Signals and Systems - Electrical Engineering

2.3 LTI Continuous-Time Systems 129

nExample 2.5

Consider constant linear capacitors and inductors, represented by differential equations

dvc(t)

dt

=

1

C

i(t)

diL(t)

dt

=

1

L

v(t)

with initial conditionsvc( 0 )=0 andiL( 0 )=0. Under what conditions are these time-invariant

systems?

Solution

Given the duality of the capacitor and the inductor, we only need to consider one of these. Solving

the differential equation for the capacitor, we get

vc(t)=

1

C

∫t

0

i(τ)dτ

Let us then find out what happens when we delay (or advance) the input currenti(t)byλsec. The

corresponding output fort> λis given by

1

C

∫t

0

i(τ−λ)dτ=

1

C

∫^0

−λ

i(ρ)dρ+

1

C

∫t−λ

0

i(ρ)dρ (2.10)

by changing the integration variable toρ=τ−λ. For Equation (2.10) to equal the voltage at the

capacitor delayedλsec, given by

vc(t−λ)=

1

C

∫t−λ

0

i(ρ)dρ

we need thati(t)=0 fort<0, so that the first integral in the right expression in Equation (2.10) is

zero. Thus, the system is time invariant if the input currenti(t)=0 fort<0. If the initial condition

v( 0 )is not zero, or if the inputi(t)is not zero fort<0, then linearity or time invariance, or both,

are not satisfied. A similar situation occurs with the inductor.

Thus, an RLC circuit is an LTI system provided that it is not energized fort<0—that is, that the

initial conditions as well as the input are zero fort<0. n

RLC Circuits
An RLC circuit is represented by an ordinary differential equation of order equal to the number of
independent inductors and capacitors (i.e., if two or more capacitors are connected in parallel, or
if two or more inductors are connected in series they share the same initial conditions and can be
simplified to one capacitor and one inductor), and with constant coefficients (due to the assumption