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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]. T 0. | I | = i 2 ( t ) dt. 1 T. ¥ n =1. ¥ n =1. ¥ n =1. i ( t ) = å a n sin n w t + b 0 + å b n cos n w t.

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6 3 6 3 effective value of distorted wave
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]


Definition of effective value

T

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


2 square mean value of distorted wave

¥

n=1

i2(t) = {I0+ åImnsin(nwt+qn)}2

ひずみ波の2乗平均値Square mean value of distorted wave

①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


2 square mean value

= 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


6 3 6 3 effective value of distorted wave 1349292

T

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


Effective value of distorted wave

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]


Example

1

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


Exercise
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.