Then slope of c(t) at t = 0 is
d
dt
c(t)
t=0
=
K
τ
e
−t/τ
t=0
=
K
τ
To say it another way the transient response would decay to zero after τ-seconds. In practice we say that
the system reaches about 63% (1 − e
−1
= .37) after one time constant and has reached steady state after
four time constants.
Example:
G(s) =
5
s + 2
=
2.5
0.5s + 1
The time constant τ = 0.5 and the steady state value to a unit step input is 2.5.
The classification of system response into
– forced response
– free response
and
– transient response
– steady state response
is not limited to first order systems but applies to transfer functions G(s) of any order.
The DC-gain of any transfer function is defined as G(0) and is the steady state value of the system to a
unit step input, provided that the system has a steady state value. This follows from the final value theorem
lim
t→∞
c(t) = lim
s→0
sC(s) = lim
s→0
sG(s)R(s)
= G(0) if R(s) = 1/s
provided sC(s) has no poles in the right half plane.
Second Order SystemsConsider a second order transfer function
G(s) =
c(s)
R(s)
=
b
0
s
2
+ a
1
s + a
0
.
The standard form of this transfer function is
G(s) = K ·
ω
2
n
s
2
+ 2ζω
n
s + ω
2
n
3