Question

An alternating voltage is given by \(V=230\sin314t\). Calculate frequency, maximum value, average value, and RMS value.

Answer 1

To analyze the alternating voltage given by the equation \( V = 230 \sin(314t) \), we can extract several key characteristics: frequency, maximum value, average value, and RMS (Root Mean Square) value.

### 1. Frequency
The voltage function is in the form \( V = V_m \sin(\omega t) \), where:
- \( V_m = 230 \, \text{V} \) (maximum value)
- \( \omega = 314 \, \text{rad/s} \)

The frequency \( f \) can be calculated from the angular frequency \( \omega \) using the relation:
\[
f = \frac{\omega}{2\pi}
\]

Substituting the value of \( \omega \):
\[
f = \frac{314}{2\pi} \approx \frac{314}{6.2832} \approx 50 \, \text{Hz}
\]

### 2. Maximum Value
The maximum value of the voltage, denoted as \( V_m \), is given directly in the equation:
\[
V_m = 230 \, \text{V}
\]

### 3. Average Value
For a sinusoidal waveform, the average value over a full cycle is zero. However, if we are interested in the average value over half a cycle (positive half), it can be calculated as:
\[
V_{avg} = \frac{V_m}{\pi}
\]

Calculating it:
\[
V_{avg} = \frac{230}{\pi} \approx \frac{230}{3.1416} \approx 73.24 \, \text{V}
\]

### 4. RMS Value
The RMS value for a sinusoidal waveform is given by:
\[
V_{rms} = \frac{V_m}{\sqrt{2}}
\]

Calculating the RMS value:
\[
V_{rms} = \frac{230}{\sqrt{2}} \approx \frac{230}{1.4142} \approx 162.63 \, \text{V}
\]

### Summary of Results
- **Frequency**: \( 50 \, \text{Hz} \)
- **Maximum Value**: \( 230 \, \text{V} \)
- **Average Value (over half cycle)**: \( 73.24 \, \text{V} \)
- **RMS Value**: \( 162.63 \, \text{V} \)

Answer 2

For the alternating voltage given by \( V = 230 \sin(314t) \), let's calculate the frequency, maximum value, average value, and RMS value.

### 1. **Frequency**

The general form of an alternating voltage can be written as:
\[ V = V_m \sin(\omega t) \]

where:
- \( V_m \) is the maximum value (or peak amplitude) of the voltage.
- \( \omega \) is the angular frequency in radians per second.

From the given equation \( V = 230 \sin(314t) \), we can see that:
\[ \omega = 314 \text{ rad/s} \]

The angular frequency \( \omega \) is related to the frequency \( f \) in hertz (Hz) by:
\[ \omega = 2 \pi f \]

So, solving for \( f \):
\[ f = \frac{\omega}{2 \pi} = \frac{314}{2 \pi} \approx 50 \text{ Hz} \]

### 2. **Maximum Value**

The maximum value \( V_m \) is the coefficient of the sine function, which is:
\[ V_m = 230 \text{ V} \]

### 3. **Average Value**

For a sinusoidal waveform over one complete cycle, the average value is zero because the positive and negative halves cancel each other out. Mathematically:
\[ \text{Average Value} = 0 \text{ V} \]

### 4. **RMS Value**

The RMS (Root Mean Square) value of a sinusoidal voltage is given by:
\[ V_{rms} = \frac{V_m}{\sqrt{2}} \]

So, substituting \( V_m = 230 \text{ V} \):
\[ V_{rms} = \frac{230}{\sqrt{2}} \approx 162.6 \text{ V} \]

### Summary

- **Frequency**: \( 50 \text{ Hz} \)
- **Maximum Value**: \( 230 \text{ V} \)
- **Average Value**: \( 0 \text{ V} \)
- **RMS Value**: \( \approx 162.6 \text{ V} \)