CHAPTER 6 Frequency Response, Bode Plots, and Resonance.
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Transcript of CHAPTER 6 Frequency Response, Bode Plots, and Resonance.
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CHAPTER 6Frequency Response,
Bode Plots, and Resonance
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6.1 Fourier Analysis, Filters, and Transfer Functions
•大部分蘊藏資訊的訊號都不是單純的sinusoidal signals 。•但是我們可以藉由將不同大小 (amplitude)、頻率 (frequency) 與相位 (phase) 的sinusoidal signals 加 (adding) 在一起而獲得與這些訊號一模一樣的訊號 。
我們可以藉由結合不同 sinusoids 成份 (component) 來組成 (construct) 所有訊號。
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一小段 music waveform
三個不同大小、頻率與相位的 sinusoidal 成份(component) 相加可獲得上面的 music waveform
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Fourier Analysis (傅利葉分析 )
All real-world signals are sums of sinusoidal components having various frequencies, amplitudes, and phases.
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Fourier Series (傅利葉序列 ) of a Square Wave
)5sin(5
44)3sin(
3
44)sin(
44)( 000 ttttvsq
方波 (square wave) Vsq(t) 可由以下之 sinusoids 來組成
稱為 fundamental angular frequency ( 基本角頻率 ).
T/20
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)5sin(5
44)3sin(
3
44)sin(
44)( 000 ttttvsq
方波為奇數倍基本角頻率 ( ) 的 sinusoidal components 組成。•component 角頻率越高時,其 amplitude( 大小 ) 越小。•所有 component 相位都是 -90o ( ) 。
000 5,3,
更多 Components 相加可更近似方波
)90cos()sin( tt
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Filters (濾波器 )Filters process the sinusoid components of an input signal differently depending on the frequency of each component. ( 濾波器對輸入訊號之不同頻率成份進行不同處理 )
Often, the goal of the filter is to retain ( 維持 )the components in certain frequency ranges and to reject ( 去除 )components in other ranges. (ex 收音機頻道、音響等化器、生理量測測儀 ,…)
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一般而言雜訊在高頻,因此低通率波器用來消除雜訊。 Ex, 可藉由 LC impedance 隨頻率改變的特性藉由RLC 來實現慮波器。
902
190
1
90290
fCCZ
fLLZ
C
L
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Transfer Functions (轉換函數 )The transfer function H(f ) of the
two-port filter is defined to be the ratio ( 比值 ) of the phasor output voltage to the phasor input voltage as a function of frequency:
in
out
V
VfH
Transfer functions 是一個複數,包含 magnitude 與phase 且 magnitude 與 phase 也都是頻率的函數(function of frequency) 。
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The magnitude of the transfer function shows how the amplitude of each frequency component is affected by the filter.
Similarly, the phase of the transfer function shows how the phase of eachfrequency component is affected by the filter.
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Example 6.1 Using the Transfer Function to Determine the Output.
Input Find the output of the filter ( ) .
)402000cos(2)(in ttv )(out tv
)()()( fHfHfH
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402)402000cos(2)(in ttv 20002 f Hz1000f
3)1000( H 30)( fH
in
outHV
V303)1000(
706402303V)1000(V inout H
)702000cos(6)(out ttv
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Determining the output of a filter for an input
with multiple components:
1. Determine the frequency and phasor representation for each input component.
2. Determine the (complex) value of the transfer function for each component.
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3. Obtain the phasor for each output component by multiplying the phasor for each input component by the corresponding transfer-function value.
4. Convert the phasors for the output components into time functions of various frequencies. Add these time functions to produce the output.
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Example 6.2 Using the Transfer Function with Several Input Components
)704000cos()2000cos(23)(in tttv Find vout(t).
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1. Determine the frequency and phasorfor each input component.
3)(in1 tv
)704000cos()2000cos(23)(in tttv
)2000cos(2)(in2 ttv
)704000cos()(in3 ttv
03Vin1
02Vin2
701Vin2
Hz0in1 f
Hz1000in2 f
Hz2000in3 f
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2. Determine the (complex) value of the transfer function for each component.
4)0( H
303)1000(H
602)2000(H
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3. Obtain the phasor for each output component 1234)0( 11 inout vHv
30602303V)1000(V 22 inout H
102701602V)2000(V 33 inout H
4. Convert the phasors into time functions of various frequencies. Add these time functions to produce the output. 12)(1 tvout
)302000cos(6)(2 ttvout )104000cos(2)(3 ttvout
)()()()( 321 tvtvtvtv outoutoutout )104000cos(2)302000cos(612 tt
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Linear circuits
1. Separate the input signal into components having various frequencies.
2. Alter ( 改變 ) the amplitude and phase of each component depending on its frequency.
3. Add the altered components to produce the output signal.
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6.2 FIRST-ORDER
LOWPASS FILTERS (一階低通濾波器 )
We can determine the transfer functions of RLC circuits by using steady-state analysis with complex impedances as a function of frequency
( 可利用 steady-state 分析並將阻抗表示為頻率函數的複數阻抗以決定 RLC 的轉換函數 )
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決定 , 並隨 frequency 改變畫出 in
out
V
VfH )(&)( fHfH
fCjCjCjC 2
111Z
)R
V(IV in
outC
CC ZZZ
πfRCjZRR
fH
C
21
1
1
1
Z
Z
V
V)(
C
C
in
out
fCjC
2Z
1
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RCfB 2
1
BffjfH
1
1
定義
Q: 如何隨 frequency 改變畫出 ?
)(&)( fHfH
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Magnitude and Phase
Plots
21
1
BfffH
B
BB
f
f
f
f
ffjfH
arctan
arctan0))/(1(
0
BffjfH
1
1
fH fH
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Magnitude and Phase
Plots 21
1
BfffH
Bf
ffH arctan
fH fH
1. For low frequency 0f 1fH 0 fH
2. For high frequency Bff 0fH 90 fH
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Magnitude and Phase
Plots fH fH
3. For Bff
707.021 BfH 45 BfH
2/12 BfH
Bff 稱為 half-power frequency
5.02
in
2
out2 V
VBfH
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Example 6.3 Calculation of RC Lowpass Output
)2000cos(5)200cos(5)20cos(5)(in ttttv
Find vout(t).
The RC lowpass filter is shown in Figure 6.9.
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Example 6.3 Calculation of RC Lowpass Output
Hz 1001010)2/1000(2
1
2
16
πRCfB
)20cos(5)(in1 ttv
)fj(ffH
B/1
1)(
05Vin1 Hz102
20
2in1
f1. The input components
)200cos(5)(in2 ttv 05Vin2 Hz100in2 f
)2000cos(5)(in3 ttv 05Vin3 Hz1000in3 f
2. The transfer function
71.59950.01.0arctan)1.0(1
1
100/101
1)10(
2)j(H
457071.0100/1001
1)100(
)j(H
29.840995.0100/10001
1)1000(
)j(H
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3. The output components
71.5975.4)05()71.59950.0( V)10(V in1out1 H
)71.520cos(975.4)(out1 ttv
45535.3)05()457071.0( V)100(V in2out2 H
)45200cos(535.3)(out2 ttv
29.844975.0)05()29.840995.0(V)1000(V in3out3 H
)29.842000cos(4975.0)(out3 ttv
)29.842000cos(4975.0)45200cos(535.3)71.520cos(975.4)(out ttttv
Note: We do not add the phasors for components with different frequencies
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6.3 DECIBELS, THE CASCADE CONNECTION, AND LOGARITHMIC FREQUENCY SCALES
0,1dB
fHfH
fHfH log20dB
Decibel (dB) 分貝
0,1dB
fHfH
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Why using dB?Very small and very large magnitudes can be displayed clearly on a single plot.
Linear scale 但圖上看不出大小
410)60( H
dB scale可清楚看出 dB85)60(
dBH
Notch filter: 消除某些頻率 component, 60Hz in this case.
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THE CASCADE CONNECTION ( 串接 )
In cascade connection, the output of one filter is connected to the input of a second filter( 前級 filter 的輸出是後級 filter 的輸入 )
fHfHfHout
out
in
out
in
out21
1
2
1
1
V
V
V
V
V
V
H(f)
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fHfHfH 21
In dBs, the individual transfer-function magnitudes are added to find the overall transfer-function magnitude
THE CASCADE CONNECTION ( 串接 )
(f)H(f)HH(f) 21log20log20
(f)H(f)H 21 log20log20 baba loglog)log(
dBdBdB(f)H(f)HH(f) 21
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Logarithmic Frequency Scales•On a logarithmic scale, the variable is
multiplied by a given factor for equalincrements of length along the axis.
•The advantage of a logarithmic frequency scale compared with linear scale is that the variations of a transfer function for a low range of frequency and the variations in a high range can be shown on a single plot.
×10 ×2
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A decade is a range of frequencies for which the ratio of the highest frequency to the lowest is 10.
1
2log decades ofnumber f
f
An octave is a two-to-one change in frequency.
2log
loglog octaves ofnumber 12
1
22
ff
f
f
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6.4 BODE PLOTSA Bode plot shows the magnitude of a network function in decibels versus frequency using a logarithmic scale for frequency.
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A Bode plot of the first-order lowpass transfer function ( 先 R 後 C)
BffjfH
1
1
222
2
2
1log101log2
201log200
1log201log201
1log20
BBB
B
B
dB
ffffff
ffff
fH
xxx log2
1loglog 2/1
2
dB1log10
Bf
ffH
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A Bode plot of the first-order lowpass transfer function- Magnitude plot
1. For low frequency )0or ( fff B
2. For high frequency Bff
01log101log102
dB
Bf
ffH
BBB f
f
f
f
f
ffH log20log101log10
22
dB
低頻漸近線 (low-frequency asymptote) 為水平直線
高頻漸近線 (high-frequency asymptote) 為斜率 -20dB/decade 之直線 , 起始於 f=fB, 故 fB 稱為 corner frequency or break frequency.
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頻率每增加 10 倍
Magnitude 減少 20dB
Bff
dB3301.0102log101log102
dB
B
B
f
ffH
3. For
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1. A horizontal line at zero for f < (fB /10).
2. A sloping line from 0 phase at fB /10 to –90° at 10fB.
3. A horizontal line at –90° for f > 10fB.
A Bode plot of the first-order lowpass transfer function- Phase plot
Bf
ffH arctan
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Exercise 6.11 Sketch approximate straight-line Bode magnitude and phase plots
A 先 R 後 C circuit
Lowpass filter
Bin
out
/11
211
)2/(1)2/(1
)(fjfRCfjfCjR
fCjfH
ππ
π
VV
0Hz10010)2/1000(2
1
2
16
πRCfB
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The magnitude plot is approximated by 1. 0 dB below 1000 Hz 2. A straight line sloping downward at 20 dB/decade above
1000 Hz.
The phase plot is approximated by 1. 0obelow 100 Hz2. -90o above 10 kHz 3. a line sloping downward between 0o at 100 Hz and -90o
at 10 kHz.
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手握拳 ( 手指彎曲度最大 )
手攤平 ( 手指彎曲度最小 )
Applications of Electronics and CircuitsHuman Computer Interface-Flex Sensor
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Applications of Electronics and CircuitsHuman Computer Interface-Flex Sensor
Low Pass Filter
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6.5 FIRST-ORDER HIGHPASS FILTERS ( 先 C 後
R)
RCfB 2
1
)R
V(IV in
outCZ
RR
)(1
)(
)(1V
V)(
in
out
B
B
C
C
C ffj
ffj
ZR
ZR
RZ
RfH
BC f
fjfRCj
R 2
Z
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Magnitude and Phase
Plots 21 B
B
ff
fffH
BB
B
f
f
ffj
ffjfH arctan90
))/(1(
)/(
fH fH
1. For low frequency 0f 0fH 90 fH
2. For high frequency Bff 1fH 0 fH
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Magnitude and Phase Plots
fH fH3. For Bff
707.021 BfH 454590 BfH
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A Bode plot of the first-order highpass transfer function- Magnitude plot
1. For low frequency )0or ( fff B
2. For high frequency Bff
0log20log20dB
BB f
f
f
ffH
低頻漸近線 (low-frequency asymptote) 為斜率 +20dB/decade 之直線
]1log[10log20)/(1
/log20
2
2
BBB
BdB f
f
f
f
ff
ffH(f)
BdB f
fH(f) log20
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Bff
dB3301.0102log1001log1002
dB
B
B
f
ffH
3. For
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與 lowpass filter 類似 , 只是加了 90o.
A Bode plot of the first-order highpass transfer function- Phase plot
BB
B
f
f
ffj
ffjfH arctan90
))/(1(
)/(
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Example 6.4 Determine the break frequency of a highpass filterDetermine the break frequency of a first-order highpass filter such that the transfer-function magnitude at 60 Hz is -30 dB.
)60
log(2030Bf
BdB f
fH(f) log20
6.311060
5.1 Bf
Hz 1900Bf
When )0or ( fff B
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Appendix-Fourier Series of a Square Wave
)5sin(5
44)3sin(
3
44)sin(
44)( 000 ttttvsq
方波 (square wave) Vsq(t) 可由以下之 sinusoids 來組成
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)5sin(5
44)3sin(
3
44)sin(
44)( 000 ttttvsq
% Matlab Code T = 1; % Period = 1st=0:0.01:3*T; % t=0-3s, delta t=0.1sOmega0 = 2*pi/T; % Omega0=2*pi*f=2*piVsq_3=[zeros(1,length(t))]; % vector for the sum of the first 3 termsVsq_9 =[zeros(1,length(t))]; % vector for the sum of the first 9 terms
for i=1:3Vsq_3=Vsq_3+(44/((2*i-1)*pi))*sin(Omega0*(2*i-1)*t);endfor i=1:9Vsq_9=Vsq_9+(44/((2*i-1)*pi))*sin(Omega0*(2*i-1)*t);end
subplot(2,1,1)plot(t, Vsq_3);xlabel('Time (sec)');ylabel('Vsq3');subplot(2,1,2)plot(t, Vsq_9);xlabel('Time (sec)');ylabel('Vsq9');
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6.6 Series Resonance (串聯諧振 )
•The resonant circuit behaves as a bandpass filter ( 帶通濾波器 ) 。
•A band of components centered at the resonant frequency is passed ( 在共振頻率附近的頻帶會通過 ) 。•The components farther from the resonant frequency are rejected/reduced .
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Bandwidth ( 頻寬 ): B
f0 :fundamental frequencyQ: quality factor ( 品質因子 )
Qs 越大通過的頻寬越窄 (B 越小 ) ,越具選擇性,品質越好
fH , fL 代表高低兩個 half-power frequency
f0
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The series resonant circuit
整體阻抗
在共振頻率出現時,電容與電感的阻抗 magnitude 相同,整體阻抗只剩電阻,電阻電壓最大 ( 輸出最大 )
LCf
22
0 4
1
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For a series circuit, the quality factor ( 品質因子 ) is defined as
( 電感電抗與電阻的比值 )
因為 ,, 所以 Qs 也可表示成
resistance
inductance theof reactance
0
2f
fRjQfLj s
f
fRjQ
fCj s
0
2
1
sQ
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整體阻抗 可改寫為
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f
f
f
fQZ ss
0
0
arctan)2(12
20
20
22
f
f
f
fQ
R
Zs
s
2. For low frequency 0f
1. For 0ff
1R
Zs 00arctan sZ
)2(12
20
20
22
f
f
f
fQ
R
Zs
s
2)arctan(arctan 0
0
f
f
f
fQZ ss
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f
f
f
fQZ ss
0
0
arctan)2(12
20
20
22
f
f
f
fQ
R
Zs
s
3. For high frequency f
)2(12
20
20
22
f
f
f
fQ
R
Zs
s
2)arctan(arctan 0
0
f
f
f
fQZ ss
Impedance magnitude 在 最小,而 越大斜率越大。
0ff sQ
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)(1
1
0
0 ff
ff
jQZ
R
Vs
V
ss
R
)2(1
1
2
20
20
22
ff
ff
QZ
R
V
V
sss
R
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Bandwidth ( 頻寬 ):
fH , fL
f0
2
1
)2(1
1
2
20
20
22
ff
ff
QV
V
ss
R
sLH Q
f
L
RffB 0
2
When Qs >>1, are given by the approximate expressions
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10sQ
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Phasor Diagram
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6.7 Parallel Resonance
整體阻抗
在共振頻率出現時,電容與電感的阻抗 magnitude 相同,整體阻抗只剩電阻
fLj
fLj 2
1
2
1
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For a parallel circuit, the quality factor is defined as
( 與 series circuit 相反 )
inductance theof reactance
resistance
sQ
亦可表示成
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整體阻抗
可改寫為
輸出 phasor voltage
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Bandwidth ( 頻寬 ):
sLH Q
fffB 0
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B
fQ
Q
fB s
s
00
pQf
RL
02
Rf
QC p
02
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Phasor diagram
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Resonance in a mechanical system
Tacoma Narrow Bridge
http://www.youtube.com/watch?v=j-zczJXSxnw