Steve C. Cripps
`AESFase.esa
`
`
`AMPLIFIERS
`FOR WIRELESS
`COMMUNICATIONS
`
`Se ee
`
`INTEL 1035
`Intel v. ParkerVision
`IPR2021-00346
`
`INTEL 1035
`Intel v. ParkerVision
`IPR2021-00346
`
`

`

`RF Power Amplifiers for
`Wireless Communications
`
`Second Edition
`
`

`

`DISCLAIMER OF WARRANTY
`
`The technical descriptions, procedures, and computer programs in this book have
`been developed with the greatest of care and they have been usefulto the authorin a
`broad range of applications; however, they are provided as is, without warranty of
`any kind. Artech House, Inc., and the author and editors of the book titled RF
`Power Amplifiers for Wireless Communications, Second Edition, make no warran-
`ties, expressed or implied, that the equations, programs, and proceduresin this book
`orits associated softwarearefree oferror, or are consistent with any particular stan-
`dard of merchantability, or will meet your requirements for any particular applica-
`tion. They should not be relied uponfor solving a problem whoseincorrect solution
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`of the programsor proceduresin this book or the associated software.
`
`Fora listing of recenttitles in the Artech House
`Microwave Library, turn to the back of this book.
`
`

`

`RF Power Amplifiers for
`Wireless Communications
`
`Second Edition
`
`Steve C. Cripps
`
`ARTECH
`
`HOUSE
`
`BOSTON | LONDON
`artechhouse.com
`
`

`

`Library of Congress Cataloging-in-Publication Data
`A catalog record for this book is available from the Library of Congress.
`
`British Library Cataloguing in Publication Data
`Cripps, Steve C.
`RF poweramplifiers for wireless communications.—2nd ed.—(Artech House microwave
`library)
`1. Amplifiers, Radio frequency 2. Power amplifiers
`I. Title
`621.3781535
`ISBN-10: 1-59693-018-7
`
`ISBN 10: 1-59693-018-7
`ISBN 13: 978-1-59693-018-6
`
`Cover design by Igor Valdman
`
`© 2006 ARTECH HOUSE,INC.
`685 Canton Street
`Norwood, MA 02062
`
`All rights reserved. Printed and boundin the United States of America. No part of this book
`maybe reproducedorutilized in any form or by any means,electronic or mechanical, includ-
`ing 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 trademarksor service marks have
`been appropriately capitalized. Artech House cannotattest to the accuracy of this informa-
`tion. Use of a term in this book should not be regarded asaffecting the validity of any trade-
`mark or service mark.
`
`International Standard Book Number: 1-59693-018-7
`
`10987654321
`
`

`

`Contents
`
`Preface to the Second Edition
`
`xi
`
`CHAPTER 1
`
`Introduction
`
`1.1
`1.2
`1.3
`1.4
`1.5
`1.6
`1.7
`
`Introduction
`Linear RF Amplifier Theory
`Weakly Nonlinear Effects: Power and Volterra Series
`Strongly Nonlinear Effects
`Nonlinear Device Models for CAD
`Conjugate Match
`RF Power Device Technology
`References
`
`Ae
`Linear Power Amplifier Design
`Class A Amplifiers and Linear Amplifiers
`2.1
`2.2
`Gain Match and Power Match
`2.3
`Introduction to Load-Pull Measurements
`Loadline Theory
`2.4
`Package Effects and Refinements to Load-Pull Theory
`2.5
`Drawing the Load-Pull Contours on CAD Programs
`2.6
`Class A Design Example
`2.7
`2.8
`Conclusions
`References
`
`Conventional High Efficiency Amplifier Modes
`3.1
`Introduction
`Reduced Conduction Angle—Waveform Analysis
`3.2
`3.3
`Output Termination
`Reduced Conduction Angle Mode Analysis—FET Model
`3.4
`Case 1: Class A
`Case 2: Class AB
`Case 3: Class B
`Case 4: Class C
`Reduced Conduction Angle Mode Analysis—BJT Model
`Effect of I-V “Knee”
`Input Drive Requirements
`
`3.5
`3.6
`3.7
`
`17
`
`17
`19
`20
`21
`
`27
`31
`31
`36
`37
`
`39
`
`39
`40
`43
`47
`48
`49
`S1
`53
`55
`59
`61
`
`

`

`vi
`
`Contents
`
`3.8 Conclusions
`References
`
`Aias
`Class AB PAs at GHz Frequencies
`4.1
`Introduction
`4.2 Class AB Using a Capacitive Harmonic Termination—TheClass J PA
`4.2.1 Theory
`4.2.2 Practicalities
`Nonlinear Device Characteristics
`Nonlinear Capacitance Effects in RF Power Devices
`4.4.1
`Introduction
`4.4.2 Nonlinear Capacitors—Characterization and Analysis
`4.4.3.
`Input Varactor Effects on Class AB PAs
`Conclusions
`References
`
`4.3
`4.4
`
`4.5
`
`Practical Design of Linear RF Power Amplifiers
`Low-Pass Matching Networks
`5.1
`Transmission Line Matching
`5.2
`Shorting the Harmonics
`5.3
`5.4
`A Generic MESFET
`A 2W Class B Design for 850 MHz
`5.5
`The Pi Section Power Matching Network
`5.6
`Pi Section Analysis for PA Design
`5.7
`Class J Design Example
`5.8
`HBTDesign Example
`5.9
`5.10 Conclusions
`References
`
`Overdriven PAs and the Class F Mode
`
`6.1
`6.2
`6.3
`6.4
`6.5
`6.6
`6.7
`6.8
`6.9
`
`Introduction
`Overdriven Class A Amplifier
`Overdriven Class AB Amplifier
`Class F: Introduction and Theory
`Class F in Practice
`The Clipping Model for the Class F Mode—Class FD
`PA_Waves
`Class F Simulations
`Conclusions
`References
`
`Switching Mode Amplifiers for RF Applications
`7.1
`Introduction
`
`65
`65
`
`67
`
`67
`68
`68
`73
`77
`81
`81
`81
`84
`89
`89
`
`91
`
`92
`100
`102
`105
`107
`112
`
`115
`122
`124
`
`129
`131
`
`133
`
`133
`134
`139
`143
`149
`155
`163
`164
`171
`172
`
`173
`
`173
`
`

`

`Contents
`
`7.2
`7.3
`7.4
`7.4
`7.5
`7.6
`7.7
`
`A Simple Switching Amplifier
`A Tuned Switching Amplifier
`The Class D Switching Amplifier
`Class E—Introduction
`Class E—Simplified Analysis
`Class E—Design Example
`Conclusions
`References
`
`Switching PA Modes at GHz Frequencies
`8.1
`Introduction
`Ignoring the Obvious: Breaking the 100% Barrier
`8.2
`Waveform Engineering
`8.3
`8.4
`PA_Waves
`Implementation and Simulation
`8.5
`8.6
`Conclusions
`References
`
`Nonlinear Effects in RF Power Amplifiers
`9.1
`Introduction
`Two-Carrier Power Series Analysis
`9.2
`Two-Carrier Envelope Analysis
`9.3
`Envelope Analysis with Variable PAR
`9.4
`AM to PM Effects
`9.5
`PA MemoryEffects
`9.6
`Digital Modulation Systems
`9.7
`9.7.1 Introduction to Digital Modulation
`9.7.2. QPSK Modulation Systems
`9.7.3 CDMA and WCDMA
`9.7.4. OFDM Modulation, 802.11/16 Standards
`30 Watt LDMOS Test Amplifier Measurements
`Conclusions
`References
`
`9.8
`9.9
`
`Aaa
`Efficiency Enhancement Techniques
`Introduction
`Efficiency Enhancement
`10.1
`The Doherty Amplifier
`10.2
`Realization of the Doherty Amplifier
`10.3
`Outphasing Techniques
`10.4
`Envelope Elimination and Restoration (EER)
`10.5
`Envelope Tracking
`10.6
`Power Converters for EER and ET
`10.7
`Pulse Width Modulation (PWM)
`10.8
`
`vii
`
`174
`178
`180
`182
`183
`192
`198
`199
`
`201
`
`201
`202
`205
`216
`225
`227
`229
`
`231
`
`231
`233
`240
`246
`250
`256
`261
`261
`262
`268
`275
`278
`282
`283
`
`285
`
`285
`286
`290
`298
`303
`309
`311
`314
`318
`
`

`

`viii
`
`Contents
`
`10.9 Other Efficiency Enhancement Techniques
`10.9.1 The Sequential Power Amplifier
`10.9.2 Pulse Position Modulation
`10.9.3. RF to DC Conversion
`10.9.4 RF Switching Techniques
`10.9.5 Smart Antennas
`10.10 Case Studies in Efficiency Enhancement
`10.11 Conclusions
`References
`
`Power Amplifier Bias Circuit Design
`11.1.
`Introduction
`11.2 Stability of RF Power Transistors
`11.3. Bias Supply Modulation Effects
`11.4 Bias Network Design
`11.5 Bias Insertion Networks
`11.6 Prime Power Supply Issues
`11.7 Bias Control Circuits
`11.8 Conclusions
`References
`
`Load-Pull Techniques
`12.1 Tuner Design for Fundamental Load-Pull
`12.2 Harmonic Load-Pull
`12.3 Active Harmonic Load-Pull
`12.4 Variations, Results, Conclusions
`References
`
`Power Amplifier Architecture
`Introduction
`13.1 Push-Pull Amplifiers
`13.2 Balanced Amplifiers
`13.3. Power Combining
`13.4 Multistage PA Design
`13.5 Conclusions
`References
`
`Power Amplifier Linearization Techniques
`Introduction
`14.1
`Introduction to PA Linearization
`14.2 Predistortion
`14.2.1
`Introduction to Predistortion Theory
`14.2.2 Digital Predistortion (DPD)
`
`323
`323
`325
`326
`328
`329
`330
`333
`334
`
`337
`337
`338
`343
`350
`353
`354
`355
`356
`357
`
`359
`359
`362
`365
`367
`369
`
`371
`371
`372
`380
`387
`391
`394
`395
`
`397
`397
`399
`401
`401
`404
`
`

`

`Contents
`
`14.2.3 Analog Predistortion
`14.2.4 Predistortion—Conclusions
`14.3 Feedforward Techniques
`14.3.1 Feedforward, Introduction
`14.3.2 Feedforward—Gain Compression
`14.3.3 Feedforward—Effect of the Output Coupler
`14.3.4 Feedforward—Adaptive Controls
`14.3.5 Feedforward—Practical Issues, Conclusions
`14.4 Feedback Techniques
`14.4.1
`Introduction, Direct Feedback Techniques
`14.4.2
`Indirect Feedback Techniques—Introduction
`14.4.3. The Cartesian Loop
`14.4.4 The Polar Loop
`14.5 Other Linearization Methods
`14.6 Conclusions
`References
`
`PA_Waves
`
`Spectral Analysis Using Excel IQ Spreadsheets
`
`Bibliography
`Introductory Texts on RF and Microwave Techniques
`Wireless Communications
`Digital Modulation
`Nonlinear Techniques and Modeling
`Power Amplifier Techniques
`Recommended Reading
`
`Glossary
`About the Author
`
`Index
`
`ix
`
`407
`410
`410
`410
`411
`414
`417
`418
`419
`419
`420
`421
`423
`424
`425
`426
`
`429
`
`433
`
`435
`435
`435
`435
`435
`435
`436
`
`437
`441
`
`443
`
`

`

`Switching Mode Amplifiers for RF
`Applications
`
`7.1
`
`Introduction
`
`This chapter will consider the possibilities offered to the RFPA designer by switch-
`ing modecircuits. Circuits of this kind have been used for many years in DC to DC
`converter applications and undoubtedly offer some possibilities for higher fre-
`quency use. This applies especially in the broader interpretation of “RF”; high
`powerapplications in the MHz and tens of MHz frequency ranges have certainly
`benefited from infusions of techniques and practice from the switching power
`converter industry.
`But at GHz frequencies there remains a stubborn andirrefutable centralissue,
`whichis that RF powertransistors at these frequencies cannotrealistically be mod-
`eled as simple switching elements. At these frequencies, and in any currently avail-
`able powertransistor technology,the device will not sweep throughits linear region
`fast enough to behavelike a switch, and the behaviorin the linear region must be
`includedin the circuit simulation,just as it is when simulating conventional PA cir-
`cuits. For these reasons, RF designers most frequently move on and dismiss switch-
`ing modesas being unrealizable for higher frequency applications.In fact, although
`the problem of “slow” switching speed can neverbefully resolved at higher RF fre-
`quencies, it appears that it can sometimes be judiciously out-maneuvered and that
`useful RF applications do exist. This can lead to some useful applications in the
`“gray” area, when a very high frequency technologyis being used at a low relative
`frequency; a GaAs PHEMTor HBT below 2 GHz would be such an example.
`Probably the most well-known, and certainly most widely touted, switching
`modefor RF applications is the Class E mode. Althoughin its most basic and origi-
`nal form, a Class E device needs to have the same unreachable switching properties
`as any other switching modeapplication, there appearto be derivatives oftheclassi-
`cal switch-mode Class E PA which allow for slower switching butstill appear to be
`distinct from conventional PA modes. This metamorphosis is an important topic
`and has been given a chapterto itself, Chapter 8; the present chapter will focus
`mainly onthe classical switching modes which can be used quite effectively below
`1000 MHz.
`The chapter starts by considering a simple switching device with a broadband
`resistive load, and then considers the effect of tuning the load. Such an amplifier,
`even with anideal switching element, has surprisingly little to offer the RF designer
`in comparison to the Class B or Class F modes considered in the previous chapters,
`
`173
`
`

`

`174
`
`Switching Mode Amplifiers for RF Applications
`
`but servesas a useful introduction to the subject. The Class D amplifier is considered
`next. In its ideal switching mode form, a “perfect” RF amplifier is created, with a
`halfwaverectified sinewave of current and a squarewave of voltage. This produces
`maximum possible fundamental RF power at 100% efficiency, but is difficult to
`realize at higher RF frequencies.
`The bulk of the chapter considers the classical Class E switching mode in some
`detail. As a lead in to the following chapter, a simple design example illustrates how
`in the “gray area,” at 800 MHz and using a GaAs MESFETdevice, a Class E ampli-
`fier can be realized which uses a design procedure based almost entirely on the
`results from theclassical analysis.
`Switching mode power amplifiers, without qualification, are highly nonlinear
`devices. Thisreality leadsto further rejection by the wireless communicationsindus-
`try, on the basis that variable envelope signals cannot be passed without hopeless
`levels of distortion. It should howeverbe bornein mindthat the samerejection could
`equally be applied to a Class C amplifier, but such amplifiers were at one time com-
`monly used in AM andSSB broadcasting. In order to overcomethe distortion prob-
`lem, techniques such as envelope restoration and outphasing were developed, and
`will be described in Chapter 10.
`
`7.2 A Simple Switching Amplifier
`
`Figure 7.1 shows the schematic diagram for a simple switching mode amplifier. The
`key element is the switch, which in this analysis will be considered as ideal. By
`“ideal” it meansthat the switchis either completely on (short circuit) or completely
`off (open circuit) and can switch between these twostates instantaneously. Another
`key aspectof the idealization is that the timing of switch opening andclosureis con-
`trolled by the input signal; the “conduction angle” is assumedto be discretionary in
`the analysis. In practice, such control of the switch conduction angle will be
`achieved by varying the drive level and bias point at the input of a transistor. This
`will usually mean that the device will be heavily overdriven in comparison to normal
`linear operation, andit is almost inevitable that the overall gain will be many dBs
`lower than a conventional linear amplifier. In any event, it will not be possible to
`estimate the powergain of such a circuit, without more detailed knowledge of the
`specific device and the drive arrangements. Useful expressions can, however, be
`derived for the output power andefficiency.
`
`
`
`Figure 7.1
`
`Basic RF switching amplifier.
`
`

`

`7.2 A Simple Switching Amplifier
`
`175
`
`Figure 7.1 shows a circuit which has some familiar features; a DC-blocked RF
`load resistor and a choked DC supply. The current and voltage waveforms are
`quite trivial in the symmetrical case where the switching duty cycle is 50%, and
`Figure 7.2 shows a more general case with a conduction angle 2a, corresponding to
`the switch closure period. The maximum current in the switch is now controlled
`entirely by the supply voltage and the RF load resistor, and the voltage toggles
`between zero and V,,, where V,, will be a function of the conduction angle a@ and
`the supply voltage V,. Before analyzing the waveforms in moredetail, it is clear
`that this circuit converts DC energy to RF energy at 100% efficiency; at no point in
`the RF cycle is there a nonzero voltage and current simultaneously, so no energyis
`wastedas heat in the switch. This is sometimes interpreted erroneously as a perfect
`andfinal solution to RF power generation.In fact, this particular switching mode
`amplifier has some undesirable characteristics in that substantial energy is gener-
`ated at harmonic frequencies, and the maximum fundamentalefficiency is only just
`over 80%.
`It is assumed that the DC supply voltage remains constant as the conduction
`angle a is varied; this will result in an asymmetrical voltage waveform whose peak
`value will be proportionately greater or less than twice the supply voltage as a is
`varied above or below /2. This circuit can now be analyzed for power andeffi-
`ciency as a function of conduction angle. With a simple broadbandresistive load,
`the relationship betweenthe peak voltage V,, and the DC supply voltageV,, is given
`by the meanvalueintegral,
`
`1
`
`px
`
`Vac = 5 F- esw(0)- do
`== J. V,, 40
`Vue _ (a-a)
`Pp
`Vir
`a
`
`1
`
`px
`
`(7.1)
`
`
`
`Figure 7.2 Basic RF switch waveforms.
`
`

`

`176
`
`Switching Mode Amplifiers for RF Applications
`
`The voltage waveform can therefore be considered to be an alternating voltage
`with zero meanvalueif it is offset by V,. So the peak-to-peak current swing will be
`
`Tie = Vo /R,
`
`(7.2)
`
`It should be noted here that an ideal switch differs from a real device in the
`important respect that the peak currentis entirely a function of the DC supply and
`the load resistor R,; this resistor can be arbitrarily reduced in value to give any
`desired amount of RF power. This analysis will therefore consider primarily the
`powerandefficiency for a given valueof I,,, which can then be used asarational
`equivalent for comparison to a transistor having a maximum,or saturated, current,
`Drax» CQual to I,,.
`Meanvalue considerationsfor the current waveform give, by simple inspection
`
`a
`Ty. == 1,
`us
`
`The fundamental even Fourier coefficient of current, I,, is given by
`
`1, =~ J" igy ()-c0s(0)- dbs
`nh a
`where isy (0) =1,,, -a<0< a;
`=0,-1<O0<aa<0<a
`2 pa
`
`= =| I, -cos(@)- dé
`I, 2 -sin(@)
`ipk
`wv
`
`Similarly, the fundamental Fourier coefficient for the voltage waveform is
`
`Vi _ _2-sin(@)
`Vir
`A
`
`(7.3)
`
`(7.4)
`
`(7.5)
`
`Combining equations (7.1) through (7.5), the RF power, P_,, can be expressed in
`terms of the DC supply terms, V,, and I,::
`
`VY, __ 2-sina
`Vi.
`m-a
`I
`“Si
`4, 2 sing
`I,
`a
`P, =V,
`
`ic
`
`~~
`
`+2
`
`2-sin° @
`de “a(x — a)
`
`So that the outputefficiency, 7, is given by
`
`(7.6)
`
`

`

`7.2 A Simple Switching Amplifier
`
`_ 2-sin? a
`a(a — a)
`
`177
`
`(7.7)
`
`So in the symmetrical squarewave case, theefficiency is (8/z’), about 81%. Fig-
`ure 7.3 showsaplot of efficiency versus conduction angle. The RFefficiency peaks
`at the symmetrical case of a@ = 2/2, corresponding to the generation of a maximum
`proportion of RF energy at the fundamental. Figure 7.3 also shows the RF output
`powerat the fundamental and at the second and third harmonics. These power
`curvesare calibrated relative to the RF powerfromaClass A linear amplifier having
`the same peak RFcurrent(i.e., [,,,, = I,,). Substituting for I,, in (7.3), the expression
`for fundamental RF output power(7.6) becomes:
`
`PL = Vy. Ly aos
`
`and defining the linear power,P,,,, as
`
`the relative poweris
`
`
`
`‘, = Vic “Lyn4
`
`PL 8-sin* (a)
`P
`a(t - a)
`
`lin
`
`(7.8)
`
`(7.9)
`
`This functionis also plotted in Figure 7.3. The striking feature of the fundamen-
`tal power curve is the peak which occurs at a conduction angle of about 113° (@=
`0.63). This peak represents an RF power which is 2.7 dB higher than a Class A
`amplifier having the same peak RF current and DC supply voltage. Although the
`
`Effcy
`
`100%
`
`50%
`
`Figure 7.3.
`
`RF powerandefficiency of basic RF switch.
`
`Conduction angle (a)
`
`

`

`178
`
`Switching Mode Amplifiers for RF Applications
`
`formulation used does allow for much higher peaksof voltage than in the compara-
`ble Class A case, this condition nevertheless represents a theoretically feasible ampli-
`fier. The peak power shownin Figure 7.3 could be regarded as somethingof a global
`maximum for RF powerobtainable from any kind of device using a stipulated DC
`supply.
`Theefficiency curve, shown dashedin Figure 7.3, represents somethingof a dis-
`appointment; despite having a device which dissipates no heat, the efficiency peaks
`at about 81%, corresponding to the symmetrical squarewave condition. The prob-
`lem, as shownby the harmonic powerplots, is that power is being wasted in har-
`monic generation andthe nextstep is to eliminate this using a resonator.
`
`7.3 A Tuned Switching Amplifier
`
`Other than the possibility for frequency multiplication, the harmonic energy indi-
`cated in Figure 7.3 is undesirable, and can be most easily removed byplacing a har-
`monic short across the load, as shownin Figure 7.4. This is now more familiar
`territory, since the voltage waveform will now assumea sinusoidal form, as shown
`in Figure 7.5. This makes analysis quite straightforward.In this analysis it will again
`be convenientto relate the current waveformsto a specific peak value,I,,, in order to
`comparethe results with that of a conventional reduced conduction mode amplifier
`having the same maximum peak RFcurrent.
`Therelationships for current are the sameas in (7.3) and (7.4),
`
`I, | 2- sin(a)
`Dik
`a
`
`I, _ a
`I,
`=
`
`The sinusoidal voltage gives the simple relationship
`
`V, = Vac
`
`so the fundamental RF poweris
`
`i circuit
`
`Figure 7.4 Tuned RF switching amplifier.
`
`

`

`7 .3 A Tuned Switching Amplifier
`
`179
`
`Figure 7.5 Tuned switch waveforms.
`
`(7.10)
`
`.
`
`(7.11)
`
` a
`
`vy, 2m
`
`dc
`
`p =r,
`
`1 # £™.dé
`
`giving an outputefficiency
`
`=
`
`a
`
`The relative power can be determined,as before, by expressing P,, in (7.10), in
`terms of the peak currentI,,, using the relationship of (7.3):
`PL =I Vic sin(a)a
`
`sin(a
`
`so the ratio of fundamental RF power toP,,, is
`
`P
`lin
`
`a Ty Vac Ol, = ne]
`
`4-sin(a@
`
`Di Vac
`
`(7.12)
`
`Theefficiency (7.11) and relative power (7.12) are plotted as a function of con-
`duction angle in Figure 7.6. As would be expected from the analysis of a Class C
`amplifier, as the switched current pulse gets very short, the efficiency increases to
`high values, but there is a corresponding reduction in relative RF power. The peak
`in RF poweroccurs at a conduction angle of 2/2, and the corresponding RF poweris
`(4/zt) times the linear equivalent, or about 1 dB higher. But the efficiency at this
`point is a rather modest 63% (2/m). Probably the mostsignificant positive feature of
`the ideal switch performance in this circuit is the efficiency of about 87% at the
`point wherethe relative powercrosses unity ontheleft side, but thisstill falls short
`of the Class F results presented in Chapter6.
`
`

`

`180
`
`Switching Mode Amplifiers for RF Applications
`
` 100%
`
`Effcy
`
`50%
`
`0
`
`WA
`wl2
`0
`aa
`Conduction angle (<)
`
`Figure 7.6
`
`RF output powerandefficiency of basic RF switch with harmonic short.
`
`Theresults of the last two sections show that an ideal switch does not,by itself,
`give any instantaneous improvements in the efficiency of more conventional high
`efficiency circuits, and alternative configurations have to be soughtin order to make
`best use of a switching device. Two such configurations will now be considered, the
`Class D and Class E modes.
`
`7.4 The Class D Switching Amplifier
`
`Figure 7.7 shows a schematic diagram for a switching mode amplifier in which a
`series resonantcircuit is connected across a two-way switch arrangement, whereby
`the LCR resonator is switched between a bypassed DC voltage generator and
`groundforalternate half cycles. It is assumed that the repetition cycle matches the
`resonant frequency of the LCRcircuit and that the Q factoris high.
`Figure 7.8 shows the waveformsresulting from such a circuit. The key issue is
`that the current in the LCR branchis constrained to remain sinusoidal, with no DC
`offset, due to the inertia effect of the resonator and the DC blocking ofthe series
`capacitor. The result is that switch “A” conducts a positive half sinewave, and
`switch “B” conducts a negative half sinewave, which add together to form the neces-
`
`
`
`Figure 7.7 Switching Class D amplifier.
`
`

`

`7.4 The Class D Switching Amplifier
`
`181tESE-/‘pk
`
`Figure 7.8 Switching Class D amplifier waveforms.
`
`sary full sinewaveof current in the LCR branch.Since the switch voltage waveform
`is a squarewave,it is clear that no poweris being dissipated in the switch and that no
`harmonic energy is being generated; the half sinewaves of current flowing in each
`switch half-cycle contain no odd harmonics.
`Theanalysis is therefore quite straightforward. The peak current,I, is given by
`
`Tie =1,,-a
`
`The fundamental RF current flowing round the LCR branch,I,, is simply
`
`The fundamental componentof RF voltage appearing across the LCR branchis
`
`2
`V, = Vac
`
`dueto the the fact that the voltage appearing across the LCR branchis a squarewave
`of peak value V,,. Once again, with ideal switches, the peak current, and conse-
`quently the power, can be madearbitrarily high by suitable reductionin the value of
`R,. The RF power can be expressed in termsof I,,, and V,,, from these relations:
`
`

`

`182
`
`Switching Mode Amplifiers for RF Applications
`
`P. =
`'
`
`and the DC supply poweris
`
`
`=
`
`2
`
`a
`
`
`V,.-1
`Ps = Vi. . Ty, = “
`ma
`
`Clearly, the efficiency is 100%. The relative power, compared to a Class A
`device having the same I,, and V,,, is
`
` -4 , about 1 dB
`
`uA
`
`This switching version of a Class D amplifier has one significant difference from
`the RF version discussed in Chapter 5. In the switching version described here, the
`peak-to-peak voltage swing is equal to the DC supply voltage, and the peak-to-peak
`RF currentis twice the peak currentof each individual switching device. The switch-
`ing action in some respects produces a similar overall effect to having a pair of
`devices in a push pull configuration. But the overall issue is realizability. The “A”
`switch is a high-side (ungrounded) device, which poses problemsat higher frequen-
`cies both in terms of parasitic reactance and drive requirements. Such amplifiers
`have been reported at low RF frequencies, in the 10 MHzregion [1], and offer some
`attractive possibilities for achieving high powerand highefficiency without generat-
`ing excessive voltage swing. But the possibilities for higher power microwave
`applications seem marginal with present technology.
`
`7.4 Class E—Introduction
`
`The Class E mode has been vigorously touted by its inventors [2-5] for over 30
`years, to a frequently ambivalent microwave community.It has also, for most of
`that time, been patented, and the expiry of the patent [U.S. 3,919,656, 1975] has
`resulted in some renewedinterest in the possibilities it offers [6, 7]. The next few sec-
`tions introduce the Class E modeinits classical form, and the possibilities for higher
`RFderivatives are considered further in Chapter8.
`The previoussections in this chapter have already highlighted the difficulty of
`harnessing anideal switch in a productive manner.Indeed,the results in Section 7.2
`indicate that the sharp, rectangular pulses of current can be as much a hindrance in
`obtaining higherefficiency as they are beneficial. The Class E mode in some ways
`represents a halfway house betweenthe analog world of conventional reduced con-
`duction angle mode amplifiers and the digital world of ideal switches. Although
`mosteasily introduced as a switching type of amplifier, it will become apparent that
`the waveformsare distinctly analog in appearance, and can be approximately sup-
`ported by a device with a slower switching characteristic which includes a linear
`region.
`
`

`

`7.5 Class E—Simplified Analysis
`
`183
`
`Furthermore, simulations and verification tests on actual amplifiers seem to
`support the view that Class E does truly represent an alternative to conventional
`reduced conduction angle operation, giving higher efficiency without the circuit
`complexity of more advanced Class F designs. Direct comparisonis troublesome,in
`that Class F can, as has been demonstrated in Chapter 6, achieve very highefficien-
`cies, but show substantially reduced power output under optimum efficiency opera-
`tion. Class E has its own disadvantage in terms of peak voltage levels, and the PUF
`has to be traded againstthis limitation. But the final judgment maybe that Class E
`appears to beeasier to realize in practice using solid state transistors than a short
`conduction angle Class C design.
`Asstated in the introductionto this chapter, a Class E amplifier is unquestion-
`ably nonlinear, in the sense that variations in input power amplitude will not be
`reproduced at the output in any acceptable form. This also applies to a Class C
`amplifier, and techniques have been available for many decades to remodulate the
`output. For reasonsthat are not entirely clear, these techniques have fallen into dis-
`use somewhere along the road of transition from vacuum tubes to solid state
`devices. One possible reason for this change of emphasis from remodulationtolin-
`ear, or quasi-linear, RF amplification may be that there has never beenareliable
`method of achievingefficiencies in the 90% region using solid state devices, such as
`were quite normally obtained from simple tube amplifier designs, and as canstill
`be found even in amateurradioliterature. Another reasonis the virtual disappear-
`ance of amplitude-only modulation systems. As will be discussed in Chapter 10,
`although most remodulation techniques can be adapted to preserve phase modula-
`tion, this will usually incur more system complexity. The Class E mode may just
`represent a possibility for achieving high enoughefficiencies from solid state
`devices that envelope restoration and outphasing may become mainstream tech-
`niques once again.
`
`7.5 Class E—Simplified Analysis
`
`The analysis presented in this section is idealized, and assumesinitially that the
`active device can be represented as a switch. This results in analytical expressions
`for the device waveforms. Thirty years ago it was conventional practice to pursue
`the analysis in symbolic form, in order to relate the external component values to
`the stipulated waveform parameters. This process can be followed, for example, in
`a classical analysis published by Raab [3]. Many authors have takenthis basic anal-
`ysis further, allowing for device andcircuit parasitics, although the particular issue
`of transmogrifying a switch into an otherwise staple transconductive transistor
`seems to have been vigorously avoided. The continuation of the Class E theme in
`Chapter8 of this book offers an alternative path to practical realization at GHzfre-
`quencies, in particular making the importanttransition from a switching device to a
`conventional transistor which operates in a linear transconductive mode aboveits
`threshold point. By way of introduction, however, wewill stick to convention and
`analyze the Class E circuit using an ideal switch as the active element. As with the
`analysis of Class AB waveformsin Chapter 3, the symbolic analysis will be curtailed
`at strategic points, where direct computation enables a clearer path to the required
`
`

`

`184
`
`Switching Mode Amplifiers for RF Applications
`
`goalof a quantitative design strategy.’ Figure 7.9 showsthe basic elements of a Class
`E circuit. An ideal switch is shuntedby a capacitor C,; the value of C, will be seen to
`be an importantpart of the RF network and at GHz frequencies is usually neither
`small enough to be a device parasitic nor large enough to be a harmonic short. The
`RF matching networkconsists additionally of a series resonantcircuit, which incor-
`poratesthe final RF loadresistor. It is assumedin this analysis that this resistor will
`probably be the input impedance of an additional matching section which trans-
`forms the required value up to a 50 ohm interface. As usual, the DC is supplied
`through a high reactance choke whosevalueis such that no variations of current can
`be supported within the timeframe of an RFcycle. Similarly, the Q-factor of the
`resonatoris assumedto be high enoughthat the “flywheel”effect forces a sinusoidal
`current to flow in the LCR branch.
`Asbefore, it is assumed that conditions on the input can be controlled such that
`the open andclosure timing of the switch during an RF cycle can bearbitrarily pre-
`determined, and are effectively independent variables in the problem.
`Looking now at the branch currents in Figure 7.9, it can be seen that two
`“immovable” currents combine to flow into the switch-capacitor combination; the
`DC supply current, J, and the sinusoidal“flywheel” current in the LCR branch, [,,.
`cos wt.
`So the current flowing into the switch-capacitor combinationis an offset cosine
`wave of current,
`
`(0) =1,,-cos(@)+1,,
`
`(@=@-t)
`
`(7.13)
`
`where the values of I, andI, have yet to be determined. An important normalization
`is now defined, writing (7.13) in the form
`
`i(0) = 1,, «(1+ mcos(@))
`
`(7.14)
`
`
`
`Figure 7.9 Basic Class E amplifier schematic.
`
`t.
`
`For better or for worse, lengthy symbolic analysis is not the flavor of the 00’s. The foregoing analysisis a sig-
`nificant consolidation from thatpresented in thefirst edition of this book, and enables direct computation
`for expressions which cannotbe further simplified using the symbolic approach [9].
`
`

`

`7.5 Class E—Simplified Analysis
`
`where
`
`
`a
`
`185
`
`(7.15)
`
`Figure 7.10(a) shows the offset cosine wave of current i(9) and also showsthe
`positions selected for switch opening andclosure. The switch will be closed for the
`duration of the interval between the positive-moving zero crossing of i(0), -a,, and
`an arbitrarily selected angle a, during the positive portion of i(9). Clearly, when the
`switch is closed, i(@) will flow entirely through the switch, and when the switchis
`opened,i(9) will flow entirely into the capacitor. This is the definitive aspect of ideal
`Class E operation. The “conduction angle” @ can be defined as the sum of a, anda,,
`
`p=a,t+a,
`
`(7.16)
`
`The current waveforms for switch and capacitor can therefore be easily drawn,
`and are shownin Figure 7.10(b, c). As already noted, the key effect is the instanta-
`neoustransfer of current from switch to capacitor at the point