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ECE201 Lect-21 1
Second-Order Circuits (7.3)
Dr. Holbert
April 19, 2006
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ECE201 Lect-21 2
2nd Order Circuits
Any circuit with a single capacitor, a single
inductor, an arbitrary number of sources,
and an arbitrary number of resistorsis acircuit of order 2.
Any voltage or current in such a circuit is
the solution to a 2nd order differentialequation.
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ECE201 Lect-21 3
Important Concepts
The differential equation
Forced and homogeneous solutions The natural frequency and the damping
ratio
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ECE201 Lect-21 4
A 2nd Order RLC Circuit
The source and resistor may be equivalent
to a circuit with many resistors and sources.
R
Cvs(t)
i(t)
L
+
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ECE201 Lect-21 5
Applications Modeled by a 2nd
Order RLC Circuit
Filters
A lowpass filter with a sharper cutoffthan can be obtained with an RC circuit.
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ECE201 Lect-21 6
The Differential Equation
KVL around the loop:
vr(t) + vc(t) + vl(t) = vs(t)
R
Cvs(t)
+
vc(t)
+
vr(t)
L
+vl(t)
i(t)
+
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ECE201 Lect-21 7
Differential Equation
)()(1)(
)( tvdxxiCdt
tdiLtRi s
t
dt
tdv
Lti
LCdt
tdi
L
R
dt
tid s )(1)(1)()(
2
2
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ECE201 Lect-21 8
The Differential Equation
Most circuits with one capacitor and inductor
are not as easy to analyze as the previous
circuit. However, every voltage and currentin such a circuit is the solution to a differential
equation of the following form:
)()()(
2)( 2
002
2
tftidt
tdi
dt
tid
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ECE201 Lect-21 9
Important Concepts
The differential equation
Forced and homogeneous solutions The natural frequency and the damping
ratio
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ECE201 Lect-21 10
The Particular Solution
The particular (or forced) solution ip(t)is
usually a weighted sum off(t)and its first
and second derivatives.
Iff(t)is constant, then ip(t)is constant.
Iff(t)is sinusoidal, then ip(t)is sinusoidal.
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ECE201 Lect-21 11
The Complementary Solution
The complementary (homogeneous) solution
has the following form:
Kis a constant determined by initial conditions.
sis a constant determined by the coefficients of
the differential equation.
st
c Keti )(
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ECE201 Lect-21 12
Complementary Solution
02 2002
2
ststst
Kedt
dKe
dt
Ked
02 2002 ststst KesKeKes
02 2002 ss
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ECE201 Lect-21 13
Characteristic Equation
To find the complementary solution, we
need to solve the characteristic equation:
The characteristic equation has two roots-
call thems1ands2.
02 2002 ss
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ECE201 Lect-21 14
Complementary Solution
Each root (s1ands2) contributes a term to
the complementary solution.
The complementary solution is (usually)
tsts
c eKeKti 21
21)(
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ECE201 Lect-21 15
Important Concepts
The differential equation
Forced and homogeneous solutions
The natural frequency and the damping
ratio
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ECE201 Lect-21 16
Damping Ratio () and
Natural Frequency (0) The damping ratiois .
The damping ratio determines what type of
solution we will get:
Exponentially decreasing (>1)
Exponentially decreasing sinusoid (< 1)
The natural frequencyis 0
It determines how fast sinusoids wiggle.
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ECE201 Lect-21 17
Roots of the Characteristic
EquationThe roots of the characteristic equation
determine whether the complementary
solution wiggles.
12001 s
12002 s
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ECE201 Lect-21 18
Real Unequal Roots
If > 1,s1ands2are realand not equal.
This solution is overdamped.
tt
c eKeKti
1
2
1
1
200
200
)(
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ECE201 Lect-21 19
Overdamped
0
0.2
0.4
0.6
0.8
1
-1.00E-06
t
i(t)
-0.2
0
0.2
0.4
0.6
0.8
-1.00E-06
t
i(t)
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ECE201 Lect-21 20
Complex Roots
If < 1,s1ands2are complex.
Define the following constants:
This solution is underdamped.
tAtAeti ddtc sincos)( 21
02
0 1 d
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ECE201 Lect-21 21
Underdamped
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1.00E-05 1.00E-05 3.00E-05
t
i(t)
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ECE201 Lect-21 22
Real Equal Roots
If = 1,s1ands2are realand equal.
This solution is critically damped.
tt
c teKeKti 00
21)(
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ECE201 Lect-21 23
Example
This is one possible implementation of the
filter portion of the IF amplifier.
10W
769pFvs(t)
i(t)
159mH
+
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ECE201 Lect-21 24
More of the Example
dt
tdv
Lti
LCdt
tdi
L
R
dt
tid s )(1)(1)()(
2
2
)()()(
2)( 2
002
2
tftidt
tdi
dt
tid
For the example, what are and 0?
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ECE201 Lect-21 25
Even More Example
= 0.011
0= 2p455000
Is this system over damped, under damped,
or critically damped?
What will the current look like?
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ECE201 Lect-21 26
Example (cont.)
The shape of the current depends on the
initial capacitor voltage and inductor
current.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1.00E-05 1.00E-05 3.00E-05
t
i(t)
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ECE201 Lect-21 27
Slightly Different Example
Increase the resistor to 1kW
What are and 0?
1kW
769pFvs(t)
i(t)
159mH
+
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ECE201 Lect-21 28
More Different Example
= 2.2
0= 2p455000
Is this system over damped, under damped,
or critically damped?
What will the current look like?
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ECE201 Lect-21 29
Example (cont.)
The shape of the current depends on the
initial capacitor voltage and inductor
current.
0
0.2
0.4
0.6
0.8
1
-1.00E-06
t
i(t)
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ECE201 Lect-21 30
Damping Summary
Roots (s1, s2) Damping
>1 Real and unequal Overdamped
=1 Real and equal Critically damped
0
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ECE201 Lect-21 31
Class Example
Learning Extension E7.9
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