【免费】Class-E类功率放大器的波形推导原始论文

功率放大器

需积分: 0 48 浏览量 2023-02-25 10:28:31 上传评论收藏 1008KB PDF 举报

资源推荐

资源详情

资源评论

IEEE TRANSAClTONS ON CIRCUITS AND SYSTEMS, VOL.

CAS-24,

NO.

12,

DECEMBER

1977

725

Idealized Operation of the Class E Tuned

Power Amplifier

FREDERICK H. RAAB,

MEMBER, IEEE

Absfracr-The class E tuned power amplifier consists of a load network

and a single transistor that is operated as a switch at tbe carrier frequency

of tbe output signal. ‘zbe most simple type of load network consists of a

capacitor shunting tbe transistor and a series-tuned output circuit, which

may bave a residual reactance. Circuit operation is determined by tbe

transistor when it is on, and by the transient response of the load network

when tbe transistor is off. The basic equations governing amplifier opera-

tion are derived using Fourier series techniques and a high-Q assumption.

These equations are then used to determine component values for opti-

mum operation at an efficiency of 100 percent. Other combinations of

component values and duty cycles which result in lOO-percent efficiency

are also determined. ‘fbe barmouic structure of tbe collector voltage

waveform is analyzed and related amplifier configurations are discussed.

While tbis analysis is directed toward the design of bigbefficiency power

amplifiers, it also provides insight into tbe operation of modern solid-state

VHF-UHF tuned power amplifiers.

INTRODUCTION

HE “class E” concept recently introduced by the

Sokals [l], [2] offers a new means of highly efficient

power amplification. This paper expands upon the Sokals’

work by providing an analytical basis for class E opera-

tion and by deriving additional amplifier configurations.

Before entering the technical discussion, some definitions

must be clarified.

As used here, “class E” refers to a tuned power ampli-

fier composed of a single-pole switch and a load network.

The switch consists of a transistor’ or combination of

transistors and diodes that are driven ‘on and off at the

carrier frequency of the signal to be amplified. In its most

basic form, analyzed here, the load network consists of a

resonant circuit iii series with the load, and a capacitor

which shunts the switch (Fig. l(a)). Note that the total

shunt capacitance is due to what is inherent in the transis-

tor (Cl) and added by the load network (C2). The collec-

tor voltage waveform is then determined by the switch

when it is on, and by the transient response of the load

network when the switch is off.

Class E amplification is easily differentiated from other

classes of power amplification. Classes A, B, and C refer

Manuscript received October 30, 1975; revised May 1976, January

1977, and July 1977. This paper represents a considerable expansion

upon work supported under U.S. Army Contract DAAB07-71-0220 with

Cincinnati Electronics Corporation [6].

The author is with Polhemus Navigation Sciences, Inc., A Subsidiary

of the Austin Company, Burlington, VT 05401.

‘Since bipolar transistors represented the state of the art when this

paper was written,

“transistor” is often used here in place of “active

device.” The theory is generally applicable to vacuum tubes, field-effect

transistors, and other devices by substitution of analogous terms (e.g.,

“drain current” for “collector current.“)

(b)

Fig. 1. Class E amplifier. (a) Basic circuit. (b) Equivalent circuit.

to amplifiers in which the transistors act as current

sources; sinusoidal collector voltages are maintained by

the parallel-tuned output circuit. If the transistors are

driven hard enough to saturate, they cease to be current

sources; however, the sinusoidal collector voltage remains.

Classes D and S are characterized by two (or more) pole

switching configurations that define either a voltage or

current waveform without regard for the load network.

Class D employs bandpass filtering, while class S employs

low-pass filtering2 (with respect to the carrier or switching

frequency). This author has used class F to designate

multiple-resonator power amplifiers in which the transis-

tor acts as a (possibly saturating) current source, and a

collector voltage waveform consisting of a sum of

sinusoids is maintained by the output circuit. Further

discussion and comparisons are given elsewhere [l], [3]

and in Table I.

The definition of class E operation used by the Sokals

[l], [2] allowed a variety of circuit topologies containing a

switch and a load network, and stated three specific

objectives for the collector voltage and current wave-

forms: a) the rise of the voltage across the transistor at

turn-off should be delayed until after the transistor is off,

b) the collector voltage should be brought back to zero at

*Some authors use “class D” and “class s” interchangeably, while

others reverse the author’s definitions.

Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on June 22,2022 at 06:00:57 UTC from IEEE Xplore. Restrictions apply.

726

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-24, NO. 12. DECEMBER 1977

TABLE I

COMPARISON OF DIFFERENT AMPLIFIERS PRODUCING SINEWAVE

OUTFUT

two devices

nfinite nu:?ber of

!$ote :

411 cases assume ideal transistors: zero saturation

v01tag4,

zero saturation resistance. infinite off

resistance. and zero switching

time.

the time of transistor turn-on, and c) the slope of the

collector voltage should be zero at the time of turn-on.

This paper broadens the definition of class E as follows. 1)

An amplifier containing a switch and a load network and

meeting criteria a), b), and c) is called “optimum class E.”

An amplifier containing a switch and a load network, but

not meeting these three criteria is called “suboptimum

class E.” This definition allows an amplifier which is

mistuned or not optimized [4] to be described as class E.

(For operation with real transistors, their nonzero satura-

tion voltage replaces the zero voltage in objective b).).

All class E power amplifiers (as well as class D and

saturating class C power amplifiers) might more ap-

propriately be called power converters. In these circuits,

the driving signal causes switching of the transistor, but

there is no relationship between the amplitudes of the

driving signal and the output signal. Nonetheless, these

circuits are colloquially referred to as “power amplifiers,”

and it is possible to define a power gain and to note that

the frequency and phase of the output signal track those

of the driving signal. The colloquial term “power ampli-

fier” is more readily recognized in radio-frequency ap-

plications, and is therefore used in this paper.

11:

BASIC

EQUATIONS

The characteristics of a class .E power amplifier can be

determined by finding its steady-state waveforms. The

least complex configuration includes a single transistor

switch, a shunt capacitor, a series-tuned output circuit,

and a RF choke (Fig. 2). A driving waveform capable of

producing the desired switching action is assumed. This

driving waveform includes the frequency and phase mod-

% (;;p

is@Joy+

b i Zn

Fig. 2.

Waveforms in class E amplifier.

ulation desired in the output signal, but not the amplitude

modulation. (Amplitude modulation may be accom-

plished by variation of the collector supply voltage.)

A simple equivalent circuit of this amplifier is based on

the following five assumptions.

The RF choke allows only a constant (dc) input current

and has no series resistance.

The Q of the series-tuned output circuit is high enough

that the output current is essentially a sinusoid at the

carrier frequency.

The switching action of the transistor is instantaneous

and lossless (except when discharging the shunt capaci-

tor); the transistor has zero saturation voltage, zero

saturation resistance, and infinite off resistance.

The total shunt capacitance is independent of the col-

lector voltage.

The transistor can pass negative current and withstand

negative voltage. (This is inherent in MOS devices, but

requires a combination of bipolar transistors and diodes.)

This equivalent circuit is shown in Fig. l(b). The series

reactance jX is actually produced by the difference in the

reactances of the inductor and capacitor of the series-

tuned circuit. Note that thejX reactance applies on& to

the fundamental frequency; the reactance is assumed to

be infinite at harmonic frequencies. The voltage u,(e) is

actually fictitious; however, it is a convenient reference

point to use in the analysis.

Analysis of the class E amplifier is straightforward but

quite tedious. There is no clear source of voltage or

current, as in classes A, B, C, and D amplifiers. The

collector voltage waveform is a function of the current

charging the capacitor, and the current is a function of the

voltage on the load, which is in turn a function of the

collector voltage. All parameters are interrelated. The

analysis begins by determining the collector voltage wave-

form as a function of the dc input current and the

sinusoidal output current. Next, the fundamental

frequency component of the collector voltage is related to

the output current;and the dc component of the collector

voltage is related to the supply voltage. These relation-

ships result in a nonlinear equation which can be solved

analytically or numerically. Finally, input and output

power and efficiency can be calculated.

Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on June 22,2022 at 06:00:57 UTC from IEEE Xplore. Restrictions apply.

RAAB: OPERATION OF CLASS E TUNED PA

727

A. Basic Relationships

The output voltage and current are sinusoidal and have

the forms

u,(~)=c sin (wt+cp)=c sin (0+cp)

(2.1)

and

iO(f3)= $

sin (e+cp)

(2.2)

where 0 is an “angular time” used for mathematical

convenience. The parameters c and rp are to be de-

termined; cp is defined in Fig. 2.

The hypothetical voltage u,(0) is also a sinusoid, but

has a different phase because of reactance X

u1(e)=%(e)+%(e)

(2.3)

sin

(e+q)+x$ cos (e+cp)

(2.4)

=c, sin

(e+q,)

(2.5)

where

c,=c

1,s =pc

and

Since the RF choke forces a dc input current and the

series-tuned output circuit forces a sinusoidal output cur-

rent, the difference between those two currents must flow

into (or out of) the switch-capacitor combination. When

switch S is open, the difference flows into capacitor C;

when switch S is closed, the current difference flows

through switch S. If a capacitor voltage of other than zero

volts is present at the time the switch closes, the switch

discharges that voltage to zero volts, dissipating the stored

energy of $ CV2.

For purposes of this analysis, the discharge of the

capacitor may be assumed to be a current impulse

qS (0 -

e,), where 0, is the time at which the switch closes.

However, in a real amplifier, discharge of the capacitor

requires a nonzero length of time, during which the collec-

tor voltage and current are simultaneously nonzero. In

any case, however, the total dissipation is the energy

stored in the capacitor, and does not depend on the

particulars of the discharge waveforms. The model re-

mains valid as long as the time required to discharge the

capacitor is a relatively small fraction of the RF cycle.

When S is off, the collector voltage is produced by the

charging of capacitor C by the difference current, hence,

where 0, indicates the time at which S opens.

Since the nonzero collector voltage appears when the

transistor is off, it is convenient to describe the waveforms

in terms of the half off-time y, which is in radians. The

center of the off-time is arbitrarily defined as 7r/2 (Fig. 2).

The switching instants are now 0, = a/2-y and 0, = 7r/2

+y. Note that in the event that 0, < 0 and is therefore

outside of the 277 interval corresponding to one cycle, it

can be repaced by &,+27, with appropriate modification

to the integral in (2.8). Equivalently, the interval of a cycle

can be redefined when necessary. The collector voltage at

time 0 can now be evaluated by expanding (2.8)

(2.9)

+$e+&

cOs(e+cp)

where

B=wC.

(2.11)

Since the tuned circuit has zero impedance to funda-

mental frequency current, there can be no fundamental

frequency voltage drop across it. This means that the

fundamental frequency component of the collector volt-

age waveform must be the hypothetical voltage u,(B). The

magnitude of this component can be calculated by a

Fourier integral. Unfortunately, the collector voltage is

not symmetrical around 7r/2, which makes the phase cp an

unknown.

B. Fourier Analysis

The magnitude c, of the fundamental frequency compo-

nent of the collector voltage is then

c,= -

~]02”o(e)

sin

(e+cpl)

(2.12)

4 -$(t-Y+‘P~)+--$

My-d]

.coscp, siny+$

[

-2 sin ‘pi siny

-&[sin(2’p+$) sin2y-2y sin$]. (2.13)

It is now possible to solve for c by substituting pc for c,

and collecting terms

pc+c

sin (2~ + Ir/) sin 2y - 2y sin I+L

2rBR

2 sin (y - cp) cos cpr sin y

?TBR

=$c-

77+2y-29,+7r+2r+7,) cos cp, siny

+(2y cosy-2 siny) sinq,]. (2.14)

Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on June 22,2022 at 06:00:57 UTC from IEEE Xplore. Restrictions apply.

剩余10页未读，继续阅读

评论收藏

内容反馈