chap5c oscillator, filter, and resonant circuit · 衰減係數）, and is a measure of how fast...

1

CHAP5c CHAP5c Oscillator, Filter, and Resonant circuit

謝志誠

2012/05/30

2

Oscillation Circuit

Solution of Second-Order Linear Differential Equations A second-order linear differential equation has the form

工程數學

where P(t), Q(t), R(t) and G(t) are constant functions.

(a)

If G(t) = 0 for all t, Equation (a) is homogenous linear equation. Thus, the form of a second-order linearhomogeneous differential equation is

(b)

)t(Gy)t(Rdtdy)t(Q

dtyd)t(P 2

2

0y)t(Rdtdy)t(Q

dtyd)t(P 2

2

二階線性微分方程式

線性齊式方程式線性齊式方程式

4

Solution of Second-Order Linear Differential Equations

TheoremTheorem If y1 and y2 are both solutions of the linear linear homogeneoushomogeneous equation (b) and c1 and c2 are any constants, then the function is also a solution of Equation (b).

工程數學

)t(yc)t(yc)t(y 2211

線性齊式方程式原理之一線性齊式方程式原理之一

若 y1與 y2 兩者都是方程式（b）的解，則y也是方程式（b）的解

Solution… y1、y2

Where a, b, and c are constants and a ≠ 0

where P(t), Q(t), and R(t) are constant functions.

General solution ?

Let y = ert. We have y' = rert y'' = r2ert

(c)

工程數學

0y)t(Rdtdy)t(Q

dtyd)t(P 2

2

Special case，求解…

假設equation (c) 具有指數函數解

……代入方程式（c）

6

Solution…

ert 不為零。因此，y = ert 要真的為equation (c) 的解，r 必須是ar2+br+c = 0 的根。

(d)

Equation (d) is called the Equation (d) is called the auxiliary equationauxiliary equation or or characteristic characteristic equationequation of the differential equation of the differential equation ayay””+by+by’’+c+c = 0.= 0.

工程數學

將 y = ert y’ = rert y” = r2ert 代入 equation (c)

0e)cbrar(or0cebreear

rt2

rtrtrt2

0cbrar2

這個要為零，才能符合！

r ????

（d）為（c）的輔助方程式或稱之為特徵方程式

Solution…

The roots r1 and r2 of the auxiliary equationauxiliary equation

CASE 1. bCASE 1. b22--4ac > 04ac > 0In this case the roots of the auxiliary equation are real and distinct, so y1 = er1t and y2 = er2t are two linearly independent solutions.

If the roots r1 and r2 of the auxiliary equationauxiliary equation ar2 + br + c = 0 are real and unequal, then the general solution of ay” + by’ + cy = 0 is y = c1er1t + c2er2t

(e)

工程數學

因此根r為…

第一種可能….

輔助方程式有兩個不同的實根…

方程式（c）的解可以如此寫……

88

Solution…

CASE 2. bCASE 2. b22--4ac = 04ac = 0In this case, r1 = r2 ; that is, the roots of the auxiliary equationauxiliary equationare real and equal. Let’s denote by r the common value of r1 and r2. From equation (e), we have

a2br

y2 = tertWe know that y1= ert is one solution.Another solution ?

If the auxiliary equationauxiliary equation ar2 + br + c = 0 has only one real root r, then the general solution of ay” + by’ + cy = 0 is y = c1ert + c2tert

工程數學

第二種可能….

輔助方程式有兩個相同的實根…

方程式（c）的解可以如此寫……

二個解？

9

Solution…

a2bac4

a2b 2

CASE 3. bCASE 3. b22--4ac < 04ac < 0In this case the roots of the auxiliary equation are complex numbers. We can writer1=α+i β r2=α-i βWhere α and β are real number

The solution of the differential equation

y = c1er1t + c2er2t =…=eαt (c1cos βt+c2sin βt)

工程數學

第三種可能….

輔助方程式有兩個相異複數根

二個解？

Solution…

If the roots of the auxiliary equationauxiliary equation ar2 + br + c = 0 are the complex numbers r1=α+i β, r2=α-i β, then the general solution of ay” + by’ + cy = 0 isy = c1er1t + c2er2t =…=eαt (c1cos βtx+c2sin β t)

工程數學

LC＆ RLC振盪電路

vL(t) vC(t)

t=0

VL(t)VC(t)

VR(t)

t=0

更新

輸入電壓源？沒有！

1212

合理懷疑合理懷疑

前面已經講過了嗎？前面已經講過了嗎？

前面，有輸入電壓源。前面，有輸入電壓源。

現在，沒有輸入電壓源。電路是靠著預先儲存在現在，沒有輸入電壓源。電路是靠著預先儲存在電容電容的電量來啟動。。

現在，我們要了解啟動後，現在，我們要了解啟動後，電路電流將如何變電路電流將如何變動？它的變動與電路上的動？它的變動與電路上的RR、、LL、、CC必然有關必然有關……

更新

LC振盪電路1/2

假設開關未接通時電容器上的電量

為Q0，開關接通後迴路電流為i(t)

By KVL and KCL vL(t) vC(t)

t=0

0)t(idt

)t(idLC1Let0)t(i

LC1

dt)t(id

dt)t(dvCi

dt)t(diLv

0ii0vv

202

2

202

2

CC

LL

CLCL

LC振盪電路2/2

vL(t) vC(t)

t=0

0)t(idt

)t(id 202

2

The complete solution to the differential equation is

tjtj 00 BeAe)t(i

A and B can be solved by considering the initial conditions.

?dt

)0t(di?)0t(i

LC振盪電路實際上並不存在。一般的振盪電路多少都會含有電阻，使得系統的振盪造成阻尼衰減。

15

RLC振盪電路1/4

假設開關未接通時電容器上的電量為Q0，開關接通後迴路電流為i(t)

VL(t)VC(t)

VR(t)

t=0

0)t(idt

)t(di2dt

)t(idLC1

L2RLet

0)t(iLC1

dt)t(di

LR

dt)t(id

0dt)t(iC1

dt)t(diL)t(iR

0vvv

202

2

20

2

2

CLR

(3)

(2)

(1)

RLC振盪電路2/4

αis called the neper frequency（納頻率）, or attenuation（衰減係數）, and is a measure of how fast the transient response of the circuit will die away after the stimulus has been removed .

ω0 is the angular resonance frequency（諧振頻率）. Define

LC

2R

0

Which is the damping factor.

在擾動移除之後，暫態反應消失的速度…

1717

RLC振盪電路3/4

The differential equation (3) has the characteristic equation

20

22

20

21

20

2

rr

0r2r

The general solution of equation (3)tr

2tr

121 eAeA)t(i

The coefficients A1 and A2 are determined by the initial conditions of the specific problem being analyzed. That is, they are set by the values of the currents and voltages in the circuit at the onset of the transient and the presumed value they will settle to after infinite time

係數由initial condition決定

1818

RLC振盪電路4/4

由於R、L、C值的不同，會有三種不同振盪的情形。

ζ>1 overdamped responseζ

Overdamped Response

The overdamped response is a decay of the transient current without oscillation.

t)12(020 eAeA)t(i

LC1

L2R1 2

t)1(1

LC1

L2R 2

0

LC

2R

0

沒振盪的衰減

Underdamped Response

The underdamped response is a decaying oscillation at frequency ωd . The oscillation decays at a rate determined by the attenuation α. The exponential in α describes the envelope of the oscillation.

20

220d

dt

3dt

2dt

1

1

)tsin(eB)t(i)tsin(eB)tcos(eB)t(iLC1

L2R1

called the damped resonance frequency called the damped resonance frequency or the damped natural frequency or the damped natural frequency

有振盪的衰減

阻尼諧振頻率－衰退震盪的頻率

Underdamped Response

http://upload.wikimedia.org/wikipedia/commons/0/0a/Underdamped_oscillation_xt.png

te

)tsin(e dt

Critically Damped ResponseCritically Damped Response

The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation.

This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting.

D1 and D2 are arbitrary constants determined by boundary conditions .

t2

t1 teDeD)t(iLC

1L2

R1

Critical damping vs. Over dampingCritical damping vs. Over damping

A critically damped system is one in which the system does not oscillate and returns to its equilibrium position without oscillating.

In an overdamped system the system does not oscillate and returns to its equilibrium position without oscillating but at a slower rate compared to a critically damped system.

回到平衡位置的速率，critically damped system比over damped system快

ωωdd vs. vs. ωω00damped resonance frequency damped resonance frequency ωωddthe frequency the circuit will naturally oscillate at if the frequency the circuit will naturally oscillate at if

not driven by an external sourcenot driven by an external source.. resonance frequency or resonance frequency or undampedundamped resonance resonance

frequency frequency ωω0 0 the frequency at which the circuit will resonate the frequency at which the circuit will resonate

when when driven by an external oscillation.driven by an external oscillation.

http://www.hydraulicspneumatics.com/200/Issue/Article/False/80810/Issue

EXAMPLE by EXAMPLE by MATHLABMATHLAB

1 mH

0.1 μF

300 Ω200 Ω100 Ω

LC

2R

0

使用三種不同的電使用三種不同的電阻，比較阻，比較responseresponse

2727

SinusoidalSinusoidalFrequency ResponseFrequency Response

2828



前面，輸入電壓源的頻率是固定。前面，輸入電壓源的頻率是固定。

現在，我們想藉由一個工具現在，我們想藉由一個工具（表示式），（表示式），通盤了通盤了解不同解不同頻率下頻率下，電路的輸出與輸入關係，電路的輸出與輸入關係。。

更新

29

Frequency ResponseFrequency Response

電路的「電路的「 sinusoidal frequency responsesinusoidal frequency response （或（或frequency responsefrequency response））」用來衡量一個電路在承受」用來衡量一個電路在承受到任意頻率（到任意頻率（振幅不變，頻率有變振幅不變，頻率有變）的輸入信號）的輸入信號

（（ sinusoidal input sinusoidal input ）時，會有怎樣的反應（）時，會有怎樣的反應（測量測量電路相對輸出端的響應電路相對輸出端的響應）？）？是動詞是動詞，，也是名詞也是名詞。。

只要知道電路的frequency responsefrequency response，只要知道，只要知道inputinput（包括（包括amplitude, phase, and frequencyamplitude, phase, and frequency ）），就，就可以推算可以推算outputoutput。。

30

定義定義 Frequency ResponseFrequency Response

以響應的幅度（用分貝）和相位（用弧度）來表示

)j(E)j(V)j(H

S

LV

就叫它為電路的frequency response

HVV )j(H)j(H

3131

表達表達 Frequency ResponseFrequency Response

以響應的幅度（用分貝）和相位（用弧度）來表示

HVV )j(H)j(H

32

EXAMPLE EXAMPLE 1/21/2

33

EXAMPLE EXAMPLE 2/22/2

把把loadload放回放回TheveninThevenin equivalent source circuitequivalent source circuit可以得到可以得到

S1S221SL

2L

THTL

LL

E)ZZ(Z)ZZZ(Z

ZZ

EZZ

ZV

)ZZ(Z)ZZZ(ZZZ)j(H

)j(E)j(V

1S221SL

2LV

S

L

The frequency response of the circuitThe frequency response of the circuit：：

3434

解讀解讀 Frequency ResponseFrequency Response

VL(jω) 與Es(jω)間的大小（magnitude）與相位角關係分別為amplitude-scaled與phase-shifted。

只要Hv(jω) 與Es(jω) 已知，就可以求VL(jω) 。

)j(E)j(V)j(H

S

LV

SHLSVL EHV

HVV )j(H)j(H

3535

EXAMPLEEXAMPLE：：Computing the Computing the Frequency Response of a CircuitFrequency Response of a Circuit

3636

SolutionSolution

3737

FiltersFilters

3838

Low Pass Filter Low Pass Filter 1/41/4

3939


4040


在濾波器中，會將此頻率點以上在濾波器中，會將此頻率點以上（（或以下或以下））頻率成份濾掉。頻率成份濾掉。

在該頻率附近，在該頻率附近，filterfilter開始開始filter out the higher filter out the higher frequency signal.frequency signal.

4141


4242

Frequency Response PlotFrequency Response Plot

The frequency response plots are The frequency response plots are commonly employed to describe commonly employed to describe the frequency response of a the frequency response of a circuit, since they can provide circuit, since they can provide the effect of a filter on an the effect of a filter on an exciting signalexciting signal（（The frequency The frequency plotplot可以顯示可以顯示filterfilter的功用）。的功用）。

This RC filter has the property of This RC filter has the property of ““PASSINGPASSING”” signals at low signals at low frequencyfrequency（（1/RC）。）。>>>>>>>>因此，這種因此，這種filterfilter稱為稱為LOW LOW PASS FILTERPASS FILTER。。

Value of HValue of H（（jj））at the cutoff frequency is 0.707.at the cutoff frequency is 0.707.

4343

High Pass Filter High Pass Filter 1/31/3

4444


4545


4646

BAND Pass Filter BAND Pass Filter 1/101/10

Passing mainly those frequencies within a Passing mainly those frequencies within a certain frequency rangecertain frequency range.

是濾波器的一種，它只留下是濾波器的一種，它只留下Cutoff FrequencyCutoff Frequency左右頻範圍左右頻範圍的部分，而將其他的頻率去的部分，而將其他的頻率去掉；反之，亦然。掉；反之，亦然。


A band-pass filter is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. An example of an analogue electronic band-pass filter is an RLC circuit

http://en.wikipedia.org/wiki/Band-pass_filter


http://www.electronics-tutorials.ws/filter/filter_4.html

4949


The related frequency response function for The related frequency response function for the filter isthe filter is：：

)j(VV)j(H

i

O

LC)j(CRj1CRj)j(V

LjCj

1R

R)j(V)j(V

2i

iO

其中，

5050


The frequency response of the RLC filter isThe frequency response of the RLC filter is：：

LC1j

LR

jLR

LC)j(CRj1CRj)j(

VV

22

i

O

222

LR

LC1

LR

)j(H

2

o

LC1

LR

arctan90)j(H H


At resonance frequency, the frequency response frequency response function function will be real.

LC21f

LC10

Cj1Lj 00

)j(H will be Max. at )j(H 0

1

LR

LC1

LR

)j(H2

0

22

0

0

0


Set to find cutoff frequencies.max21 )j(H

2

C

22

C

C

LR

LC1

LR

21

22

c

22

c LC1

L2R

L2R

LC1

L2R

L2R

21

LC1

21 cc0


Bandwidth

The Quality factor QLRBW

12 cc

CL

R1

BWQ 0

Where Where ωωc1c1 and and ωωc2c2 are the two frequencies that determine the are the two frequencies that determine the passpass--band band （（bandband--widthwidth））of the filterof the filter（即可以通過（即可以通過input input signalsignal的頻率範圍）。的頻率範圍）。

5454


55


當當ωω00，，capacitorcapacitor形同形同openopen--circuitcircuit，，the response of the the response of the filter is ZEROfilter is ZERO！！

當當ωω∞∞，，inductorinductor形同形同openopen--circuitcircuit，，the response of the the response of the filter is ZEROfilter is ZERO！！

因此，低頻部分與高頻部分的因此，低頻部分與高頻部分的input signalinput signal是無法過過是無法過過bandband--pass filterpass filter。。

5656

Q vs. Freq. ResponseQ vs. Freq. Response

QQ值越大，值越大，the sharpness of the the sharpness of the resonanceresonance越大，越大，bandband--pass filterpass filter變得變得更具選擇性，能通過更具選擇性，能通過bandband--pass filterpass filter的的input signalinput signal的頻率將僅侷限於的頻率將僅侷限於resonant frequencyresonant frequency附近。附近。

BandwidthBandwidth，或，或width of the passwidth of the pass--bandband，一個用來衡量選擇範圍的指，一個用來衡量選擇範圍的指標：標：

QQ越大，越大，BWBW越小；越小；QQ越小，越小，BWBW越大。越大。

LR

QBW 0

57

Example 5.14Example 5.14

58


59


60

Bode PlotBode Plot

Bode plotBode plot是另外一種用是另外一種用來表示來表示frequency responsefrequency response的方法，其水平軸為的方法，其水平軸為

frequency on logarithm frequency on logarithm scalescale，垂直軸為，垂直軸為amplitude of the frequency amplitude of the frequency response in units of response in units of decibelsdecibels（（dBdB）。）。

LowLow--pass and highpass and high--pass filterpass filter的的dB magnitude plotsdB magnitude plots

61

WhatWhat’’s Decibel s Decibel （（dBdB））

The ratio The ratio is given in units of dBis given in units of dBin

out

VV

in

out10

dBin

out

VVlog20

VV

Where

62

Bode PlotBode Plot

Low frequency partLow frequency part（（lowlow--pass pass filterfilter）及）及high frequency parthigh frequency part（（highhigh--pass filterpass filter）為一水平）為一水平線，在該範圍內其線，在該範圍內其amplitude amplitude responseresponse大小為大小為11。此外，在。此外，在cutoff frequencycutoff frequency，，wwoo，附近，，附近，ResponseResponse開始下降或上升的斜率開始下降或上升的斜率為為±±20dB/decade20dB/decade（－為（－為lowlow--pass pass filterfilter，＋為，＋為highhigh--pass filterpass filter）。）。

63

WhatWhat’’s Decades Decade

EXAMPLE by EXAMPLE by MATHLABMATHLAB

R=143.25 Ω , L=2.533 mH, C=1μF, Plot f=50~100 KHz. 改變R？

R=143.25 R=143.25 ΩΩ , L=2.533 , L=2.533 mHmH, C=1, C=1μμF, F, Q = 0.3513, Q = 0.3513, ffoo = 1.987= 1.987××101044 HzHz

R=50 R=50 ΩΩ , L=2.533 , L=2.533 mHmH, C=1, C=1μμF, F, Q = 1.006578 , Q = 1.006578 , ffoo = 1.987= 1.987××101044 HzHz

R=250 R=250 ΩΩ , L=2.533 , L=2.533 mHmH, C=1, C=1μμF, F, Q = 0.201315 , Q = 0.201315 , ffoo = 1.987= 1.987××101044 HzHz

6868

Resonant Circuit

6969

Series RLC Circuit Series RLC Circuit

)XX(jRZ CL

7070



前面，輸入電壓源的頻率是固定。前面，輸入電壓源的頻率是固定。

現在，輸入電壓源的頻率非固定，導致現在，輸入電壓源的頻率非固定，導致total total impedance Zimpedance Z與電流均非固定，且與頻率有關。與電流均非固定，且與頻率有關。

現在，諧振電路，就是要找到一個特定的頻率，現在，諧振電路，就是要找到一個特定的頻率，讓電路進入諧振狀態。在該狀態下，讓電路進入諧振狀態。在該狀態下， total total impedance Zimpedance Z最小（串聯）或最大（並聯）。最小（串聯）或最大（並聯）。

更新

7171

ImpedanceImpedance

2CL

2

CL

)XX(RZ

)C

1L(jRZ

)XX(jRZ

某個ω，Z最小。

7272

Impedance vs. FrequencyImpedance vs. Frequency

-150

-100

-50

0

50

100

150

1000

2000

3000

4000

5000

6000

7000

8000

9000

1000

011

000

1200

013

000

1400

015

000

1600

017

000

1800

019

000

2000

021

000

2200

023

000

2400

025

000

Frequency

Impe

danc

e

XL XC

XL-XC

7373

CurrentCurrent

)tsin(I2)t(iZE)j(I

)C

1L(R)C

1L(jRZ

0VE

m

22

22

m

C1LR

VI

)

RC

1L(tan 1

7474

Current vs. Frequency

7575

Resonance Series RLC CircuitResonance Series RLC Circuit

The resonance of a series RLC circuit occurs when the inductive and capacitive reactances are equal in magnitude but cancel each other because they are 180 degrees apart in phase. 串聯RLC電路發生共振的條件？電容與電感的reactance相等！

7676

Resonance Series RLC CircuitResonance Series RLC Circuit

0

7777

Resonant Frequency

The condition of resonance

7878

At Resonance

The impedance of the network

The power delivered

共振條件下，電路的阻抗共振條件下，電路的阻抗（虛部為零），即（虛部為零），即net net reactancereactance為零，即為零，即power power factor=1factor=1

RE

ZEI

Tmax

7979

BANDWIDTH

電流降低到電流降低到0.7070.707××peak valuepeak value水準時的頻率水準時的頻率ff11與與ff22稱為稱為halfhalf--powerpower或或cutoffcutoff或或band frequenciesband frequencies。。ff 11、、ff 22與共振頻率與共振頻率ff00等距等距離。兩者的間距稱為離。兩者的間距稱為BWBW。電流降低到。電流降低到0.7070.707××peak valuepeak value水準水準時的功率；為共振頻率下的功率的一半。時的功率；為共振頻率下的功率的一半。

共振條件下，共振條件下，impedanceimpedance最小最小

Selectivity curveSelectivity curve：選擇：選擇可以通過的頻率範圍。可以通過的頻率範圍。

頻率越低，電容的電抗增頻率越低，電容的電抗增加的幅度，高於電感的電加的幅度，高於電感的電抗，整體阻抗增加，電流抗，整體阻抗增加，電流降低。降低。

頻率越高，電感的電抗增加的頻率越高，電感的電抗增加的幅度，高於電容的電抗，整體幅度，高於電容的電抗，整體阻抗增加，電流降低。阻抗增加，電流降低。

8080

BANDWIDTHBANDWIDTH

0.7070.707的位置的位置

8181

Quality factor Q Quality factor Q －－ selectivityselectivity

The quality factor Q is defined by

where Δω is the width of the resonant power curve at half maximum. Since that width turns out to be Δω = R/L, the value of Q can also be expressed as

R越大，Q越小

LR

12

8282

Q vs. SelectivityQ vs. Selectivity

A "quality factor" Q is a measure of that selectivity, and we speak of a circuit having a "high Q" if it is more narrowly selective.

QQ值越大，值越大，the sharpness of the resonancethe sharpness of the resonance越大，越大，bandband--pass pass filterfilter變得更具選擇性，也就是說能通過變得更具選擇性，也就是說能通過bandband--pass filterpass filter的的input signalinput signal的頻率將僅侷限於的頻率將僅侷限於resonant frequencyresonant frequency附近。附近。

8383

Selectivity vs. RSelectivity vs. R

The selectivity of a circuit is dependent upon the amount of resistance in the circuit.

High Q

R越大，Q越小

8484


8585


8686

Parallel RLC CircuitParallel RLC Circuit

R1G

ZVI

)BB(jG1

)L

1C(jG

1Y1Z

LC

Parallel RLC CircuitParallel RLC Circuit 更新

...RCjLC

LjR

CjLjR

11Z 2

Focus on the condition

10R

XQ LP

8888

ImpedanceImpedance 更新

...RCjLC

LjR

CjLjR

11Z 2

RXQ LP

8989

Voltage Voltage

9090

Resonance Parallel RLC CircuitResonance Parallel RLC Circuit

The resonance of a parallel RLC circuit occurs when the inductive and capacitive reactancesare equal in magnitude but cancel each other because they are 180 degrees apart in phase. 並聯RLC電路發生共振的條件？電容與電感的reactance相等！

9191

BANDWIDTHBANDWIDTH

LC21fp

9292


9393


9494


chap5c oscillator, filter, and resonant circuit · 衰減係數）, and is a measure of how fast...

Documents