A sinusoidal voltage is given by 200 sin(ωt + π/3). What is the average value of the voltage?

Explanation:

The given sinusoidal voltage is represented by the equation:

V(t) = 200 sin(ωt + π/3)

To determine the average value of this sinusoidal voltage over one complete cycle, we need to use the properties of the sinusoidal function. The average value of a pure sinusoidal waveform (sinusoidal function) over one complete cycle is zero. However, what we often look for in AC (Alternating Current) voltage analysis is the Root Mean Square (RMS) value because it represents the effective value of the voltage that can deliver the same power as a DC (Direct Current) voltage.

Average Value Calculation:

The average value of a sinusoidal function like sin(ωt + φ) over one complete cycle is zero. This is because the positive and negative halves of the sine wave cancel each other out.

Mathematically, this can be expressed as:

Average Value = (1/T) ∫₀^T V(t) dt

where T is the period of the sinusoidal function. For a function of the form V(t) = V_m sin(ωt + φ), the average value over one complete cycle is:

Average Value = (1/T) ∫₀^T V_m sin(ωt + φ) dt = 0

since the integral of a sinusoidal function over a complete cycle is zero.

RMS Value Calculation:

The RMS value of a sinusoidal function is more relevant for AC voltage analysis. The RMS value gives us a measure of the effective voltage. For a sinusoidal voltage of the form V(t) = V_m sin(ωt + φ), the RMS value is given by:

V_RMS = V_m / √2

where V_m is the peak voltage (amplitude).

Given V(t) = 200 sin(ωt + π/3), the peak voltage V_m is 200 V. The RMS value is therefore:

V_RMS = 200 / √2 = 200 / 1.414 ≈ 141.42 V

Correct Option Analysis:

The correct option is:

Option 1: 127.32 V

Although the RMS value calculated above is approximately 141.42 V, it appears there might be a misunderstanding in the question's context. If the average value over one complete cycle is expected, it would be zero, which is not listed among the options. However, if we consider the possible misunderstanding and look for effective (RMS) values, the closest value given is 127.32 V, though 141.42 V is more accurate.

Important Information:

To further understand the analysis, let’s evaluate the other options:

Option 2: 100 V

100 V does not correspond to the RMS value of the given sinusoidal voltage. The RMS value of the given voltage is higher than this value.

Option 3: 250 V

250 V is greater than the peak voltage (amplitude) of the sinusoidal voltage, which is 200 V, hence it cannot be the correct RMS value.

Option 4: 314.14 V

This value is significantly higher than the peak voltage of 200 V and does not correspond to the RMS value of the given sinusoidal voltage.

Conclusion:

The correct analysis involves recognizing that the average value of a sinusoidal voltage over one complete cycle is zero. However, if the question intends to refer to the RMS value, which is a common effective value in AC analysis, the calculated RMS value is approximately 141.42 V. Given the options, 127.32 V is the closest, though not entirely accurate. This discrepancy suggests a possible misinterpretation or mistake in the provided options. Understanding the properties of sinusoidal functions and the significance of RMS values is crucial for accurate AC voltage analysis.