CHAPTER 6Frequency Response,

Bode Plots, and Resonance

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。

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

Fourier Analysis (傅利葉分析 )

All real-world signals are sums of sinusoidal components having various frequencies, amplitudes, and phases.

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

)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

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 ,…)

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

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) 。

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.

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

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

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.

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.

Example 6.2 Using the Transfer Function with Several Input Components

)704000cos()2000cos(23)(in tttv Find vout(t).

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

2. Determine the (complex) value of the transfer function for each component.

4)0( H

303)1000(H

602)2000(H

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

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.

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 )

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

RCfB 2

1

BffjfH

1

1

定義

Q: 如何隨 frequency 改變畫出 ?

)(&)( fHfH

Magnitude and Phase

Plots

21

1

BfffH

B

BB

f

f

f

f

ffjfH

arctan

arctan0))/(1(

0

BffjfH

1

1

fH fH

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

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

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.

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

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

6.3 DECIBELS, THE CASCADE CONNECTION, AND LOGARITHMIC FREQUENCY SCALES

0,1dB

fHfH

fHfH log20dB

Decibel (dB) 分貝

0,1dB

fHfH

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.

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)

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

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

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

6.4 BODE PLOTSA Bode plot shows the magnitude of a network function in decibels versus frequency using a logarithmic scale for frequency.

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

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.

頻率每增加 10 倍

Magnitude 減少 20dB

Bff

dB3301.0102log101log102

dB

B

B

f

ffH

3. For

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

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

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.

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

Applications of Electronics and CircuitsHuman Computer Interface-Flex Sensor

Low Pass Filter

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

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

Magnitude and Phase Plots

fH fH3. For Bff

707.021 BfH 454590 BfH

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

Bff

dB3301.0102log1001log1002

dB

B

B

f

ffH

3. For

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(

)/(

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

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

)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');

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 .

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

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

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

整體阻抗 可改寫為

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

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

)(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

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

10sQ

Phasor Diagram

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

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

整體阻抗

可改寫為

輸出 phasor voltage

Bandwidth ( 頻寬 ):

sLH Q

fffB 0

B

fQ

Q

fB s

s

00

pQf

RL

02

Rf

QC p

02

Phasor diagram

Resonance in a mechanical system

Tacoma Narrow Bridge

http://www.youtube.com/watch?v=j-zczJXSxnw