IEEE TRANSACTIONS ON MICROWAVE THEORY AND...

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 8, AUGUST 2011 2037

Analysis of High-Efficiency Power Amplifiers WithArbitrary Output Harmonic Terminations

Michael Roberg, Student Member, IEEE, and Zoya Popovic, Fellow, IEEE

Abstract—This paper presents an analysis of ideal power ampli-fier (PA) efficiency maximization subject to a finite set of arbitrarycomplex harmonic terminations, extending previous results whereonly purely reactive harmonic terminations were treated. Max-imum efficiency and corresponding fundamental output power andload impedance are analyzed as a function of harmonic termina-tion(s). For a PA restricted to second harmonic drain waveformshaping, maximum efficiency as a function of second harmonic ter-mination is treated for cases of both purely real and complex fun-damental frequency impedances. For the case of a PA restrictedto second and third harmonic drain waveform shaping, peak effi-ciency as a function of third harmonic impedance with an idealsecond harmonic termination is analyzed. Additionally, the sen-sitivity of PA efficiency with respect to the magnitude and phaseof the second and third harmonic load reflection coefficients is ex-amined. The analysis is extended to include device and packageparasitics. The paper concludes with a discussion of how the pre-sented general analysis method provides useful insights to the PAdesigner.

Index Terms—Amplifier efficiency, Fourier analysis, harmonicterminations, load impedance, microwave power amplifiers (PAs),waveform shaping.

I. INTRODUCTION

P OWER amplifier (PA) efficiency is traditionally controlledby the current conduction angle of the transistor, which re-

sults in sinusoidal voltage and clipped sinusoidal current time-domain waveforms at the virtual drain (collector) [1]. In the fre-quency domain, the clipped sinusoidal current waveforms cor-respond to generation of harmonic current components by thetransistor. Given that the voltage waveform is a pure sinusoid, noharmonic voltage components are produced, requiring that har-monic shorts are presented to the transistor at the virtual drain.Other classes of PAs, such as class-F and class-F , present spe-cific harmonic impedances at the virtual drain, using either har-monic voltage or current components to shape the drain wave-forms, therefore improving efficiency. Nonlinearities inherentto the transistor such as knee voltage, etc. are viable methodsof producing harmonic voltage components [2]. At higher mi-crowave frequencies, parasitics inherent to the transistor, as wellas its package can limit the ability to present specific harmonicimpedances.

Manuscript received March 04, 2011; accepted April 04, 2011. Date of pub-lication May 23, 2011; date of current version August 17, 2011.

The authors are with the Department of Electrical, Computer and Energy En-gineering, University of Colorado at Boulder, Boulder, CO 80309-0425 USA(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2011.2144995

Investigation of PAs via Fourier analysis was published asearly as 1932 [3]. In [4], Raab analyzed efficiency and outputpower capability of an ideal PA having a finite number of re-active harmonic terminations. This method maximized PA ef-ficiency through optimization of the drain voltage and currentwaveform Fourier coefficients and fundamental frequency re-actance, under the restriction of a finite set of harmonic ter-minations. In [5], an analytical treatment of ideal class-F am-plifiers subject to finite harmonic terminations was presentedin which purely reactive harmonic terminations were also as-sumed. Using a technique similar to [4], a set of PA classes withclass-B efficiency has been treated [6]. In [7], an analytical so-lution to finite harmonic class-C PA maximum efficiency wasderived, which is equally applicable to analysis of a finite har-monic class-C PA. Cripps recently discussed the waveformanalysis of a second-harmonic only PA in [8].

To the best of the authors’ knowledge, a general analysis witharbitrary sets of resistive and reactive harmonic terminations hasnot been presented to date. However, in practice, harmonic ter-minations are complex due to resistive loss of the PA outputnetwork. Therefore, it is of interest to understand the impact ofarbitrary impedances at harmonic frequencies on PA efficiency,output power, and load impedance. This paper generalizes theclassical method of [4] in order to investigate several issues ofpractical interest:

1) fidelity of harmonic terminations required to achieve aspecified efficiency;

2) impact of resistive, reactive, and complex harmonic termi-nations on maximum PA efficiency and corresponding fun-damental output power and load resistance;

3) impact of fundamental frequency reactance on maximumPA efficiency and corresponding fundamental outputpower and load resistance.

In Section II, a generalized theoretical analysis of PAefficiency, output power, and load impedance is presented.Sections III and IV detail the results for PAs having up tothird harmonic voltage and current components for real andcomplex fundamental load impedances, respectively. Section Vpresents a practical application of the theoretical analysis fora 2.14-GHz PA with a constant device output capacitance as-sumed. Section VI contains an example of harmonic load–pulldata using a practical 50-W GaN HEMT demonstrating trendspredicted by the theory.

II. HARMONICALLY TERMINATED PA ANALYSIS APPROACH

Fig. 1 depicts an ideal PA, which will be described in termsof a field-effect transistor (FET) without loss of generality, andwith the following assumptions.

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2038 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 8, AUGUST 2011

Fig. 1. Circuit diagram of ideal common source FET-based PA. The dc block�� and RFC are assumed to be ideal, therefore � is equivalent to theimpedance � presented to the virtual drain at the fundamental and harmonicfrequencies.

• The device has zero on-state resistance . Whilenonzero may be considered, the intention of the fol-lowing analysis is to investigate the performance ceilingrather than the performance as a function of .

• The drain is biased through an ideal RF choke (RFC).• The load is dc isolated from the supply by an ideal

blocking capacitor that acts as a short circuit at the op-erating frequency and corresponding harmonics.

• A sinusoidal voltage at is applied to the gate. Harmonicsare present only at the output of the transistor, and they areassumed to be generated by the various nonlinearities ofthe transistor, which include nonlinear capacitances suchas gate-to-source capacitance.

• The dc drain voltage and dc drain current arestrictly greater than or equal to zero.

• The drain current and voltage waveforms have fixedmaximum values and , regardless of harmoniccontent. This allows a fair comparison of output powerand fundamental load impedance for amplifiers havingdifferent harmonic terminations. Alternate parametersmay be fixed, as will be discussed in the Appendix.

The time–harmonic drain voltage and currentwaveforms of an ideal PA limited to harmonics can be repre-sented by Fourier series as

V (1)

A (2)

where is angular time [9]. In principle, may ex-tend to , but practically is finite due to the device gain roll-offin frequency, and in the case of this paper, up to is an-alyzed. The minimum values of the drain waveforms definedin (1) and (2) are required to be greater than or equal to zero,where the case when the waveforms have minima of zero cor-responds to (17) being maximized. This corresponds to the dcpower being minimized for the given fundamental output power.Under this restriction, the ideal PA with no harmonic voltage or

current components corresponds to a class-A PA with full 360conduction angle, as expected. By inspection of Fig. 1, the loadnetwork voltage and current waveforms are givenby

V (3)

A (4)

Due to the definitions of the load voltage in (3) and currentwaveforms in (4), the voltage and current fundamental fre-quency and harmonic components are expressed as

V (5)

A (6)

The load impedance at frequency is given by

(7)

The time average power delivered to the load at frequencyis given by

W (8)

where denotes the complex conjugate operator. For analysis ofan ideal PA, it is convenient to define the fundamental frequencydrain voltage Fourier coefficient as

V (9)

Any other choice of would simply shift the phase of thefundamental component of the drain voltage waveform with re-spect to the higher order components, therefore requiring thesame phase shift to higher order components to restore wave-form alignment. Consequently, the time average power deliv-ered to the load at the fundamental frequency simplifies to

W (10)

In order to perform a generalized analysis of an ideal PA, it isconvenient to normalize the drain and load waveforms by themaximum drain voltage , which can be withstood withoutdevice breakdown, and the maximum drain current thedevice can support, such that

V (11)

A (12)

V (13)

A (14)

ROBERG AND POPOVIC: ANALYSIS OF HIGH-EFFICIENCY PAs WITH ARBITRARY OUTPUT HARMONIC TERMINATIONS 2039

where and are scaling factors defined as

V (15)

A (16)

The maximum values of the drain voltage and current wave-forms are fixed at and , respectively, regardless ofthe harmonic content and are easily scaled. PA efficiency is ex-pressed as

(17)

where , , and are the fundamental power, dcdrain voltage, and dc drain current, respectively, calculated viaoptimization of the normalized equations (11)–(14). Equation(17) refers to drain efficiency, or an upper bound to power-addedefficiency. Equation (17) shows that PA efficiency can be max-imized using the normalized equations. Once efficiency opti-mization using the normalized equations is performed, the cor-responding fundamental output power and fundamental loadimpedance are calculated. Let the normalized fundamental fre-quency voltage Fourier coefficient and normalized fundamentalfrequency load impedance be defined as

V (18)

(19)

where the normalized fundamental load resistance isunity and is the normalized fundamental load reactance.Defining and in this manner results in 1-W normal-ized output power when . The freedom to chooseconvenient definitions of these quantities is due to the normal-ized equations given in (11)–(14). Alternate definitions wouldsimply result in different values for the normalized quantities

, , , and without impactingthe efficiency, fundamental output power, and fundamental loadimpedance. Given the definitions of (18) and (19), the normal-ized fundamental frequency Fourier coefficients of the drain cur-rent are derived as

A (20)

A (21)

Substituting (18) and (21) into (10) yields an expression for thenormalized fundamental frequency average power given by

W (22)

The fundamental frequency average power is then expressed as

W (23)

which through substitution of (15) and (16) simplifies the fun-damental frequency average power to

W (24)

In order to compare PAs having different sets of arbitrary har-monic terminations with constant and , the outputpower is normalized by the output power of a class-A ampli-fier, which is given by

W (25)

Dividing (24) by (25) produces an expression for fundamentaloutput power normalized to class-A output power given by

(26)

The fundamental frequency resistance is expressed as

(27)

which by substitution of (15) and (16) becomes

(28)

To compare the impact of harmonic terminations on the fun-damental frequency resistance, it is useful to normalize by theclass-A load line resistance given by

(29)

Dividing (28) by (29) produces an expression for fundamentalload resistance normalized to the class-A load line resistancegiven by

(30)

Similarly, the fundamental load reactance normalized to theclass-A load line resistance is given by

(31)

In general, maximization of efficiency given a finite set of har-monic terminations must be performed numerically via a globaloptimization procedure, although special cases exist in whichexplicit expressions have been obtained [10], [11]. A generalprocedure for numerical maximization of efficiency given a fi-nite set of harmonic terminations is as follows.

• Define a finite set of normalized harmonic load imped-ances where voltage andcurrent harmonics greater than are not generated by theactive device.


• Apply a global optimization algorithm to maximize theefficiency by optimizing the normalized fundamental fre-quency reactance and the normalized drain wave-form Fourier coefficients given the set of normalized har-monic load impedances. Either the current or voltage co-efficients may be optimized as convenient, given the re-maining coefficients are explicitly defined by the harmonicimpedances.

• Once efficiency optimization is complete, use (26) to cal-culate the fundamental frequency output power normalizedto class-A output power. Use (30) and (31) to calculate thefundamental frequency resistance and reactance, respec-tively, normalized to the class-A load line resistance.

Maximization of efficiency is performed with the MATLAB

optimization toolbox using the function, which im-plements the direct search Nelder–Mead simplex method de-scribed in [12]. The method does not require a gradient of thefunction being minimized, and it is well suited for problems thatexhibit discontinuities. Since the function may re-sult in a local solution, multiple sets of drain waveform Fouriercoefficients and initial conditions are used to ensure theglobal maximum efficiency is found. Given that isa minimization function, the cost function is defined as nega-tive efficiency in order to solve for maximum efficiency.Solved efficiency, Fourier coefficients, and results forspecific sets of harmonic impedances were in agreement withthose found by the authors of this paper when using alternate op-timization methods, e.g., genetic and simulated annealing [13],[14].

III. REAL FUNDAMENTAL LOAD IMPEDANCE

In this section, the method is applied to optimizing a PA withreal valued fundamental load impedance with arbitrary com-plex harmonic terminations. The parameters of interest are effi-ciency, fundamental output power, and fundamental load resis-tance. In particular, their sensitivity to harmonic terminations isinvestigated.

A. Second-Harmonic Only PA

The real fundamental load impedance represents the PA con-figuration that will deliver maximum power given fixed valuesof peak voltage and current. From (22), the normalized power

remains constant at 1 W, independent of harmonic ter-mination(s). Efficiency optimization is performed for secondharmonic terminations spaced uniformly over the Smith chart.For each harmonic termination, either the drain voltage coeffi-cients or the drain current coefficients areselected for optimization. In the case where , thevoltage coefficients cannot be optimized because they are nec-essarily zero. Therefore, the current coefficients must be opti-mized instead. Similarly, when , the current coef-ficients cannot be optimized because they are necessarily zero.

Fig. 2(a) shows efficiency contours as a function of secondharmonic impedance. Maximum efficiency of 70.71% occursunder conditions of an ideal short or open circuit in agreementwith [7] with degraded efficiency elsewhere. The ideal short-circuit case corresponds to a second harmonic class-C PA, while

Fig. 2. (a) Ideal second harmonic PA efficiency contours versus � �� . Theminimum efficiency of 53.76% occurs when � �� . The maximum ef-ficiency of 70.71% occurs when � �� or � �� . (b) Idealsecond harmonic PA efficiency for purely resistive and purely reactive � �� .The fundamental frequency impedance � �� is purely real.

the ideal open circuit case corresponds to a second harmonicclass-C PA.

A subset of the data shown in Fig. 2(a) is plotted in Fig. 2(b),which shows efficiency as a function of when

, and when is purely real. For a purelyreactive second harmonic termination (i.e., ),70% efficiency is achieved within 22.5 of a short and opencircuit. For a purely real second harmonic termination (i.e.,

, 180 ), 70% efficiency is achieved when.

In practice, this analysis gives insight to the PA designer con-cerning how closely the fabricated second harmonic terminationmust be to an ideal short or open circuit to achieve a desired ef-ficiency. For example, must be achieved for an


Fig. 3. (a) Ideal second harmonic PA normalized power � �� contoursversus � �� . (b) Ideal second harmonic PA � �� for purely resistiveand purely reactive � �� . The fundamental frequency impedance � �� ispurely real.

approximate short circuit and for an approximateopen circuit given an ideal termination phase to achieve 70%efficiency, which is rather restrictive. A more reasonable goalwould be to achieve , which would result ingreater than 68% efficiency. This corresponds to foran approximate short circuit and for an approximateopen circuit, which is a more practical.

Fig. 3(a) shows contours of fundamental frequency outputpower, normalized to class-A. Fig. 3(b) shows from(26) as a function of when , and when

is purely real. An output power of is realizedunder conditions of maximum efficiency. A minimum outputpower of occurs when , correspondingto a 0.56-dB reduction relative to class-A output power. It isevident that output power may not be improved over class-Aoutput power with only a second harmonic termination whenpeak current and voltage are held fixed.

Fig. 4. (a) Ideal second harmonic PA normalized load resistance � �� con-tours versus � �� . (b) Ideal second harmonic PA � �� for purely re-sistive and purely reactive � �� . The fundamental frequency impedance� �� is purely real.

Fig. 4(a) shows contours of fundamental frequency load re-sistance, normalized to the class-A load line. Fig. 4(b) shows

as a function of when , andwhen is purely real. The load resistance exhibits a dis-continuity about the imaginary axis. The load resistance underconditions of maximum efficiency is when

and when . This is dueto the swapping of normalized current and voltage waveformswhen the normalized load is changed from 0 to , which effec-tively inverts (30). For example, when , the peakvoltage is increased over the class-A case, while the peak cur-rent stays the same, implying the load resistance must decreaseto maintain constant and . In general, if the load resis-tance for one half of the Smith chart (e.g., the left half) is calcu-lated, the load resistance for the other half can be directly calcu-lated by inverting the data, resulting in a discontinuity about theimaginary axis. In other words, the solutions from the left half of


the Smith chart correspond to current peaking drain waveforms,while the right half solutions correspond to voltage peaking.

B. Second and Third Harmonic PA

As demonstrated, second harmonic short and open circuitsare optimal terminations from the standpoint of PA efficiency,which is in agreement with [4]. Efficiency is next re-optimizedfor an added third harmonic under conditions of an ideal secondharmonic termination. When , a peak efficiency of81.65% is realized when , as shown in Fig. 5(a).This set of harmonic terminations corresponds to a second andthird harmonic class-F amplifier. Under the short-circuit con-dition , an efficiency of 80.90% issimulated, which is in agreement with analysis of a finite har-monic class-C amplifier [7]. Fig. 5(b) shows the correspondingcontours of output power normalized to class-A output power.The class-F output power is improved by 0.5 dB relative toclass-A output power, while class-C output power is reduced0.69 dB relative to class-A output power. Fig. 5(c) shows thecorresponding contours of fundamental load resistance normal-ized to class-A load resistance. The class-F and class-C loadresistance are and , respectively.

Alternatively, efficiency optimization can be performedunder the condition that . In this case, the resul-tant efficiency and power contours may be obtained by rotatingFig. 5(a) and (b) 180 about the imaginary axis. The resultantresistance contours are obtained by inverting the contour valuesshown in Fig. 5(c) and rotating them 180 about the imagi-nary axis. The transformed contours may be used to extractthe efficiency, output power, and load resistance of secondand third harmonic class-F and class-C amplifiers. Theefficiencies and output powers of the inverted amplifier classesare equivalent to those quoted for the noninverted classes,while the load resistance is found to be the reciprocal of thatfor the noninverted class. Consequently, the second and thirdharmonic class-F and class-C amplifiers require a smallerload resistance than their class-F and class-C counterparts toproduce equivalent output power with maximum efficiencyunder conditions of fixed peak voltage and current. Table Isummarizes the efficiency, fundamental output power, and fun-damental load resistance for second-harmonic only and secondand third harmonic class-F, class-C, class-F , and class-Camplifiers.

In practice, it is important to understand the sensitivity ofPA efficiency to the phase and magnitude of the third har-monic termination under the condition of an ideal secondharmonic termination. Fig. 6 shows PA efficiency as a functionof for the analyzed second and third harmonic caseswhen . The maximum and minimum efficiencywhen are 81.65% and 76.30%, respectively,demonstrating the importance of terminating the third harmonicin an appropriate angle. Greater than 81% efficiency is achievedwithin 30.0 of a third harmonic short when or athird harmonic open when . For the case of a purelyresistive , greater than 81% efficiency is achieved when

for and for, as shown in Fig. 7. Based upon the simulated

results, it is concluded that PA efficiency is less sensitive to

Fig. 5. (a) Ideal PA efficiency contours versus � �� , � �� .The maximum efficiency of 81.65% occurs when � �� . Note howterminating � �� of the third harmonic PA in any impedance improvesPA efficiency over the second harmonic only PA. (b) Ideal second andthird harmonic PA normalized power � �� contours versus � �� ,� �� . (c) Ideal second and third harmonic PA normalized loadresistance � �� contours versus � �� , � �� . The fundamentalfrequency impedance � �� is purely real.

both and than and .This is a fortunate result, given that harmonic terminations be-


TABLE IHARMONICALLY TERMINATED AMPLIFIER PARAMETERS

Fig. 6. Ideal PA efficiency with fixed second harmonic versus � �� ,�� . Greater than 81% efficiency is achieved within �30.0 of theideal termination. The fundamental frequency impedance � �� is purely real.

Fig. 7. Ideal PA efficiency with fixed second harmonic versus � �� when � �� is purely real. Greater than 81% efficiency is achieved when� �� for � �� and � �� for � �� .The minimum efficiency of 71.52% occurs when � �� . Thefundamental frequency impedance � �� is purely real.

come increasingly sensitive and difficult to realize as frequencyincreases.

IV. COMPLEX FUNDAMENTAL LOAD IMPEDANCE

This section investigates optimization of the normalized fun-damental frequency reactance in order to maximize PAefficiency. Further optimization of PA efficiency is done at the

Fig. 8. Ideal PA efficiency contours versus � �� when the normalized loadreactance � �� is optimized in addition to the harmonic Fourier coefficients.The minimum efficiency and maximum efficiency are equivalent to those calcu-lated without optimizing � �� . However, the regions for which a given highefficiency is obtained are significantly expanded.

Fig. 9. Ideal PA � �� contours versus � �� when � �� is optimizedin addition to the harmonic Fourier coefficients. Power is reduced in order toimprove efficiency for much of the Smith chart. In the worst case, power isreduced by 1.63 dB relative to class-A output power.

expense of normalized output power, given (22). The efficiencyof an ideal PA restricted to only second harmonic content isagain analyzed. Unlike the analysis in Section III, the normal-ized fundamental load power will not remain constant at1 W for all second harmonic impedances. Efficiency optimiza-tion is performed as described in Section III. However, in thiscase, is an additional optimization parameter, resultingin three optimization parameters for a second harmonic only PA.

Figs. 8–10(b) show efficiency, fundamental output power,fundamental resistance, and fundamental reactance contours,respectively, as a function of second harmonic impedance when

is an additional optimization parameter. The efficiencycontours are significantly different than those shown in Fig. 2(a)and are no longer symmetric about the imaginary axis. For thecase when , an efficiency of 70.71% is obtained


Fig. 10. (a) Ideal PA � �� contours versus � �� when � �� is op-timized in addition to the harmonic Fourier coefficients. Note that the funda-mental frequency resistance differs significantly from that shown in Fig. 4(a).(b) Ideal PA � �� contours versus � �� when � �� is additionallyoptimized. Fundamental frequency reactance is used to restore PA efficiencyunder nonideal harmonic termination conditions.

regardless of . It is evident that efficiency is signifi-cantly improved for much of the Smith chart at the expense offundamental output power, where power is reduced by 1.63 dBin the worst case. When comparing Fig. 9 with Fig. 3(a), oneconcludes that the fundamental frequency reactance results inan output power reduction. It is evident that the fundamentalfrequency resistance shown in Fig. 10(a) is also impacted bytuning . Investigation of Fig. 10(b) reveals that theoptimal fundamental frequency reactance is zero for purelyreal second harmonic terminations. Given an inductive secondharmonic termination, a capacitive fundamental frequencyreactance is used to improve PA efficiency. Similarly, underthe condition of a capacitive second harmonic termination, aninductive fundamental frequency reactance is used to improvePA efficiency.

Fig. 11. Circuit diagram of ideal common source FET-based PA with incorpo-ration of a constant output capacitance � . The �-parameters of � repre-sent a transformation between the virtual drain and the measurable plane corre-sponding to the loading network � .

In practice, the improvement in efficiency by tuningsupports class-J amplifier theory [15]. Tuning has utilityin cases where the PA designer has limited or no control overthe harmonic terminations, and is forced to improve efficiencyat the expense of output power. However, in the case where idealharmonic terminations are enforceable at the virtual drain, careshould be taken to do so in order to achieve maximum outputpower for a given PA efficiency. Note that the presented analysisdoes not directly apply to a class-E PA, where the equivalentoutput capacitance at the drain is a part of the wave-shapingcircuit [16].

V. EXTENSION TO PRACTICAL PA WITH

PARASITIC OUTPUT NETWORK

The analysis and results presented in previous sectionscorrespond to an ideal PA model. In particular, the efficiencyand power contours as a function of harmonic impedance arereferenced to the impedance at the virtual drain of the transistor,rather than a measurable impedance plane. As shown in [17],the parasitics between the virtual drain and loading network notonly transform the impedance, but may significantly increasethe phase sensitivity of harmonic terminations at the virtualdrain. As a first step toward developing a more realistic model,consider the PA block diagram that includes a constant parasiticdevice output capacitance shown in Fig. 11. Thoughcan be nonlinear in some devices, it has been shown that thedrain-to-source capacitance for a GaN HEMT device isnearly constant [18]. Since the gate-to-drain capacitance issignificantly smaller than , can be considered to beapproximately linear. Another device technology having anapproximately constant is the GaAs high-voltage HBT(HVHBT) [19]. Device technologies that have a nonlinearoutput capacitance would require an analysis procedure dif-ferent from that presented in this section.

In general, the assumed constant can be replaced byany two-port -parameter model corresponding to an arbitrarylinear transformation describing the parasitics. The impedanceseen at the virtual drain is no longer equivalent to the impedancepresented by the load [i.e., ]. For the


th harmonic, the normalized impedance at the virtual drainis translated to an actual impedance given

by

(32)

where is defined in (27). Note that (27) refersto ; however, in the ideal analysis of Section II,

so the expression is applicable. Addition-ally, note that is a function of the set of harmonicterminations enforced, and in general, will differ for eachunique set of harmonic terminations. Alternatively,may be fixed at a specific value without impacting the efficiencycontours shown in Fig. 2. However, this will result in noncon-stant maximum voltage and current, along with different powercontours than those shown in Fig. 3. The reflection coefficient

at the virtual drain is calculated by

(33)

Define the two-port -parameters at the th harmonic that rep-resent the parasitic transformation between the virtual drain anda measurable reference plane as , where the character-istic impedance of the ports is . The reflection coeffi-cient looking into the two-port network is expressedas

(34)

where is the reflection coefficient at the measurementreference plane. The reflection coefficient at the mea-surable plane is calculated by rearranging (34) and is given by

(35)

where is given by

(36)

The normalized impedance at the measurable plane isthen given by

(37)

Using (35)–(37), the efficiency, fundamental power, funda-mental resistance, and fundamental reactance contours at thevirtual drain can be translated to the measurable referenceplane. This method enables a theoretical analysis of efficiencyand power sensitivity to device and packaging parasitics.

As an explicit example, consider the circuit diagram of Fig. 11where exists between the virtual drain and . The -pa-rameters of assuming the characteristic impedance of theports is are given by

(38)

(39)

Fig. 12. (a) Ideal PA efficiency contours versus � �� . (b) Ideal PA � �� contours versus � �� . (c) Ideal PA � �� contours versus � �� . Thecontour shapes shown in Figs. 2–4 are severely distorted given the transforma-tion due to the 5-pF � .

Equations (35)–(37) may now be used along with the -param-eters in order to calculate PA contours at the loading networkplane rather than the virtual drain. Under the assumptions that

GHz, , , and pF,the efficiency, power, and resistance contours are calculated asa function of and are shown in Fig. 12. Due to the dis-continuity in about the imaginary axis, as discussed in


Fig. 13. (left) Measured device and (right) HFSS model of transformation. Thetransistor die is 824 �m� 2482 �m in size.

Section III-A, the contours shown in Fig. 12 were not evaluatednear . The efficiency, power, and resistancecontours shown in Figs. 2–4 experience a significant transfor-mation due to .

This simple example illustrates a fundamental problem thePA designer must overcome in order to achieve a high-efficiencyPA design. The transformation between the virtual drain and ameasurable plane must be understood and carefully quantifiedin order to achieve a high-efficiency PA via proper harmonictermination.

VI. DISCUSSION AND CONCLUSIONS

The goal of this paper has been to develop a generalized theo-retical description of harmonically terminated PAs. Full exper-imental verification of the theory requires variation of supplyvoltage and current in addition to fundamental and harmonicload impedances at the virtual drain of the device. A qualita-tive experimental validation was performed through load–pullmeasurements on a Triquint TGF2023–10 50 W Discrete PowerGaN on SiC HEMT [20]. A fundamental frequency load–pullwas performed for six unique second harmonic loads at a fixedquiescent supply voltage and current of 28 V and 300 mA, re-spectively. The measured results are then de-embedded to thevirtual drain of the transistor by accounting for the intrinsicoutput capacitance and the extrinsic impedance transformationdue to the bond wires and fringing capacitance, as shown inFig. 13. The extrinsic transformation network was obtained byfull-wave HFSS simulations. The optimal drain efficiency andassociated output power are shown in Fig. 14.

The measured data reveals several trends predicted by thetheory.

• As the second harmonic termination is swept from a ca-pacitive to inductive, the optimal fundamental impedancesweeps from inductive to capacitive. This is in agreementwith the results shown in Fig. 10(b).

• The trend of the optimal fundamental resistance as a func-tion of second harmonic termination is in agreement withFig. 10(a).

• When the second harmonic is nearest an open circuit, thedrain efficiency is maximum with an approximately realfundamental load impedance.

Fig. 14. Results from harmonic load–pull of Triquint 50-W device on a 10-�Smith chart. The points denoted by diamonds mark the second harmonic ter-mination at the virtual drain along with resultant peak drain efficiency and as-sociated fundamental output power. The points denoted by an � mark the fun-damental load impedance resulting in peak drain efficiency for each respectivesecond harmonic termination.

• High efficiency is achieved over a range of second har-monic termination phase, as predicted in Fig. 8. For in-stance, points C and D in Fig. 14 show the same measuredpower, high efficiency, and optimal fundamental imped-ances, which are very close.

A practical experiment providing full validation of the optimaldrain efficiency, fundamental output power, and fundamentalload impedance presented in the previous sections would requirea large set of measurements, which are not always necessary forpractical design. Nevertheless, the presented experimental datashows that the theory is useful for providing design insight andpredicting trends.

In summary, this paper presents a generalized analysis of PAefficiency maximization with associated fundamental outputpower and fundamental load impedance given arbitrary har-monic terminations. To the best of the authors’ knowledge, ageneral treatment has not been previously presented in litera-ture. Efficiency, fundamental output power, and fundamentalload resistance contours as a function of second and thirdharmonic terminations are calculated in Section III given a realfundamental load impedance, and provide useful guidelines forPA design.

More importantly, the analysis (Section III) gives the designerinformation about the sensitivity of PA efficiency with respectto the magnitude and phase of the second and third harmonicload reflection coefficients. It is shown that PA efficiency is lesssensitive to both and than and

. This is important for practical PA design when theharmonic frequencies are high and the harmonic impedancespresented at the virtual drain are sensitive to parasitics. Ad-justing the fundamental frequency load reactance al-lows the PA designer to improve efficiency at a reduced outputpower given nonideal harmonic terminations. Section IV givesquantitative results for efficiency, output power, resistance, andreactance contours for a second-harmonic only PA.


In practical microwave PAs, device and package parasiticscan prevent the PA designer from presenting optimal harmonicterminations at the virtual drain. In that case, the results ofanalysis presented here quantify the fundamental output powerand load impedance tradeoffs associated with improving PAefficiency via tuning the fundamental frequency reactancepresented to the virtual drain of a given device.

The method in this paper can be extended to include anal-ysis of PA efficiency, output power, and output load given anarbitrary set of input and output harmonic terminations. InSection V, the theory was extended to include a linear transfor-mation due to and was further extended to include bondwires in the experimental validation. In general, the methodmay be applied to a PA having an arbitrary parasitic outputnetwork, which is linear and characterized by -parameters.An interesting result from the theory is the limited sensitivityof efficiency and power to harmonic termination phase, whichpoints to the possibility of extending bandwidth through use ofappropriate resonant circuits.

APPENDIX

ALTERNATE NORMALIZATION CONDITIONS

It is important to note that the normalization conditions de-fined by (11)–(16) may be redefined in many different waysif desired without impacting the maximal efficiency. However,the fundamental output power and load impedance will be im-pacted. For instance, it may be useful to normalize to a constantsupply voltage and supply current . In this case, thenormalization equations are restructured to the forms given by

V (40)

A (41)

V (42)

A (43)

where and are the redefined scaling factors given by

V (44)

A (45)

The normalized output power, fundamental load resistance, andfundamental load reactance defined in (26), (30), and (31), re-spectively, are now given by

(46)

(47)

(48)

It is evident that the calculated fundamental output power,load resistance, and load reactance will differ from those calcu-lated when normalizing to peak voltage and current. However, itis necessary to understand that this normalization scheme placesno limits on the peak voltage and current. Care should be takento determine which normalization scheme is more appropriatefor the problem at hand. The presented fundamental load power,resistance, and reactance results in this manuscript correspondto normalization to peak voltage and current.

ACKNOWLEDGMENT

The authors would like to acknowledge Dr. F. Raab, GreenMountain Radio Research, Colchester, VT, for providinguseful suggestions and a detailed review of the manuscript,and Dr. J. Hoversten, National Semiconductor Corporation,Longmont, CO, and Dr. S. Pajic, Urban RF Inc., Boulder, CO,for the useful critiques , which greatly improved this paper.

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[8] S. Cripps, “Grazing zero [microwave bytes],” IEEE Microw. Mag., vol.11, no. 7, pp. 24–34, Jul. 2010.

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Michael Roberg (S’09) received the B.S. degreein electrical engineering from Bucknell University,Lewisburg, PA, in 2003, the M.S.E.E. degree fromthe University of Pennsylvania, Philadelphia, in2006, and is currently working toward the Ph.D.degree at the University of Colorado at Boulder.

From 2003 to 2009, he was an Engineer withLockheed Martin–MS2, Moorestown, NJ, wherehe was involved with advanced phased-array radarsystems. His current research interests include mi-crowave PA theory and design, and high-efficiency

radar and communication system transmitters.

Zoya Popovic (F’02) received the Dipl.Ing. degreefrom the University of Belgrade, Serbia, Yugoslavia,in 1985, and the Ph.D. degree from the California In-stitute of Technology, Pasadena, in 1990.

Since 1990, she has been with the Universityof Colorado at Boulder, where she is currently theHudson Moore Jr. Chaired Professor of Electricaland Computer Engineering. In 2001, she was aVisiting Professor with the Technical Universityof Munich. Her research interests include high-ef-ficiency, low-noise, and broadband microwave

and millimeter-wave circuits, quasi-optical millimeter-wave techniques forimaging, smart and multibeam antenna arrays, intelligent RF front ends, RFoptics, and wireless powering for batteryless sensors.

Dr. Popovic is a foreign member of the Serbian Academy of Sciences andArts. She was the recipient of the 1993 and 2006 Microwave Prizes presentedby the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) forbest journal paper. She was the recipient of the 1996 URSI Issac Koga GoldMedal, and the Humboldt Research Award for Senior U.S. Scientists from theGerman Alexander von Humboldt Stiftung. She was also the recipient of the2001 HP/ASEE Terman Medal for combined teaching and research excellence.

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