Why Do We Use RMS Voltage?

 

Electrical Engineering

Why Do We Use RMS Voltage?

We use RMS voltage because it gives the correct average power dissipated in a resistor supplied by an AC source.

Given Input Voltage

v(t) = 12 sin(ωt) V

Peak voltage = 12 V

1

Why Not Peak Voltage?

Peak voltage is the maximum instantaneous voltage. An AC voltage continuously changes with time, so it does not remain at its peak value.

Incorrect average-power calculation

P = V_peak² / R

P = 12² / 1000

P = 144 mW

This is the maximum instantaneous power, not the average power over a complete cycle.

2

Why Not Average Voltage?

Over one complete cycle, the positive and negative halves of a pure sine wave cancel each other.

V_avg = 0 V

If we square this average voltage to calculate power, we get zero. That is incorrect because the resistor heats up during both positive and negative half-cycles.

Remember: Negative voltage does not mean negative power.

Instantaneous power in a resistor

p(t) = v²(t) / R

Squaring the voltage makes the power nonnegative.

3

Why RMS Voltage Works

RMS means Root Mean Square. We square the voltage values, calculate their average over time, and then take the square root. This gives the equivalent DC voltage for the same heating effect in a resistor.

V_rms = √[(1/T) ∫₀^T v²(t) dt]

Step 1: Find RMS voltage for a sine wave

V_rms = V_peak / √2

V_rms = 12 / √2

V_rms = 8.49 V

Step 2: Calculate average power

P_avg = V_rms² / R

P_avg = 8.49² / 1000

72 mW

Voltage Comparison

Quantity	Meaning	Value
Peak voltage	Maximum voltage	12 V
Average voltage	Mean over a full cycle	0 V
RMS voltage	Effective voltage for heating	8.49 V

Exam Tip: Remember These Formulas

Peak to RMS: V_rms = V_peak / √2

Average power: P_avg = V_rms² / R

Using peak voltage directly: P_avg = V_peak² / (2R)

RMS is preferred because it makes AC power calculations straightforward. Both power formulas give the same answer.