Response of Relay Amplifiers with Feedback

J. E. GIBSON ASSOCIATE MEMBER AIEE

F. B. TUTEUR ASSOCIATE MEMBER AIEE

THE RELAY AMPLIFIER with feedback is a simple and economical solution to many power amplifica­

tion problems. The configuration of the primitive relay amplifier is shown in Fig. 1. The relay is assumed to have both deadband and hysteresis as shown. Inasmuch as the low-pass filter is the only predominant lag in the feed­back, it might be surmised that the relay amplifier is stable under normal conditions and indeed this may be shown.

The operation of the amplifier may be described by referring to the waveshapes given in Fig. 2. It is assumed that a step function of voltage Er is applied to the input at time t0. It is assumed that initially there is no charge on the capacitor and that the relay is open. The capacitor begins to charge along the usual exponential to Er volts. However, at e2 volts the deadband voltage of the relay is exceeded and the relay closes to Eb volts. Now as the re­sult of the action of the subtractor, the capacitor begins to charge down to Er — Eb volts. When the capacitor volt­age reaches e± volts, the relay opens and the capacitor must once again charge toward Er volts, and the process repeats itself. During the period that the relay is closed, a pulse of Eb volts is present at the output and is applied to the load.

The response of the amplifier obviously is not a faith­ful reproduction of the input step function. However, it may be shown that the average value of the output pulses is essentially linearly related to the input magnitude. Thus, if the load presents a smoothing or low-pass filter effect, its response would be an adequate representation of the input wave. Because most of the loads driven by power amplifiers do incorporate the low-pass effect, the relay amplifier is quite generally useful. The relation for the average output voltage may be found to be

Eb In 1 -

Er-Eh

Er-Eh J ^m —

In 1 - Eb

Er-ei

1 - Eb

Er-e2 J

(1)

where the symbols are as defined in Fig. 2. The frequency response of the relay amplifier is found

to depend upon the amplitude of the input signal. How­ever, as an approximation, it may be said that the re­sponse is generally flat to about the break frequency of the low-pass filter. The frequency response of the ampli­fier may be improved up to a point by raising the break frequency of the filter. If the filter frequency is made too high, however, the other lags in the loop that could pre-

wmn 4- *

- \

LOW PASS N FILTER e r

r RELAY

L OUTPUT

e«

em

*b it

1Î -F βΓ ^b

Fig. 1. The basic configuration of the relay amplifier. ^ b OUTPUT

RELAY CLOSING VOLTAGE e^

RELAY « FALL OPEN*«

VOLTAGE

-OUTPUT PULSES

L HYSTERESIS

— r — DEADBAND

Ito t W , t « \ \ time

Fig. 2. Voltage wave shapes in response to a step-function input.

viously be neglected, such as the time constant of the re­lay coil and the time required for the relay to close, cause the amplifier to become unstable.

C O N C L U S I O N

T H E OPERATION of the relay amplifier has been dis­cussed and has been shown to be satisfactory for loads that incorporate a low-pass filter effect. The relay ampli­fier may be shown to be stable for the usual values of the filter time constant. The static response is a linear func­tion of input step magnitude and the sinusoidal fre­quency response has been determined. The responses are normalized to deadband and hysteresis so that the results may be applied to any configuration. The responses de­rived are independent of the type of load driven.

Digest of paper 57-183, recommended by the AIEE Committee on Feed­back Control Systems and approved by the AIEE Technical Operations Department for presentation at the AIEE Winter General Meeting, New York, N. Y.f Jan. 21-25, 1957. Scheduled for publication in AIEE Applica­tions and Industry, 1957. J. E. Gibson and F. B. Tu teur are with Yale University, New Haven, Conn.

JUNE 1957 Gibson, Tuteur—Relay Amplifiers 475