gen-define-rms-voltage-and-current-and-give-one-short-worked-exa-7d0e

Define RMS voltage and current and give one short worked example.

# RMS Voltage and Current ## Definition **RMS (Root Mean Square)** voltage or current is the equivalent DC value that would deliver the same average power to a resistive load as the AC signal. For a periodic signal, RMS is calculated by: 1. Squaring the instantaneous values 2. Finding the Mean (average) of the squared values 3. Taking the Square Root **Mathematical formula:** $$V_{rms} = \sqrt{\frac{1}{T}\int_0^T v^2(t)\,dt}$$ For a **sinusoidal AC signal**: $v(t) = V_0 \sin(\omega t)$ $$V_{rms} = \frac{V_0}{\sqrt{2}} \approx 0.707 \times V_0$$ Similarly for current: $I_{rms} = \frac{I_0}{\sqrt{2}}$ ## Worked Example **Problem:** An AC voltage source has a peak voltage of 170 V. Find the RMS voltage and the power dissipated in a 50 Ω resistor. **Solution:** RMS voltage: $$V_{rms} = \frac{170}{\sqrt{2}} = \frac{170}{1.414} = 120\text{ V}$$ Power dissipated: $$P = \frac{V_{rms}^2}{R} = \frac{(120)^2}{50} = \frac{14400}{50} = 288\text{ W}$$ > **Note:** Household AC voltage (120 V in North America, 230 V in Europe) is always given as RMS values.