Sinusoidal voltages and currents only; root mean square, peak and…

The idea

For a sinusoidal alternating voltage or current, the value changes continuously, so a single representative value is needed for power calculations. The root mean square value is the steady direct value that would dissipate the same average power in a given resistor. For a sinusoid, V_rms = V₀/√2 and I_rms = I₀/√2, where V₀ and I₀ are peak values. The peak-to-peak value is the difference between the maximum positive and maximum negative values, so V_pp = 2V₀. Average power is P = I_rms²R = V_rms²/R = I_rms V_rms for a resistor. These relationships apply only to sinusoidal waveforms; other waveforms need their own rms calculation.

What earns the marks

• Defines the root mean square value of an alternating current or voltage as the steady direct value that produces the same average power in a resistor.
• States and applies the sinusoidal relationships V_rms = V₀/√2 and I_rms = I₀/√2, where V₀ and I₀ are peak values.
• Calculates peak-to-peak values from peak values using V_pp = 2V₀, and converts between peak, peak-to-peak and rms values for sinusoidal waveforms.
• Uses rms values to calculate average power in a resistor with P = I_rms²R, P = V_rms²/R or P = I_rms V_rms.
• Explains that the √2 factor arises from averaging the square of a sinusoidal function over a complete cycle, and that it applies only to sinusoidal waveforms.

Common mistakes and how to put them right

• Using peak values directly in power equations such as P = I₀²R.

  Put it right: Use rms values, so P = I_rms²R = I₀²R/2 for a sinusoid.

• Confusing peak-to-peak with peak and dividing by √2 twice.

  Put it right: Halve the peak-to-peak value first to obtain the peak, then divide by √2 for the rms value.

• Applying the √2 relationships to non-sinusoidal waveforms such as square or triangular waves.

  Put it right: Use the defining average-power method or the specific rms value for that waveform.

Exam tips

• Write down whether a quoted value is peak, peak-to-peak or rms before substituting it, since mixing them is the most common source of error.
• Show the conversion steps explicitly, for example peak-to-peak to peak by halving, then peak to rms by dividing by √2.
• When calculating power, choose the form of the equation that matches the quantity given, such as P = V_rms²/R when the rms voltage is known.