Understanding Half-Wave Rectifier Output Voltage

Let's analyze the output voltage of an ideal half-wave rectifier under no-load conditions. An ideal half-wave rectifier is a circuit that allows only one-half cycle (either positive or negative) of an input AC voltage signal to pass through to the output, effectively blocking the other half cycle. In this case, we usually consider the positive half-cycle to be passed.

Ideal Half-Wave Rectifier Output Waveform

If the input AC voltage is a sine wave given by \(v_{in}(t) = V_{peak} \sin(\omega t)\), where \(V_{peak}\) is the maximum value and \(\omega\) is the angular frequency, the output voltage \(v_{out}(t)\) of an ideal half-wave rectifier under no load will be:

• \(v_{out}(t) = V_{peak} \sin(\omega t)\) when \(v_{in}(t) > 0\) (for \(0 \le \omega t \le \pi\))
• \(v_{out}(t) = 0\) when \(v_{in}(t) \le 0\) (for \(\pi < \omega t < 2\pi\))

This waveform repeats every period \(T = 2\pi/\omega\).

Calculating RMS Voltage of Half-Wave Rectifier Output

The Root Mean Square (RMS) value of a periodic voltage waveform \(v(t)\) with period \(T\) is a measure of its effective voltage. It is defined as:

\(V_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} v(t)^2 dt}\)

For the half-wave rectified output voltage \(v_{out}(t)\), the period is \(T = 2\pi/\omega\). We need to integrate the square of the output voltage over one full period:

\(V_{rms} = \sqrt{\frac{1}{2\pi/\omega} \int_{0}^{2\pi/\omega} v_{out}(t)^2 dt}\)

Let's perform the integration over the angular period \(0\) to \(2\pi\):

\(V_{rms} = \sqrt{\frac{1}{2\pi} \int_{0}^{2\pi} v_{out}(\omega t)^2 d(\omega t)}\)

Since \(v_{out}(\omega t) = V_{peak} \sin(\omega t)\) for \(0 \le \omega t \le \pi\) and \(0\) for \(\pi < \omega t < 2\pi\), the integral becomes:

\(V_{rms} = \sqrt{\frac{1}{2\pi} \left[ \int_{0}^{\pi} (V_{peak} \sin(\omega t))^2 d(\omega t) + \int_{\pi}^{2\pi} (0)^2 d(\omega t) \right]}\)

\(V_{rms} = \sqrt{\frac{1}{2\pi} \int_{0}^{\pi} V_{peak}^2 \sin^2(\omega t) d(\omega t)}\)

\(V_{rms} = \sqrt{\frac{V_{peak}^2}{2\pi} \int_{0}^{\pi} \sin^2(\omega t) d(\omega t)}\)

Using the trigonometric identity \(\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}\), we substitute \(\theta = \omega t\):

\(V_{rms} = \sqrt{\frac{V_{peak}^2}{2\pi} \int_{0}^{\pi} \frac{1 - \cos(2\omega t)}{2} d(\omega t)}\)

\(V_{rms} = \sqrt{\frac{V_{peak}^2}{4\pi} \int_{0}^{\pi} (1 - \cos(2\omega t)) d(\omega t)}\)

\(V_{rms} = \sqrt{\frac{V_{peak}^2}{4\pi} \left[ \omega t - \frac{\sin(2\omega t)}{2} \right]_{0}^{\pi}}\)

Now, we evaluate the definite integral at the limits \(\pi\) and \(0\):

\(V_{rms} = \sqrt{\frac{V_{peak}^2}{4\pi} \left[ (\pi - \frac{\sin(2\pi)}{2}) - (0 - \frac{\sin(0)}{2}) \right]}\)

Since \(\sin(2\pi) = 0\) and \(\sin(0) = 0\):

\(V_{rms} = \sqrt{\frac{V_{peak}^2}{4\pi} \left[ (\pi - 0) - (0 - 0) \right]}\)

\(V_{rms} = \sqrt{\frac{V_{peak}^2 \pi}{4\pi}}\)

\(V_{rms} = \sqrt{\frac{V_{peak}^2}{4}}\)

Taking the square root:

\(V_{rms} = \frac{V_{peak}}{2}\)

Comparing the Result with the Options

We found that the RMS output voltage of an ideal half-wave rectifier is half of the peak input voltage (\(V_{peak}\)). Let's check which option matches this relationship:

The derived relationship \(V_{rms} = V_{peak} / 2\) exactly matches Option 3.

Revision Table: Ideal Rectifier Voltage Characteristics

Here is a summary of voltage characteristics for ideal rectifiers under no load:

Additional Information: No-Load vs. Loaded Rectifiers

The calculations above are for the "no-load" condition, meaning no external resistance is connected to the output of the rectifier. When a load is connected, especially a filter capacitor, the output voltage waveform changes significantly, and the average and RMS values will be different. A filter capacitor attempts to smooth out the pulsating DC output, ideally bringing the DC output voltage closer to the peak voltage for full-wave rectifiers under light load. However, the question specifically asks for the no-load output voltage characteristics, which are derived directly from the rectified waveform without any filtering or loading effects.