Nonlinear Microwave and RF Circuits Second Edition For a listing of recent titles in the Artech House Microwave Library, turn to the back of this book Nonlinear Microwave and RF Circuits Second Edition Stephen A Maas Artech House Boston • London www.artechhouse.com Library of Congress Cataloging-in-Publication Data Maas, Stephen A Nonlinear microwave and RF circuits / Stephen A Maas.—2nd ed p cm Rev and updated ed of: Nonlinear microwave circuits, 1988 and reprinted in 1997 Includes bibliographical references and index ISBN 1-58053-484-8 (alk paper) Microwave circuits I Maas, Stephen A Nonlinear microwave circuits II Title TK7876.M284 2003 621.381'32—dc21 2002043664 British Library Cataloguing in Publication Data Maas, Stephen A Nonlinear microwave and RF circuits — 2nd ed.— (Artech House microwave library) Microwave circuits Radio circuits Electronic networks, Nonlinear I Titles 621.3'8132 ISBN 1-58053-484-8 Cover design by Gary Ragaglia © 2003 ARTECH HOUSE, INC 685 Canton Street Norwood, MA 02062 All rights reserved Printed and bound in the United States of America No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized Artech House cannot attest to the accuracy of this information Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark International Standard Book Number: 1-58053-484-8 Library of Congress Catalog Card Number: 2002043664 10 This is a sample dedication Contents Preface Chapter xix Introduction, Fundamental Concepts, and Definitions 1.1 Linearity and Nonlinearity 1.2 Frequency Generation 1.3 Nonlinear Phenomena 13 1.3.1 Harmonic Generation 13 1.3.2 Intermodulation Distortion 14 1.3.3 Saturation and Desensitization 14 1.3.4 Cross Modulation 15 1.3.5 AM-to-PM Conversion 15 1.3.6 Spurious Responses 16 1.3.7 Adjacent Channel Interference 16 1.4 Approaches to Analysis 17 1.4.1 Load Pull 17 1.4.2 Large-Signal Scattering Parameters 18 1.4.3 Time-Domain (Transient) Analysis 19 1.4.4 Frequency-Domain Methods 19 1.4.5 The Quasistatic Assumption 20 vii viii Chapter Nonlinear Microwave and RF Circuits 1.5 Power and Gain Definitions 21 1.6 Stability 26 Reference 27 Solid-State Device Modeling for Quasistatic Analysis 29 2.1 Nonlinear Device Models 29 2.2 Nonlinear Lumped Circuit Elements and Controlled Sources 31 2.2.1 The Substitution Theorem 33 2.2.2 Large-Signal Nonlinear Resistive Elements 34 2.2.3 Small-Signal Nonlinear Resistive Elements 35 2.2.4 Large-Signal Nonlinear Capacitance 38 2.2.5 Small-Signal Nonlinear Capacitance 39 2.2.6 Relationship Between I/V, Q/V and G/V, C/V Expansions 41 2.2.7 Multiply Controlled Nonlinear Capacitors 43 2.2.8 Nonlinear Inductance 47 2.3 Numerical and Human Requirements for Device Models 2.3.1 48 Continuous Derivatives in I/V or Q/V Expressions 48 2.3.2 Accuracy of Derivatives 49 2.3.3 Range of Expressions 49 2.3.4 Transient-Analysis Models in HarmonicBalance Analysis 50 2.3.5 Matrix Conditioning 50 2.3.6 Limiting the Range of Control Voltages 51 2.3.7 Use of Polynomials 52 2.3.8 Loops of Control Voltages 53 2.3.9 Default Parameters 53 568 Nonlinear Microwave and RF Circuits sinusoidal and noise cases on the basis of power For a sinusoid, the RMS and peak values are related as 2∆φ R MS = ∆φ sin (12.33) so the carrier to noise ratio L(fm) becomes V ssb L ( f m ) = = - ∆φ RM S Vs (12.34) where ∆φ R MS represents the RMS value of either the sinusoid or the noise process Sinusoid and Noise In many systems, noise is added to a carrier and the combination is limited in amplitude The limiting removes the amplitude component of the noise, but not the phase component The resulting phase noise can be found easily Figure 12.14 shows the combined signal and noise phasors From Figure 12.14, the phase deviation is n(t) n(t) ∆φ ( t ) = acos ≈ - v ( t ) v ( t ) (12.35) The mean square noise can be defined by a noise factor, F: n(t) v(t) v(t) + n(t) ∆φ(t) Figure 12.14 When a signal plus noise process is limited, amplitude variations are removed and only phase variations remain Since the oscillator’s transistor is driven into hard saturation, it acts as a limiter, removing most AM noise Transistor Oscillators n ( t ) = FKT R 569 (12.36) where K is Boltzmann’s constant, 1.37⋅10–23 J/K; T0 = 290K, by definition; and R is the load resistance at which n(t), which has units of voltage, is measured The signal power is v ( t ) = PR (12.37) where P is the power dissipated in R Substituting (12.35) through (12.37) into (12.34) gives FKT L ( fm) = - ∆φ RM S = - -2 P (12.38) L ( fm ) = – 174 + F – P – (12.39) or, in dBC, Noise Spectrum and Leeson’ s Model The previous relations assume that the noise is white In reality, the dominant noise process is the upconversion of 1/f noise by the oscillator’s nonlinearities We can assume that this power spectrum is centered on the carrier and has the form f v n2 ( fm ) = FKT0 + cfm (12.40) where fc is the corner frequency of the noise and, as before, fm is the deviation from the carrier in either a positive or negative direction The power spectrum of the phase fluctuations, S( fm), is FKT f S ( f m ) = ∆φ RM S = ∆φ = + cP fm (12.41) Plotted on a logarithmic scale, the noise spectrum has the shape shown in Figure 12.15 570 Nonlinear Microwave and RF Circuits Vn2(fm) fc fm Figure 12.15 1/f noise spectrum showing the corner frequency, fc In 1966, Leeson [12.11] proposed a simple model of a noisy oscillator by treating it as a phase-feedback system with added noise The added noise is a high-frequency noise spectrum, which consists of both broadband noise and upconverted 1/f noise The model does not treat the upconversion process, so it is valuable only for its qualitative, not quantitative predictions Even so, it provides considerable insight into oscillator operation The oscillator model is shown in Figure 12.16(a) It consists of an amplifier, a resonator, and feedback Noise is added at the amplifier’s input, and the oscillator’s output port is the amplifier’s output port Leeson showed that this circuit can be represented as the baseband circuit in Figure 12.16(b), in which the variable quantity is the oscillator’s phase In Figure 12.16(b), the resonator becomes a low-pass filter and the “output” is the phase, not the signal itself We now can apply ordinary feedback theory to the circuit of Figure 12.16(b) The transfer function of the low-pass filter, T( fm), is T ( fm ) = -fm + j2Q L f0 (12.42) where QL is the loaded Q of the resonator and f0 is the frequency of the oscillator The transfer function between the phase of the noise and that of the oscillator’s output is ∆φ = - ∆θ – T ( fm) (12.43) Transistor Oscillators 571 s(t) n(t) Resonator (a) Oscillator model ∆φ(t) ∆θn(t) (Noise phase) LPF (b) Phase-feedback loop Figure 12.16 (a) Leeson’s model of a noisy oscillator; (b) the equivalent circuit, in which phase is the variable Substituting (12.42) into (12.43) and using (12.41) and (12.38), we obtain f 1 FKT L ( fm ) = - S φ ( fm ) = - + c2 P fm f 02 1 + -2 f m 4Q L (12.44) The loop acts as a kind of filter on the phase noise According to (12.44), there are two break points in the phase noise spectrum: one at the corner frequency, fc, and another at fm = f0/2QL At frequencies well below both break points, the phase-noise spectrum has a slope of 30 dB per decade; at higher frequencies, regions can exist where the slope is either 20 dB per decade or 10 dB per decade, depending upon the relative values of fc and f0/2QL The possible spectra are shown in Figure 12.17 Note that these depend on the assumption that the dominant noise source has a 1/f spectrum; often the spectrum is not precisely 1/f, so the phase-noise spectrum may deviate from this ideal case Other Sources of Phase Noise It is important to recognize that phase noise can arise from sources other than the noise in the transistor Some important sources are the following: 572 Nonlinear Microwave and RF Circuits L(fm) L(fm) 30 30 20 10 fc f0/2QL fm (a) f0/2QL fc fm (b) Figure 12.17 Phase noise spectra: (a) “low-Q” case, in which f0 / 2QL > fc; (b) “highQ” case, in which f0 / 2QL < fc The former corresponds to VCOs and oscillators having microstrip resonators; the latter, to DROs and oscillators using resonant cavities • Power supply noise can easily modulate the phase of an oscillator The power supply must be well filtered to remove such noise For measurements, a battery can be used to eliminate this noise source • Coupling from the ac line is invariably evident in measurements of the phase-noise spectrum of low-noise oscillators Peaks at the ac line frequency, and its harmonics, are invariably present If the peaks are not too great, they can simply be ignored; however, large peaks can degrade the accuracy of a phase noise measurement and make it difficult to interpret In many cases, it may be necessary to shield the oscillator during the measurement • Mechanical vibration can generate phase fluctuations that appear as phase noise Ambient mechanical vibration has frequency components from a few hertz to a few kilohertz; this is just the right range to corrupt most phase-noise measurements • The noise of a buffer amplifier, especially if it uses active biasing, can degrade the oscillator’s phase noise 12.3.5.2 Frequency Multiplication Since frequency is the time derivative of phase, frequency multiplication is, in fact, phase multiplication Multiplying the frequency by a factor, n, Transistor Oscillators 573 multiplies ∆φ by n as well From (12.38), we see that the phase noise increases by n2 or, in decibels, 20 log(n) 12.3.6 Pushing and Pulling In Section 12.1 we saw that changes in the load impedance could affect the oscillation frequency by changing the phase of Zs This phenomenon is called pulling To some degree, pulling is inevitable; it occurs because the feedback necessary to make oscillation possible increases S1,2, and thus increases the sensitivity of Z s to the load impedance Nevertheless, pulling can be minimized Beyond the obvious solutions of using an output isolator or buffer amplifier, a high-Q resonator is effective in reducing pulling Similarly, changes in dc bias voltage can change the transistor’s S parameters and Zs, thus changing the oscillation frequency This phenomenon is called pushing The straightforward way to minimize pushing is to maintain adequate regulation in the oscillator’s bias circuits As with pulling, pushing is minimized by a high-Q resonator Pushing is not always undesirable; it is sometimes used as a means to obtain voltagetuning capability in a narrowband VCO 12.3.7 Post-Tuning Drift When the frequency of a VCO is changed, the RF current and voltage waveforms throughout the oscillator also change, as well as the dc bias current As a result, the heat dissipated in the transistor and the tuning varactor, in blocking capacitors (which dissipate heat because of finite Q), and in coupling inductors all change as well A small time interval is required before the circuit returns to thermal equilibrium and steady-state conditions During this time the frequency may drift; this phenomenon is called post-tuning drift It is most significant in fast-tuning, wide-range VCOs In a well-designed oscillator, the primary cause of post-tuning drift is heat dissipation in the varactor Thus, careful thermal design of the varactor can reduce post-tuning drift significantly If the varactor is mounted in a package, it should be mounted on a large metal surface; a beam-lead or chip device mounted on a substrate should be bonded to the substrate metallization over as large an area as possible 12.3.8 Harmonics and Spurious Outputs A well designed transistor oscillator should be free of spurious outputs that are not harmonically related to the frequency of oscillation However, 574 Nonlinear Microwave and RF Circuits because the transistor is driven into saturation, most oscillators have significant harmonic outputs Harmonic distortion can also occur in a buffer amplifier, which may be driven into saturation to level the output of a VCO In most cases the designer has little control of the harmonic levels unless an output filter is used References [12.1] R W Rhea, Oscillator Design and Computer Simulation, Second Edition, New York: McGraw-Hill, 1997 [12.2] K Kurokawa, “Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits,” Bell Sys Tech J., Vol 48, 1969, p 1937 [12.3] W Wagner, “Oscillator Design by Device Line Measurement,” Microwave J., Vol 22, Feb 1979, p 43 [12.4] V Rizzoli, A Lipparini, and E Marazzi, “A General-Purpose Program for Nonlinear Microwave Circuit Design,” IEEE Trans Microwave Theory Tech., Vol MTT-31, 1983, p 762 [12.5] E Ngoya et al., “Steady-State Analysis of Free or Forced Oscillators by Harmonic Balance and Stability Investigation of Periodic and Quasiperiodic Regimes,” Int J Microwave and Millimeter-Wave Computer Engineering, Vol 5, 1995, p 210 [12.6] D Kajfez and P Guillon (eds.), Dielectric Resonators, Norwood, MA: Artech House, 1986 [12.7] S J Fiedziuszko, “Microwave Dielectric Resonators,” Microwave J., Vol 29, Sept 1986, p 189 [12.8] S J Fiedziuszko, “Dielectric Resonators Raise your High-Q,” IEEE Microwave Magazine, Sept 2001, p 51 [12.9] N Elmi and M Radmanesh, “Design of Low-Noise, Highly Stable GaAs Dielectric Resonator Oscillators,” Microwave J., Vol 39, Nov 1996, p 104 [12.10] M Regis et al., “Design of a Low Phase Noise Ku-Band Oscillator Using a SiGe HBT,” Microwave J., Vol 44, Oct 2001, p.136 [12.11] D B Leeson, “A Simple Model of Feedback Oscillator Noise Spectrum,” Proc IEEE, Vol 54, Feb 1966, p 329 About the Author Steve Maas received BSEE and MSEE degrees in electrical engineering from the University of Pennsylvania in 1971 and 1972, respectively, and a Ph.D in electrical engineering from UCLA in 1984 He joined the National Radio Astronomy Observatory in 1974, where he designed the low-noise receivers for the Very Large Array radio telescope Subsequently, at Hughes Aircraft Company and TRW, he developed low-noise microwave and millimeter-wave systems and components, primarily FET amplifiers and diode and FET mixers, for space communication He also has been employed as a research scientist at The Aerospace Corporation, where he worked on the optimization of nonlinear microwave circuits and the development of circuit-design software based on harmonic-balance, Volterra-series, and time-domain methods He joined the UCLA Electrical Engineering Faculty in 1990 and left it in 1992 Since then, he has worked as an independent consultant and currently is chief scientist of Applied Wave Research, Inc Dr Maas is the author of two other books, Microwave Mixers (1986 and 1992) and The RF and Microwave Circuit Design Cookbook (1998), both published by Artech House From 1990 until 1992 he was the editor of the IEEE Transactions on Microwave Theory and Techniques, and from 1990 to 1993 he was an Adcom member and publications chairman of the IEEE MTT Society He received the MTT Society’s Microwave Prize in 1989 for his work on distortion in diode mixers and its Application Award in 2002 for his invention of the FET resistive mixer He is a fellow of the IEEE 575 576 Nonlinear Microwave and RF Circuits Index Active mixer 497 balanced 515 bipolar 505 dual gate 510 Gilbert cell 506 Adjacent channel interference 16 Adjacent channel power 274 Admittance matrix 126 Amplifier bias 405 broadband 406 conditionally stable 397 gain 401 harmonic generation 417 nonlinear transfer function 421 push-pull 456 small signal 396 stability 397 AM-to-PM conversion 15, 416, 465 Angelov HBT model 104 Anholt HBT model 439 Antiparallel connection 301 Antiseries connection 306 APFT 203 Artificial frequency mapping 206 Available gain 26 Avalanche breakdown 63, 436 Back off 468 Balanced components amplitude balance 296 effect of imperfect balance 294 intercept point 299 phase balance 296 spurious signal rejection 296 Balanced FET resistive mixer 530 Balanced mixer 515 Balanced multiplier 391 Ballast 472 Beam lead diode 318 Bias bipolar power devices 470 Bipolar transistor 95 See also BJT BJT 95 base transit time 99 current gain/ft 99 Early effect 102 HBT 100 Kirk effect 103 model 101 operation 96 parameter extraction 115 Boltzmann’s constant 62 Box truncation 207, 464 577 578 Nonlinear Microwave and RF Circuits Branch line hybrid 285 Bridge quad 392 BSIM 89 power devices 437 Burkhardt 362 Capacitance bipolar 100 diode 61 FET 93 model 91 varactor 70 Cascaded stages 232 Class-A amplifier 439 Class-B amplifier 443 Class-D amplifier 448 Class-E amplifier 448 CMOS 79 Commutating mixer 345 Condition number 154 Conduction angle 480 Consistency problem 108 Continuation methods 158 Control voltage 3, 51 Conversion matrix 165 Cross modulation 15 Crossbar mixer 342 Current error vector 133 Depletion region 58 Derivative 48 continuity 48 IM simulation 49 Desensitization 14 Device line measurement 554 DFT 200 Diamond truncation 207, 464 Dielectric resonator 551, 563 Diffused epitaxial varactor 69 Diffusion charge storage 68 Diffusion potential 58 Diode anode structure 57 beam lead 319 conversion-loss degradation factor 334 cutoff frequency 66 flip chip 321 ideality factor 62, 335 parameter extraction 109 planar 323 series resistance 323 Discrete Fourier transform 200 Dispersion 85 Division by capacitance 90 Division by charge 91 Drain dispersion 85 DRO 551, 563 Duty cycle 477 Early effect 102 Effective gate width/length 88 Envelope analysis 209 in power amplifiers 465 Error trapping 54 Euclidean norm 159 Feedback oscillator 537 Feedforward linearization 470 FET 73 capacitance 85 capacitance model 93 parameter extraction 111 FET capacitance 90 FET frequency multiplier 477 FET Mixer gain 504 FET mixer analysis 501 FET resistive mixer 525 balanced 530 MOSFETs 535 ring FET mixer 533 Index FFT 200, 464 two-dimensional 204 Flip chip diode 321 Fourier transform 200 Frequency generation 4, 224 Frequency multiplier active doubler design 483 balanced active 491 bipolar 490 broadband 487 efficiency 482 FET 477 high-order, active 495 noise 493 phase noise degradation 356 stability 494 Frequency stability 562 Gain circle 403 Gain definitions 21 Gilbert cell 506 GMRES 153 Gummel-Poon model 101 Harmonic 13 generation in small-signal amps 417 Harmonic-balance analysis 120 current error vector 133 inexact Newton 153 initial estimate 164 Jacobian 140 Krylov methods 153 linear subcircuit 135 matrix conditioning 154 multitone 187 Newton’s method 140 nodal formulation 161 nonlinear capacitance 146 nonlinear subcircuit 129 single tone 124 termination 162 579 Harmonic distortion 13 Harmonic input method 241 HBT Angelov model 104 Anholt model 439 model 104 operation 100 UCSD model 104 HEMT 73 capacitance 85 model 86 operation 78 structure 78 HICUM 438 Hybrid 278 Hyperabrupt varactor 564 Ideality factor 61 Inductance 47 Intercept point 225 Intermodulation distortion 5, 14, 418 frequency spectrum 220 in FET amplifiers 410 intercept point 225 the 10-dB rule 419 Intermodulation intercept point 225, 299 Isolation 279 Jacobian 140 multitone case 207 Kirk effect 103 Krylov subspace techniques 153 Kurokawa 542 Lange coupler 286 Large-signal/small signal analysis 119 Large-signal/small-signal analysis 164 nodal formulation 185 S parameters 184 580 Nonlinear Microwave and RF Circuits LDMOS 80 Leeson’s model 569 Liapunov stability 27 Linear subcircuit 126, 135 Linearity definition Load pull 17 LU decomposition 151 Majority carrier device 59 Manley-Rowe equations 357 Marchand balun 353 Matrix conditioning 50, 154 Maximum norm 154 MESFET 73 capacitance model 85 operation 73 See also FET Method of nonlinear currents 254 MEXTRAM 438 Meyer model 89 Microwave hybrid 279 Minority carrier lifetime 73 Mixer active 497 active balanced 515 active doubly balanced 520 balanced 339 balun 348 bipolar 505 diode types 318 doubly balanced 345 frequencies 325 ring mixer 345 single-diode design 328 singly balanced 339 star 352 Mixing frequency 224 Mixing product Model BJT 102 diode 58 Gummel-Poon 102 HBT 104 HEMT 86 MESFET 81 MOSFET 88 thermal 104 varactor 70 Modulation index 567 MOSFET 79 model 88 MOSFET capacitance 94 MOSFET mixer 510 Multiplier balanced 391 drive level 362 Manley-Rowe equations 357 noise 369 resistive (Schottky diode) 382 SRD 370 stability 368 varactor 361 Multitone analysis 198 power amplifier 463 Negative resistance 545 Negative resistance oscillator 542 Newton’s method 139 Nodal formulation 161 Noise 369, 493 phase noise 566 Nonlinear capacitance 146 Nonlinear current method 254 Nonlinear subcircuit 129 Nonlinear transfer function 241 Nonlinearity capacitance 38 conductance 35 derivatives 48 in bipolars 413 in small-signal FETs 413 incremental 31 inductance 47 Index large signal 34 multiterminal capacitance 45 voltage and current control 31 weak 35, 231 Norm reduction methods 155 Normal equation 52 NU norm 159 Odd mode oscillation 467 Optimization 138 Oscillator 537 device line measurement 556 Eigenvalue formulation 560 feedback 537 frequency stability 562 negative resistance 542 nonlinear analysis 555 oscillation condition 544 post-tuning drift 573 pulling 554 pushing and pulling 573 resonator 540 stability condition 544 p+n varactor 69 Parameter extraction 108 BJT 115 cold FET 111 diode 109 direct extraction 108 FET 111 weak nonlinearity 113 Parametric instability 500 Parametric models 160 Phase noise 566 pHEMT 78 See also HEMT Polynomial 52 Post-tuning drift 573 Power added efficiency 446 Power amplifier class A 439 581 class B 443 harmonic termination 456 HBT 457 multitone analysis 463 odd-mode oscillation 467 push-pull 456 self heating 434 thermal design 471 Power device 434 avalanche breakdown 436 bipolar models 438 MESFET models 438 modeling 434 scaling 435 Power gain 25 Power series analysis 216 system model 216 Predistortion linearization 470 Prematching 454, 471 Pulling 573 Pushing 573 Quasistatic assumption 20, 30 Rat race hybrid 281 Recombination time 73 Richardson constant 63 Ring mixer 345 S parameter 18, 396 large signal 18 large-signal/small-signal analysis 184 See also Scattering parameter Saturation 14, 416 Scattering parameter 18, 396 amplifier design 400 Schottky-barrier diode 56 I/V characteristic 62 junction capacitance 58 mixer 317 mixer diode 65 582 Nonlinear Microwave and RF Circuits nonlinear model 58 series resistance 59 structure 57 varactor 68 Schottky-barrier varactor 68 Self heating 434 Series resistance 59, 70 Short channel effects 88 Source stepping 158 Spectral regrowth 274 SPICE 50 Gummel-Poon model 101 models 50 MOSFET models for power 437 Spurious response 16 SRD 72, 370 Stability 26, 368, 494 Liapunov 27 Stability circle 399 Star mixer 352 Step-recovery diode 71 Substitution theorem 33 Substrate current 88 Substrate impedance 71 Superposition Taylor series 35, 217 Terminal charge 94 Thermal collapse 434 Thermal instability 434 Thermal model 104 Thermal resistance matrix 107 Transconductance mixer 498 Transducer gain 25, 401 Transformer 280 Transient analysis 19 Transit time 99 Two-tone test 226 UCSD HBT model 104 Unilateral amplifier 397 Varactor 68 C/V characteristic 366 diffused epitaxial 69 dynamic cutoff frequency 67 frequency multiplier 356 hyperabrupt 564 model 70 p+n 70 punch through 70 VBIC 438 VCO 552, 573 Volterra functional 237 Volterra-series analysis 19, 235 nonlinear current method 254 time domain 238 Weak nonlinearity 35 Weakly nonlinear circuit 232, 236 Wilkinson hybrid 283