Question

What is meant by root mean square value of alternating current? Derive an expression for r.m.s. value of alternating current.

Or

Define peak and root mean square value of alternating current. Derive an expression for the root mean square value of alternating current or voltage.

Answer 1

Peak Value: Peak Value of a.c. is the maximum value of alternating current, denoted by I₀.

Let I be instantaneous value of current at any instant t. This varies from 0 to I₀, I₀ to -I₀ and -I₀ to 0.

Root Mean Square Value of a.c.

[I_rms = I_eff = I_v]

Root mean square value of a.c. is that steady current; which would produce the same amount of heat in a given resistance in a given time as is done by a.c. when passed through the same resistance for the same time. It is also called effective or virtual value of a.c.

Expression for virtual value of current (Relation between virtual and peak value)

The instantaneous value of alternating current is given by

I = I₀ sin ωt ...........(1)

Let this current be passed through a resistance R for a small time dt.

The heat produced dH is given by

dH = I² Rdt = RI²di

Using Eq. (1), we get

dH = RI₀2 sin² ωt dt

Total heat produced when the current is passed for time T (say) can be obtained by integrating the above equation within the limits t = 0 and t = T. 

If Iv is the virtual (or rms) value of a.c., then by definition of virtual current, we have

H = I_v² RT .............(3)

From Eq.(2) and (3), we get

I_v² RT = \(\frac{I_0^2RT}{2}\)

or I_v = \(\frac{I_0}{\sqrt2}\)  = 0.707 I₀

or I_v = 0.707 I₀

i.e. Virtual value of alternating current is 0.707 times (or 70.7%) of its peak value.