Physics · Alternating Current · NEET
Over a full cycle the current is positive for the first half and negative for the second half. The two halves cancel exactly, so the full-cycle average is zero. But in a half cycle you only take one hump (say the positive one). There is nothing negative to cancel it, so a real non-zero average survives: I_avg = 2im/pi = 0.637 im.
I_avg = 2im/pi = 0.637 im for current, and V_avg = 2vm/pi = 0.637 vm for voltage. Here im and vm are the peak (maximum) values. The factor 2/pi comes from averaging sin(wt) over the half period from 0 to T/2.
Average = (1/(T/2)) times the integral of im sin(wt) from t=0 to T/2. The integral of sin(wt) is -cos(wt)/w. Evaluated from 0 to T/2 (where wT/2 = pi) it gives (1 - cos pi)/w = 2/w. Multiply by (2/T) times im: I_avg = (2/T)(im)(2/w). Since w = 2pi/T, this simplifies to I_avg = 2im/pi = 0.637 im.
No. Mean (half-cycle average) = 0.637 im, but RMS = im/sqrt(2) = 0.707 im. RMS is always larger. RMS is used for power and heating (it is the DC-equivalent value); the half-cycle mean is mainly used for rectified AC and moving-coil meter theory.
Always the PEAK value im (or vm), not the RMS value. If a problem gives you RMS instead, first convert: im = sqrt(2) times I_rms, then put that im into I_avg = 2im/pi.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
It is 2im/pi = 0.637 times the peak current im. For voltage it is 2vm/pi = 0.637 vm.
Because the full-cycle average is zero and gives no useful information. A half-cycle average is non-zero and is needed for half-wave rectified currents and for understanding moving-coil (DC) meters.
RMS/Mean = (0.707 im)/(0.637 im) = pi/(2 sqrt(2)) = 1.11. This number 1.11 is called the form factor of a sine wave.
No. The positive half gives +0.637 im and the negative half gives -0.637 im - same size, opposite sign. The magnitude of the mean over any half cycle is 0.637 im.
