18 High Frequency Electronics

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POWER AMPLIFIERS

Power Amplifier DesignFundamentals: More Notesfrom the Pages of History

By Andrei GrebennikovBell Labs Ireland

In a previous excur-sion through the his-tory of high-efficiency

power amplifier design—published in High Fre-quency Electronics, Sept.2008—it was found thatnew results very often

can represent long-forgotten ones. Sometimesthose who were pioneering those technicalideas are undeservedly passed over, especiallyif it was long ago, perhaps almost a century.

In some cases, due to the difficulties withaccessing the particular periodical publica-tions, especially in different languages, theresults are simply unknown. But it is alwaysinteresting to know how, where and whensome technical idea or design principle firstappeared. And, of course, history also shouldbe learned in order not to repeat it. For exam-ple, most of the aspects of the power amplifierdesign basics described so long ago are stillrelevant and very useful nowadays in simpli-fying the entire design procedure, even withmodern circuit design simulators.

Operation Modes and Conduction AngleAs it was established at the end of the

1910s, the amplifier efficiency may reach quitehigh values when suitable adjustments of thegrid and anode voltages are made [1]. Withresistive load, the anode current is in phasewith the grid voltage, whereas it leads withthe capacitive load and it lags with the induc-tive load. On the assumption that the anodecurrent and anode voltage both have sinu-soidal variations, the maximum possible out-put of the amplifying device would be just halfthe DC supply power, or anode efficiency of

50%. However, by using a pulsed-shapedanode current, it is possible to achieve anodeefficiency considerably in excess of 50%,potentially as high as 90%, by choosing theproper operation conditions.

By applying the proper negative bias volt-age to the grid terminal to provide the pulsedanode current of different width, for angle θand angles greater than θ, the anode currentbecomes equal to zero, where the double angle2θ represents a conduction angle of the ampli-fying device [2]. In this case, a theoreticalanode efficiency approaches 100% when theconduction angle, during which the anode cur-rent flows, reduces to zero starting from 50%which corresponds to the conduction angle of360° or 100% duty cycle.

Generally, power amplifiers can be classi-fied in three classes according to their mode ofoperation: linear mode when its operation isconfined to the substantially linear portion ofthe active device characteristic curve; criticalmode when the anode current ceases to flow,but operation extends beyond the linear por-tion up to the saturation and cutoff regions;and nonlinear mode when the anode currentceases to flow during a portion of each cycle,with a duration that depends on the grid bias[3]. When high efficiency is required, poweramplifiers of the third class are employedsince the presence of harmonics contributes tothe attainment of high efficiencies. In order tosuppress harmonics of the fundamental fre-quency to deliver a sinusoidal signal to theload, a parallel resonant circuit can be used inthe load network which bypasses harmonicsthrough a low impedance path and, by virtueof its resonance to the fundamental, receivesenergy at that frequency.

This article is a thoroughreview of the behaviors of

current, voltage and powerthat are the foundations

for understanding poweramplifier operation

From May 2010 High Frequency ElectronicsCopyright © 2010 Summit Technical Media, LLC

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POWER AMPLIFIERS

At the very beginning of 1930s, power amplifiers oper-ating in the first two classes with 100% duty cycle werecalled Class A power amplifiers, whereas the poweramplifiers operating in the third class with 50% dutycycle were assigned to Class B power amplifiers [4]. TheClass A power amplifier with sinusoidal anode voltageand current waveforms can include a tuned or untunedload network with the load resistance RL = Vm/Im, whereVm and Im are the anode voltage and current amplitudes,respectively. For an input cosine voltage, the operatingpoint must be fixed at the middle point of the linear partof the device transfer characteristic which can be ideallyrepresented by a piecewise-linear approximation. As aresult, the output current is written as

(1)

with the anode quiescent current Ia greater or equal tothe anode current amplitude Im. In this case, the outputanode current contains only two components—DC andcosine—and the averaged current magnitude is equal tothe quiescent current Iq. If the anode voltage amplitudeVm approaches the anode supply voltage Va and the cur-rent amplitude Im approaches the anode DC current Ia asa limit, the anode efficiency η = (VmIm / 2) / (VaIa) is equalto 50% for a sinusoidal output signal.

Class B power amplifiers had been defined as thosewhich operate with a negative grid bias such that theanode current is practically zero with no excitation gridvoltage, and in which the output power is proportional tothe square of the excitation voltage [5]. In this case, noanode current flows during negative swing of the signalbut the anode current is essentially proportional to theinstantaneous value of the grid voltage on positiveswings. As a result, the peak anode current is limited onlyby the emission, load resistance, and anode voltage.

Figure 1 shows the basic circuit schematic and voltageand current waveforms in a Class B operation with θ =90° when the active device is biased to approximately thecutoff point. Here, the active device behaves as an idealvoltage-controlled current source having zero saturationresistance. The DC supply voltage Va is applied to bothplates of the DC-blocking capacitor having zero reactanceat the operating frequency f0 and being constant duringentire signal period, and RL is the load resistor. Byincreasing the current ratio Im / Ia, the anode efficiencycan be increased, as follows from the expression for anodeefficiency η for a Class A mode. This leads to a step-by-step nonlinear transformation of the current cosine wave-form to a pulsed waveform when the magnitude of theanode current exceeds zero during only a part of the sig-nal period. In this case, an active device (vacuum tube) isoperated in the active region followed by operation in thecutoff region when the anode current is zero. In this case,

the frequency spectrum at the device output will general-ly contain the second, third, and higher-order harmonicsfor different conduction angles. However, due to the highquality factor of the parallel resonant LC-circuit tuned tothe fundamental frequency f0, only the fundamental-fre-quency signal flows into the load, while the higher-orderharmonic components are effectively short-circuited.Therefore, ideally the anode voltage represents a purelysinusoidal waveform with the voltage amplitude V ≤ Va. Ifthe anode voltage amplitude Vm approaches the anodesupply voltage Va as a limit, then the anode efficiency η =0.5(Vm Im / 2) / (Va Im / π) is equal to π/4 or 78.5% for half-cosine-wave output current [4].

Equation (1) for the output current can be rewrittenthrough the ratio between the quiescent anode current Iaand the current amplitude Im when ia = 0 as

(2)

As a result, the basic definitions for nonlinear opera-tion modes of a power amplifier through half the conduc-tion angle θ can be introduced as

• when θ > 90°, then cosθ < 0 and Ia > 0 correspondingto Class AB operation

• when θ = 90°, then cosθ = 0 and Ia = 0 correspondingto Class B operation

• when θ < 90°, then cosθ > 0 and Ia < 0 correspondingto Class C operation.

Class C power amplifiers had been defined as thosethat operate with a negative grid bias more than suffi-

cosθ = − II

a

m

i I I ta a m= + cosω

Figure 1 · Voltage and current waveforms in Class Boperation mode.

22 High Frequency Electronics

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POWER AMPLIFIERS

cient to reduce the anode current to zero with no excita-tion grid voltage, and in which the output power varies asthe square of the anode voltage between limits [5]. Themain distinction between Class B and Class C is in theduration of the output current pulses which are shorterfor Class C when the active device is biased beyond thecutoff point.

The periodic pulsed output current ia(ωt) can be rep-resented as a Fourier series expansion

(3)

where the DC, fundamental-frequency, and nth harmoniccomponents are calculated by

(4)

(5)

(6)

where γn(θ) are called the nth-harmonic coefficients ofexpansion of the output current waveform or the nth-har-monic current coefficient [6, 7]. They can be analyticallydefined as

(7)

(8)

(9)

where n = 2, 3, … . As a result, for a Class B mode with θ= 90° corresponding to a half-cosine anode current wave-form, the odd-harmonic current coefficients γn(θ) areequal to zero, as it follows from Eq. (9) for odd n. It shouldbe noted that, for the device transfer characteristic, whichcan be ideally represented by a square-law approxima-tion, the odd-harmonic current coefficients γn(θ) are notequal to zero in this case, although there is no significantdifference between the square-law and linear cases [8]. Toobtain the maximum anode efficiency in Class C, theactive device should be biased (negative) considerablypast the cutoff point to provide the sufficiently low con-duction angles [9].

From Eqs. (4), (5), (7), and (8) it follows that the ratioof the fundamental-frequency component of the anodecurrent to the DC current is a function of θ only,

(10)

which means that, if the operating angle is maintainedconstant, the fundamental component of the anode cur-rent will replicate linearly to the variation of the DC cur-rent, thus providing the linear operation of the Class Cpower amplifier when DC current is directly proportionalto the grid voltage [10].

Load Line and Output ImpedanceThe graphical method of laying down a load line on

the family of the static curves representing anode currentversus anode voltage for various grid potentials wasalready well known in the 1920s [11]. If an active deviceis connected in a circuit in which the anode load is a pureresistance, the performance may be analyzed by drawingthe load line where the lower end of the line representsthe anode supply voltage and the slope of the line is estab-lished by the load resistance, that is, the resistance isequal to the value of the intercept on the voltage axisdivided by the value of the intercept on the current axis.

In Class A, the output voltage va across the deviceanode (collector or drain) represents a sum of the DC sup-ply voltage Va and cosine voltage across the load resis-tance RL. Consequently, the greater output current ia, thegreater voltage across the load resistance RL and thesmaller output voltage va. Thus, for a purely real loadimpedance (ZL = RL) the anode voltage va is shifted by180° relative to the grid voltage vg and can be written as

(11)

Substituting Eq. (1) into Eq. (11) yields

(12)

where RL = Vm/Im is the load resistance. In this case, thepower dissipated in the load and the power dissipated inthe tube is equal when Va = Vm, and the load resistanceRL is equal to the tube output resistance Rout [5].

Equation (12) can be rewritten in the form

(13)

which determines a linear analytical dependence of theanode current versus anode voltage. In practical applica-tions, because of the device nonlinearities, it is necessaryto connect a parallel LC-circuit with resonant frequencyequal to the operating frequency to suppress any possibleharmonic components.

In a pulsed operation mode (Classes AB, B, or C), sincethe parallel LC-circuit is tuned to the fundamental fre-

i IVR

vRm a

a

L

a

L

+⎛⎝⎜

⎞⎠⎟

−

v V i I Ra a a a L= − −( )

v V V t V V tm a m a m= + +( ) = −cos cosω ω180

II

1

0

= −=

θ θ θθ θ θsin cos

sin cos

γ θπ

θ θn

nn n

nn n

( ) =−( )−( ) −

+( )+( )

⎡

⎣⎢

⎤

⎦⎥

1 11

11

sin sin

γ θπ

θ θ1

1 22

( ) = −⎛⎝⎜

⎞⎠⎟

sin

γ θπ

θ θ θo ( ) = −( )1sin cos

I I t n t d t In m n m= −( ) = ( )−∫

1π

ω θ ω ω γ θθ

θ

cos cos cos

I I t t d t Im m1 1

1= −( ) = ( )−∫π

ω θ ω ω γ θθ

θ

cos cos cos

I I t d t Io m m= −( ) = ( )−∫

12 0π

ω θ ω γ θθ

θ

cos cos

i t I I t I t I ta oω ω ω ω( ) = + + +1 2 32 3cos cos cos ...

24 High Frequency Electronics

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POWER AMPLIFIERS

quency, ideally the voltage across the load resistor RL rep-resents a cosine waveform. In this case, the relationshipbetween the anode current ia and voltage va during a timeperiod of –θ ≤ ωt < θ can be expressed using Eqs. (2) and(11) as

(14)

where the fundamental current coefficient γ1 as a func-tion of θ is determined by Eq. (8) and the load resistanceis defined by RL = Vm/I1 where I1 is the fundamental-fre-quency current amplitude. Equation (14) determining thedependence of the anode current on the anode voltage forany values of conduction angle in the form of a straightline function with different slope angles β = tan–1(1/γ1RL)is called the load line of the active device in a generalform. For a Class A operation mode with θ = 180° whenγ1 = 1, Eq. (14) becomes identical to Eq. (13). The formulasfor the amplitudes of the harmonic components can beobtained in terms of points taken from the load line onthe anode voltage-current characteristics at equallyspaced intervals of grid voltage [12].

The load resistance RL for the active device as a func-tion of θ which is required to terminate the device outputin order to deliver maximum output power to the load canbe written in a general form as

(15)

which is equal to the tube equivalent output resistanceRout at the fundamental frequency [5]. The term “equiva-lent” means that this is not a real physical tube resistanceas in a Class A mode, but its equivalent output resistance,the value of which determines the optimum load, whichshould terminate the tube output to deliver maximumfundamental-frequency output power. The equivalent out-put resistance is calculated as a ratio between the ampli-tudes of the anode cosine voltage and fundamental-fre-quency anode current component, which depends on theangle θ. In a Class B mode when θ = 90° and γ1 = 0.5, theload resistance RL

Bis defined as RLB= 2Va / Imax.

Class B power amplifiers are often driven hardenough to cause the active devices to enter saturationduring a portion of each RF cycle, resulting in a depres-sion in the top part of the cosine anode waveform. Similarresults can be achieved by increasing the value of RLwhen the load line is characterized by smaller slope angleβ. The anode current waveform becomes asymmetrical forthe complex load, the impedance of which is composed ofthe load resistance and capacitive or inductive reactance.In this case, the Fourier expansion of the pulsed anodecurrent given by Eq. (3) includes a particular phase for

each harmonic components. Then, generally the anodevoltage of the active device is written as

(16)

where In is the amplitude of nth output current harmon-ic component, |Zn| is the magnitude of the load networkimpedance at nth output current harmonic component,and φn is the phase of nth output current harmonic com-ponent. Assuming that Zn is zero for n = 2, 3, …, which ispossible for a resonant load network having negligibleimpedance at any harmonic component except the funda-mental, Eq. (16) can be rewritten as

(17)

As a result, for the inductive load impedance, thedepression in the anode current waveform reduces andmoves to the left-hand side of the waveform, whereas thecapacitive load impedance causes the depression to deep-en and shift to the right-hand side of the anode currentwaveform [13]. This effect can simply be explained by thedifferent phase conditions for fundamental and higherorder harmonic components composing the anode currentwaveform and is illustrated by the different load lines for(a) inductive and (b) capacitive load impedances shown inFigure 2. Note that now the load line represents the two-dimensional curve with a complicated behavior.

v V I Z ta a= − +( )1 1 1cos ω φ

v V I Z n ta a n nn

n= − +( )=

∞

∑1

cos ω φRV

ILm

m

θγ θ

( ) = ( )1

i IVR

vRa a

a

L

a

L

= +⎛⎝⎜

⎞⎠⎟

−γ γ1 1

Figure 2 · Load lines for (a) inductive and (b) capaci-tive load impedances.

26 High Frequency Electronics

High Frequency Design

POWER AMPLIFIERS

Power Gain and Output MatchingIn order to extract the maximum power

from a generator, it is well known fact that theexternal load should have a vector value whichis conjugate of the internal impedance of thesource [14]. The power delivered from a gener-ator to a load, when matched on this basis, willbe called the available power of the generator[15]. In this case, the power gain of the four-terminal network is defined as the ratio of the powerdelivered to the load impedance connected at the outputterminals to the power available from the generator con-nected to the input terminals, usually measured in deci-bels, and this ratio is called “power gain” irrespective ofwhether it is greater or less than one [16, 17].

Figure 3 shows the basic block schematic of the single-stage power amplifier circuit which includes an activedevice, an input matching circuit to match with thesource impedance, and an output matching circuit tomatch with the load impedance. Generally, the two-portactive device can be characterized by a system of immit-tance W-parameters, which is any system of impedance Z-parameters, hybrid H-parameters, or admittance Y-parameters [18, 19]. The input and output matching cir-cuits transform the source and load immittances WS andWL into specified values between points 1-2 and 3-4,respectively, by means of which the optimal design opera-tion mode of the power amplifier is realized.

The transducer power gain GT defined as the ratio ofthe power delivered to the load immittance WL to thepower available from the source having an immittanceWS can be expressed in terms of the immittance W-parameters as

(18)

The operating power gain GP defined as the ratio ofpower delivered to the load immittance WL to the powerdelivered to the input port of the active device with animmittance Win can be expressed in terms of the immit-tance W-parameters as

(19)

where the input and output immittances Win and Wout aredefined as

(20)

(21)

The operating power gain GP does not depend on thesource parameters and characterizes only the effective-ness of the power delivery from the input port of theactive device to the load. This power gain helps to evalu-ate the gain property of a multistage amplifier when theoverall operating power gain GP(total) is equal to the prod-uct of each stage GP. The transducer power gain GTincludes an assumption of conjugate matching of both theload and the source. In this case, the denominator of Eq.(18) becomes minimal since the reactive parts of theimaginary parts of the device input and outputimpedances are compensated for, thus resulting in a max-imum available transducer power gain GT.

It should be noted that Eqs. (18) and (19) are given ingeneral immittance forms without indication of whetherthey are used in a small-signal or large-signal applica-tion. In the latter case, this only means that the deviceimmittance W-parameters are fundamentally averagedover large signal swing across the device equivalent cir-cuit parameters and that the conjugate-matching princi-ple is valid in both the small-signal application and thelarge-signal application where the optimum equivalentdevice output resistance is matched to the load resistanceand the effect of the device output reactance is eliminat-ed by the conjugate reactance of the load network.

Note that the load resistance RL defined by the gener-al load-line dependence given by Eq. (14) is a ratio of theoutput cosine voltage and fundamental-frequency currentamplitudes which in turn determines the optimum deviceequivalent output resistance when maximum fundamen-tal-frequency output power is delivered to the load, andload line represents only a graphical interpretation ofreal physical process. Therefore, it is generally incorrectto introduce separately the concepts of the gain matchwith respect to the linear power amplifiers only and thepower match in nonlinear power amplifier circuits sincethe maximum large-signal power gain being a function ofthe angle θ corresponds to the maximum fundamental-frequency output power delivered to the load due to large-signal conjugate output matching. It is very important toprovide a conjugate matching at both input and outputdevice ports to achieve maximum power gain in a large-signal mode. It also implies an efficiency match for thespecified conduction angle since the maximum achievableanode (collector or drain) efficiency is a function of the

W WW W

W WSout = −

+2212 21

11

W WW W

W WLin = =

+1112 21

22

GW W

W W WP

L

L

=+21

2

222

Re

Re in

GW W W

W W W W W WT

S L

S L

=+( ) +( ) −

4 212

11 22 12 21

2

Re Re

Figure 3 · Generalized single-stage power amplifier circuit.

28 High Frequency Electronics

High Frequency Design

POWER AMPLIFIERS

angle θ as well. In a Class A mode, the maximum small-signal power gain ideally remains constant regardless ofthe output power level.

The transistor characterization in a large-signal modecan be done based on equivalent quasi-harmonic nonlin-ear approximation under the condition of sinusoidal portvoltages [20]. In this case, the large-signal impedancesare generally determined in the following manner. Thedesigner tunes the load network (often by trial anderrors) to maximize the output power to the required levelusing a particular transistor at a specified frequency andsupply voltage. Then, the transistor is removed from thecircuit and the impedance seen by the collector is mea-sured at the carrier frequency. The complex-conjugate ofthe measured impedance then represents the equivalentlarge-signal output impedance of the transistor at thatfrequency, supply voltage, and output power. Similardesign process is used to measure the input impedance ofthe transistor and to maximize power-added efficiency ofthe power amplifier.

To deliver maximum power to the load, it is necessaryto use some impedance matching network which can mod-ify the load as viewed from the generator [14]. The designof reactance networks to connect a resistive load effi-ciently to a source of power can be carried out most con-veniently by the theory of image impedances. In this case,if the image impedances of such a network are pure resis-tances and it is connected between a generator and a loadwhose impedances are equal to these image impedances,the impedances will match at both junctions. Under theseconditions, with an assumption of pure reactances of thearms of the connecting network, no power will dissipateduring transmission and so the maximum power will bedelivered to the load. If the terminating impedances arenot pure resistances, they can be made so at any singlefrequency by placing additional reactance in series. Suchreactance networks can provide not only for high efficien-cy but can also attenuate undesired harmonics. A varietyof configurations can be designed to accomplish thedesired result. Both T-type and π-type configurations aremost popular for matching networks, depending on con-venience of the circuit implementation and particularapplication technology. They can be implemented in bothlow-pass and high-pass topologies.

Load Pull CharacterizationLoad pull is a computer-controlled technique for large-

signal characterization of microwave power transistorsused to map contours of constant power and efficiency ona Smith chart for dynamic matching of both input andoutput circuits. This was originally developed for theinterstage matching between a varactor multiplier and atransistor power stage, and then successfully applied tothe broadband optimization of Class A and Class C tran-

sistor power amplifiers [21, 22]. The power-load contoursconsist of a series of curves on a Smith chart representingconstant output power, approaching the point of maxi-mum output power, as shown in Figure 4. When drawn atseveral frequencies over the band of interest, they repre-sent loci of required output impedance for various outputpower levels. If constant efficiency contours are overlaidon the constant power contours, the efficiency is alsoknown at each of the load impedances.

A power transistor operated in the Class C mode hasan output equivalent circuit which is represented by anonlinear multiharmonic current source and a nonlinearreactance. To obtain maximum output power, the loadimpedance must produce the maximum current and volt-age swing. The relationship between the required loadimpedance and nonlinear output impedance determinesthe shape of the power-load contour which can vary withinput drive level. For low drive levels, the contour isalmost elliptical. As the drive level increases toward thepoint where the transistor current saturates, the power-load contour expands into a circle with a constant trans-mission loss circle [23]. To get a variable load, in additionto the traditional passive-network technique with vari-able elements, an active load-pull characterization can beused where the reflection coefficient is obtained using anauxiliary signal derived from the same test generator toinject into the output port [24, 25]. One of the advantagesof this technique lies in the inherent simplicity of the cal-ibration procedure necessary to correct the transmission-line losses in the measurement system. However, its accu-racy critically depends on the effective directivities of thedirectional couplers in the system [26]. Independent tun-ing of the fundamental frequency and its second harmon-ic had become possible by using a scheme with frequency-selective tuners [27].

Generally, tuning with a load-pull system (mechanical

Figure 4 · Constant-power load-pull contours.

30 High Frequency Electronics

High Frequency Design

POWER AMPLIFIERS

or fully automatic) is a very complicated procedure, espe-cially we must take into account several harmonics of thefundamental frequency. Besides, it can only give the opti-mum input and output impedances, as are usually incor-porated in the datasheet for power transistors for speci-fied output power, supply voltage and frequency range. Ina monolithic implementation, however, this is difficult tophysically realize and can be done based on the load-pullsetup incorporated into the simulation tool. But, in allcases, the measurement should be followed by the subse-quent load-network design with real elements. Therefore,in most cases where the device nonlinear model is avail-able, the device equivalent output circuit can be repre-sented by the output admittance where the equivalentoutput resistance can be estimated from Eq. (15) for fixedconduction angle and supply voltage equal ideally to thecosine voltage amplitude (the corresponding device satu-ration voltage can be subtracted). For example, in a ClassB mode with θ = 90°, it is written as RL = Va/I1 =Va

2/ 2Pout, where Pout is the maximum fundamental-fre-quency power delivered to the load, and the output shuntreactance is represented by the sum of the output andfeedback capacitances.

Next MonthThis article concludes next month with discussions of

stability and and the effects of higher-order harmonics.

Author InformationAndrei Grebennikov received the MSc degree in elec-

tronics from Moscow Institute of Physics and Technology,and the Ph.D. degree in radio engineering from MoscowTechnical University of Communications and Informatics.He can be reached by e-mail at: grandrei@ ieee.org

References1. J. H. Morecroft and H. T. Friis, “The Vacuum Tubes as

a Generator of Alternating-Current Power,” Trans. AIEE, vol.38, pp. 1415-1444, Oct. 1919.

2. D. C. Prince, “Vacuum Tubes as Power Oscillators, PartI,” Proc. IRE, vol. 11, pp. 275-313, June 1923.

3. A. A. Oswald, “Power Amplifiers in Trans-AtlanticRadio Telephony,” Proc. IRE, vol. 13, pp. 313-324, June 1925.

4. L. E. Barton, “High Audio Power from Relatively SmallTubes,” Proc. IRE, vol. 19, pp. 1131-1149, July 1931.

5. C. E. Fay, “The Operation of Vacuum Tubes as Class Band Class C Amplifiers,” Proc. IRE, vol. 20, pp. 548-568, Mar.1932.

6. P. H. Osborn, “A Study of Class B and C Amplifier TankCircuits,” Proc. IRE, vol. 20, pp. 813-834, May 1932.

7. A. I. Berg, Theory and Design of Vacuum-TubeGenerators (in Russian), Moskva: GEI, 1932.

8. F. E. Terman and J. H. Ferns, “The Calculation of ClassC Amplifier and Harmonic Generator Performance of Screen-

Grid and Similar Tubes,” Proc. IRE, vol. 22, pp. 359-373, Mar.1934.

9. L. B. Hallman, “A Fourier Analysis of Radio-FrequencyPower Amplifier Wave Forms,” Proc. IRE, vol. 20, pp. 1640-1659, Oct. 1932.

10. W. L. Everitt, “Optimum Operating Conditions forClass C Amplifiers,” Proc. IRE, vol. 22, pp. 152-176, Feb. 1934.

11. C. E. Kilgour, “Graphical Analysis of Output TubePerformance,” Proc. IRE, vol. 19, pp. 42-50, Jan. 1931.

12. D. C. Espley, “The Calculation of Harmonic Productionin Thermionic Valves with Resistive Load,” Proc. IRE, vol. 21,pp. 1439-1446, Oct. 1933.

13. V. I. Kaganov, Transistor Radio Transmitters (inRussian), Moskva: Energiya, 1976.

14. W. L. Everitt, “Output Networks for Radio-FrequencyPower Amplifiers,” Proc. IRE, vol. 19, pp. 725-737, May 1931.

15. H. T. Friis, “Noise Figure of Radio Receivers,” Proc.IRE, vol. 32, pp. 419-422, July 1944.

16. S. Roberts, “Conjugate-Image Impedances,” Proc. IRE,vol. 34, pp. 198-204, Apr. 1946.

17. S. J. Haefner, “Amplifier-Gain Formulas andMeasurements,” Proc. IRE, vol. 34, pp. 500-505, July 1946.

18. R. L. Pritchard, “High-Frequency Power Gain ofJunction Transistors,” Proc. IRE, vol. 43, pp. 1075-1085, Sept.1955.

19. A. R. Stern, “Stability and Power Gain of Tuned PowerAmplifiers,” Proc. IRE, vol. 45, pp. 335-343, Mar. 1957.

20. L. S. Houselander, H. Y. Chow, and R. Spense,“Transistor Characterization by Effective Large-Signal Two-Port Parameters,” IEEE J. Solid-State Circuits, vol. SC-5, pp.77-79, Apr. 1970.

21. E. F. Belohoubek, A. Rosen, D. M. Stevenson, and A.Pressser, “Hybrid Integrated 10-Watt CW Broad-Band PowerSource at S Band,” IEEE J. Solid-State Circuits, vol. SC-4, pp.360-366, Dec. 1969.

22. A. Presser and E. F. Belohoubek, “1-2 GHz High PowerLinear Transistor Amplifier,” RCA Rev., vol. 33, pp. 737-751,Dec. 1972.

23. J. M. Cusack, S. M. Perlow, and B. S. Perlman,“Automatic Load Contour Mapping for Microwave PowerTransistors,” IEEE Trans. Microwave Theory Tech., vol. MTT-12, pp. 1146-1152, Dec. 1974.

24. Y. Takayama, “A New Load-Pull CharacterizationMethod for Microwave Power Transistors,” 1976 IEEE MTT-S Int. Microwave Symp. Dig., pp. 218-220.

25. G. P. Bava, U. Pisani, and V. Pozzolo, “Active LoadTechnique for Load-Pull Characterization at MicrowaveFrequencies,” Electronics Lett., vol. 18, pp. 178-180, Feb.1982.

26. C. Rauscher and H. A. Willing, “Simulation ofNonlinear Microwave FET Performance Using a Quasi-Static Model,” IEEE Trans. Microwave Theory Tech., vol.MTT-27, pp. 834-840, Oct. 1979.

27. R. B. Stancliff and D. D. Poulin, “Harmonic Load-Pull,”1979 IEEE MTT-S Int. Microwave Symp. Dig., pp. 185-187.