課件考題電工基礎(chǔ)

第8章非正弦周期電流電路重點(diǎn)周期函數(shù)分解為付里葉級(jí)數(shù)非正弦周期函數(shù)的有效值和平均功率非正弦周期電流電路的計(jì)算返

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頁(yè)8.1

非正弦周期信號(hào)生產(chǎn)實(shí)際中不完全是正弦電路，經(jīng)常會(huì)遇到非正弦周期電流電路。在電子技術(shù)、自動(dòng)控制、計(jì)算機(jī)和無(wú)線電技術(shù)等方面，電壓和電流往往都是周期性的非正弦波形。非正弦周期交流信號(hào)的特點(diǎn)不是正弦波按周期規(guī)律變化例1半波整流電路的輸出信號(hào)f

(t

)

=

f

(t

+

kT

)返回上頁(yè)下頁(yè)例2示波器內(nèi)的水平掃描電壓周期性鋸齒波返回上頁(yè)下頁(yè)脈沖電路中的脈沖信號(hào)Tt例2返回上頁(yè)下頁(yè)交直流共存電路Es+V例4返回上頁(yè)下頁(yè)函數(shù)同頻）二次諧波（2倍頻）直流分量高次諧波￥f

(t)

=

A0

+

Akm

cos(kw

1t

+fk

)k

=18.2周期函數(shù)分解為付里葉級(jí)數(shù)基波（和原f

(t

)

=

A0

+

A1m

cos(w

1t

+f1)

++

A

cos(2w

1t

+f2

)

+2m+

Anm

cos(nw

1t

+fn

)

+周期函數(shù)展開(kāi)成付里葉級(jí)數(shù)：返回上頁(yè)下頁(yè)￥f

(

t)

=

a0

+

[ak

cos

kw

1

t

+bk

sin

kw

1t]Akm

cos(kw

1

t

+

jk

)

=

ak

cos

kw

1

t

+

bk

sin

kw

1

t也可表示成：kkkkm

k

ka-

bA

=

a2

+

b2k

=1A0

=

a0f

=

arctanbk

=

-

Akm

sinfkak

=

Akm

cosfk系數(shù)之間的關(guān)系為返回上頁(yè)下頁(yè)p02p02p11000p=

1=

1=

1f

(

t

)

sin

kw

1

td(w

1

t

)btd(w

t

)f

(t

)

cos

kwaf

(

t

)d

tTA

=

akkT求出A0、ak、bk便可得到原函數(shù)f(t)的展開(kāi)式。系數(shù)的計(jì)算：返回上頁(yè)下頁(yè)利用函數(shù)的對(duì)稱性可使系數(shù)的確定簡(jiǎn)化（1）偶函數(shù)－T/2tT/2f(t)kf

(t

)

=

f

(-t

)

b

=

0－T/2tT/2f(t)ak

=

0f

(t)

=

-

f

(t)（2）奇函數(shù)（3）奇諧波函數(shù)a2k

=

b2k

=

0Tf

(t

)

=

-

f

(t

+

)2tf

(t)T/2T返回上頁(yè)下頁(yè)tT/2

TiSmI周期性方波信號(hào)的分解例1解圖示矩形波電流在一個(gè)周期內(nèi)的表達(dá)式為：<

t

<

TTTS20i

(t

)

=

Im

0

<

t

<

2100TmSOTT=

1T

/

2

II

dt

=

mi

(t)

dt

=直流分量：Ip2p0=

12i

(w

t)

sin

kw

td

(w

t)諧波分量：

bSK=pk=

2Im

kp0

K為偶數(shù)

K為奇數(shù)Im

1(-

cos

kw

t

)p0返回上頁(yè)下頁(yè)00=

2pp1

sin

kwt

p

=

0k2pi

(w

t

)

cos

kwtd

(w

t

)a=

2ImSkKKb2=

2Im+

a2

=

bK

KA

=（K為奇數(shù)）KKbkpy

=

arctan

aK

=

0is

的展開(kāi)式為：3 5i=

ImS+

2Im

(sinw

t

+

1

sin

3w

t

+

1

sin

5w

t

+)2 p返回上頁(yè)下頁(yè)ttt基波直流分量三次諧波五次諧波七次諧波周期性方波波形分解返回上頁(yè)下頁(yè)直流分量+基波三次諧波直流分量基波直流分量+基波+三次諧波返回上頁(yè)下頁(yè)3

5Si

=

Im

+

2Im

(sin

wt

+

1

sin

3wt

+

1

sin

5w

t

+)2

ptT/2

TiSImIS0is1

is3

is5等效電源IS0is1is3is5返回上頁(yè)下頁(yè)3

5Si

=

Im

+

2Im

(sin

wt

+

1

sin

3wt

+

1

sin

5w

t

+)2

ptT/2

TiSImw

3w

5w

7wAkm0矩形波的頻譜圖返回上頁(yè)下頁(yè)給定函數(shù)

f(t)的部分波形如圖所示。為使f(t)

的傅立葉級(jí)數(shù)中只包含如下的分量：Of(t)tT/4OT/4

t(1)正弦分量；f(t)T/2-

T/2

-

T/4(1)

正弦分量； (2)

余弦分量；(3)正弦偶次分量；(4)余弦奇次分量。試畫出f(t)的波形。例2解返回上頁(yè)下頁(yè)(2)余弦分量；tT/4Of(t)T/2-

T/2

-

T/4(3)正弦偶次分量；(4)余弦奇次分量。tT/4Of(t)T/2-

T/2

-

T/4tT/4Of(t)T/2-

T/2

-

T/4返回上頁(yè)下頁(yè)02p

2p0cos

kwtd(w

t

)

=

0sin

kwtd(w

t

)

=

08.2有效值、平均值和平均功率1.

三角函數(shù)的性質(zhì)（1）正弦、余弦信號(hào)一個(gè)周期內(nèi)的積分為0。k整數(shù)（2）sin2、cos2

在一個(gè)周期內(nèi)的積分為p。2p022p02wt

)

=

pwtd(cos

kwtd(w

t

)

=

psin

k返回上頁(yè)下頁(yè)（3）

三角函數(shù)的正交性0sin

pwtd(w

t

)

=

0

cos

pwtd(w

t

)

=

0

sin

pwtd

(w

t

)

=

0cos

kwtcos

kwtsin

kwt02p02p2p(k

?

p)返回上頁(yè)下頁(yè)2.

非正弦周期函數(shù)的有效值￥k

=1i(t

)

=

I0

+

Ikm

cos(kw

t

+

jk

)若則有效值:T021

1TT

0

2T

=I

=￥k

=1

I0

+

Ikm

cos(kw

t

+

jk

)

d

(t

)i

(w

t

)d

(t

)返回上頁(yè)下頁(yè)￥+0

2k

=1

km2I

2I

=

I周期函數(shù)的有效值為直流分量及各次諧波分量有效值平方和的方根。利用三角函數(shù)的正交性得：I

=

I

2

+

I

2

+

I

2

+0

1

2結(jié)論返回上頁(yè)下頁(yè)3.

非正弦周期函數(shù)的平均值00TIAV=

1Ti(w

t

)dt

=

I正弦量的平均值為0￥k

=1則其平均值為：i(t

)

=

I0

+

Ik

cos(kw

t

+

jk

)若返回上頁(yè)下頁(yè)4.

非正弦周期交流電路的平均功率Tu

idtP

=0

1T￥u(t

)

=

U0

+

Ukm

cos(kw

t

+

juk

)￥k

=1i(t

)

=

I0

+

Ikm

cos(kw

t

+

jik

)k

=1利用三角函數(shù)的正交性，得：￥(j

k

=

j

uk

-jik

)P

=

U0

I0

+

Uk

Ik

cosj

kk

=1=

P0

+

P1

+

P2

+

......返回上頁(yè)下頁(yè)平均功率＝直流分量的功率＋各次諧波的平均功率P

=

U0

I0

+

U1

I1

cosj1

+

U2

I2

cosj

2

+結(jié)論返回上頁(yè)下頁(yè)8.4

非正弦周期交流電路的計(jì)算1.

計(jì)算步驟利用付里葉級(jí)數(shù)，將非正弦周期函數(shù)展開(kāi)成若干種頻率的諧波信號(hào)；利用正弦交流電路的計(jì)算方法，對(duì)各諧波信號(hào)分別應(yīng)用相量法計(jì)算；（注意:交流各諧波的XL、XC不同，對(duì)直流C相當(dāng)于開(kāi)路、L相于短路。）將以上計(jì)算結(jié)果轉(zhuǎn)換為瞬時(shí)值迭加。返回上頁(yè)下頁(yè)2.

計(jì)算舉例例1方波信號(hào)激勵(lì)的電路。求u,已知：R

=20W

、L

=1mH

、C

=1000pFIm

=157μ

A、T

=6.28mStT/2TiSImRLCuiS解（1）已知方波信號(hào)的展開(kāi)式為：532i=

Im

+

2ImS+

1

sin

5w

t

+)(sinw

t

+

1

sin

3w

t

+pIm

=

157μ

A,T

=

6.28μ

s代入已知數(shù)據(jù)：返回上頁(yè)下頁(yè)直流分量220I=

157

=

78.5μ

A=

Im3.14p=

2

·1.57

=

100

mA1m=

2ImI基波最大值5=

20μA5

m

1mI

=

1

Irad/sw

=

2p

=

2

·3.14

=

10

6角頻率三次諧波最大值五次諧波最大值31m3m=

33.3mAI

=

1

IIS

0

=

78.5

mAT

6.28

·10-6電流源各頻率的諧波分量為：6is1

=

100

sin10

t

mA3s

3i

=

100

sin

3

106

t

mA5s5i

=

100

sin

5

106

t

mA返回上頁(yè)下頁(yè)-6U0

=

RI

S

0

=

20

·78

.5

·10

=

1.57

mV（2）對(duì)各種頻率的諧波分量單獨(dú)計(jì)算：（a)直流分量IS0

作用IS0R

u0電容斷路，電感短路:IS

0

=

78.5mA（b)基波作用

is1=

100

sin106t=

1kW

1

=

1

=

1kW-361w

L

=

10

·106

-12w

1C

10

·1000

·10RLCuSi=?=

50kW1RCLRR

+

j(

X

-

X

))

(-

jX

)

X

X(

R

+

jXL

CL

CCLZ

(w

)

=LX

>>R返回上頁(yè)下頁(yè)Z(w

1

)

=

50KΩ2

250

=

mV100

·10-6

5000U1

=

I

Z

(w

）=1

1is1=

100

sin106

t

μ

A(c)三次諧波作用3i

=

100

sin

3

106

t01=

374.5—

-

89.19

WR

+

j(

XL3

-

XC

3)(

R

+

jXL3)(-

jXC

3)Z

(3w

)

=s

3111·1000

·103

·10=6

-12=

0.33KW6

-33w

1

L

=

3

·10

·10

=

3kΩ3w

C02·374.5—

-

89.1910-6Z

(3w

1

)=

33.3

·U

3

=

IS

3=

12.47

—

-

89.20

mV2返回上頁(yè)下頁(yè)(d)五次諧波作用5s5=

208.3—

-

89.53

W1R

+

j(5

XL5

-

XC

5)(

R

+

jXL5)(-

jXC

5)Z

(5w

)

=i

=

100

sin

5

106

t

mA111=

0.2(KΩ)=5

·106

·1000

·10-126

-35w

1

L

=

5

·10

·10

=

5kΩ5w

C21015-6=

4.166

—

-

89.53

mV2208.3—

-

89.53w

)=

20

·5

s=

I

Z

(5U返回上頁(yè)下頁(yè)(3)各諧波分量計(jì)算結(jié)果瞬時(shí)值迭加：u

=

U0

+

u1

+

u3

+

u5?

1.57

+

5000

sin

wt+

12.47

sin(

3w

t

-

89.2

)+

4.166

sin(

5w

t

-

89.53

)

mVU0

=1.57

mV3=

12.47

—

-89.2

mVU21=

5000mVU224.1665—

-89.53

mVU

=返回上頁(yè)下頁(yè)已知:u

=30

+120

cos1000t

+60

cos(2000t

+π

)V.4求圖示電路中各表讀數(shù)(有效值)及電路吸收的功率。例2L11C2L2C

40mH10mHu_25mF25mF+abc30W

dA3A2V2VV11A1返回上頁(yè)下頁(yè)L1C1C2L240mH10mHu25mF_25mF30Wa

+bcdiiC1iL2解C1C2(1)

u0=30V作用于電路，L1、

L2

短路，

C1

、

C2開(kāi)路。L1

L2u030Wa

+_bcdi0iC10iL20i0=

iL20

=

u0/R

=30/30=1A,

iC10=0,uad0=

ucb0

=

u0

=30V返回上頁(yè)下頁(yè)1111

22(2)

u1=120cos1000t

V作用31=

40Ω==w

C

1000

·

25

·10-6w

Cw

L

=

1000

·10

·10-3

=

10Ωw

L

=

1000

·40

·10-

=

40ΩL1、C1

發(fā)生并聯(lián)諧振。+_30Wabcd1U1IC

11IL21Ij40W-j40W-j40Wj10W1=

120—

0

VUI

=

I

=

01

L21U

cb1

=

0=

3—

90

A-

j40IC

11

=

jωC1U

1

=1120—

0ad1=

U

=

120—

0

VU返回上頁(yè)下頁(yè)(3)

u2=60cos(2000t+p

/4)V作用1

11

221=

20Ω=

=2ωC

2000

·

25

·10-62ωC=

80Ω,

2ωL

=

2000

·10

·10-3

=

20Ω12ωL

=

2000

·

40

·10-3L2、C2

發(fā)生并聯(lián)諧振。_30Wa

+bcdU

12IIC

12L22Ij80W-j20W-j20Wj20WU

2

=

60—

45

Vj202=

3—

-

45

A60—

45==j2ωLIL22I2

=

I

=

0C

12ad2=

0,U

1=

U

=

60—

45

Vcb2

2UU返回上頁(yè)下頁(yè)i=i0+

i1

+

i2

=1A所求的電壓、電流的瞬時(shí)值為：iC1=

iC10

+iC11

+iC12

=3cos(1000t+90 )

AiL2=

iL20

+iL21

+iL22

=1+3cos(2000t-

45 )

Auad=

uad0

+

uad1

+

uad2

=30+120cos1000tVucb=

ucb0

+

ucb1

+

ucb2

=30+60cos(2000t+45 )

V2

=

2.12A電流表A1的讀數(shù)：

I

=

1

A

電流表A2的讀數(shù)：3

/12

+

(3

/

2

)2

=

2.35A302+

(120

/

2

)2=

90V302+

(60

/

2

)2=

52.0V電流表A3的讀數(shù)：電壓表V1的讀數(shù)：電壓表V2的讀數(shù)：返回上頁(yè)下頁(yè)例3求Uab、i、及功率表的讀數(shù)。I+–60Wj20W–U1+abU

2*

W*2u

=

2202

cos(3w

t

+

300

)V2

cosw

tV2

cosw

t

+

100已知u1

=220解ab(1)=

440—

00V一次諧波作用時(shí)：U(1)=

440

=

22

=

6.96—

-

18.40

A60

+

j20

3

+

jIab(

3)=

100—

300V三次諧波作用時(shí)：U(

3)60

+

j60

3

+

j3100—

300

5—

300=

=

=

1.18—

-1