Signals and Systems - Electrical Engineering

374 C H A P T E R 6: Application to Control and Communications

Second-Order System

A series RLC circuit with the input a voltage source,vs(t), and the output the voltage across the

capacitor,vc(t), has a transfer function

Vc(s)

Vs(s)

=

1 /Cs

R+Ls+ 1 /Cs

=

1 /LC

s^2 +(R/L)s+ 1 /LC

If we define

Natural frequency:n=

1

√

CL

(6.9)

Damping ratio:ψ=0.5R

√

C

L

(6.10)

we can write

Vc(s)

Vs(s)

=

^2 n

s^2 + 2 ψns+^2 n

(6.11)

A feedback system with this transfer function is given in Figure 6.11 where the feedforward transfer

function is

G(s)=

^2 n

s(s+ 2 ψn)

Indeed, the transfer function of the feedback system is given by

H(s)=

Vc(s)

Vs(s)

=

G(s)

1 +G(s)

=

^2 n

s^2 + 2 ψns+^2 n

The dynamics of a second-order system can be described in terms of the parametersnandψ, as these

two parameters determine the location of the poles of the system and thus its response. We adapted

the previously given script to plot the cluster of poles and the time response of the second-order

system.

Assumen=1 rad/sec and let 0≤ψ≤1 (so that the poles ofH(s)are complex conjugate for 0≤

ψ <1 and double real forψ=1). Let the input be a unit-step signal so thatVs(s)= 1 /s. We then

have:

FIGURE 6.11

Second-order feedback system.

vs(t) e(t)

G(s)

vc(t)

−

+