Question

If a direct current of value a ampere is superimposed on an alternating current `1 = b sin omega t` flowing through a wire, what is the effective value of the resulting current in the circuit?

Answer 1

A current at any instant in the ciruit will be,
`I = I_(dc) + I_(ac) = a + b sin omega t`
So, `I_("eff") = [(int_(0)^(T) I^(2) dt)/(int_(0)^(T) dt)]^((1)/(2)) = [(1)/(T) int_(0)^(T) (a + b sin omega t)^(2) dt]^(1//2)`
i.e., `I_("eff") = [(1)/(T) int_(0)^(T) (a^(2) + 2 b sin omega t + b^(2) + sin^(2) omega t) dt]^((1)/(2))`
But as
`(1)/(T) int_(0)^(T) sin omega t dt = 0` and `(1)/(T) int_(0)^(T) omega dt = (1)/(2)`
So, `I_("eff") = [a^(2) + (1)/(2) b^(2)]^(1//2)`