ECE 2100
Circuit Analysis
Lesson 26
Chapter 14
Sec 14.1, 14.5, 14.7
Series RLC Circuit
RC Low Pass Filter
Daniel M. Litynski, Ph.D.
http://homepages.wmich.edu/~dlitynsk/
ECE 2100
Circuit Analysis
Lesson 25
Chapter 9 & App B:
Passive circuit elements
in the phasor representation
Daniel M. Litynski, Ph.D.
http://homepages.wmich.edu/~dlitynsk/
Sinusoids and Phasor
Chapter 9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
Motivation
Sinusoids’ features
Phasors
Phasor relationships for circuit elements
Impedance and admittance
Kirchhoff’s laws in the frequency domain
Impedance combinations
3
9.5 Impedance and Admittance (1)
• The impedance Z of a circuit is the ratio of the phasor
voltage V to the phasor current I, measured in ohms Ω.
V
I
Z
R
jX
where R = Re, Z is the resistance and X = Im, Z is the
reactance. Positive X is for L and negative X is for C.
• The admittance Y is the reciprocal of impedance,
measured in siemens (S).
Y
1
Z
I
V
4
9.5 Impedance and Admittance (2)
Impedances and admittances of passive elements
Element
R
L
C
Impedance
Z
Admittance
Y
R
Z
j L
Z
1
j C
Y
Y
1
R
1
j L
j C
5
9.5 Impedance and Admittance (3)
0; Z
Z
j L
0
;Z
0; Z
Z
1
j C
;Z
0
6
9.5 Impedance and Admittance (4)
After we know how to convert RLC components
from time to phasor domain, we can transform
a time domain circuit into a phasor/frequency
domain circuit.
Hence, we can apply the KCL laws and other
theorems to directly set up phasor equations
involving our target variable(s) for solving.
7
9.5 Impedance and Admittance (5)
Example 8
Refer to Figure below, determine v(t) and i(t).
vs
5 cos(10t )
Answers: i(t) = 1.118cos(10t – 26.56o) A; v(t) = 2.236cos(10t + 63.43o) V
8
9.6 Kirchhoff’s Laws
in the Frequency Domain (1)
• Both KVL and KCL hold in the phasor domain
or more commonly called frequency domain.
• Moreover, the variables to be handled are
phasors, which are complex numbers.
• All the mathematical operations involved are
now in the complex domain.
9
9.7 Impedance Combinations (1)
• The following principles used for DC circuit
analysis all apply to AC circuit.
• For example:
a. voltage division
b. current division
c. circuit reduction
d. impedance equivalence
e. Y-Δ transformation
10
9.7 Impedance Combinations (2)
Example 9
Determine the input impedance of the circuit in figure below
at ω =10 rad/s.
Answer: Zin = 32.38 – j73.76
11
Frequency Response
Chapter 14
14.1 Introduction
14.5 Series Resonance
14.7 Passive Filters
12
14.1 Introduction (1)
What is Frequency Response of a Circuit?
It is the variation in a circuit’s
behavior with change in signal
frequency and may also be
considered as the variation of the gain
and phase with frequency.
13
14.2 Transfer Function (1)
• The transfer function H(ω) of a circuit is the
frequency-dependent ratio of a phasor output
Y(ω) (an element voltage or current ) to a phasor
input X(ω) (source voltage or current).
H( )
Y( )
X( )
| H( ) |
14
14.2 Transfer Function (2)
• Four possible transfer functions:
H( )
Voltage gain
Vo ( )
Vi ( )
H( )
H( )
Current gain
Io ( )
Ii ( )
H( )
Y( )
X( )
Transfer Impedance
Vo ( )
Ii ( )
| H( ) |
H( )
Transfer Admittance
Io ( )
Vi ( )
15
14.2 Transfer Function (3)
Example 1
For the RC circuit shown below, obtain the transfer
function Vo/Vs and its frequency response.
Let vs = Vmcosωt.
16
14.2 Transfer Function (4)
Solution:
The transfer function is
H( )
Vo
Vs
1
j C
R 1/ j C
1
1 j RC
,
The magnitude is
The phase is
1
H( )
1 ( /
tan
1
o
o
2
)
o
1/RC
Low Pass Filter
17
14.2 Transfer Function (5)
Example 2
Obtain the transfer function Vo/Vs of the RL circuit
shown below, assuming vs = Vmcosωt. Sketch its
frequency response.
18
14.2 Transfer Function (6)
Solution:
Vo
Vs
H( )
The transfer function is
j L
R j L
1
R
1
j L
High Pass Filter
,
The magnitude is
The phase is
90
1
H( )
1 (
tan
o
)2
1
o
o
R/L
19
14.3 Series Resonance (1)
Resonance is a condition in an RLC circuit in which
the capacitive and inductive reactance are equal in
magnitude, thereby resulting in purely resistive
impedance.
Resonance frequency:
o
Z R
j( L
1
)
C
fo
1
rad/s
LC
1
Hz
2 LC
or
20
14.3 Series Resonance (2)
The features of series resonance:
Z R j( L
1
)
C
The impedance is purely resistive, Z = R;
• The supply voltage Vs and the current I are in phase, so
cos = 1;
• The magnitude of the transfer function H(ω) = Z(ω) is
minimum;
• The inductor voltage and capacitor voltage can be much
more than the source voltage.
21
14.3 Series Resonance (3)
Bandwidth B
The frequency response of the
resonance circuit current is
I |I|
Z R j( L
1
)
C
Vm
R 2 ( L 1/
C) 2
The average power absorbed
by the RLC circuit is
P( )
The highest power dissipated
occurs at resonance:
1 2
IR
2
P(
o
)
1 Vm2
2 R
22
14 3 Series Resonance (4)
Half-power frequencies ω1 and ω2 are frequencies at which the
dissipated power is half the maximum value:
P( 1 )
P(
2
)
1 (Vm / 2 ) 2
2
R
Vm2
4R
The half-power frequencies can be obtained by setting Z
equal to √2 R.
1
R
2L
(
Bandwidth B
R 2
)
2L
1
LC
B
2
2
R
2L
(
R 2
)
2L
1
LC
o
1
2
1
23
14.3 Series Resonance (5)
Quality factor,
Q
The relationship
between the B, Q
and ωo:
Peak energy stored in the circuit
Energy dissipated by the circuit
in one period at resonance
B
R
L
o
Q
2
o
L
R
o
1
o CR
CR
• The quality factor is the ratio of its
resonant frequency to its bandwidth.
• If the bandwidth is narrow, the
quality factor of the resonant circuit
must be high.
• If the band of frequencies is wide,
the quality factor must be low.
24
14.3 Series Resonance (6)
Example 3
A series-connected circuit has R = 4 Ω
and L = 25 mH.
a. Calculate the value of C that will produce a
quality factor of 50.
b. Find ω1 and ω2, and B.
c. Determine the average power dissipated at ω
= ωo, ω1, ω2. Take Vm= 100V.
25
14.4 Parallel Resonance (1)
It occurs when imaginary part of Y is zero
Y
1
R
j( C
1
)
L
Resonance frequency:
o
1
rad/s or f o
LC
1
2
LC
Hz
26
14.4 Parallel Resonance (2)
Summary of series and parallel resonance circuits:
characteristic
Series circuit
ωo
1
LC
Q
ωo L
1
or
R
ωo RC
B
ω1, ω2
Q ≥ 10, ω1, ω2
o
1 (
o
Parallel circuit
1
LC
R
or
L
o
o
o
o
Q
Q
1 2
)
2Q
B
2
o
2Q
o
1 (
1 2
)
2Q
o
RC
o
2Q
B
2
27
14.4 Parallel Resonance (3)
Example 4
Calculate the resonant frequency of the
circuit in the figure shown below.
Answer:
19
2
2.179 rad/s
28
14.5 Passive Filters (1)
• A filter is a circuit
that is designed to
pass signals with
desired frequencies
and reject or
attenuate others.
Low Pass
High Pass
• Passive filter consists
of only passive
element R, L and C.
Band Pass
• There are four types
of filters.
Band Stop
29
14.5 Passive Filters (2)
Example 5
For the circuit in the figure below, obtain the transfer function
Vo(ω)/Vi(ω). Identify the type of filter the circuit represents
and determine the corner frequency. Take R1=100 =R2
and L =2mH.
Answer:
25 krad/s
30