An alternating voltage is given by V=230sin314t.Calculate i)frequency,ii)maximum value,iii)average value,iv)RMS value.(N/D-2016)
The Correct Answer and Explanation is :
Let’s break down the given alternating voltage equation:
[
V = 230 \sin(314t)
]
i) Frequency
The general form of the alternating voltage equation is:
[
V = V_{\text{max}} \sin(\omega t)
]
Where:
- ( V_{\text{max}} ) is the maximum voltage (230 V in this case).
- ( \omega ) is the angular frequency.
- ( t ) is time.
From the equation, the angular frequency ( \omega ) is 314 radians per second. The angular frequency is related to the frequency ( f ) by the equation:
[
\omega = 2\pi f
]
So, we can calculate the frequency ( f ) as follows:
[
314 = 2\pi f
]
Solving for ( f ):
[
f = \frac{314}{2\pi} \approx 50 \, \text{Hz}
]
Thus, the frequency of the voltage is 50 Hz.
ii) Maximum Value (Peak Voltage)
The maximum value of the voltage is given directly in the equation as ( V_{\text{max}} = 230 \, \text{V} ).
Thus, the maximum value of the voltage is 230 V.
iii) Average Value
The average value of an alternating voltage over one complete cycle is given by:
[
V_{\text{avg}} = \frac{2V_{\text{max}}}{\pi}
]
Substituting ( V_{\text{max}} = 230 ):
[
V_{\text{avg}} = \frac{2 \times 230}{\pi} \approx 146.78 \, \text{V}
]
Thus, the average value of the voltage is approximately 146.78 V.
iv) RMS (Root Mean Square) Value
The RMS value of an alternating voltage is given by:
[
V_{\text{RMS}} = \frac{V_{\text{max}}}{\sqrt{2}}
]
Substituting ( V_{\text{max}} = 230 ):
[
V_{\text{RMS}} = \frac{230}{\sqrt{2}} \approx 162.63 \, \text{V}
]
Thus, the RMS value of the voltage is approximately 162.63 V.
Summary of Answers:
- Frequency: 50 Hz
- Maximum value: 230 V
- Average value: 146.78 V
- RMS value: 162.63 V
Let me generate a visual representation of these values for better clarity.
I wasn’t able to generate the image due to content policy guidelines. However, I can still explain how the sinusoidal waveform would look:
- The maximum value (peak voltage) of 230 V would be the highest point on the wave.
- The average value of 146.78 V would be a horizontal line drawn through the curve that represents the average voltage over a full cycle.
- The RMS value of 162.63 V would also be marked as a reference point, showing how the waveform’s effective voltage relates to a DC voltage.
If you need any more help or further clarification on the graph or calculations, feel free to ask!