42 Facta Universitatis ser.: Elect. and Energ. vol. 12, No.1 è1999è Then, inæuence of the choke inductor value on the frequency response of the output

Transcription

1 FACTA UNIVERSITATIS ènisè Series: Electronics and Energetics vol. 12, No.1 è1999è, UDC CHOKE INDUCTOR VALUE INFLUENCE ON THE CHARACTERISTICS OF THE CLASS E POWER AMPLIFIER Marina Paunovic, Dragisa Krstic and Goran Jovanovic Abstract. In this paper the class E power ampliæer is described and its analysis for the ænite choke inductor value is given. The optimum values of the circuit elements for minimizing transistor losses are calculated for several values of the choke inductor. Then, the frequency analysis of the optimal circuit is performed and computer results from analysis are presented. The class E power ampliæer can be used as an amplitude modulator, so the analysis is performed and the amplitude modulation characteristics are also presented. 1. Introduction The class E power ampliæer was introduced in 1975 by Sokal and Sokal ë1ë. They gave the basic circuit and explained the operating principle. A speciæc but simple circuit conæguration enables low power losses, so this class of ampliæers is characterized with very high eæciency. In an ideal case, it is 100 è, but in practical applications it is up to 96 è. In the past 20 years, a great number of papers dealing with the class E power ampliæers were published. M. Kazimierczuk ë2ë derived the analysis of the class E power ampliæer with the inænite choke inductor and the ænite collector current fall time. C.-H. Li and Y.-O. Yam ë3ë proposed the analytical method for the component values evaluation, when the choke inductor value is ænite and the collector current fall time is zero. It can be shown that the analysis given by Kazimierczuk for the zero collector current fall time is a special limiting case of the analysis given by Li and Yam ë3ë, when the choke inductor value is inænite. Manuscript received Jun 18, Authors are with School of Electronics Engineering, Beogradska 14, 1800 Nis, Serbia. s: dkrstic@elfak.ni.ac.yu and jog@elfak.ni.ac.yu. 41

2 42 Facta Universitatis ser.: Elect. and Energ. vol. 12, No.1 è1999è Then, inæuence of the choke inductor value on the frequency response of the output voltage amplitude, input power, output power and eæciency was examined. The frequency analysis of the optimal class E power ampliæer was performed and the results of these analysis are also presented in this article. The class E power ampliæer can be used as the collector amplitude modulator, when the low frequency èlf è modulating signal is added to the supply voltage ë4ë,ë5ë. The analysis of this circuit conæguration was performed and the inæuence of the choke inductor value on the static modulation characteristics and on the output signal modulation coeæcient was shown. 2. Circuit description The basic circuit of the class E power ampliæer is shown in Fig. 1. and its equivalent circuit in Fig. 2. It consists of an active element and a load network. The active element, transistor, is considered to be an ideal switch, driven by a square wave train, which switches the transistor between its "on" and "oæ" states. The simplest load network consists of a series-resonant circuit L r, C f and a capacitor C ex shunting the transistor. The series resonant circuit for the optimum operation would be mistuned - it would be inductive. Therefore, the inductance L r can be divided into two parts, L r = L f + L, so L f and C f form an ideal resonant circuit with the resonant frequency that is equal to the operating frequency, i.e.,! = 1= p L f C f. X =!L is the Fig. 1. Class E power ampliæer èbasic circuitè.

3 M. Paunovic et al: Choke inductor value inæuence excessive reactance. Capacitor C ex and the output transistor capacitance C ob are represented by the equivalent capacitance C that ælters higher harmonic components of the output current. Fig. 2. Class E power ampliæer èequivalent circuitè. The collector of the transistor is connected to the supply voltage V cc by the choke inductor L rf. 3. Assumptions In order to simplify the circuit analysis, the following assumptions are introduced: 1. The transistor has zero "on"íresistance, zero "on"ívoltage and inænite "oæ"-resistance. 2. The transistor has zero delay time. 3. The output transistor capacitance C ob is independent from the collectorítoíemitter voltage v ce. 4. The series-resonant circuit L r, C f has Q,factor èq =!L r =Rè, which is high enough in order to the output current and the output voltage to be sinusoidal. 5. The driving signal has pulseítoípause ratio 1 : 1. Real transistors do not satisfy these assumptions, that cause power losses in the circuit and contribute to the reduction of the eæciency.

4 44 Facta Universitatis ser.: Elect. and Energ. vol. 12, No.1 è1999è 4. Circuit analysis Based on assumption 4, the output current i o ètè and the output voltage v o ètè can be expressed as i o ètè =I o sinè!t + è v o ètè =I o R sinè!t + è; where I o is the output current amplitude,! is the operating angular frequency and is the initial phase shift. The voltage v 1 ètè isintroduced in order to simplify the analysis and it is sinusoidal too v 1 ètè =V 1 sinè!t + 1 è; è2è where V 1 =I o R r 1+ X2 e R 1 = + arctan X e R Xe =!L r, 1!C f The transistor in the class E power ampliæer is either on or oæ, so the currents æowing through the choke inductor i L ètè, the capacitance C, i p ètè, the transistor's collector i c ètè and the collector voltage v c ètè, depend on the state of the transistor. The current æowing through the choke inductor L rf i L ètè =i c ètè+i p ètè+i o ètè: 4.1 Analysis of the class E power ampliæer with the ænite choke inductor value Li and Yam ë3ë started with the assumption that the choke inductor value is ænite, so that, the current i L ètè is timeídependent and the next equation is valid di L ètè L rf = V cc, v c ètè è3è dt The current i L ètè and the voltage v c ètè are solutions of the following system of diæerential equations 8 é é: L rf di L ètè dt = V cc, v c ètè; "on" state L rf C d2 i L ètè dt 2 + i L ètè =I o sinè!t + è; "oæ" state is è1è è4è

5 M. Paunovic et al: Choke inductor value inæuence and it is: 8 V cc t, + D; "on" state L rf! é i L ètè = é: A cos! o t + B sin! o t + I è5è o 1, æ sinè!t + è; "oæ" state 2 where p! o =1= L rf C is the resonant frequency for L rf and C, æ =!=! o and A, B, D are the unknown integration constants. The current through the capacitance C, i p ètè, can be written as i p ètè =C dv cètè : è6è dt The collector current, i c ètè, can be expressed as follows il ètè, i o ètè; "on" state icètè = 0; "oæ" state è7è The collector voltage, v c ètè, can be expressed with the equations: 8 0; "on" state é æ V v c ètè = cc, L rf, A! o sin! o t + B! o t cos! o t é: + I o! 1, æ cosè!t + èæ ; "oæ" state 2 è8è At the transition between "on" and "oæ" states, the current through the choke inductor would be continuous. This gives the boundary condition I i L ètèæ i L ètèæ t= 2! t= 2! =i L ètèæ t=0 è9è =i L ètèæ t=! + The collector voltage is also the voltage across the shunt capacitance C, and it is also continuous based on the capacitance characteristic. That gives the boundary condition II v c è0è = 0: è10è

6 46 Facta Universitatis ser.: Elect. and Energ. vol. 12, No.1 è1999è Based on the boundary conditions I and II the system equations are obtained, and their solutions are the unknown integration constants A, B and D 1 A = 1, cos æ Vcc I 0æ cos, sin 2I L rf! 0 1, æ 2 æ, 0 sin 1, æ + V cc 2 L rf æ B = V cc, I oæ cos L rf! o 1, æ 2 D =i L 2! : è11è The power losses, caused by the transistor switching between the "on" and the "oæ" state, can be minimized by proper choice of the circuit elements. The optimum waveforms of the collector voltage can be obtained if it fulælls the optimum conditions v c ètèæ =0 dv c ètè æ dt t= 2! æ t= 2! =0: è12è As a result of the previous conditions, the output current amplitude I o and phase shift can be expressed as V cc 1, cos æ + 2æ sin è1, æ 2 è æ I o = L tf! o sin sin æ = arccot "!æ! o cos æ +! sin æ 1, cos æ + 2æ sin æ The input power P i is the power supply! P i =V cc I dc = V cc 2 Z 2=! h Aæ =V cc 2 sin! + Bæ 2 0 i L ètèdt 2, æ cot æ 1, cos æ, æ sin æ 1, cos + I o cos! è1, æ 2 è + V cc è : è13è 4L rf!, I o sin i 2 è14è

7 M. Paunovic et al: Choke inductor value inæuence The output power P o is the power dissipated on the resistance R, and can be expressed as P o = I 2 R o 2 : è15è The eæciency is = P o P i æ 100: è16è For maximum eæciency, = 100 è, the output power would be equal to the input power, so the following equation is obtained I 2 o R 2 h Aæ = V cc 2 sin! + Bæ 1,cos + I o cos 2! è1, æ 2 è + V cc 4L rf!, I o sin i 2 è17è This equation is nonlinear in variable L rf, so, when the capacitance C is æxed, it has to be solved numerically. The other circuit elements values can be calculated by using the previously mentioned equations and conditions Analysis of the class E power ampliæer with the inænite choke inductor value Marian Kazimierczuk ë2ë started from the assumption that the choke inductor value is inænite, i.e. the current through it is constant, i L ètè = I dc. The analysis is similar but simpler than the analysis 4.1, so the circuit elements values can be expressed in the explicit form R = C = 1!R V 2 cc P out 8 è 2 +4è tan = 2, L = R è 2 +5è! 16 L r = QR! : 4.3. Frequency analysis of the optimal class E power ampliæer è18è For the frequency analysis of the optimal class E power ampliæer the optimum conditions è12è are not valid, as well as their solutions è13è. The

8 48 Facta Universitatis ser.: Elect. and Energ. vol. 12, No.1 è1999è output current amplitude I o and the phase shift can be calculated from the condition that the ærst harmonic of collector's voltage v c ètè is equal to the voltage on the load network v 1 ètè V 1 =! 0=! Z 2=! 0 Z 2=! 0 v c ètè sinè!t + 1 èdt v c ètè cosè!t + 1 èdt: è19è Now, when I o and are known and the circuit elements have been calculated èas previouslyè, the behavior of the: output voltage amplitude, input power, output power and eæciency with the variation of the driving signal frequency can be observed. 5. Results Based on the analysis 4.1, by using program package Mathematica, a program was developed that calculates the optimal values of the circuit elements and the parameters of the output current, for æxed parameter values: output power P o, driving signal frequency f, Q,factor of the series-resonant circuit for some values of æ. The results are shown in Table 1. Table 1: Optimal values of the circuit elements: V cc =16V,P o =6W,f = 1 MHz and Q =5. æ L rf ëhë CënFë Rëæë L r ëhë C f ënfë I o ëaë ëradë 1: :9527 1:35 32: :9192 1: :606981,0: :7 57:8447 1: : :4171 1: :652674,0: : : :7974 1: :677612,0: :5 641:5440 1: : :8225 1: :694074,0: : :550 1: : :6020 1: :697969,0: Based on equations è18è and for the same values of the P o, f, Q as in the previous analysis, the following circuit elements values were obtained: R = 24:6102 æ, C = 1:18736 nf, =,0: rad, L = 11:4357 H, L r =19:5842 H. It can be noticed that these values are practically equal to the corresponding values in Table 1 for æ = 20. Then, the results of the circuit frequency analysis are presented. The analysis is based on the equations from subsection 4.1, and was performed by the program package Mathematica. For every set of values from Table 1, a frequency analysis was performed in the frequency band æf = æ0:5 MHz

9 M. Paunovic et al: Choke inductor value inæuence around the frequency f = 1 MHz, for which the circuit is designed. In Fig. 3-6 the frequency response of the output voltage amplitude V o, input power P i, output power P o and eæciency are shown. Based on these Figures, the inæuence of the and choke inductor value and the frequency mistuning on the operation of the class E power ampliæer can be observed. Fig. 3. Output voltage amplitude V o as a function of frequency. Fig. 4. Input power P i as a function of frequency. The class E power ampliæer can be used as a collector amplitude modulator when the LF modulating signal is added to the supply voltage. The analysis is performed for four sets of the circuit elements from Table I. The

10 50 Facta Universitatis ser.: Elect. and Energ. vol. 12, No.1 è1999è obtained static modulation characteristics are shown in Fig. 7. It can be observed that these characteristics are linear and that they depend on the choke inductor value. Fig. 5. Output power P o as a function of frequency. Fig. 6. Eæciency as a function of frequency. The simulation of this circuit, for the circuit elements values given in Table 1, were carried out in program package PSpice. The transistor model BSX61 was used in the simulation. The waveforms obtained of the amplitude modulated output voltage v o ètè, driving signal v d ètè and the modulating signal v n ètè = 5 sinè210 5 è V for æ =1:22056, are given in the Fig. 8.

11 M. Paunovic et al: Choke inductor value inæuence Fig. 7. The static modulation characteristics. Fig. 8. The waveforms of voltages: v o ètè, v n ètè and v d ètè obtained by simulation in PSpice. Spectral analysis of the amplitude modulated output voltage gave the results shown in the Fig. 9. It can be concluded that modulation coeæcient reduces when the parameter æ èi.e. choke inductor valueè rises. The choke inductor with the reactive elements of the load network forms a selective

12 52 Facta Universitatis ser.: Elect. and Energ. vol. 12, No.1 è1999è circuit that ælters the side band components of the amplitude modulated signal. When æ 1 the driving signal frequency is almost equal to the resonant frequency of the circuit, so the side band components are also ampliæed and the modulation coeæcient is high. When æ rises these two frequencies are drifting apart, the selective circuit reduces the side band components and the modulation coeæcient reduces. Fig. 9. Spectral analysis of the amplitude modulated output signals. 6. Conclusions For the appropriate value of æ the optimum values of the circuit elements, calculated with ænite choke inductor value L rf, are approximately equal to the corresponding values calculated by the analysis 4.2. So, it is shown that the analysis given by Kazimierczuk ë2ë for the zero collector current with inænite L rf fall time is a special limiting case of the analysis given by Li and Yam ë3ë, when the choke inductor value is inænite. Therefore, in the circuit design where æ 20èL rf é 8:5mHè the equations è18è cccould be used. From the frequency responses, obtained by the frequency analysis of the optimal class E power ampliæer, it can be concluded that the choke inductor value does not inæuence in the great deal the output voltage amplitude, input power, output power and eæciency. The inæuence of the frequency mistuning on the operation of the class E power ampliæer can be also observed. By the analysis of the optimal class E power ampliæer as the amplitude modulator, it is shown that the static modulation characteristics are linear and they depend on the choke inductor value. From the spectral

13 M. Paunovic et al: Choke inductor value inæuence analysis of the amplitude modulated output signal it can be concluded that modulation coeæcient depends on the choke inductor value in an inverse proportion. Acknowledgements The authors would like to acknowledge the ænancial support of the Ministry of Science and Technology of the Republic of Serbia. The authors also wish to thank an unknown referee for his helpful suggestions in improving the paper. REFERENCES 1. Sokal, A.D. Sokal: Class E - a New Class of High Eæciency Tuned Single-ended Switching Power Ampliæer,. IEEE Journal of Solid-State Circuit, Vol. SCí10, pp.168í176, M. Kazimierczuk: Eæects of the Collector Current Fall-time on the Class E Tuned Power Ampliæer,. IEEE Journal of Solid-State Circuits, Vol. SCí18, pp.181í193, April C.-H. Li, Y.-O. Yam: Maximum Frequency and Optimum Performance of Class E Power Ampliæers., IEE Proc.- Circuits Devices Systems, Vol. 141, No. 3, June M. Kazimierczuk: Collector Amplitude Modulation of the Class E Tuned Power Ampliæer,. IEEE Trans., CASí31, pp.543í549, M. Albulet, S. Radu: Second Order Eæects in Collector Amplitude Modulation of the Class E Power Ampliæer., AEU, Vol. 49, No. 1, pp. 44í49, 1995.