TE (AS) mode. Since this frequency is of some importance, it istabulated in Table 2. When b � 1 the frequency checks the firstroot of J1��k� and when b approaches zero, the value of 2.150 isextrapolated.

The TM modes are also important in the vibration of mem-branes. The lowest TM frequency (fundamental frequency) corre-sponds to TM (SS) and is tabulated in Table 3. When b � 0 it isthe first root of J0�k�. When b approaches zero, it is infinite. In fact,the frequencies of all TM modes (and all TE (SA) modes) becomeinfinite when the thickness b approaches zero.

It is seen that for b close to one (near circle) numerous modescan be excited. However, as b becomes smaller (narrower shape),the modes become more separated. For an aspect ratio of 2:1 (b �0.5), there are seven modes for k below 6. The field lines are shown

in Figure 2. If the waveguide is made narrower, say b � 0.2 (5:1ratio), there are only three widely separated modes for k below 6.5(see Fig. 3). There is no improvement for even narrowerwaveguides.

In conclusion, we find the Ritz method is efficient and accuratefor this problem. Our results should be useful in the design ofwaveguides.

REFERENCES

1. J.A. Edminster, Shaum’s outline of electromagnetics, McGraw-Hill,New York, 1979.

2. R.F. Harrington, Time harmonic electromagnetic fields, Wiley, NewYork, 2001.

3. F.L. Ng, Tabulation of methods for the numerical solution of the hollowwaveguide problem. IEEE Trans Microwave Theory Tech 22 (1974),322–329.

4. C.Y. Wang, Frequencies of a truncated circular waveguide-method ofinternal matching. IEEE Trans Microwave Theory Tech 48 (2000),1763–1765.

5. S.L. Lin, L.W. Li, T.S. Yeo, and M.S. Leong, Cutoff wavenumbers intruncated waveguides. IEEE Microwave Wireless Comp Lett 11 (2001),214–216.

6. R. Weinstock, Calculus of variations, McGraw-Hill, New York, 1952.

© 2007 Wiley Periodicals, Inc.

ANALYSIS AND DESIGN OF CLASS EPOWER AMPLIFIER WITH NONLINEARPARASITIC CAPACITANCE AT ANYDUTY RATIO

Hao Zhang,1 Xikui Ma,2 Siu Chung Wong,3 and Chi K. Tse3

1 State Key Laboratory of Electrical Insulation & Power Equipment,Xi’an Jiaotong University, Xi’an 710049, China2 School of Electrical Engineering, Xi’an Jiaotong University, Xi’an710049, China3 Department of Electronics & Information Engineering, Hong KongPolytechnic University, Hong Kong, China

Received 27 August 2006

ABSTRACT: The nominal operating condition of Class E power ampli-fiers with parallel-connected nonlinear parasitic capacitance and linearcapacitance at any duty ratio has been studied in this article. Basic de-sign equations are derived analytically. Using these equations, the com-ponent values for ensuring nominal operation at any given duty ratiocan be conveniently found. Furthermore, the design equations highlightseveral important practical constraints, which help develop effectivedesigns for improving the performance of practical Class E amplifiers.Finally, a design example is provided for verification. Both PSPICEsimulation and experimental measurements are presented. © 2007 WileyPeriodicals, Inc. Microwave Opt Technol Lett 49: 920–923, 2007;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.22312

Key words: Class E power amplifier; nonlinear parasitic capacitance;circuit design

1. INTRODUCTION

Because of its high efficiency and compactness, the Class Eamplifier [1] operating at high frequency has gained widespreadacceptance in various practical applications such as mobile andportable communications equipment, fluorescent lamp ballasts,induction and dielectric heating, induction generation of plasma,lasers, glass, and lens coating, portable light sources, DC/DC

Figure 2 The first seven modes for b � 0.5. Only 1⁄4 of the waveguideis shown. Heavy lines are when w � 0. A: k � 2.051 (TE), B: k � 3.604(TE), C: k � 3.610 (TE), D: k � 3.934 (TM), E: k � 4.993 (TE), F: k �5.119 (TE), G: k � 5.387 (TM)

Figure 3 The first three modes for b � 0.2. H: k � 2.132 (TE), I: k �3.758 (TE), J: k � 5.347 (TE)

920 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 4, April 2007 DOI 10.1002/mop

converters and medical implanted systems, frequency multipliers,etc. This circuit has been extensively analyzed [2, 3] with theassumption that the shunt capacitance is linear, i.e., constant. Inpractice, however, the parasitics of the switching device maybecome dominant as a shunt capacitance at high operating fre-quencies. In the extreme case, the shunt capacitance is entirelyformed by the MOSFET output capacitance. Since the nonlinearparasitic capacitance of the switching device varies with the drain-to-source voltage, the operation condition of an actual Class Eamplifier can be very different from that designed assuming alinear shunt capacitance. Thus, the design of the Class E poweramplifier with nonlinear parasitic capacitance has become one ofthe most interesting subjects of on-going research in this field.

For the duty ratio of 0.5, the analysis and design of the Class Eamplifier with nonlinear parasitic capacitance have been per-formed in the past few years [4–7]. Chudobiak [4] derived someanalytical equations, which can be applied to the design of a ClassE amplifier using a power MOSFET. Alinikula et al. [5] obtaineddesign equations in closed-form for grading coefficient m � 0.5and the design equations were solved numerically otherwise. Re-cently, Suetsugu and Kazimierczuk [6, 7] presented analysis anddesign procedures for the Class E power amplifier with a shuntcapacitance composed of both a transistor nonlinear output capac-itance and a linear external capacitance. However, all these meth-ods [4–7] are restricted for duty ratio being 0.5, which may notsatisfy performance specifications in some practical applications.Therefore, the purpose of this article is to present an analyticalanalysis of the Class E amplifier at any duty ratio D.

In this article, we will consider the Class E power amplifierwith nonlinear parasitic capacitance under nominal operation.Some basic equations are derived to analyze its performance. Thenominal conditions are then substituted into these equations. Ex-pressions of the component values are obtained in terms of thespecifications of the amplifier for designing the circuit undernominal operation. Finally, PSPICE simulation and experimentalresults show an excellent agreement between the observed circuitperformances and the theoretical predictions.

2. CIRCUIT ANALYSIS

The basic circuit of the Class E amplifier and its equivalent circuitare illustrated in Figures 1(a) and 1(b), respectively. The shuntcapacitance of the amplifier consists of the MOSFET output ca-pacitance Ci and an external linear capacitance Ce. Because thepower MOSFET contains a p-n junction body diode, the parasiticcapacitance between the drain and source Ci can be expressed forgrading coefficient m � 0.5 as [4]

Ci �Cj0

�1 �vs

Vbi

(1)

where Vbi is the built-in potential typically ranges from 0.5 to 0.9V, vs is the drain-to-source voltage and Cj0 is the capacitance at vs

� 0. The analysis below is based on the equivalent circuit of theamplifier shown in Figure 1(b). The following assumptions aremade

1. The RF choke LRFC allows only a constant dc input currentICC and has no series resistance.

2. The loaded quality factor QL of the output resonant circuit ishigh enough such that the output current can be consideredas a sine wave, i.e., iR��� � Im sin�, where Im is theamplitude of the output current and � � �t.

3. The MOSFET has zero saturation voltage, zero saturationresistance and infinite off resistance. The switching action isinstantaneous and lossless (except when discharging theshunt capacitance).

4. There are no loss in the circuit except in the load R.The circuit operation can be described in terms of the twoswitched states, namely, ON state and OFF state correspondingto the switch being on and off, respectively.

2.1. ON StateFor �1 � � � �2 � �1 � 2�D, the switch is turned on and thedrain-to-source voltage is given by vs��) � 0, the currents throughthe two shunt capacitances are expressed as

ic��� � ��Ci � Ce�dvs

d�� 0, (2)

is(�) � ICC � iR(�) � ICC � Im sin�. (3)

2.2. OFF StateFor �2 � � � 2� � �1, the switch is turned off and the currentthrough the MOSFET is equal to zero, i.e., is��� � 0. Hence, thetotal current through the two shunt capacitances is

iC(�) � ICC � iR(�) � ICC � Im sin� (4)

Knowing the shunt capacitances current, one can then calculate thedrain-to-source voltage using

iC(�) � ��Ci � Ce�dvs

d�(5)

Integrating both sides of Eq. (5) with respect to � gives a relation-ship between the switch voltage and the shunt capacitance current,i.e.,

Figure 1 Class E power amplifier: (a) basic circuit; (b) equivalent circuit

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 4, April 2007 921

��0

vs

�Cj0

�1 �vs

Vbi

� Ce�dvs � ��1

�2

iC(�) d� (6)

Substituting Eq. (4) into Eq. (6) and integrating both sides of Eq.(6) yield a quadratic equation in vs, i.e.,

�1 �vs

Vbi�

1

2�Vbivs � g��� (7)

whereg��� � 1 � ICC�� � �2) � Im(cos� � cos�2)/2�Cj0Vbi and �

� Cj0/Ce.Solving this quadratic equation and rejecting the solution that

does not satisfy the initial condition vs(�2) � 0, we get

vs � 2�Vbi[g��� � � � ��2 � 2�g��� � 1] (8)

3. OPTIMUM OPERATION

To obtain the optimum performance of a Class E amplifier, twonominal conditions [1, 2] should be substituted into the equationsderived. These two nominal conditions are

1. vs (2� � �1) � 0, i.e.,

ICC(2� � �1 � �2) � Im(cos�1 � cos�2) � 0 (9)

2.dvs

d��

� � 2���1

� 0, i.e.,

is�2� � �1) � 0 (10)

From the above two nominal conditions, we can obtain

�1 � �tan�1� 1 � cos2�D

2�(1 � D) � sin2�D� (11)

and

ICC � �Imsin�1 (12)

Since we assume no loss, the input and output powers are equal,i.e.,

ICCVCC �1

2Im

2 R (13)

Now, putting Eq. (12) into Eq. (13), one obtains the amplitude ofthe current through the series resonant circuit Im, the amplitude ofthe output voltage Vm, the DC supply current ICC, and the outputpower Po as

Im �2VCC

Rsin�1 (14)

Vm � 2VCCsin�1 (15)

ICC �2VCC

Rsin2�2 (16)

Po �2VCC

2

Rsin2�1 (17)

The fundamental of vs(�) applied to the switch terminals consistsof two quadrature components whose amplitudes can be foundusing the Fourier formulas as

�1

2�(1 � D) ��2

2���1

vs(�)sin� d� � ImR, (18)

�1

2�(1 � D) ��2

2���1

vs(�)cos� d� � XIm. (19)

TABLE 1 The Theoretical and Experimental Values of CircuitComponents

Circuit ParametersTheoretical

ValuesExperimental

Values

VCC 15 V 15 VR 50 � 51.2 �LRFC 100 H 120 HCe 25 F 22 FC 137 pF 140 pFL 13.9 H 14.2 H

Figure 2 Simulation waveforms of switch voltage vs and driver signal. [Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]

922 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 4, April 2007 DOI 10.1002/mop

Note that Eq. (18) and Eq. (19) do not have analytical solutions.The external capacitance Ce and the series reactance X can beobtained by means of numerical integration of the two equations.After X is determined, we can calculate the corresponding induc-tance or capacitance, using

L �RQL

�(20)

where QL is the load quality factor of the resonant circuit.If X is inductive, C is determined by

C �1

�2(L � L�0)(21)

where L�0 is the corresponding inductance of X. Moreover, if X iscapacitive, C is determined by

C � C�0 �1

�2L(22)

where C�0 is the corresponding capacitance of X.The above design equations allow very efficient computation of

the required parameter values for nominal operating conditions.

4. DEVICE STRESS

The switch peak current ISM can be obtained as a sum of the dcsupply current I0 and the output current amplitude IR. Using Eq.(12) and Eq. (14), we get

ISM � ICC � Im �2VCC sin�1

R(sin�1 � 1). (23)

The switch peak voltage VSM occurs when the shunt capacitancecurrent iC is zero. The switch peak voltage VSM is equal to theswitch voltage at � � � � �1, i.e.,

VSM � vs(� � �1) (24)

Equations (23) and (24) provide quick insights into the stressproblem of the Class E amplifier.

5. SIMULATION AND EXPERIMENTAL RESULTS

To verify the analytical results given in the previous sections, aClass E amplifier operating at 4 MHz and duty ratio 0.5 (themethod works equally well for other duty ratio) is designed usingthe afore-developed method. After the specifications are set, therequired component values can be calculated directly. The circuitparameters of the amplifier and the calculated component valuesare shown in column 2 of Table 1. The waveform of the switchvoltage vs is obtained through PSPICE simulation, as shown inFigure 2. From Figure 2, we can observe that the Class E amplifierworks in nominal condition.

The 4 MHz Class E amplifier was implemented based on thetheoretical component values and a power MOSFET IRF510 wasused as the active device. The experimental component values areshown in column 3 of Table 1. Figure 3 depicts the experimentalwaveform of the switching drain-to-source voltage and driversignal. It has been found that the predicted values of the amplifierand those obtained experimentally are in perfect agreement.

6. CONCLUSIONS

A theoretical analysis of the Class E power amplifier with nonlin-ear parasitic capacitance and linear capacitance operation has beenpresented for any duty ratio. Design formulas have been derived toallow the designer to select the appropriate parameter for achiev-ing nominal operating conditions. Moreover, as the formulas are inclosed form, the required circuit component values can be calcu-lated directly from the equations. In addition, quick insights can beobtained from the closed-form expressions about the operation ofthe circuit.

ACKNOWLEDGMENT

This work was supported in part by Hong Kong PolytechnicUniversity Project A-P A2R, Natural Science Foundation of China(Grant No. 50577047), and Specialized Research Fund for theDoctoral Program of Higher Education (Grant No. 20050698004).

REFERENCES

1. N.O. Sokal, and A.D. Sokal, Class E—A new class of high-efficiencytuned single-ended switching power amplifiers, IEEE J Solid stateCircuits SC-10 (1975), 168–176.

2. F.H. Raab, Idealized operation of the Class E tuned power amplifier,IEEE Trans Circ Syst CAS-24 (1977), 725–735.

3. C.H. Li, and Y.O. Yam, Analysis and design of the Class E amplifierwith nonzero on resistance, Microwave Opt Technol Lett 7 (1994),337–341.

4. M.J. Chudobiak, The use of parasitic nonlinear capacitors in Class Eamplifiers, IEEE Trans Circ Syst I 41 (1994), 941–945.

5. P. Alinikula, K. Choi, and S.L. Long, Design of class E power amplifierwith nonlinear parasitic output capacitance, IEEE Trans Circ Syst II 46(1999), 114–119.

6. T. Suetsugu and M.K. Kazimierczuk, Comparison of class-E amplifierwith nonlinear and linear shunt capacitance, IEEE Trans Circ Syst I 50(2003), 1089–1097.

7. T. Suetsugu and M.K. Kazimierczuk, Analysis and design of class Eamplifier with shunt capacitance composed of nonlinear and linear,IEEE Trans Circ Syst I 51(2004), 1261–1267.

© 2007 Wiley Periodicals, Inc.

Figure 3 Experimental waveforms of switch voltage vs and driver signal.[Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 4, April 2007 923