Signals and Systems - Electrical Engineering

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

is transformed into heat, and so RLC circuits spend the energy given to them. This characteristic

is calledpassivity, indicating that RLC circuits can only use energy, not generate it. Clearly, RLC

circuits are also causal systems as one would not expect them to provide any output before they

are activated.

According to Kirchhoff’s voltage law, the RL circuit is represented by a first-order differential

equation

vs(t)=i(t)R+L

di(t)

dt

=i(t)+

di(t)

dt

To find its impulse response we would need to solve this equation with inputvs(t)=δ(t)and zero

initial condition,i( 0 )=0. In the next chapter, the Laplace domain will provide us an algebraic way

to solve the differential equation and will confirm our intuitive solution given here. Intuitively, in

response to a large and sudden impulsevs(t)=δ(t), the inductor tries to follow it by instanta-

neously increasing its current. But as time goes by and the input is not providing any additional

energy, the current in the inductor goes to zero. Thus, we conjecture that the current in the inductor

isi(t)=h(t)=e−tu(t)whenvs(t)=δ(t)and initial conditions are zero,i( 0 )=0. It is possible to

confirm that is the case. Replacingvs(t)=δ(t)andi(t)=e−tu(t)in the differential equation, we get

δ(t)

︸︷︷︸

vs(t)

=e−tu(t)

︸ ︷︷︸

i(t)

+[e−tδ(t)−e−tu(t)]

︸ ︷︷ ︸

di(t)/dt

=e^0 δ(t)=δ(t)

which is an identity, confirming that indeed our conjectured solution is the solution of the dif-

ferential equation. The initial condition is also satisfied by remembering that there is no initial

current at the source—that is,δ(t)is zero right before we close the switch—and that physically the

inductor remains at that point for an extremely short time before reacting to the strong input.

Thus, the RL circuit whereR= 1 andL=1 H has an impulse response of

h(t)=e−tu(t)

indicating that it is causal sinceh(t)=0 fort<0; that is, the circuit output is zero given that the

initial conditions are zero, and that the inputδ(t)is also zero before 0. We can also show that the

RL circuit is stable. In fact,

∫∞

−∞

|h(t)|dt=

∫∞

0

e−tdt= 1

n

nExample 2.17

Consider the causality and BIBO stability of an echo system (or a multipath system). See

Figure 2.17. Let the outputy(t)be given by

y(t)=α 1 x(t−τ 1 )+α 2 x(t−τ 2 )