Alexander-Sadiku
Fundamentals of Electric Circuits
Chapter 8
Second-Order Circuits
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1
Second-Order Circuits
Chapter 8
8.1
8.2
8.3
8.4
8.5
Examples of 2nd order RCL circuit
The source-free series RLC circuit
The source-free parallel RLC circuit
Step response of a series RLC circuit
Step response of a parallel RLC
2
1
8.1 Examples of Second
Order RLC circuits (1)
What is a 2nd order circuit?
A second-order circuit is characterized by a secondorder differential equation. It consists of resistors
and the equivalent of two energy storage elements.
RLC Series
RL T-config
RLC Parallel
RC Pi-config
3
8.2 Source-Free Series
RLC Circuits (1)
The solution of the source-free
series RLC circuit is called as the
natural response of the circuit.
The circuit is excited by the energy
initially stored in the capacitor and
inductor.
The 2nd
order of
expression
d 2i
dt 2
R di
L dt
i
0
LC
How to derive and how to solve?
4
2
8.2 Source-Free Series
RLC Circuits (2)
Method will be
illustrated
during the lecture
5
8.2 Source-Free Series
RLC Circuits (3)
There are three possible solutions for the following
2nd order differential equation:
d 2i
dt 2
=>
d 2i
di
2
2
dt
dt
2
0
R di
L dt
i 0
i
0
LC
where
R
2L
and
0
1
LC
General 2nd order Form
The types of solutions for i(t) depend
on the relative values of
and
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3
8.2 Source-Free Series
RLC Circuits (4)
There are three possible solutions for the following
2nd order differential equation:
d 2i
di
2
2
dt
dt
1. If
>
i (t ) A1e
2. If
=
i (t ) ( A2
3. If
i (t ) e
<
t
o,
s1t
o,
2
0
over-damped case
A2 e s2t
( B1 cos
2
2
where s1, 2
0
critical damped case
A1t )e
o,
i 0
t
where
s1, 2
under-damped case
d
t B2 sin
d
t ) where
d
2
0
2
7
8.2 Source-Free Series
RLC Circuits (5)
Example 1
If R = 10 , L = 5 H, and
C = 2 mF in 8.8, find ,
0, s1 and s2.
What type of natural
response will the circuit
have?
Please refer to lecture or textbook for more detail elaboration.
Answer: underdamped
8
4
8.2 Source-Free Series
RLC Circuits (6)
Example 2
The circuit shown below
has reached steady state
at t = 0-.
If the make-before-break
switch moves to position b
at t = 0, calculate i(t) for
t > 0.
Please refer to lecture or textbook for more detail elaboration.
Answer: i(t) = e
2.5t[5cos1.6583t
7.538sin1.6583t] A
9
8.3 Source-Free Parallel
RLC Circuits (1)
0
Let
i ( 0) I 0
1
v(t )dt
L
v(0) = V0
Apply KCL to the top node:
t
v 1
dv
vdt C
0
R L
dt
Taking the derivative with
respect to t and dividing by C
The 2nd
order of
expression
d 2v
dt 2
1 dv
RC dt
1
v 0
LC
10
5
8.3 Source-Free Parallel
RLC Circuits (2)
There are three possible solutions for the following
2nd order differential equation:
d 2v
dt 2
1. If
dv
dt
2
2
0
v
>
o,
A1 e
s1t
2. If
=
o,
v(t )
( A2
3. If
<
o,
t
( B 1 cos
v(t )
v (t )
e
0
1
2 RC
where
and
1
LC
0
over-damped case
A2 e s2t where
2
2
s1, 2
0
critical damped case
A1t ) e
where
t
s1, 2
under-damped case
d
t
B 2 sin
d
t ) where
d
2
0
2
11
8.3 Source-Free Parallel
RLC Circuits (3)
Example 3
Refer to the circuit shown below.
Find v(t) for t > 0.
Please refer to lecture or textbook for more detail elaboration.
Answer:
v(t) = 66.67(e
10t
e
2.5t)
V
12
6
8.4 Step-Response Series
RLC Circuits (1)
The step response
is obtained by the
sudden application
of a dc source.
d 2v
dt 2
The 2nd
order of
expression
R dv
L dt
v
LC
vs
LC
The above equation has the same form as the equation for
source-free series RLC circuit.
The same coefficients (important in determining the
frequency parameters).
Different circuit variable in the equation.
13
8.4 Step-Response Series
RLC Circuits (2)
The solution of the equation should have two components:
the transient response vt(t) & the steady-state response vss(t):
v (t )
v t (t )
v ss ( t )
The transient response vt is the same as that for source-free case
vt (t ) A1e s1t
A2 e s2t
vt (t ) ( A1 A2t )e
vt ( t ) e
t
(over-damped)
t
( A1 cos
(critically damped)
d
t
A2 sin
d
t ) (under-damped)
The steady-state response is the final value of v(t).
vss(t) = v( )
The values of A1 and A2 are obtained from the initial conditions:
v(0) and dv(0)/dt.
14
7
8.4 Step-Response Series
RLC Circuits (3)
Example 4
Having been in position for a long time, the
switch in the circuit below is moved to position b
at t = 0. Find v(t) and vR(t) for t > 0.
Please refer to lecture or textbook for more detail elaboration.
Answer: v(t) = {10 + [( 2cos3.464t
vR(t)= [2.31sin3.464t]e
1.1547sin3.464t)e 2t]} V
2t
V
15
8.5 Step-Response Parallel
RLC Circuits (1)
The step response
is obtained by the
sudden application
of a dc source.
The 2nd
order of
expression
d 2i
dt 2
1 di
RC dt
i
LC
Is
LC
It has the same form as the equation for source-free parallel
RLC circuit.
The same coefficients (important in determining the
frequency parameters).
Different circuit variable in the equation.
16
8
8.5 Step-Response Parallel
RLC Circuits (2)
The solution of the equation should have two components:
the transient response vt(t) & the steady-state response vss(t):
i (t ) it (t ) iss (t )
The transient response it is the same as that for source-free case
it (t ) A1e s1t
A2 e s2t
it (t ) ( A1 A2t )e
it (t ) e
t
( A1 cos
(over-damped)
t
(critical damped)
d
t
A2 sin
d
t)
(under-damped)
The steady-state response is the final value of i(t).
iss(t) = i( ) = Is
The values of A1 and A2 are obtained from the initial conditions:
i(0) and di(0)/dt.
17
8.5 Step-Response Parallel
RLC Circuits (3)
Example 5
Find i(t) and v(t) for t > 0 in the circuit shown in
circuit shown below:
Please refer to lecture or textbook for more detail elaboration.
Answer:
v(t) = Ldi/dt = 5x20sint = 100sint V
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