电学代写-ENGN 110
时间:2022-05-25
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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


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