Signals and Systems - Electrical Engineering

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

according to Faraday’s induction law. Solving this differential equation for the current we obtain

iL(t)=

1

L

∫t

0

v(τ)dτ+iL( 0 ) (2.7)

Like the capacitor, the inductor is not a linear system unless the initial current in the inductor is

zero. The inductor can be considered an incrementally linear system.

Notice that an explicit relation between the input and the output was necessary to determine

linearity. n

Op-Amps and Feedback

Operational amplifiers, or op-amps, are high-gain amplifiers typically used with feedback. In the 1930s, Harold S. Black

developed the principles of feedback amplifiers—that is the application of a portion of the output back to the input to

reduce the overall gain. By doing so, the characteristics of the amplifier are greatly enhanced. In the late 1930s, George A.

Philbrick developed a vacuum-tube circuit that performed some of the op-amp functions. Professor John Ragazzini, from

Columbia University, coined the name of “operational amplifier” in 1947. Early op-amps were vacuum-tube based, and thus

bulky and expensive. The trend to cheaper and smaller op-amps began in the 1960s [50, 72].

The Op-Amp

An excellent example of a device that can be used as either a nonlinear or a linear system is the

operational amplifier orop-amp. It is a two-port device (see Figure 2.2) with two voltage inputs:v−(t),

in theinverting terminal, andv+(t), in thenoninverting terminal. The output voltagev 0 (t)is a nonlinear

function of the difference between the two inputs—that is,

vo(t)=f[v+(t)−v−(t)]=f(vd(t))

The functionf(vd(t))is approximately linear for small values± 1 Vofvd(t), in the order of millivolts,

and it becomes constant beyond± 1 V. The output voltagev 0 (t)is, however, in the order of volts, so

that letting

vo(t)=Avd(t) − 1 V≤vd(t)≤ 1 V

FIGURE 2.2

Operational amplifier: (a) circuit

diagram, and (b) input–output

voltage relation.

−ΔV

ΔV

vo(t)

vd(t)

−Vsat

Vsat

(a)(b)

−

+

v−(t)

v+(t) vo(t)

+

+ +

−− −

i− A

i+