2.3 LTI Continuous-Time Systems 125
be a line through the origin, its slopeAis very large. If|vd(t)|> 1Vthe output voltage is a constant
Vsat. That is, the gain of the amplifier saturates. Furthermore, the input resistance of the op-amp is
large so that the currents into the negative and the positive terminals are very small. The op-amp
output resistance is relatively small.
Thus, depending on the dynamic range of the input signals, the op-amp operates in either alinear
region or anonlinearregion. Restricting the operational amplifier to operate in the linear region sim-
plifies the model. Assuming thatA→∞, and thatRin→∞, then we obtain the following equations
defining anideal operational amplifier:
i−=i+= 0
vd(t)=v+(t)−v−(t)= 0 (2.8)
These equations are called thevirtual shortand are valid only if the output voltage of the operational
amplifier is limited by the saturation voltageVsat—that is, when
−Vsat≤vo(t)≤Vsat
Later in the chapter we will consider ways to use the op-amp to get inverters, integrators, adders, and
buffers.
2.3.2 Time Invariance
A continuous-time systemSistime invariantif whenever for an inputx(t)with a corresponding output
S[x(t)], the output corresponding to a shifted inputx(t∓τ)(delayed or advanced) is the original output shifted
in timeS[x(t∓τ)](delayed or advanced). Thus,
x(t) ⇒ y(t)=S[x(t)]
x(t∓τ) ⇒ y(t∓τ)=S[x(t±τ)] (2.9)
That is, the system does not age—its parameters are constant.
A system that satisfies both the linearity and the time invariance is called alinear time-invariantor LTI
system.
Remarks
n It should be clear that linearity and time invariance are independent of each other. Thus, one can have
linear time-varying or nonlinear time-invariant systems.
n Although most actual systems are, according to the above definitions, nonlinear and time varying, linear
models are used to approximate around an operating point the nonlinear behavior, and time-invariant
models are used to approximate in short segments the system’s time-varying behavior. For instance, in
speech synthesis the vocal system is typically modeled as a linear time-invariant system for intervals of
about 20 msec, attempting to approximate the continuous variation in shape in the different parts of the
vocal system (mouth, cheeks, nose, etc.). A better model for such a system is clearly a linear time-varying
model.
