Signals and Systems - Electrical Engineering

152 C H A P T E R 2: Continuous-Time Systems

The input resistance of this circuit isRin=∞and the output resistance isRout=0 so that the

output behaves as an ideal voltage source. The voltage follower is used to isolate two circuits con-

nected in cascade, as the connected circuit at either the input or the output port does not draw any

current from the first—that is, it does not load the other circuit. This is because of the infinite input

resistance, or the behavior of the circuit as a voltage source (Rout=0). This circuit is very useful in

the implementation of analog filters.

Inverting integrator circuit.If we let the feedback element be a capacitor, we obtain the follow-

ing equation from the virtual short equations. The current through the resistorRisvi(t)/Rgiven

thatv−(t)=0 and it is the current through the capacitor as no current enters the negative terminal.

Therefore, the output voltage is

vo(t)=−vc(t)=−

1

C

∫t

0

vi(τ)

R

dτ−vc( 0 )

wherevc( 0 )is the voltage across the capacitor att=0, when the voltage source is turned on. If we

letvc( 0 )=0 andRC=1 the above equation is the negative of the integral of the voltage source.

Thus, we have a circuit that realizes an integrator with a sign inversion. Again this circuit will be

very useful in the implementation of analog filters.

Adder circuit.Since the circuit components are linear, the circuit is linear and we can use super-

position. Lettingv 2 (t)=0 the output voltage due to it is zero, and the output voltage due tov 1 (t)is

vo 1 (t)=−v 1 (t)R/R 1. Similarly, if we letv 1 (t)=0, its corresponding output is zero, and the output

due tov 2 (t)isvo 2 (t)=−v 2 (t)R/R 2 , so that when bothv 1 (t)andv 2 (t)are considered the output is

vo(t)=vo 1 (t)+vo 2 (t)=−v 1 (t)

R

R 1

−v 2 (t)

R

R 2

Using this circuit:

1. WhenR 1 =R 2 =R, we have an adder with a sign inversion:

vo(t)=−[v 1 (t)+v 2 (t)]

2. WhenR 2 →∞andR 1 =R, we get an inverter of the input

vo(t)=−v 1 (t),

3. WhenR 2 →∞andR 1 =αR, we get a constant multiplier with sign inversion:

vo(t)=−

1

α

v 1 (t)

i.e., the inverted input with a gain 1/α.

The above three circuits illustrate the realization of a buffer, an integrator, and an adder that

can be used to realize analog filters. n