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6.3 ひずみ波の実効値 6.3 Effective Value of Distorted WavePowerPoint Presentation

6.3 ひずみ波の実効値 6.3 Effective Value of Distorted Wave

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6.3 ひずみ波の実効値6.3 Effective Value of Distorted Wave

このテーマの要点

- ひずみ波の実効値の計算方法
Calculating method of effective value

教科書の該当ページ

- 6.4 ひずみ波の実効値[p.130]

0

|I|= i2(t)dt

1

T

¥

n=1

¥

n=1

¥

n=1

i(t) = åansinnwt+ b0+ åbncosnwt

= I0+ åImnsin(nwt+qn)

bn

an

qn= tan-1

Imn= an2+bn2

実効値の定義 Definition of Effective Value- 実効値の定義

- ひずみ波の展開式Fourier’s series of distorted wave

単振動の合成

I0 = b0

n=1

i2(t) = {I0+ åImnsin(nwt+qn)}2

ひずみ波の2乗平均値Square mean value of distorted wave①直流分の2乗

= I02

+Im12sin2(wt+q1)+Im22sin2(2wt+q2)+···

②n次調波の自乗

+ 2I0Im1sin(wt+q1)+2I0Im2sin(2wt+q2)+···

③直流分とn次調波の積

+ 2Im1sin(wt+q1)·Im2sin(2wt+q2)

+ 2Im1sin(wt+q1)·Im3sin(3wt+q3)+···

④異なるn次調波の積

各項について平均値を求める

Calculate mean value of each term

= I02[t] = I02

T

0

T

0

T

0

T

0

I2①= I02dt

1

T

1

T

1

T

I2②= Imn2sin2(nwt+qn)dt

1-cos2(nwt+qn)

2

= dt

Imn2

T

Imn2

2T

Imn2

2

= [t] =

T

0

2乗平均値Square mean value- ①直流分について

- ②n次調波の自乗について

cosのn周期の

積分は0

0

T

0

T

0

T

0

1

T

1

T

2I0Imn

T

= sin(nwt+qn)dt = 0

I2③= 2I0Imnsin(nwt+qn)dt

I2④= 2Imksin(kwt+qk)·Imnsin(nwt+qn)dt

= [cos{(k-n)wt+qk-qn}

-cos{(k+n)wt+qk+qn}]dt

=0

ImkImn

T

③直流分とn次調波の積についてsinのn周期の

積分は0

- ④異なるn次調波の積について

cosのk-n, k+n周期の

積分は0

Imn2

2

¥

n=1

¥

n=1

= I02+å|In|2

= I02+å

sin波では

Im

2

|I|=

= I02+|I1|2+|I2|2+|I3|2+···(6.17)

¥

n=1

|I| = I02+å|In|2

ひずみ波の実効値 Effective Value of Distorted wave- ひずみ波の2乗平均値

I2= I2①+I2②+I2③+I2④

- ひずみ波の実効値

各調波の実効値の自乗和のルート

Root-square [sum of (each haromonic’s effective value)2]

3

1

5

4A

p

i(t) = (sinwt+sin3wt+sin5wt+···)

1

5 2

1

5 2

1

3 2

1

3 2

4A

p

1

2

1

2

- 基本波：

4A

p

- 3次調波：

4A

p

Fundamental

3rd

5th

- 5次調波：

()2+( )2+( )2+···

4A

p

|I| =

4A

p

0.575…

= 0.97A (n=5まで)

=

例題 Example- 方形波の実効値 Effective value of square wave

- 展開式Fourier’s series

- 各調波の実効値は |I| of each harmonics

- 方形波の実効値は |I| of square wave

演習 Exercise

No. Name :

図の方形波をフーリエ級数展開し、第5次調波までの成分で実効値を計算せよ

Calculate Fourier’s series and effective value by using under 5th harmonics.

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