurement errors and enable a separation of the overall values into numbers attributable to the device, independent of the measurement system used to characterize it. These DUT properties (gain, loss, re ection coef cient, etc.) are famil-iar, intuitive, and important [2]. Another key property of S-parameters is that the S-parameters of a composite system are completely determined from knowledge of the S-parameters of the constituent components and their connectivity. S-parameters provide the complete speci cation of how a linear component responds to an arbitrary

signal. Therefore designs of linear systems with S-parameters are predictable with absolute certainty. S-parameters de ne a complete behavioral description of the linear component at the external terminals, independent of the detailed physics or speci cs of the realization of the component. S-parameters can be shared between component vendors and system integrators freely, without the possibility that the component

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implementation can be reverse engineered, protecting IP and promoting sharing and reuse. Indeed, one may ask the question, “are S-parameters measurements, or do they constitute a model?” The answer is really “ both.”

S-parameters need not come only from measurements. They can be calculated from physics by solving Maxwell’s equations, by linearizing the semiconductor equations, or

computed from matrix analysis of linear equivalent circuits. In this way, the many bene ts of S-parameters can be realized, starting from a more detailed representation of

the component from rst principles or from a complicated linear circuit model. Graphical methods based around the Smith chart were invented to visualize and interpret S-parameters, and graphical design methodologies soon followed for circuit design [2][3]. These days, electronic design automation (EDA) tools provide simulation components – S-parameter blocks – and design capabilities using the familiar

S-parameter analysis mode.

One of the great utilities of S-parameters is the interoperability among the measurement, modeling, and design capabilities they provide. One can characterize the component with measured S-parameters, use them as a high- delity behavioral model of the component with complete IP protection, and design systems with them in the EDA environment.

1 1..3

3 W

Wa

avve

e vva

arriia

ab

blle

ess

The term “scattering” refers to the relationship between incident and scattered (re ected

and transmitted) traveling waves.

By convention, in this text the circuit behavio r is described using generalized power waves [2]. There are alternative wave de nitions used in the industry. These are brie y reviewed in Appendix A, together with the general notations used in this text.

The wave variables, A and B, corresponding to a speci c port of a network, are de ned as simple linear combinations of the voltage and current, V and I , at the same port, according to Figure 1.1 and equations (1.1):

A

_¼

V

þ

Z 0 I 2

p

_{ffiffiffiffiffi}

Z 0 , B

_¼

V



Z 0 I 2

p

_{ffiffiffiffiffi}

Z 0 :

ð

1:1

_Þ

The reference impedance for the port, Z 0, is, in general, a complex value. For the

purpose of simplifying the concepts presented, the reference impedance is restricted to real values in this text.

Figure 1.1

Figure 1.1 Wave de nitions. 2 S-paraS-parametermeters s – – a a conciconcise se revierevieww

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The currents and voltages can be recovered from the wave variables, according to equations (1.2): V

_¼

p ð

_{ffiffiffiffiffi}

Z 0 A

þ

B

Þ

, I

_¼

1

ffiffiffiffiffi

Z 0

p

A

_ð

_

B

_Þ

:

ð

1:2

Þ

Here, A and B represent the incident and scattered waves, V and I are the port voltage and current, respectively, and Z 0is the reference impedance for the port. A typical value of Z 0is

50Ω by convention, but other choices may be more practical for some applications. A value for Z 0_{closer to 1}_Ω_{is more appropriate for S-parameter measurements of power transistors,}

for example, given that power transistors typically have very small output impedances. The variables in equations (1.1) and (1.2) are complex numbers representing the RMS-phasor description of sinusoidal signals in the frequency domain (see Appendix A

for a more detailed discussion of the notations used). Later this will be generalized to the envelope domain by letting these complex numbers vary in time.

A, B, V , and I can be considered RMS vectors, the components of which indicate the values associated with sinusoidal signals at particular ports labeled by positive integers. Thus A j is the incident wave RMS phasor at port j and I k is the current RMS phasor at

port k . For now, Z 0 is taken to be a xed real constant, in particular, 50 Ω.

A graphical representation of the wave description is given in Figure 1.2.

To retrieve the time-dependent sinusoidal voltage signal at the ith port, the complex value of the phasor and also the angular frequency, ω, to which the phasor corresponds, must be known. The voltage is then given by

v i

ð

t

Þ ¼

Re

f

V i

ð Þ

pk e j ωt

g

,

ð

1:3

Þ

and similarly for the other variables, where V pk

ð Þ

_i are peak values. Equation (1.3) for the voltage, and a similar equation for the time-dependent current, can be used to de ne real, time-dependent “wave” quantities using the same linear combinations as in (1.1):

a

_ð

t

_{Þ ¼}

1 2

p

_{ffiffiffiffiffi}

Z 0



v

_ð

t

_{Þ þ}

Þ



;

ð

1:4

_Þ

I1 I2 A1 DUT + -+ -+ -+ -Port 1 Port 2 B1 A2 B2 V1 V2 Figure 1.2

Figure 1.2 Incident and scattered waves of a two-port device.

3

1.3

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v

_ð

t

_{Þ ¼}

_{ffiffiffiffiffi}

p

Z 0 a



ð

t

Þ þ

b

ð

t

Þ



,

i

_ð

indexing notation is made explicit:

Ai

ð

ω

Þ ¼

1 2

p

_{ffiffiffiffiffi}

Z 0 V



Þ 

Z 0 I i

ð

ω

Þ

ð

ω

X

N

j

_¼

Þ

The set of complex coef cients,S ij (ω), in(1.8)de nes the S-parameter matrix or, simply, the

S-parameters at that frequency. Equation (1.8), for the xed set of complex S-parameters, determines the output phasors for any set of input phasors. The summation is over all port indices, so that incident waves at each port, j , contribute in general to the overall

scattered wave at each output port, i. For now we consider all frequencie s to be positive (ω>0). Note that contributions to a scattered wave at frequency ω come only from incident waves at the same frequency. This is not the case for the more general X-parameters, where a stimulus at one frequency can lead to scattered waves at different frequencies.

The set of equations (1.8)represents a model of the network under study. However, this model is valid only if the network has the topological connections shown in Figure 1.2. For example, the model might not accurately represent the behavior of the network when connected as shown in Figure 1.3 because potential losses between the reference pins of the two ports are not individually identi ed in the set of S-parameters in (1.8). For the purpose of creating a model for the network, all ports should be referenced to the same pin, as shown in Figure 1.4.

Such connectivity is the natural option for the measurement and modeling process of a three-pin network (like a transistor), but it has to be extended in the general case of an arbitrary network, and it is necessary for all networks, linear and/or nonlinear. This connectivity convention is considered by default (unless otherwise speci ed) for the remainder of this text.

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Using the topological connection in Figure 1.4, the set of equations (1.8) represents a complete model of the network under test.

From (1.8) we note that a stimulus (incident wave) at a particular port j will produce a response (scattered wave) at all ports, including the port at which the stimulus is applied.

Equation (1.8) shows that the scattered waves are linear functions of the complex amplitudes (the phasors) of the incident waves. The dependence on frequency of the S-parameter matrix elements, S ij (ω), can be highly nonlinear, even for a linear device.

For example, an ideal band-pass lter response is linear in the incident wave variable, but the lter response is a nonlinear function of the frequency of the incident wave.

This is shown in Figure 1.5.

1

1.4 .4

S

S--p

pa

arra

am

me

ette

er

r m

me

ea

assu

urre

em

me

en

ntt

By setting all incident waves to zero in (1.8), except for A j , one can deduce the simple

relationship between a given S-parameter (S-parameter matrix element) and a particular ratio of scattered to incident waves according to (1.9):

_6¼

0 j :

ð

1:9

_Þ

I1 I2 A1 DUT + -+ -+ -+ -Port 1 Port 2 B1 A2 B2 V1 V2 Figure 1.3

Figure 1.3 Potential losses between reference pins are not individually identi ed bythe model in(1.8).

I1 I2 A1 DUT + -+ + -+ -Port 1 Port 2 B1 A2 B2 V1 V2 Figure 1.4

Figure 1.4 All ports should be referenced to the same pin for modeling purposes.

5

1.4

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Equation (1.9) corresponds to a simple graphical representation shown in Figure 1.6 for the simple case of a two-port component. In Figure 1.6(a), the stimulus is a wave incident at port 1. The fact that A2 is not present ( A2

¼

0) is interpreted to mean that the B2 wave

scattered and traveling away from port 2 is not re ected back into the device at port 2. Under this condition, the device is said to be perfectly matched at port 2. Two of the four complex S-parameters, speci callyS 11andS 21, can be identi ed using(1.9)for this case of

exciting the device with only A1. Figure 1.6(b) shows the case where the device is

stimu-lated with a signal, A2, at port 2, and assumed to be perfectly matched at port 1 ( A1

¼

0).

The remaining S-parameters, S 12 and S 22, can be identi ed from this ideal experiment.

Output power vs. frequency @ 0 dBm input power Output power vs. input power @ 2.5 GHz

50 10 P_1Tone PORT1 Num=1 z=50 Ohm P=polar(dbmtow(Pwr_dBm),0) Freq=RFfreq Term Term2 Num=2 Z=50 Ohm DUT Ref 1 2 + – + – 8 6 4 2 0 –2 0 –50 –100 –150 –200 –250 –300 –350 0 01 23 4 5 67 8 9 10 Input power (dBm) O u t p u t p o w e r ( d B m ) 1 2 3 45 6 7 8 91 0 Frequency (GHz) O u t p u t p o w e r ( d B m ) Figure 1.5

Figure 1.5 A linear network has a linear behavior when plotted versus input power level, but the dependence on frequency is usually not linear.

Figure 1.6

Figure 1.6 S-parameter experiment design: (a) forward transmission; (b) reverse transmission. 6 S-paraS-parametermeters s – – a a conciconcise se revierevieww

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It is important to note that the ratio on the right-hand side of (1.9) can be computed from independent measurements of incident and scattered waves for actual components corresponding to any non-zero value for the incident wave, A j . The value of this ratio,

however, will generally vary with the magnitude of the incident wave. Therefore, the identi cation of this ratio with “the S-parameters” of the component is valid for any

particular value of incident A j only if the component behaves linearly, namely according





Ak

₈

_k

¼

_6¼

_j0 :

ð

1:10

_Þ

That is, for a general component, bias at a constant DC stimulus, the S-parameters are related to ratios of output responses to input stimuli in the limit of small input signals. This emphasizes that S-parameters properly apply to nonlinear compone nts only in the small-signal limit.

1

1.5 .5

S

S--p

pa

arra

am

me

ette

errs

s a

as

s a

a ssp

pe

ec

cttrra

al l m

ma

ap

p

If there are multiple frequencies present in the input spectrum, one can represent the output spectrum in terms of a matrix giving the contributions to each output frequency from each input frequency.

2

4

3

5

¼

S

ð Þ

ω1 0 0 0 S

ð Þ

ω2 0 0 0 S

ð Þ

ω3

2

4

3

Þ

2

4

3

5

:

ð

1:11

Þ

Here we assume a single port, for simplicity, and therefore drop the port indices. It is clear from (1.11) that S-parameters are a diagonal map in frequency space. This means that each output frequency contains contributions only from inputs at that same fre-quency. Or, in other words, each input frequency never contributes to outputs at any different frequency.

A graphical representation is given in Figure 1.7 for the case of forward transmi ssion through a two-port network with matched terminations at both ports.

The interpretation of Figure 1.7, mathematically represented by (1.11), is that S-parameters de ne a particularly simple linear spectral map relating incident to scattered waves. S-parameters are diagonal in the frequency part of the map, namely they predict a response only at the particular frequencies of the corresponding input stimuli. It will be demonstrated in later chapters that X-parameters provide for richer behavior.

7

1.5

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For signals with a continuous spectrum, the diagonal nature of the S-parameter spectral map can be written as equation (1.12):

Bi

ð

ω

Þ ¼

X

N

j

_¼

1

ð

_S

up

pe

errp

po

ossiittiio

on

n

Any linear theory, such as S-parameters, enables the general response to an arbitrary input signal to be computed by superposition of the responses to unit stimuli. Superposition enables great simpli cations in analysis and measurement. Superpos-ition is the reason S-parameters can be measured by independent experiments with one sinusoidal stimulus at a time, one stimulus per port per frequency using (1.9). The general response to any set of input signals can be obtained by superposition using (1.8).

An example of superposition is shown in Figure 1.8, with all signals represented in both time and frequency domains.

Term1 A1 B2 Term2 0 –20 –40 –60 –80 –100 –120 3.0 3.2 3.4 3.6 3.8 4.0 Frequency (GHz) P o w e r o f A 1 ( d B m ) P o w e r o f B 2 ( d B m ) S (w w ₁).A(w w ₁) + 0.A(w w ₂) + 0.A(w w ₃)=B (w w ₁) 0.A(w w ₁) +S (w w ₂).A(w w ₂) + 0.A(w w ₃)=B (w w ₂) 0.A(w w ₁) + 0.A(w w ₂) +S (w w ₃).A(w w ₃)=B (w w ₃) 0 –20 –40 –60 –80 –100 –120 3.0 3.2 3.4 3.6 3.8 3.0 Frequency (GHz) DUT Ref 1 2 + – + – Figure 1.7

Figure 1.7 Linear spectral map through S-parameters matrix. 8 S-paraS-parametermeters s – – a a conciconcise se revierevieww

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This example uses two signals, each containing two frequency components, as stimuli incident at each port, independently, with the other port perfectly matched. The example shows that the response to a linear combination of the stimuli is the same linear combination of the individual responses.

As always, the caveat is that the component actually behaves linearly over the range of signal levels used to stimulate the device. There is no a-priori way to know whether a component will behave linearly without precise knowledge about its composition or physical measured characteristics.

Signal 1 - incident (a) (c) (b) m a g n i t u d e o f A 1 ( d B m ) m a g n i t u d e o f B 1 ( d B m ) a 1 ( s q r t ( W ) ) b 1 ( s q r t ( W ) * 1 e 3 )

Signal incident- 1 Signal transmitted- 1

time (ns) Signal 1 - transmitted frequency (GHz) 1DUT2 Ref 1DUT2 Ref 1DUT2 + + Ref 0 37 38 0.08 b 2 ( s q r t ( W ) ) m a g n i t u d e o f B 2 ( d B m ) 0.04 0.00 –0.04 –0.08 39 40 41 42 43 –90 1 2 3 4 –70 –50 –30 –10 10 Signal 2 - reflected time (ns) Signal 2 - reflected frequency (GHz) 0 37 38 0.08 b 2 ( s q r t ( W ) ) m a g n i t u d e o f B 2 ( d B m ) 0.04 0.00 –0.04 –0.08 39 40 41 42 43 –90 1 2 3 4 –70 –50 –30 –10 10 Signal 2 - incident

Signal 1+ Signal 2− scattered @ port 2 Signal 1+ Signal 2− scattered @ port 2

Signal 1+ Signal 2− scattered @ port 1 Signal 1+ Signal 2− scattered @ port 1

time (ns) Signal 2 - incident frequency (GHz) 0 37 38 0.08 b 2 ( s q r t ( W ) ) b 2 ( s q r t ( W ) ) b 2 ( s q r t ( W ) ) m a g n i t u d e o f A 2 ( d B m ) m a g n i t u d e o f B 2 ( d B m ) m a g n i t u d e o f A 2 ( d B m ) 0.04 0.00 –0.04 –0.08 39 40 41 42 43 –90 1 2 3 4 –70 –50 –30 –10 10 Signal 1 - reflected Signal 1 - reflected

10 –10 –30 –50 –70 –90 10 –10 –30 –50 –70 –90 0123 frequency (GHz) time (ns) time (ns) 4 0123 frequency (GHz) 4 37 38 –0.08 –0.04 0.00 0.04 0.08 –0.4 –0.2 0.0 0.2 0.4 39 40 41 42 43 37 38 39 40 41 42 0.004 Signal 2 - transmitted

Signal 1 - incident Signal 1 - incident

Signal 2 - incident Signal 2 - incident 10 10 –10 –30 –50 –70 –90 0.4 0.3 0.2 0.1 0.0 –0.1 –0.2 –0.3 –0.4 373 83 94 04 14 24 3 0 1 2 3 4 frequency (GHz) 10 –10 –30 –50 –70 –90 01234 frequency (GHz) time (ns) 0.3 0.2 0.1 0.0 –0.1 –0.2 –0.3 37 38 39 40 41 42 43 time (ns) m a g n i t u d e o f B 1 ( d B m ) m a g n i t u d e o f A 1 ( d B m ) b 1 ( s q r t ( W ) ) –10 –30 –50 –70 –90 0 10 –10 –30 –50 –70 –90 01234 0.15 a 1 ( s q r t ( W ) )0.10 0.05 0.00 –0.05 –0.10 –0.15 37 38 39 40 41 42 43 1 2 3 frequency (GHz) frequency (GHz) m a g n i t u d e o f B 1 ( d B m ) 10 –10 –30 –50 –70 –90 01234 frequency (GHz) time (ns) 0.015 b 1 ( s q r t ( W ) )0.010 0.005 0.000 –0.005 –0.010 –0.015 37 38 39 40 41 42 43 time (ns) time (ns) 4 Signal 2 - transmitted 0.003 0.002 0.001 0.000 –0.001 –0.002 –0.003 –0.004 37 38 39 40 41 42 43 43 Figure 1.8

3 (response 2).

9

1.6

(28)

1.

1.7

7 Ti

Time

me in

inva

vari

rian

ance

ce of

of co

comp

mpon

onen

ents

ts de

desc

scri

ribe

bed

d by

by S-

S-pa

para

ramet

meter

erss

A DUT description in terms of S-parameters de ned by (1.8) naturally embodies an important principle known as time invariance. Time invariance states that if y(t ) is the DUT response to an excitation x(t ), the DUT response to the time-shifted excitation, x(t

_

τ ), must be y(t

_

τ ). This must be true for all time shifts, τ . That is, if the input is

_

:

ð

1:13

_Þ

Time invariance is a property of common linear and nonlinear components, such as passive inductors, capacitors, resistors, and diodes, and active devices, such as

transis-tors. Examples of components not time invariant (in the usual sense) are oscillators and other autonomous systems.

The proof follows from elementary properties of the Fourier transform, where a phase shift by e j ωτ

in the frequency domain corresponds to a time shift of τ in the time domain. The time-domain waves incident at the ports (the stimuli) are ak (t ), and their Fourier

transforms are A

ð Þ

_k pk ω

_{ð Þ}

, as in equation (1.14):

F ak

ð

t

Þ





¼

A

ð Þ

_k pk

A

ð Þ

k pk

ð

ω

Þ ¼

X

N k

_¼

1 S ik

ð

ð

ω

Þ

F



ak

ð

t



τ

Þ



(

)

¼

F



1

_X

N k

_¼

1 S ik

ð

ω

Þ

F



ak

ð

t

Þ



e j ωτ

(

)

¼

F



1

n

B

ð Þ

_i pk

_ð

ω

_Þ

e j ωτ

o

_¼

bi

ð

t



τ

Þ

:

ð

1:16

Þ

Equation (1.16) proves that S-parameters are automatically consistent with the principle of time invariance. Therefore, any set of S-parameters describes a time-invariant system.

Unlike the case for S-parameters, a more general (e.g. nonlinear) relationship between incident A waves and scattered B waves is not automatically consistent with the property (1.13) of time invariance. This will be demonstrated in Chapter 2. There-fore, in order to have a consistent representation of a nonlinear time-invariant DUT, the time-invariance property is manifestly incorporated into the mathematical formulation of X-parameters relating input to output waves. A representation of a time-invariant DUT by equations not consistent with (1.13) means the model is fundamentally wrong, and can yield very inaccurate results for some signals, even if the model “ tting”

(or identi cation) appears good at time t .

(29)

1 1..8

8 C

Ca

assc

ca

ad

da

ab

biilliittyy

Another key property of S-parameters is that a linear circuit or system can be designed with perfect certainty knowing only the S-parameters of the constituent components and their interconnections. The overall S-parameters of the composite design can be calcu-lated by using (1.8) in conjunction with Kirchhoff ’s voltage law (KVL) and Kirchhoff ’s current law (KCL), applied at the internal nodes created by connections between two or more components.

This is illustrated for a cascade of two 2-ports in Figure 1.9(a). Each component is characterized by its own S-parameter matrix, S (1)

and S (2)

, respectively. The output port (subscript number 2) of the rst component is connected to the input port (subscript number 1) of the second component, creating an internal node.

It is a straightforward exercise to write the two equations that follow from applying KVL and KCL at the internal node. These are given by equations (1.17) and (1.18): B₂

ð

1

Þ

_¼

Þ

B₁

ð

2

Þ

:

ð

1:18

_Þ

Each 2-port relates two input and two output variables. When cascaded, equations (1.17) and (1.18) can be used to compute the overall S-parameter matrix of the composite. The result is given in equation (1.19):

(a)

(b)

Figure 1.9

Figure 1.9 The cascade of two 2-ports. (a) Cascade of two DUTs. (b) The equivalent network, yielding the same response as the cascade of two DUTs.

11

₂₂ 1 S 2

ð Þ

₁₁

þ

S 2

ð Þ

22

0 B

B

@

1 C

C

A

:

ð

1:19

_Þ

A network characterized by the S compositeis equivalent to the cascade of the two DUTs, yielding the same response under the same stimuli. This is shown in Figure 1.9(b).

1 1..9

9 D

DC

C o

op

pe

erra

attiin

ng

g p

po

oiin

ntt

For devices such as diodes and transistors, the S-parameter values depend very strongly on the DC bias conditions de ning the component operating conditions. For example, a eld-effect transistor (FET) biased in the saturation region of operation will have S-parameters appropriate for an active ampli er (where

_j

S 21

j 

1

 j

S 12

j

), whereas

when biased “off ” (say V DS

¼

0) the S-parameters will represent the characteristics of a

passive switch (S 21

¼

S 12 with

j

S ij

j

<1).

1.

1.10

10 S-pa

S-

para

rame

mete

ters

rs of

of a

a no

nonl

nlin

inea

ear

r de

devi

vice

ce

1

It is possible to derive the S-parameters of a transistor starting from a simple nonlinear model of the intrinsic device. The process illustrates the concept of linearizing a non-linear mapping about an operating point. This concept, suitably generalized, is used as a foundation for X-parameters in Chapter 2.

The equivalent circuit of a simple, unilateral FET model is shown in Figure 1.10. More complete model topologies and equations are discussed in [ 4].

A current source, i DS , represents the nonlinear bias-dependent channel current from

drain to source, and a simple nonlinear two-terminal capacitor, qGS , represents the

nonlinear charge storage between gate and source terminals.

The large-signal model equations corresponding to this equivalent circuit are given by (1.20) and (1.21), where the stimulus applied to the device is the set of port voltages,

v GS and v DS , and the device response is the set of port currents, iGS and i DS :

iG

ð Þ ¼

t dqGS



v GS

ð Þ

t



dt ,

ð

1:20

Þ

i D

ð Þ ¼

t i DS



v GS

ð Þ

t ,v DS t

ð Þ



:

ð

1:21

Þ

Equations (1.20) and (1.21) are evaluated for time-varying voltages assumed to have a xed DC component and a small sinusoidal component at a single RF or microwave

1

This section can be omitted on rst reading. 12 S-paraS-parametermeters s – – a a conciconcise se revierevieww

(31)

δi1

ð Þ

t

¼

d dt qGS v DC

ð Þ

1





_þ

dqGS dV 1



v ð₁ DC Þ,v ð DC _Þ 2 δV 1 cos ω

ð

t

þ

ϕ1

Þ

0 @

1 A

¼

cGS



v

ð Þ

1 DC



d dt



δV 1 cos ω

ð

t

þ

ϕ1

2





þ

g _m v

ð Þ

₁ DC ,v DC

ð Þ

2





δV 1cos ω

ð

t

þ

ϕ1

Þþ

g DS v DC

ð Þ

1 ,v DC

ð Þ

2





δV 2cos ω

ð

t

þ

ϕ2

Þ

:

ð

1:24

_Þ

The following de nitions of the “linearized equivalent circuit elements” have been used:

cGS



v

ð Þ

1 DC



ð

1 : 25

_Þ

By equating the zeroth-order terms in δV i, the operating point conditions are

2





:

ð

1 :26

_Þ

G Port 1 Port 2 S D ₊ + -qGS iDS Figure 1.10

Figure 1.10 Simple nonlinear equivalent circuit model of a FET.

13

1.10

(32)

Equating the rst-order terms in δV i leads to δi1

ð Þ ¼

t cGS



v

ð Þ

1 DC



d dt



δV 1 cos ω

ð

t

þ

ϕ1

Þ



, δi2

ð Þ ¼

t g m v DC

ð Þ

1 ,v DC

ð Þ

2





δV 1 cos ω

ð

δV 1

ð Þ

ω δV 2

ð Þ

ω

2





g _DS v

ð Þ

₁ DC ,v DC

ð Þ

2





2

4

3

5

:

ð

1:30

Þ

Equation (1.30) de nes the (commo n source) admit tance matrix of the model. The matrix elements are evidently functions of the DC operating point (bias conditions) of the transistor, and also the (angular) frequency of the excitation.

Since the phasors representing the port voltage and currents in (1.29) can be re-expressed as linear combinations of incident and scattered waves using (1.7), it is possible to derive the expression for the S-parameters in terms of the Y-parameters (admittance matrix elements). This results in the well-known conversion formula (1.31) [2]:

S

_¼

I

½

_

Z 0Y



½

I

þ

Z 0Y





1:

ð

1:31

_Þ

Here I is the 2

_

2 unit matrix, Z 0 is the reference impedance used in the wave

de nitions in (1.6), Y is the two-port admittance matrix, and S is the corresponding S-parameter matrix. Substituting (1.30) into (1.31) results in an explicit expression,

(1.32), for the S-parameters corresponding to this simple model in terms of the linear equivalent circuit element values given in (1.25):

2





Z 0



₁

þ

g DS v DC

ð Þ

1 ,v DC

ð Þ

2





Z 0



1

þ

j ωcGS



v

ð Þ

1 DC



Z 0



1

_

g DS v DC

ð Þ

1 ,v DC

ð Þ

2





Z 0 1

_þ

g DS v DC

ð Þ

1 ,v DC

ð Þ

2





Z 0

0 B

B

@

1 C

C

A

:

ð

1:32

_Þ

(33)

In summary, the S-parameters of a nonlinear component can be derived or computed by linearizing the full nonlinear characteristics of the circuit equations around a static (DC) operating point de ned by the voltage or current bias conditions. The S-parameters de ne a linear relationship between the incident and scattered waves at a xed DC operating point of the device and xed frequency for the incident waves. The S-parameters are an accurate description of how the device responds to signals, provided the signal amplitude is suf ciently small that the DC operating point is not signi cantly affected by the signal.

This will almost always be the case for signals of suf ciently small amplitude.

1.

1.11

11 Ad

Addi

diti

tion

onal

al be

bene

neﬁt

ﬁts

s of

of S-

S-pa

para

rame

mete

ters

rs

1.1

1.11.1

1.1 S-p

S-para

aramet

meters

ers are

are app

applic

licabl

able t

e to d

o dist

istrib

ribute

uted c

d comp

ompone

onents

nts at

at hig

high f

h freq

requen

uencie

ciess

S-parameters can accommodate an arbitrary frequency dependence in the linear spectral mapping. S-parameters therefore apply when describing distribut ed components for which lumped approximations are not very accurate or ef cient. This is especially true for high-frequency microwave components when the typical wavelengths of the stimu-lus approach and become smaller than the physical size of the component. The simplest example is the case of linear transmission lines. Another common example is the case of an active device, for which measured S-parameters of a transistor can be much more accurate than those computable from the linearized lumped nonlinear model. This is especially true as the frequency approaches and exceeds the device cutoff frequency, f T ,

beyond which a distributed representation is generally required.

1.

1.11

11.2 .2

S-

S-pa

para

rame

mete

ters

rs ar

are e

e eas

asy t

y to m

o mea

easu

sure

re at

at hi

high

gh fre

frequ

quen

enci

cies

es

S-parameters contain no more information than the familiar Y- and Z-parameters of elementary linear circuit theory, yet they have great practical advantages. S-parameters are much easier to measure at high frequencies. Y- and Z-parameters require short- and open-circuit boundary conditions, respectively, on the components for a direct meas-urement. Short- and open-circuit conditions are hard to achieve at microwave frequen-cies, and so are impractical. Moreover, such conditions presented to a power transistor can create oscillations that can destroy the component.

It will be demonstrated that X-parameters combine the accuracy and ease of meas-urement of a frequency-domain approach based on wave variables with the ability to handle nonlinearities that go beyond the linear relationship assumed by equation (1.8).

1.

1.11

11.3 .3

In

Inte

terp

rpre

reta

tati

tion

on of

of tw

two-

o-po

port

rt S-

S-pa

para

rame

mete

ters

rs

S-parameters relate to familiar quantities, such as gain, return loss, and output match. They provide insight into the component behavior.Table 1.1lists the four complex S-parameters and their corresponding interpretation for generic linear 2-ports [ 2]. The third column expresses common ampli er quantities in terms of the corresponding S-parameters.

15

1.11

X-Parameters Characterization Modeling and Design of Nonlinear RF and Microwave Components_Root.pdf

X-Parameters

X-Parameters

X-Parameters

X-Parameters

Characterization, Modeling, and Design of

Characterization, Modeling, and Design of

Nonlinear RF and Microwave Components

Nonlinear RF and Microwave Components

DAVID E. ROOT

DAVID E. ROOT

JAN VERSPECHT

JAN VERSPECHT

JASON HORN

JASON HORN

MIHAI MARCU

MIHAI MARCU

For Marilyn, with thanks for her patience, support, and, most of all, her love.

For Marilyn, with thanks for her patience, support, and, most of all, her love.

David

David

In memory of Petrus Verspecht.

In memory of Petrus Verspecht.

Jan

Jan

To Jessica, Jonathan, and Elise, my inspiration.

To Jessica, Jonathan, and Elise, my inspiration.

Jason

Jason

To Domnica, for the patience shown in the many evenings and weekends

To Domnica, for the patience shown in the many evenings and weekends

that I have spent away from her.

that I have spent away from her.

Mihai

Contents

Contents

Preface

Preface

Acknowledgments

Acknowledgments

1

1 S-p

S-para

aramet

meters

ers –

– a

a con

concis

cise

e rev

review

iew

1

1..1

1

IIn

nttrro

od

du

uc

cttiio

on

n

1

1..2

2

S

S--p

pa

arra

am

me

ette

errss

1

1..3

3

W

Wa

_¼

_{ffiffiffiffiffi}

_¼

_{ffiffiffiffiffi}

_Þ

_¼

_{ffiffiffiffiffi}

_¼

_ð

_

_Þ

_ð

_{Þ ¼}

_{ffiffiffiffiffi}

_ð

_{Þ þ}

_ð

_{Þ ¼}

_{ffiffiffiffiffi}

_ð

_{Þ }

_Þ

_ð

_{Þ ¼}

_{ffiffiffiffiffi}

_ð