Proving the relationship between the effective value and the peak value of the sinusoidal alternating current by integration Such as the title
Let I = asin (2 π T / T)
According to the valid value definition, there are
In one cycle
T
RTI ^ 2 = ∫ R [asin (2 π T / T)] ^ 2DT [0, t] integral
0
T
Ti ^ 2 = (a ^ 2) ∫ [sin (2 π T / T)] ^ 2DT [0, t] integral
0
Substitution x = (2 π T / T)
π/2
Integral in I ^ 2 = 4 (a ^ 2) (1 / 2 π) ∫ [sin (x)] ^ 2DX [0, π / 2]
0
According to the integral formula
π/2
In=∫(sinx)^ndx=[(n-1)/n]I_ (n-1)
0
π/2
There is ∫ [sin (x)] ^ 2DX = (1 / 2) (π / 2) = π / 4
0
I^2 = (A^2)/2
I=A(√2)/2
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