What is the rms value of an ac voltage that has a 480 v peak?
The Correct Answer and Explanation is:
To calculate the root mean square (RMS) value of an AC voltage, we can use the relationship between the peak voltage and the RMS voltage. The RMS voltage is a measure of the effective value of an alternating voltage or current, representing the equivalent DC voltage that would produce the same power dissipation in a resistive load.
The formula to calculate the RMS value from the peak value ( V_{\text{peak}} ) is:
[
V_{\text{RMS}} = \frac{V_{\text{peak}}}{\sqrt{2}}
]
Given:
- Peak voltage, ( V_{\text{peak}} = 480 \, \text{V} )
Step 1: Apply the formula
[
V_{\text{RMS}} = \frac{480}{\sqrt{2}}
]
We know that ( \sqrt{2} \approx 1.414 ).
[
V_{\text{RMS}} = \frac{480}{1.414} \approx 339.41 \, \text{V}
]
Thus, the RMS value of the AC voltage is approximately 339.41 V.
Explanation:
The RMS value of an AC signal is important because it quantifies the voltage in terms of the amount of heat or power it would dissipate in a resistive load (like a heating element or light bulb). The RMS voltage gives a way to compare AC voltages to their DC counterparts since the power delivered by both would be the same in an equivalent resistive circuit.
For a sinusoidal waveform (which is typical in AC systems), the peak voltage ( V_{\text{peak}} ) represents the maximum value of the waveform, while the RMS voltage is the equivalent DC voltage that delivers the same power.
The factor of ( \frac{1}{\sqrt{2}} ) arises from the mathematical properties of sinusoidal waveforms, where the RMS value is calculated as the square root of the mean of the squares of all instantaneous values of the waveform over one full cycle. This results in the RMS voltage being about 0.707 times the peak voltage for a sine wave.