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

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  1. 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]

  2. 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

  3. ¥ 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

  4. = 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

  5. 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

  6. 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]

  7. 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

  8. 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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