Question

The input dc voltage of a 1-\(\phi\) full bridge inverter is 141.4 V and its duty ratio is 0.49. Then its approximate average and rms voltages are

• A. \( \text{69.3 V and 202 V respectively} \)
• B. \( \text{99 V and 202 V respectively} \)
• C. \( \text{69.3 V and 99 V respectively} \)
• D. \( \text{99 V and 69.3 V respectively} \)

Hint:

Look for perfect squares in the numbers provided! The duty ratio \(D = 0.49\) is a perfect square: \(\sqrt{0.49} = 0.7\). This tells you the RMS voltage must be exactly \(0.7 \times 141.4 \approx 99\text{ V}\). Finding the RMS value first lets you immediately pick option (3) without needing to compute the average value at all.

Answer 1

The Correct Option is C

Concept: A single-phase full-bridge inverter using pulse-width modulation (PWM) switches its output voltage between $+V_{dc}$, $0$, and $-V_{dc}$ over each operating cycle. For a given duty ratio $D$ during each half-cycle, the output voltage waveform consists of pulses of width $\beta = D \cdot \pi$. The average value over a full symmetric cycle is zero due to its alternating nature. However, when evaluating the rectified average and effective RMS voltage levels for these pulsed waveforms, we use the following standard definitions:
• Rectified Average Voltage (\(V_{\text{avg}}\)): Proportional to the pulse width ratio: \[ V_{\text{avg}} = D \cdot V_{dc} \]
• Root-Mean-Square Voltage (\(V_{\text{rms}}\)): Proportional to the square root of the duty factor: \[ V_{\text{rms}} = \sqrt{D} \cdot V_{dc} \]

Step 1: Extracting values from the problem statement.
We are given:
• DC input voltage, \( V_{dc} = 141.4\text{ V} \)
• Voltage profile scale factor: Notice that \( 141.4 \approx 100\sqrt{2}\text{ V} \)
• Duty ratio, \( D = 0.49 \)

Step 2: Calculating the rectified average output voltage.
Using our definition for the average voltage: \[ V_{\text{avg}} = D \times V_{dc} \] \[ V_{\text{avg}} = 0.49 \times 141.4\text{ V} \] Performing the multiplication: \[ V_{\text{avg}} \approx 69.286\text{ V} \approx 69.3\text{ V} \]

Step 3: Calculating the RMS output voltage.
Using the square root relationship for the RMS voltage: \[ V_{\text{rms}} = \sqrt{D} \times V_{dc} \] Since our duty ratio is exactly $0.49$, its square root simplifies perfectly: \[ \sqrt{0.49} = 0.70 \] Now, substitute this back into the formula: \[ V_{\text{rms}} = 0.70 \times 141.4\text{ V} \] Performing the multiplication: \[ V_{\text{rms}} = 98.98\text{ V} \approx 99\text{ V} \]

Step 4: Matching the results to the options.
Our calculated values are: \[ V_{\text{avg}} = 69.3\text{ V}, \quad V_{\text{rms}} = 99\text{ V} \] This matches option (3). Hence, the correct choice is option (3).