Answer MET Question 22

Question:  A. Explain the significance of the root mean square value of an alternating current or voltage wave form; Define the form factor of such a wave form.

Answer: Root mean square (RMS) Value:
The RMS (or effective) value of an alternating current is given by that steady (D.C) current which when flowing through a given circuit for a given time produces the same heat as produced by the alternating current when flowing through the same circuit for the same time.
RMS value is the value which is taken for power of any description. This value is obtained byb finding the square root of the mean value of the squared ordinates for a cycle or half-cycle.
Equation of sinusoidal alternatong current is given by:
 $\displaystyle \small \mathrm{i = I_{m}sin\theta }$
The mean of squares of the instantenous value of current over half cycle is
$\displaystyle \small \mathrm{I^2 = \int_{0}^{\pi }\frac{i^2\ d\theta }{\pi -0} }$
$\displaystyle \small \mathrm{I^2 = \frac{1}{\pi }\int_{0}^{\pi }i^2\ d\theta }$
$\displaystyle \small \mathrm{I^2 = \frac{1}{\pi }\int_{0}^{\pi }(I_{m}\ sin\theta )^2\ d\theta }$
thus,
$\displaystyle \small \mathrm{I_{r.m.s} = \frac{I_m}{\sqrt{2}}= 0.707 I_m}$

Form Factor: It is the ratio of the root mean square value to the average value of an alternating quantity (current or voltage) is called Form Factor.
Mathematically, it is expressed as:
$\displaystyle \small \mathrm{Foam\ factor =\frac{I_{rms}}{I_{av}}=\frac{V_{rms}}{V_{av}}}$
For the current varying sinusoidally, by putting the average and rms values the Form Factor is given as:
$\displaystyle \small \mathrm{Foam\ factor =\frac{\frac{I_m}{\sqrt{2}}}{\frac{2I_m}{\pi}}=1.11}$