For the same peak value which of the following wave will have the highest r.m.s. value?

Question

Examoogle · Accepted Answer

square wave

Explanation: The Root Mean Square (RMS) value represents the effective power-producing capacity of a waveform. For a fixed peak value ($V_m$), a square wave maintains that maximum amplitude for the entire duration, whereas other waveforms like sine, triangular, or half-wave rectified waves spend time at lower amplitude levels, resulting in a lower average power output.

Working Principle: The RMS value is calculated as the square root of the mean of the squared instantaneous values over one period: $V_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} v(t)^2 dt}$. Since a square wave is constant at $\pm V_m$, its square $v(t)^2$ is constant at $V_m^2$, making the mean value $V_m^2$ and the square root $V_m$. All other waveforms result in a factor (Form Factor) less than 1 when multiplied by the peak value.

Key Formulas: $V_{rms(square)} = V_m$; $V_{rms(sine)} = \frac{V_m}{\sqrt{2}} \approx 0.707 V_m$; $V_{rms(triangular)} = \frac{V_m}{\sqrt{3}} \approx 0.577 V_m$; $V_{rms(half-wave)} = \frac{V_m}{2} = 0.5 V_m$

Key Points: RMS value is the equivalent DC value that produces the same heat in a resistance.; The Square wave has the highest Form Factor, which is the ratio of RMS value to Average value.; For the same peak value, the square wave occupies the maximum area under the squared-waveform curve.

Advantages: Square waves provide maximum RMS output for a given peak voltage.; Useful in power electronics for switching applications.

Disadvantages: Square waves contain high-frequency harmonic content causing EMI issues.; Switching square waves causes high $dv/dt$ stress on insulation.

Applications: Inverters; Digital signal processing; Switch-mode power supplies

Additional Info: The Peak Factor is defined as Peak Value / RMS Value. The square wave has the lowest Peak Factor of 1.0.. Option B (half-wave rectified) has the lowest RMS value at 0.5 times the peak.

Answer: A is correct — The square wave maintains its peak value throughout the entire cycle, resulting in the highest RMS value ($V_{rms} = V_m$) compared to all other waveforms.

Exam Tip: Remember the sequence of RMS values for peak value $V_m$: Square (1.0) > Sine (0.707) > Triangular (0.577) > Half-wave Sine (0.5).

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