Frequency Response
Instructor: Chia-Ming TsaiElectronics Engineering
National Chiao Tung UniversityHsinchu, Taiwan, R.O.C.
Contents• Introduction
• Transfer Function
• The Decibel Scale
• Bode Plots
• Series Resonance
• Parallel Rosonance
• Passive Filters
• Active Filters
• Applications
Introduction• Analysis with a constant frequency is already
learned.
• To obtain the frequency response
– Keep the amplitude and the phase of the sinusoidal source constant (amplitude=1, phase=0)
– Sweep the frequency from a starting frequency to a stop frequency
– Plot the amplitude and the phase of the desired voltage or current versus frequency
Transfer Function
)(
)(AdmittanceTransfer )(
)(
)(ImpedanceTransfer )(
)(
)(gainCurrent )(
)(
)(gain Voltage)(
)()(
)()(
i
o
i
o
i
o
i
o
H
V
IH
I
VH
I
IH
V
VH
X
YH
• The transfer function H()
can be expressed as
• Zeros: the roots of N()=0
• Poles: the roots of D()=0
)(
)()(
D
NH
.... , , 21 zzj
.... , , 21 ppj
Example 1
RC
H
RCjCjR
Cj
s
o
1 where
tan1
1
1
1
1
1)(
0
0
1
20
V
VH
Phasor domainTime domain
90)(
45)(
0)0(
0
0)(
707.0)(
1)0(
0
H
H
H
Example 2
)( Lj
Example 3
1 012 :Poles
2 ,0 0)2( :Zeros
, 12
)2(
1)24(5.0
)24(5.0
)(
)(
)(5.0124
24)(
:Sol
zeros. and poles its
and )()( Find
2
2
sss
sss
jsss
ss
jj
jj
jj
j
i
o
io
io
I
I
II
II
The Thought of Bode Plots• It is quite difficult to handle the plotting of the
transfer function in a linear scale.
• If the transfer function is transformed to a logarithmic scale, then the plotting becomes much more easy.
)log()log(
)log()log(log
))((
))((log)(log
))((
))((
)(
)()( If
21
21
21
2110
21
21
pjpj
zjzj
pjpj
zjzj
pjpj
zjzj
A
AH
AD
NH
The Decibel (dB) Scale
dB 35.0log10 ,5.0 If
dB 32log10 ,2 If
(dB) log10
dB) (10 log(bel) log
bel 10
1dB 1 decibel 1
logbels ofNumber
is, that ,gain
power themeasure toused is bel The
dB12
dB12
dB1
2
1
2
1
2
1
2
GPP
GPP
GP
P
P
P
P
PG
P
PG
G
1
2dB
1
2dB
21
1
2
1
2
12
1
22
2dB
22
log 20
levels,current
compare toused be alsocan It
log 20
levels, voltage
comparing when Assume
log10log 20
log10
I
IG
V
VG
RR
R
R
V
V
RV
RVG
RIRVP
Bode Plots
2
1
22
11
1
221
211
1
10dB
)(21
zeroor )(211 pole quadraticA 4.
)1( zeroor )1(1 pole simpleA 3.
origin at the )( zeroor )( pole simpleA 2.
gain A 1.
are These function. transfer ain appear can factors basicSeven
)(21)1(
)(21)1()()(
asgiven is )( oftion representa form standard The
log20
definesplot magnitude Bode The
lnlnlnln
kk
nn
nn
kk
j
j
jj
jj
zjpj
jj
K
jjpj
jjzjjK
HH
jHeH
HeH
H
H
H
H
Bode Plots• Steps to construct a Bode plot:
– Plot each factor separately
– Additively combine all of them graphically because of the logarithms involved
• The mathematical convenience of the logarithm makes the Bode plots a powerful tool
• Straight-line plots used instead of actual plots
Bode Plots: A Gain K
0 if 180
0 if 0
log20
)(
10dB
K
K
KH
K
H
Bode Plots: Zero/Pole at Origin
90
log20
90)(
10dB
H
jH
90
log20
90)(
10dB
11
H
jH
Bode Plots: Simple Zero
, 90
, 45
0 , 0
tan
, log20
0 , 01log20
tan
1log20
1)(
11
1
110
10
dB
1
1
110dB
1
zz
zH
z
z
jH
zjH 20 dB/decade
7.5)1.0(tan 1 3.84)10(tan 1
Bode Plots: Simple Pole
, 90
, 45
0 , 0
tan
, log20
0 , 01log20
tan
1log20
11)(
11
1
110
10
dB
1
1
110dB
1
zp
pH
p
p
jH
pjH
-20 dB/decade
-45/decade
Bode Plots: Quadratic Pole
,180
,90
0 , 0
1
2tan
, log40
0 , 0
)1for polescomplex (
21log20
)(211)(
2221
10dB
2
2
210dB
22
n
n
n
n
nn
nn
H
jjH
jjH
Bode Plots: Quadratic Zero
,180
,90
0 , 0
1
2tan
, log40
0 , 0
)1for zeros(complex
21log20
)(21)(
2211
10dB
1
2
110dB
21
k
k
k
n
kk
kk
H
jjH
jjH
Summary
Summary
Summary
Example 1
)101(
1
)21(
1)(10
)101)(21(
10
)10)(2(
200)(
jjj
jj
j
jj
j
H
Example 1 (Cont’d)
)101(
1
)21(
1)(10
)101)(21(
10
)10)(2(
200)(
jjj
jj
j
jj
j
H
Example 2
2
22
)51(
1)101(
14.0
)51(
)101(4.0
)5(
10)(
jj
j
jj
j
jj
j
H
Example 2 (Cont’d)
2
22
)51(
1)101(
14.0
)51(
)101(4.0
)5(
10)(
jj
j
jj
j
jj
j
H
Example 3
2
22
)10(1061
1)1(
100
1
)10(1061
)1(1001
10060
1)(
sss
ss
s
ss
s
H
Example 3 (Cont’d)
2
22
)10(1061
1)1(
100
1
)10(1061
)1(1001
10060
1)(
sss
ss
s
ss
s
H
Series Resonance
Hz 2
1
2
)( rad/s 1
01
)Im(
iscondition resonance The
1
1)(
00
0
00
LCf
requencyresonant fLC
CL
CLjR
CjLjRs
Z
I
VHZ
When Resonance Occurs1. The impedance is purely resistive. The LC
series combination acts like a short circuit.
2. The voltage and the current are in phase, so the power factor is unity.
3. The impedance Z() is minimum.
4. The voltage across L and C can be much larger than the source voltage.
CR
VL
R
VLI m
Cm
L0
00
1
VV
Half-Power Frequencies
is, that ),( at
maximum thehalf ispower dissipated The2
1
2
1)(
is, that highest, theis
power dissipated theresonance,At 2
1)(
iscircuit the
by dissipatedpower average The
1
1
2 1,
2
2
2
0
2
22
es frequencihalf-power
R
VR
R
VP
RIP
RLC
CLR
VI
CLjR
mm
m
I
Z
LCL
R
L
R
RC
LR
R
R
VPP m
1
22
21
2)()(
4
1)()(
2
2,1
2
2,12,1
2
21
2
21
ZZ
L
RB
12
210
Quality Factor: Q
) ( 1
2
121
21
2
1
2
1
2
1
is period onein dissipatedenergy The2
1 is storedenergy peak The
resonanceat period onein
circuit by the dissipatedEnergy
circuit in the storedenergy Peak 2
0
0
0
0
0
2
2
0
22
2
L
RB
BCRR
LQ
R
Lf
fRI
LIQ
fRITRI
LI
Q
Summary
Voltage across L and C QVm
Parallel Resonance
L
RRC
BQ
RCB
LCRCRC
LC
LC
LCj
R
LjCj
R
00
0
22
2
2,1
0
00
1
1
2
1
2
1
rad/s 1
01
)Im(
iscondition resonance The
11
11)(
Y
V
IHY
When Resonance Occurs1. The impedance is purely resistive. The LC
parallel combination acts like an open circuit.
2. The voltage and the current are in phase, so the power factor is unity.
3. The admittance Y() is minimum.
4. The current flowing through L and C can be much larger than the source current.
mmCm
L QIRCIL
RI
L
V 0
00
II
ComparisonsSeries circuit Parallel circuit
Passive FiltersLowpass
Highpass
Bandpass
Bandstop
Lowpass Filter
, 09
, 54
0 , 0
, 0
, 2
1 0 , 1
)(
1 where
1
1
1
1
1
1)(
c
c
c
c
i
o
RC
jRCj
CjR
Cj
Η
V
VH
Highpass Filter
, 0
, 54
0 , 09
, 1
, 2
1 0 , 0
)(
1 where
1
1
1
1)(
c
c
c
cc
i
o
RC
j
j
RCj
RCj
CjR
R
H
V
VH
Bandpass Filter
, 90
, 0
0 , 90
, 0
, 1
0 , 0
)(
1 where
)(1
)(1
1)(
0
0
210
20
2
H
V
VH
LC
jRCj
jRC
LCjRCj
RCj
CjLjR
R
i
o
B
Bandstop Filter
, 0
, 0
0 , 0
, 1
: , 0
0 , 1
)(
1 where
)(1
)(1
)(1
)(1
1
1)(
0
0
210
20
20
2
2
H
V
VH
LC
jRCj
j
LCjRCj
LCj
CjLjR
CjLj
i
o
rejectionfrequency
B
Active Filters
i
f
i
o
Z
Z
V
VH )(
ffc
ci
f
ffi
f
i
f
ff
f
fff
ii
CRjR
R
RCjR
R
RCj
R
CjR
R
1 ,
1
1
1
1)(
1
1||
Z
ZH
Z
Z
A general 1st-order active filter
Active 1st-order lowpass filter
Active 1st-Order Highpass Filter
1
ff
iii
RCj
R
Z
Z
iic
cfi
ii
fi
ii
f
i
f
CRj
jRC
RCj
RCj
CjR
R
1 ,
1
1
1)(
Z
ZH
Active Bandpass Filter
Active Bandpass Filter
21
2210
12
21
1
1
2
)()(
1 &
1 ,
11
1)(
i
f
i
f
i
o
R
RKGainPassband
RCRCR
R
j
j
j
HH
V
VH
Bandreject (or Notch) Filter
Bandreject (or Notch) Filter
i
f
i
f
i
o
R
RKGainPassband
RCRCR
R
j
j
j
)()0(
1 &
1 ,
11
1)(
12
21
1
1
2
HH
V
VH
Applications: Radio Receiver
2055kHzrejected
2055kHz
Touch-Tone Telephone (1/2)
Touch-Tone Telephone (2/2)
Crossover Network
(Lowpass)
(Highpass)
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