Question

Derive an expression for RMS value of AC

Answer 1

\(I = I_0\sin \omega t\)

\(d = I^2R dt\)

\(dH = (I_0\sin \omega t)^2 R dt\)

\(d H = I_0^2R\sin^2 \omega t dt\)

The heat produced in a time T/2 is given by:

\(H = \int \limits_0^\frac T2 I_0^2 R\sin^2 \omega t dt\)

\(H = I_0^2 R \int \limits_0^\frac T 2 \sin^2 \omega t dt\)

\(H = \frac{I_0^2R}2[\frac T2 - 0]\)

\(= \frac{I_0^2R}2. \frac T2\)   .......(1)

The rms value of AC is represented as:

\(H = I_{rms}^2R.\frac T2\)    ......(2)

By equating equation (1) and equation (2), we get:

\( I_{rms}^2R.\frac T2 = \frac{I_0^2R}2 .\frac T2\)

\( I_{rms}^2 = \frac{I_0^2}2\)

\( I_{rms} = \frac{I_0}{\sqrt 2}\)

\( I_{rms} = 0.707 I_0\)

These values are measured by an ammeter and voltmeter that are used in the circuit.

Answer 2

Root mean square (r.m.s.) or virtual value of a.c.: It is that steady current, which when passed through a resistance for a given time will produce the same amount of heat as the alternating current does in the same resistance and in the same time. It is denoted
I_rms or I_v.

Derivation of r.m.s. value of current:

The instantaneous value of a.c. passing through a resistance R is given by
I = I₀ sin ωt

The alternating current changes continuously with time.

Suppose, that the current through the resistance remains constant for an infinitesimally small time dt. 

Then, small amount of heat produced the resistance R in time dt is given by

dH = I² R dt

 = (1₀ sin ωt)² R dt

= I₀² R sin² ωt dt. 

The amount of heat produced in the resistance in time T/2 is

If I_rms be the r.m.s. value of a.c., then by definition,

  

From equtaions (i) and (ii), we have