The value of alternating current or voltage that has the same hea

SOLUTION

• RMS, or root mean square, voltage or current is a method of representing an AC voltage/current as an equivalent voltage/current which represents the DC voltage value that will produce the same heating effect or power dissipation in the circuit as this AC voltage/current will produce.
• It is also called the effective value. The idea of effective value arises from the need to measure the effectiveness of a voltage or current source in delivering power to a resistive load.

This is explained with the help of the following diagram:

The average power absorbed by the resistor in the ac circuit is given by:

\(P = \frac{1}{T}\mathop \smallint \limits_0^T {i^2}R\;dt = \frac{R}{T}\mathop \smallint \limits_0^T {i^2}\;dt\)

The power absorbed by the resistor in the dc circuit is:

\(P=I_{eff}^2 R\)

We need to find the I_eff that will transfer the same power to resistor R as the sinusoid i, i.e. equating the above two powers, we get:

\(\Rightarrow \frac{R}{T}\mathop \smallint \limits_0^T {i^2}\;dt=I_{eff}^2R\)

\(I_{eff}^2=\frac{1}{T}\mathop \smallint \limits_0^T {i^2}\;dt\)

\(I_{eff}=\sqrt{\frac{1}{T}\mathop \smallint \limits_0^T {i^2}\;dt}\)

The effective value of the voltage is calculated the same way as current, i.e.

\(V_{eff}=\sqrt{\frac{1}{T}\mathop \smallint \limits_0^T {v^2}\;dt}\)

Observation:

• The effective value is the root of the mean (average) of the square of the periodic signal.
• Thus, the effective value is often known as the root-mean-square value or RMS value.
• I_eff = I_rms 'or' V_eff = V_rms