sibt.nsw.edu.au navitas.com
Chapter 8
Basic RL and RC Circuits
(Part 1)
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
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
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
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
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
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
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
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
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
Assuming the current through the inductor at = 0 is 0.
Applying KVL:
+ = 0
+
= 0
+
= 0
Source Free RL Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
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
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
ln
0
= −
0
= −
= 0
−
The above solution satisfies KVL equation as well yields
0 = 0.
Source Free RL Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
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
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
Plotting
0
versus .
0
=
−
/
Let
= = time constant
0
= −
When t = 0,
0
= 1
Source Free RL Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
At t = ,
0
= −1 = 0.3679
At t = 2,
0
= −2 = 0.135
At t = 3,
0
= −3 = 0.049
At t = 4,
0
= −4 = 0.018
At t = 5,
0
= −5 = 0.006
Source Free RL Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis of RL Circuit
Hence, theoretically, current will continue to flow until
= ∞ .
But, practically, current will become zero at = 5.
For small values of =
, current will decay quickly.
For large values of =
, current will decay slowly.
RL Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Power and Energy
In a source free RL circuit, power delivered by inductor is
dissipated by resistor.
= 2 = 0
2 −
2
Total energy supplied by the inductor can be calculated by
integrating power:
=
0
∞
= 0
2
0
∞
−
2
= 0
2 −
2
0 − 1
=
1
2
0
2
Source Free RL Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
Consider a simple series RC 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, charge stored in the capacitor
will provide the current for a while.
But, it will gradually slow down and eventually stops
after some time.
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
Assuming the voltage across capacitor at = 0 is 0.
Applying KCL:
+ = 0
+
= 0
+
1
= 0
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
Here, we are interested in finding a solution for () such
that it satisfies KCL equation and 0 = 0.
Separating () in KCL equation:
= −
1
0
()
= −
1
0
ln − ln 0 = −
1
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
ln
0
= −
1
0
= −
1
= 0
−
1
The above solution satisfies KCL equation as well yields
0 = 0.
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
= 0
−
1
Voltage in source free RC 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 C.
However, how quickly or slowly voltage decays from its
initial value depends upon R and C values.
Hence, our transient analysis will focus on R and C
values.
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
Plotting
0
versus .
0
= −
Let = = time constant
0
= −
When t = 0,
0
= 1
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
At t = ,
0
= −1 = 0.3679
At t = 2,
0
= −2 = 0.135
At t = 3,
0
= −3 = 0.049
At t = 4,
0
= −4 = 0.018
At t = 5,
0
= −5 = 0.006
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Transient Analysis
Hence, theoretically, voltage will continue to stay until
= ∞ .
But, practically, volatge will become zero at = 5.
For small values of = , voltage will decay quickly.
For large values of = , voltage will decay slowly.
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
Power and Energy
In a source free RC circuit, power delivered by inductor is
dissipated by resistor.
=
2
=
0
2
−
2
Total energy supplied by the inductor can be calculated by
integrating power:
=
0
∞
=
0
2
0
∞
−
2
=
0
2
−
2
0 − 1
=
1
2
0
2
Source Free RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin
sibt.nsw.edu.au navitas.com
If an RL circuit has multiple resistors and inductors which
can be solved using series and parallel combinations, the
time constant becomes:
=
Similarly, if an RC circuit has multiple resistors and
capacitors which can be solved using series and parallel
combinations, the time constant becomes:
=
General RL and RC Circuits
ENGN 110 Principles of Electric Circuits © Engineering Circuit Analysis By W.H. Hayt, J.E. Kemmerly and S.M. Durbin