3.2 TRANSIENTS IN CIRCUITS 125
3.2 Transients in Circuits
Thetotal responseof a system to an excitation that is suddenly applied or changed consists of
the sum of thesteady-stateandtransientresponses, or ofnaturalandforcedresponses. The
forced response is the component of the system response that is due to the applied excitation.
The transition from the response prior to the change in excitation to the response produced by the
excitation is indicated by the natural (transient) response, which is characteristic of all systems
containing energy-storage components.
Let us consider theRLcircuit excited by a voltage sourcev(t), as shown in Figure 3.2.1. The
KVL equation around the loop, fort>0, is given by
Ldi(t)
dt
+Ri(t)=v(t) (3.2.1)
Dividing both sides byL, we have
di(t)
dt
+
R
L
i(t)=
v(t)
L
(3.2.2)
which is a first-order ordinary differential equation with constant coefficients. Note that the highest
derivative in Equation (3.2.2) is of first order, and the coefficients 1 andR/Lare independent of
timet. Equation (3.2.2) is also linear since the unknowni(t) and any of its derivatives are not
raised to a power other than unity or do not appear as products of each other. Let us then consider
a general form of a linear, first-order ordinary differential equation with constant coefficients,
dx(t)
dt
+ax(t)=f(t) (3.2.3)
in whichais a constant,x(t) is the unknown, andf(t) is the known forcing function. By rewriting
Equation (3.2.3) as
dx(t)
dt
+ax(t)= 0 +f(t) (3.2.4)
and using the superposition property of linearity, one can think of
x(t)=xtr(t)+xss(t) (3.2.5)
wherextr(t)is the transient solution which satisfies the homogeneous differential equation with
f(t)=0,
dxtr(t)
dt
+axtr(t)= 0 (3.2.6)
andxss(t) is the steady-state solution (or a particular solution) which satisfies the inhomogeneous
differential equation for a particularf(t),
dxss(t)
dt
+axss(t)=f(t) (3.2.7)
A possible form forxtr(t) in order to satisfy Equation (3.2.6) is the exponential functionest. Note,
−
+
v(t) i(t)
= iL(t)
L
S R
t = 0
Figure 3.2.1RLcircuit excited byv(t).
