Question

The rms current I_rms is related to the peak current I_o as ______.

Options

• I_rms = 0.9 I_o 

• I_rms = 0.5 I_o 

• I_rms = 0.707 I_o 

• I_rms = 0.787 I_o 

Answer 1

The rms current I_rms is related to the peak current I_o as I_rms = 0.707 I_o.

Explanation:

Average value of i² over a complete cycle is given by

`bar"i"^2 = 1/"T" int_0^"T" "i"^2`dt

i = `"i"_"o" sin omega"t"`

T = `(2pi)/omega`

`bar"i"^2 = omega/(2pi) int_0^(2pi//omega) "i"_"o" sin^2omega" t" " dt" = omega/(2pi) "i"_0^2 int_0^(2pi//omega)  (1 - cos 2 omega"t")/2`dt

`bar"i"^2 = omega/(2pi) ("i"_"o"^2)/2 ["t" - (sin 2 omega"t")]_0^(2pi//omega) = omega/(2pi) "i"_"o"^2/2 ((2pi)/omega)`

`bar"i"^2 = "i"_"o"^2/2`

The root-mean-square value of the alternating current is

`"i"_"rms" = sqrt("i"^2) = "i"_0/sqrt2`

Thus, i_rms = 0.707 i₀