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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
Frequency Response
Chapter Objectives:
Ø Understand the Concept of Transfer Functions.Ø Be Familiar with the Decibel Scale.Ø Learn how to make Bode Magnitude and Phase plots.Ø Know Different Types of Passive and Active Filters and their
Characteristics.
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
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.
FREQUENCY RESPONSE
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
TRANSFER FUNCTIONØThe transfer function H(w) of a circuit is the is the frequency dependent ratio of the phasor output Y(w) to a phasor input X(w).
Ø Considered input and output may be either the current or the voltage variable.
Ø 4 types of possible transfer functions.
Y( )H( ) X( )
= H( ) |
wwww f
=
Ð
)(V)(V gain Voltage )(H
i
o
www == )(I
)(V ImpedanceTransfer )(Hi
o
www ==
)(I)(I gain Current )(H
i
o
www ==
)(V)(I AdmittanceTransfer )(H
i
o
www ==
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
Magnitude plot for a low-pass filter
TRANSFER FUNCTION of Low-pass RC Circuit
R=20 kΩC=1200 pF
At low frequencies At high frequencies
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
Phase plot for a low-pass filter
TRANSFER FUNCTION of Low-pass RC Circuit
At low frequencies At high frequencies
R=20 kΩC=1200
pF
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
TRANSFER FUNCTION of High-pass RC Circuit
Magnitude plot for a high-pass filter
At high frequencies At low frequencies
R=20 kΩC=1200 pF
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
TRANSFER FUNCTION of High-pass RC Circuit
Magnitude plot for a high-pass filter
Phase plot for high-pass filter
At high frequencies At low frequencies
R=20 kΩC=1200 pF
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
TRANSFER FUNCTION of a Band-pass RC Circuit
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
Frequency Response of the RC Circuit
2
1
1( ) 1( ) Transfer Function1( ) 1
Magnitude Response
Phase Res
1( )1 ( )
( ) ( ) tan
1Where
ponse
o
o
o
o
s
H
H
V j CHV j RCR C
C
j
R
w
w www w
ww
wf w ww
w
w
-
= = =++
=+
= Ð = -
=
a) Time Domain RC Circuit b) Frequency Domain RC Circuit
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
Drawing Frequency Response of RC Circuit
a) Amplitude Response b) Phase Response
2
1( )1 ( )
o
H www
=+
1( ) ( ) tan o
H wf w ww
-= Ð = -
Low Pass Filter
ØThe frequency value of wo is of special interest.
Ø Because output is considerable only at low values of frequency, the circuit is also called a LOW PASS FILTER.
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
HIGH Pass Filter
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
TRANSFER FUNCTIONØ The transfer function H(w) can be expressed in terms of its numerator polynomial N(w) and its denominator polynomial D(w).
( )( )( )NHDwww
=
Ø The roots of N(w)=0 are called ZEROS of H(w) (jw=z1, z2, z3, ….).Similarly The roots of D(w)=0 are called POLES of H(w) (jw=p1, p2, p3, ….). A zero as a root of the numerator polynomial, results in a zero value of the transfer function. A pole as a root of the denominator polynomial, results in an infinite value of the transfer function.
21 1
1
21
1
2( ) 1 1( )( )( ) 21 1
k k
n n
jj jK j zNHD jj j
p
V ww ww w wwww V ww w
w w
± é ùæ öæ ö+ + +ç ÷ ç ÷ê úè ø è øë û= =é ùæ öæ ö+ + +ç ÷ ç ÷ê úè ø è øë û
K
K
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
s=jw
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
Vx
0.5Vx0.5VxVx
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
DECIBEL SCALEØ The DECIBEL provides us with a unit of less magnitude.
210
1
10log is defined as Decibel GaindBPGP
=
22 2
2 2 2 110 10 10 102
11 1 2
1
2 2 210 10 10 2 1
1 1 1
10log 10log 10log 10log
20log 10log 20log if
dB
VP R V RG
VP V RR
V R V R RV R V
æ ö= = = +ç ÷
è ø
= - = =
2 210 10
1 1
10log 20logdBP VGP V
= =
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
DECIBEL SCALE
210 10
1
20log 20logdBVH HV
= =Magnitude H Decibel Value HdB
0.001 -600.01 -400.1 -200.5 -6
1/√2 -31 0
√2 32 6
10 2020 26
100 40
1 2 1 2
11 2
2
log log log
Some Properties of Logar
log log log
log loglog
ith s
1 0
m
n
PP P P
P P PP
P n P
= +
æ ö= -ç ÷
è ø=
=
Ø The DECIBEL value is a logarithmic measurement of the ratio of one variable to another of the same type.
ØDecibel value has no dimension.
Ø It is used for voltage, current and power gains.
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
Typical Sound Levels and Their Decibel Levels.
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
SEMILOG SCALE
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTSØ Bode plots are APPROXIMATE semilog plots of the magnitude (in Decibels) and phase (in degrees) of a transfer function versus frequency. They are much easier to plot.
High pass filter circuit magitude response.
Bode plots are APPROXIMATE plots of the magnitude and phase responses.
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTS
Actual Response
BODE PLOT
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTS
Bode plot and actual magnitude and phase responses of the RC high pass filter circuit.
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTSØ Bode plots are semilog plots of the magnitude (in Decibels) and phase (in degrees) of a transfer function versus frequency.
Bode plots carry same information. They are much easier to plot.
jH He ff= Ð =H
Logarithmic axis
Decibel values
Total Magnitude Response
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTS
21 1
1
21
1
2( ) 1 1( )( )( ) 21 1
k k
n n
jj jK j zNHD jj j
p
V ww ww w wwww V ww w
w w
± é ùæ öæ ö+ + +ç ÷ ç ÷ê úè ø è øë û= =é ùæ öæ ö+ + +ç ÷ ç ÷ê úè ø è øë û
K
K
Ø This representation is called the STANDARD FORM. It has several different factors.We can draw the Bode plots by plotting each of the terms of the transfer function separately and then adding them.The different factors of the transfer function are
1.) Gain term K.2.) A pole (jw)-1 or a zero (jw) at the origin.
3.) A simple pole or zero
4.) A quadratic pole or zero
11 j
pwæ ö+ç ÷
è ø 11 j
zwæ ö+ç ÷
è ø
2121
n n
j jV w ww w
é ùæ ö+ + ç ÷ê úè øë û
2121
k k
j jV w ww w
é ùæ ö+ + ç ÷ê úè øë û
Ø A transfer function may be written in terms of factors that have real and imaginary parts.
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTS, The Decade
Ø A DECADE is an interval between 2 frequencies with a ratio of 10 (between 10 Hz and 100 Hz or between 500 Hz and 5000 Hz). 20 dB/decade means that magnitude changes 20 dB whenever the frequency changes tenfold or one decade.
The DC value (ω→ 0) does not appear on Bode plots ( Log0 = - ∞).
Ø Slopes are expressed in dB/decade.
One Decade
20 dB
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTS
2
0.4(1STANDARD F
10)( )(1 5)
ORMjH
j jww
w w+
=+
Ø To plot the Bode plots of a given transfer function.
1.) Put the transfer function in STANDARD FORM.2.) Write the Magnitude and phase equations from the STANDARD FORM.3.) Plot the magnitude of each term separately.4.) Add all magnitude terms to obtain the magnitude transfer function.5.) Repeat 2-4 for the phase response.6.) The total magnitude response in Decibel units is the summation and subtraction of the responses of different terms.7.) The total phase response in degrees is the summation and subtraction of the phase responses of different terms.
10
10
10
10
20log 0.420log 1 10
20log
40log 1 5
dbHj
j
j
w
w
w
= +
+ -
-
+
-1
-1
0 tan ( 10)90 2 tan ( 5)
f w
w
= °+
- °-
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTSØ We examine how to plot different terms that may appear in a transfer function.
The total response will be obtained by adding all the responses.
CONSTANT TERM 1020log K
21 1
1
21
1
2( ) 1 1( )( )( ) 21 1
k k
n n
jj jj zNHD jj j
p
K V ww ww w wwww V ww w
w w
± é ùæ öæ ö+ + +ç ÷ ç ÷ê úè ø è øë û= =é ùæ öæ ö+ + +ç ÷ ç ÷ê úè ø è øë û
K
K
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOTSZERO AT THE ORIGIN ( )jw
1020log w
POLE AT THE ORIGIN ( ) 1jw -
-20 -90°
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
SIMPLE ZERO ( )11 j zw+
1010 1
10 110 1
20log 1 0 0 0 dB 020log 1
20log 0 20 log dBH j zzz
w ww
w ww w= ® £ì ì
= + = í í ³»
®¥ îî
11
1
0, =0tan 45 , =z
90 , z
wwf w
w
-
ìæ ö ï= = °íç ÷è ø ï ° ®¥î
Ø Approximate the magnitude response of a simple zero by two linear curves before and after ω=z1Approximate the phase response of a simple zero by three linear curves before ω=0.1z1 after ω=10z1 and between ω=0.1z1 and ω=10z1
CORNER FREQUENCY
BREAK FREQUENCY
3 dB FREQUENCY
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
SIMPLE POLE ( )11 j pw+
-20 -90°
10 120log 1 1 j pw+
Ø Approximate the magnitude response of a simple pole by two linear curves before and after ω=p1Approximate the phase response of a simple pole by three linear curves before ω=0.1p1after ω=10p1 and between ω=0.1p1 and ω=10p1
ØNOTICE the pole and zero responses are in opposite directions.
CORNER FREQUENCY
11
1
0, =0tan 45 , =p
90 , p
wwf w
w
-
ìæ ö ï= - = - °íç ÷è ø ï- ° ®¥î
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
EXACT RESPONSE OF QUADRATIC POLE
21
10
21
220log 1
1( )21
dB
n
n
n
n
j jH
Hj j
V w ww
w ww w
w
wV
æ ö= - +
=é ùæ ö+ + ç ÷ê ú
+ ç
ø
è
è û
÷ø
ë
1
12
2
2
tan1
n
n
V wwf
ww
-= --
BODE PLOTS
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOT OF QUADRATIC ZERO
21
1010
0 as 0220log 1 40log as dBn n
n
j jHwV w www w ww
»®ìïæ ö= + + ç ÷ í ®¥è ø ïî
1
12
2
2 0, 0tan 90 ,
1 180 ,
nn
n
V w wwf w w
www
-
®ìï= » ° =í
- ï ° ®¥î
Ø The EXACT responses can be approximated by BODE plots in terms of the corner frequency ωn
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
BODE PLOT OF QUADRATIC POLE 2
110
10
0 as 0220log 1 40log as dBn n
n
j jHwV w w
ww w ww»
®ìïæ ö= - + + ç ÷ í- ®¥è ø ïî
1
12
2
2 0, 0tan 90 ,
1 180 ,
nn
n
V w wwf w w
www
-
®ìï= - » - ° =í
- ï- ° ®¥î
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
• Express transfer function in Standard form.
• Find the Pole and Zero frequencies.
• Express the magnitude and phase responses.
• Sketch each term of the magnitude and phase responses.
• Add each term of magnitude response to find total magnitude response.
• Add each term of phase response to find total phase response.
Another Procedure• Zeros cause an increase and poles cause a decrease in the slope .
• Start with the lowest frequency of the Bode plot.
• Move along the frequency axis and increase or decrease the slope at each corner frequency.
•Repeat the same procedure for both the magnitude and the phase.
Procedure for Bode plot Construction
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
• Express transfer function in Standard form.
• Express the magnitude and phase responses.
• Two corner frequencies at ω=2, 10 and a zero at the origin ω=0.
• Sketch each term and add to find the total response.
EXAMPLE 14.3 Construct Bode plots for
( )( )200( )2 10jH
j jww
w w=
+ +
10STANDARD FORM ( )(1 2)(1 10)
jHj j
www w
=+ +
10
-1 -1
10 10 1020log 10 20log 20log 1 2 20log 1
90 tan ( 2)
10
tan ( 10)dbH j j j
f
w
w
w
w
w
= ° -
= +
-
- + - +
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
EXAMPLE 14.3 Construct Bode plots for ( )( )
200( )2 10jH
j jww
w w=
+ +
10 10 10 1020log 10 20log 20log 1 2 20log 1 10dbH j j jw w w= + - + - +
-1 -190 tan ( 2) tan ( 10)f w w= °- -
1020log 2 6dB=
26 dB
X X
XX
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
EXAMPLE 14.3 Continued: Let us calculate |H| and f at w=50 rad/sec graphically.
26 dB
10(50) (10) 20log (50 /10) 26 20 0.7 26 14 12H H dB= - = - ´ = - =
10 10 10(50) 90 45 log (1/ 0.2) 90 log (20 /1) 45 log (50 / 20)90 45 0.7 90 1.3 45 0.4 90 31.5 117 18 76.5
f = °- °´ - °´ - °´
= °- °´ - °´ - °´ = °- °- °- ° = - °
w=50
w=50
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
EXAMPLE 14.4 Construct Bode plots for ( )210( )
5jH
j jww
w w+
=+
2
0.4(1STANDARD F
10)( )(1 5)
ORMjH
j jww
w w+
=+
10 10 10 1020log 0.4 20log 1 10 20log 40log 1 5dbH j j jw w w= + + - - +-1 -10 tan ( 10) 90 2 tan ( 5)f w w= °+ - °-
• Express transfer function in Standard form.
• Express the magnitude and phase responses.
• Two corner frequencies at ω=5, 10 and a zero at ω=10.
• The pole at ω=5 is a double pole. The slope of the magnitude is -40 dB/decade and phase has slope -90 degree/decade.
• Sketch each term and add to find the total response.
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
EXAMPLE 14.4 Construct Bode plots for ( )210( )
5jH
j jww
w w+
=+
10
10
10
10
20log 0.420log 1 10
20log
40log 1 5
dbHj
j
j
w
w
w
= +
+ -
-
+
-1
-1
0 tan ( 10)90 2 tan ( 5)
f w
w
= °+
- °-
X O
X O
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‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu
PRACTICE PROBLEM 14.6 Obtain the transfer function for the Bode plot given.
2 2
2
2
A zero at 0.5,
A pole at 1,
Two poles at 10,
1 0.5 (1 0.5)(0.5 ) 200( 0.5)( )(1 1)(1 10) (1 100)(1 )(10 ) ( 1)( 10)
1 0.51
1 11
(1 10)
j
j
j j sHj j j j s s
j
w
w
w
w www
w
w
w w
w
w
=
=
=
+ + += = =
+ + + + + +
+
+
+
‹#›ELEC 24409: Circuit Theory 2 Dr. Kalyana Veluvolu