Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

7.1 
transIent state and steady-state In cIrcuIts
The mesh equations (node equations) needed to solve for the response of any circuit – memoryless or 
dynamic – are obtained by applying Kirchhoff’s Voltage Law in meshes (Kirchhoff’s Current Law at 
nodes) along with the element equations. The element equation of an inductor is 
v
L
di
dt
L
L
=
and that of 

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Transient State and Steady-State in Circuits 
7.3
a capacitor is 
v
C
i dt
t
C
C
=
−∞
∫
1
.
Therefore, the mesh equations (node equations) describing a dynamic 
circuit will be integro-differential equations. These equations are true for all t since they are obtained 
from KVL and KCL that are true on an instant to instant basis. Therefore, both sides of such integro-
differential equations can be differentiated or integrated with respect to time, if needed. We will be 
able to eliminate the integral terms in the equations by this technique. The resulting equations will be 
differential equations and the coefficients of differential equation will be decided by the values of R, 
L, C and M. Since they are constants in the circuits we consider, the resulting differential equations 
will be a set of simultaneous linear differential equations with constant coefficients. We illustrate this 
in two examples in the next sub-section.
7.1.1 
Governing differential equation of circuits – examples
(a)
v
C
v
S
(
t
)
+
+
–
–
1 
Ω
t
= 0
1 F 
(b)
i
1
i
2
v
S
(
t
)
+
–
1 
Ω
1 
Ω
t
= 0
1 H
1 H
Fig. 7.1-1 
Example circuits illustrating circuit differential equations
Consider the simple Series RC Circuit in Fig. 7.1-1(a). Assume that capacitor was initially 
uncharged. The voltage across the capacitor is taken as the response variable. Then, the current 
through the resistor from left to right can be expressed as (v
S
-
v
C
)/R. Writing KCL at the positive 
terminal of capacitor,
v
v
dv
dt
dv
dt
v
v
S
C
C
C
C
S
−
= ×
∴
+
=
1
Now, we take up the second example circuit in Fig. 7.1-1 and write the two mesh equations as,
di
dt
i
i
v
1
1
2
+ − =
S
(7.1-1)
di
dt
i
i
2
2
1
2
0
+
− =
(7.1-2)
This is a set of two first-order simultaneous ordinary differential equations with constant coefficients. 
The first mesh current variable i
2
is selected as the circuit response variable. We have to eliminate i
1
from these two equations and get a single differential equation for i
2
in order to find the differential 
equation describing the circuit. This may be done by substituting Eqn. 7.1-2 in Eqn. 7.1-1 as shown
below:
i
di
dt
i
1
2
2
2
=
+
-from Eqn. 7.1-2. Substituting this in Eqn. 7.1-1, we get,
d
dt
di
dt
i
di
dt
i
i
v
2
2
2
2
2
2
2
+




+
+




− =
S

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7.4
The Sinusoidal Steady-State Response
Therefore, the describing differential equation in Fig. 7.1-1 (b) is
d i
dt
di
dt
i
v
2
2
2
2
2
3
+
+ =
S
These two examples help us to understand the following generalisation.
A dynamic circuit is described by a linear, constant-coefficient, ordinary differential 
equation for one chosen response variable. The coefficients will be decided by circuit 
parameters. The right-hand side will contain the applied forcing function terms (including 
their derivatives in general).