Mean Value of Alternating Current

Mean or Average Value of Alternating Current

Alternating current is positive during the first half cycle and negative during another half cycle, so the mean average value of a.c. over one cycle is zero. We can find the mean or average value of a.c. over any half cycle.

Mean or average value of alternating current is that value of steady current which sends the same amount of charge through a circuit in a certain time interval as is sent by an alternating current through the same circuit in the same time interval.

To calculate its value, let an alternating current be represented by

\begin{align*} I &= I_0 \sin \omega t \\ \end{align*}The charge sent by the alternating current I in time dt is given by\begin{align*} \\ dq &= Idt \: \left [ \therefore I = \frac {dq}{dt} \right ] \\ dq &= I_0\sin \omega t dt \\ \end{align*}

The amount of charge passing through the circuit in half time period can be obtained by integrating above equation (i) from t = 0 to t = T/2.

\begin{align*} q &= \int _0^{T/2} I_0 \sin \omega t \: dt = I_0 \int _0^{T/2} \sin \omega t dt \\ &= I_0 \left [ \frac {-\cos \omega t}{\omega } \right ]_0 ^{T/2} \\ &=- \frac {I_0}{\omega } [\cos \omega t ] _0^{T/2} \\ &= - \frac {I_0}{2\pi /T} \left [ \cos \frac {2\pi }{T} t \right ]_0^{T/2} \: \left [\therefore w = \frac {2\pi }{T} \right ] \\ &= - \frac {I_0 T}{2\pi } \left [ \cos \frac {2\pi }{T}\frac T2 - \cos 0 \right ] \\ &= - \frac {I_0T}{2\pi } [\cos \pi - \cos 0] \\ &=- \frac {I_0 T}{2\pi }[-1-1] \\ &= - \frac {I_0 T}{2\pi } \times -2 \\ q &= \frac {I_0 T}{\pi } \dots (ii) \\ \end{align*}

If I_m be the mean value of an a.c. over positive half cycle, then the charge sent by it in time T/2 is given by

\begin{align*} q &= I_m \frac T2 \dots (iii) \\ \text {From equation} \: (ii) \: \text {and} \: (iii), \: \text {we get} \\ I_m \frac T2 &= \frac {I_0T}{\pi } \\\therefore I_m &= \frac {2I_0 }{\pi } \\ &= 0.637 I_0 \\ \end{align*}

Hence it means an average value of a.c. cover positive half cycle is 0.637 times the peak value of a.c. is 63.7% of the peak value.

Similarly, the mean or average value of a.c. over the negative half cycle is obtained by integrating equation (1) from t = T/2 to t = T. It comes out to be -0.637I₀. Hence the mean or average value of a.c. over one complete cycle is 0.637I₀ – 0.637I₀ = zero

Similarly, the mean value of alternating e.m.f is

$$ E_m = \frac {2E_0} {\pi} = 0.637 E_0 $$

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, et al.Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.