Signals and Systems - Electrical Engineering

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

FIGURE 2.5

RLC circuit.

v(t)

i(t)

+

−

t= (^0) RC
L
that theR,L, andCvalues are constant). If the initial conditions of the RLC circuit are zero, and the
input is zero fort<0, then the system represented by the linear differential equation with constant
coefficients is LTI.
Consider, for instance, the circuit in Figure 2.5 consisting of a series connection of a resistorR, an
inductorL, and a capacitorC. The switch has been open for a very long time and it is closed att=0,
so that there is no initial energy stored in either the inductor or the capacitor (the initial current in
the inductor isiL( 0 )=0 and the initial voltage in the capacitor isvC( 0 )=0) and the voltage applied
to the elements is zero fort<0. This circuit is represented by a second-order differential equation
with constant coefficients. According to Kirchhoff’s voltage law,
v(t)=Ri(t)+L
di(t)
dt

+

1

C

∫t

0

i(τ)dτ

and taking a derivative ofv(t)with respect totwe obtain

dv(t)

dt

=R

di(t)

dt

+L

d^2 i(t)

dt^2

+

1

C

i(t)

a second-order differential equation, with input the voltage sourcev(t)and output the currenti(t).

2.3.3 Representation of Systems by Differential Equations

Given a dynamic system represented by a linear differential equation with constant coefficients,

a 0 y(t)+a 1

dy(t)

dt

+···+

dNy(t)

dtN

=b 0 x(t)+b 1

dx(t)

dt

+···+bM

dMx(t)

dtM

t≥ 0

withNinitial conditionsy( 0 ),dky(t)/dtk|t= 0 fork=1,...,N− 1 and inputx(t)= 0 fort< 0 , itscomplete

responsey(t)fort≥ 0 has two components:

n Thezero-state response,yzs(t), due exclusively to the input as the initial conditions are zero.

n Thezero-input response,yzi(t), due exclusively to the initial conditions as the input is zero. So that

y(t)=yzs(t)+yzi(t) (2.11)

Thus, when the initial conditions are zero, theny(t)depends exclusively on the input (i.e.,y(t)=yzs(t)), and

the system is linear and time invariant or LTI.