We know that,
\(q=\int\limits_0^{T/2}dq\)
\(q=\int\limits_0^{T/2}sin\,ωt\,dt\)
\(=I_m[-\frac{cos\,ωt}{ω}]^{\frac{T}{2}}_0\)
\(=-\frac{I_m}{ω}[cos\,ω\frac{1}{2}-cos\,0°]\)
∵ ωT = 2π
\(=\frac{2I_m}{ω}\)
\(=\frac{I_mT}{\pi}\)
The average value of a.c over one half cycle is
\(I_{av}=\frac{charge}{Time}\)
\(=\frac{q}{\frac{T}{2}}\)
\(I_{av}=\frac{2q}{T}\)
\(I_{av}=\frac{2I_mT}{T\pi}\)
\(I_{av}=\frac{2}{\pi}I_m\)
\(I_{av}=0.637 I_m\)
