ENGN 110 Principles of Electric Circuits 
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 KCL and KVL can be applied to circuits containing
capacitors and inductors.
 Consider the following circuit, where is an independent
but time varying voltage source.
 Assuming mesh currents
1 and 2 in the left and right
mesh, respectively:
 Left mesh :
201 + = or 201 +
1

1 − 2 =
 Right Mesh
102 + = or 102 +
1

=
1

(2−1)
Introduction

 Similarly, applying KCL on the node above the capacitor:
1 −
20
+
1

+
1 − 2
10
= 0
 And then, the top right node:
2 − 1
10
+
1

= 0
Introduction

 The solution of these integro-differential equations will
give us detailed insight about this circuit.
 Until we know how to solve them, we can not analyze circuits
involving capacitors, inductors and time varying sources.
 However, when independent sources are active in the circuit,
capacitors and inductors store energy.
 In this situation, circuit behaviour is called Forced Response
or Steady State Response.
 When independent sources are switched off, capacitors and
inductors can provide energy over a limited period of time.
 The behaviour of source free circuit is called Natural
Response or Transient Response.
Introduction

Transient Analysis
 Consider a simple series RL circuit which was connected
to a source for a long time.
 The source is removed from the circuit at = 0.
 In the source free circuit, magnetic field in the inductor
will continue pushing the current for a while.
 But, it will gradually slow down and eventually stops
after some time.
Source Free RL Circuits

Transient Analysis
 Assuming the current through the inductor at = 0 is 0.
 Applying KVL:
+ = 0
+


= 0


+


= 0
Source Free RL Circuits

Transient Analysis
 Here, we are interested in finding a solution for () such
that it satisfies KVL equation and 0 = 0.
 Separating () in KVL equation:


= −




0
()


= −



0


ln − ln 0 = −



Source Free RL Circuits

Transient Analysis
ln

0
= −




0
= −


= 0
−


 The above solution satisfies KVL equation as well yields
0 = 0.
Source Free RL Circuits

Transient Analysis
= 0
−


 Current in source free RL circuit starts from 0 and decays
exponentially with time.
 0 mainly depends upon the source connected to the
circuit at t < 0 and is not really a function of R and L.
 However, how quickly or slowly current decays from its
initial value depends upon R and L values.
 Hence, our transient analysis will focus on R and L
values.
Source Free RL Circuits