1 Chapter 14 Frequency Response 電路學 ( 二 ). 2 Frequency Response Chapter 14 14.1 Introduction...
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Transcript of 1 Chapter 14 Frequency Response 電路學 ( 二 ). 2 Frequency Response Chapter 14 14.1 Introduction...
1
Chapter 14Chapter 14
Frequency ResponseFrequency Response
電路學電路學 (( 二二 ))
2
Frequency ResponseFrequency ResponseChapter 14Chapter 14
14.1 Introduction14.2 Transfer Function14.3 The Decible Scale14.4 Bode Plots14.5 Series Resonance14.6 Parallel Resonance14.7 Passive Filters14.8 Active Filters14.9 Scaling
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What is Frequency Response of a Circuit?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.
14.1 Introduction (1)14.1 Introduction (1)
4
14.2 Transfer Function (1)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).
|)(|
)()(
)( HXY
H
5
14.2 Transfer Function (2)14.2 Transfer Function (2)
• Four possible transfer functions:
)()(
gain Voltage )(i
o
VV
H
)()(
gain Current )(i
o
II
H
)()(
Impedance Transfer )(i
o
IV
H
)()(
Admittance Transfer )(i
o
VI
H
|)(|
)()(
)( HXY
H
6
14.2 Transfer Function (3)14.2 Transfer Function (3)
)()(
)(
DN
H
The transfer function can be expressed in terms The transfer function can be expressed in terms of its numerator polynomial of its numerator polynomial NN(() and ) and denominator polynomial denominator polynomial DD(() )
zero zero 零點零點 : a root of the numerator : a root of the numerator polynomial.polynomial.
pole pole 極點極點 : a root of the denominator : a root of the denominator polynomial.polynomial.
7
14.2 Transfer Function (3)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.
8
14.2 Transfer Function (4)14.2 Transfer Function (4)
Solution:
The transfer function is
,
The magnitude is 2)/(1
1)(
o
H
The phase iso
1tan
1/RCo
jωω R/ jω j Rjωω
11
1
1
)(s
o
VV
H
Low Pass FilterLow Pass Filter
9
14.2 Transfer Function (5)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.
10
14.2 Transfer Function (6)14.2 Transfer Function (6)
Solution:
The transfer function is
,
The magnitude is 2)(1
1)(
o
H
The phase iso
1tan90
R/Lo
jωω Rjωω R
Ljω
1
1)(
s
o
VV
H
High Pass FilterHigh Pass Filter
11
14.3 The Decibel Scale (1)14.3 The Decibel Scale (1)
210
1
Number of bels logP
GP
2dB 10
1
10logP
GP
210
1
20logV
V
• log log PP11PP22 = log = log PP11 + log + log PP22
• log log PP11//PP22 = log = log PP11 - log - log PP22
• log log PPnn = n = n log log PP
• log 1 = 0log 1 = 0
belbel 為功率增益的單位為功率增益的單位
dB (decibel) dB (decibel) 為為 belbel 的的1/101/10
210
1
20logI
I
12
14.4 Bode Plots14.4 Bode Plots (1)(1)
Bode plotsBode plots are are semilog plots of the semilog plots of the magnitude (in dB) and magnitude (in dB) and phase (in degree) of a phase (in degree) of a transfer function vs transfer function vs frequency.frequency.
13
14.4 Bode Plots14.4 Bode Plots (2)(2)
jH He H
ln ln ln lnjH e H j H
1 21 1
21 2
( ) (1 / )[1 2 / ( / ) ]( )
(1 / )[1 2 / ( / ) ]k k
n n
K j j z j j
j p j j
H
dB 10
10 10 10 1
10 1
20log
20log | | 20 log | | 20 log |1 / |
20 log |1 / |
H H
K j j z
j p
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14.4 Bode Plots14.4 Bode Plots (2)(2)
1.1. A gain A gain KK
2.2. A pole (A pole (jj))-1-1 or zero ( or zero (jj) at origin) at origin
15
14.4 Bode Plots14.4 Bode Plots (3)(3)
3.3. A simple pole 1/(1+A simple pole 1/(1+jj/p/p11) or zero (1+) or zero (1+jj/z/z11))
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3.3. A quadratic pole 1/[1+A quadratic pole 1/[1+jj2222//nn+(+(jj//nn))22] or ] or zero [1+zero [1+jj2211//kk+(+(jj//kk))22]]
14.4 Bode Plots14.4 Bode Plots (4)(4)
17
14.4 Bode Plots14.4 Bode Plots (5)(5)
18
14.4 Bode Plots14.4 Bode Plots (6)(6)
19
14.4 Bode Plots14.4 Bode Plots (7)(7)Example 3Construct the Bode plots for the transfer function
200( )
( 2)( 10)
j
j j
H
20
14.4 Bode Plots14.4 Bode Plots (8)(8)Example 4Draw the Bode plots for the transfer function
5( 2)( )
( 10)
j
j j
H
21
14.4 Bode Plots14.4 Bode Plots (9)(9)Example 5Obtain the Bode plots for
2
10( )
( 5)
j
j j
H
22
14.4 Bode Plots14.4 Bode Plots (10)(10)Example 6Draw the Bode plots for
2
1( )
60 100
s
s s
H
23
14.3 Series Resonance (1)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.
)C
1L ( jRZ
Resonance frequency:
HzLC2
1f
rad/sLC
1
o
oro
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14.3 Series Resonance (2)14.3 Series Resonance (2)
The features of series resonance:
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.
)C
1L ( jRZ
25
14.3 Series Resonance (3)14.3 Series Resonance (3)
Bandwidth B
The frequency response of the resonance circuit current is )
C
1L ( jRZ
22
m
)C /1L (R
V|I|I
The average power absorbed by the RLC circuit is
RI2
1)(P 2
The highest power dissipated The highest power dissipated occurs at resonance:occurs at resonance: R
V
2
1)(P
2mo
26
14 3 Series Resonance (4)14 3 Series Resonance (4)
Half-power frequencies ω1 and ω2 are frequencies at which the dissipated power is half the maximum value:
The half-power frequencies can be obtained by setting Z
equal to √2 R.
4R
V
R
)2/(V
2
1)(P)(P
2m
2m
21
LC
1)
2L
R(
2L
R 21
LC
1)
2L
R(
2L
R 22 21 o
Bandwidth Bandwidth BBLR
ω ωB 12
27
14.3 Series Resonance (5)14.3 Series Resonance (5)
Quality factor,RLω
RILI o
02
21
221
/resonance at period one in
circuit the bydissipated Energycircuit the in stored energy Peak
Q
• 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.
The relationship between the B, Q and o:
CRωQω
LR
B oo 2
28
14.3 Series Resonance (6)14.3 Series Resonance (6)
Example 7
A series-connected circuit has R = 2 Ω, L = 1 mH, and C = 0.4 F.
a. Find the resonant frequency and the half-power frequencies.
b. Calculate the quality factor and bandwidth.
c. Determine the amplitude of the current at 0
, 1 and 2.
29
14.4 Parallel Resonance (1)14.4 Parallel Resonance (1)
Resonance frequency:
HzLCπ
fLC
ωo 2
11 o or rad/s
)L
1C ( j
R
1Y
It occurs when imaginary part of Y is zeroIt occurs when imaginary part of Y is zero
30
Summary of series and parallel resonance circuits:Summary of series and parallel resonance circuits:
14.4 Parallel Resonance (2)14.4 Parallel Resonance (2)
LC
1
LC
1
RC
1or
R
L
o
o
ω
ωRCor
L
Ro
o
Qo
Qo
2Q )
2Q
1( 1 2 o
o
2Q
)2Q
1( 1 2 o
o
2
Bo
2
Bo
characteristic Series circuit Parallel circuit
ωo
Q
B
ω1, ω2
Q ≥ 10, ω1, ω2
31
14.4 Parallel Resonance (3)14.4 Parallel Resonance (3)Example 8
In a parallel RLC circuit, R = 8 kΩ, L = 0.2 mH, and C = 8 F (a) Calculate 0, Q, and B. (b) Find 1 and 2 (c) Determine the power dissipated at 0, 1 and 2
32
14.4 Parallel Resonance (4)14.4 Parallel Resonance (4)Example 9
Calculate the resonant frequency of the circuit in the figure shown below.
rad/s2.1792
19AnswerAnswer::
33
14.5 Passive Filters (1)14.5 Passive Filters (1)• A filter is a circuit that
is designed to pass signals with desired frequencies and reject or attenuate others.
• Passive filter consists of only passive element R, L and C.
• There are four types of filters.
Low Pass
High Pass
Band Pass
Band Stop
34
14.5 Passive Filters (2)14.5 Passive Filters (2)Example 10
For the circuit in the figure below, obtain the transfer function For the circuit in the figure below, obtain the transfer function
VVoo(()/V)/Vii((). Identify the type of filter the circuit represents and ). Identify the type of filter the circuit represents and
determine the corner frequency. Take determine the corner frequency. Take RR1 1 = 100 = 100 = = RR22
and and LL =2 mH. =2 mH.
krad/s25AnswerAnswer::