Question

A 3 phase, 50Hz, 8 pole alternator has star connected winding with 120 slots and 8 conductors per slot. The flux per pole is 0.05 Wb, sinusoidally distributed. Determine the phase and line voltages.

Answer 1

To determine the phase and line voltages of the alternator, we need to go through a few calculations. Let’s break down the problem step by step.

### Given Data:
1. **Frequency (\(f\))**: 50 Hz
2. **Number of Poles (\(P\))**: 8
3. **Number of Slots**: 120
4. **Number of Conductors per Slot**: 8
5. **Flux per Pole (\(\Phi\))**: 0.05 Wb (sinusoidally distributed)

### Step 1: Calculate the Number of Phases

For a 3-phase alternator, the number of phases is 3.

### Step 2: Determine the Number of Slots per Phase

Since the winding is star-connected and there are 120 slots, these slots are distributed among the 3 phases. In a balanced star-connected system, the slots are evenly divided. So:

- **Slots per Phase** = \(\frac{120}{3}\) = 40 slots

### Step 3: Calculate the Number of Poles per Phase

Each phase winding will span several slots, but let's focus on calculating the phase voltage first, which involves the number of poles directly.

### Step 4: Calculate the Speed of the Alternator

The speed of the alternator in revolutions per minute (RPM) is given by the formula:

\[
N = \frac{120 \times f}{P}
\]

where:
- \( f \) = frequency (50 Hz)
- \( P \) = number of poles (8)

Substituting the values:

\[
N = \frac{120 \times 50}{8} = 750 \text{ RPM}
\]

### Step 5: Calculate the Voltage per Phase

To find the phase voltage, we use the formula for the induced voltage in an alternator. For sinusoidal flux distribution, the induced EMF per phase is given by:

\[
E_{\text{ph}} = \frac{4.44 \times f \times \Phi \times N_{\text{ph}}}{P}
\]

where:
- \( \Phi \) = flux per pole (0.05 Wb)
- \( N_{\text{ph}} \) = number of conductors per phase

First, we need to calculate the number of conductors per phase winding.

### Number of Conductors per Phase Winding

Each slot has 8 conductors, and there are 40 slots per phase.

\[
N_{\text{ph}} = 40 \text{ slots} \times 8 \text{ conductors/slot} = 320 \text{ conductors}
\]

Now, substituting the values into the formula:

\[
E_{\text{ph}} = \frac{4.44 \times 50 \times 0.05 \times 320}{8}
\]

\[
E_{\text{ph}} = \frac{4.44 \times 50 \times 0.05 \times 320}{8} = \frac{4.44 \times 50 \times 16}{8}
\]

\[
E_{\text{ph}} = \frac{3552}{8} = 444 \text{ V}
\]

### Step 6: Calculate the Line Voltage

In a star-connected system, the line voltage (\(V_{\text{L}}\)) is related to the phase voltage (\(V_{\text{ph}}\)) by:

\[
V_{\text{L}} = \sqrt{3} \times V_{\text{ph}}
\]

So:

\[
V_{\text{L}} = \sqrt{3} \times 444 \approx 1.732 \times 444 = 769.3 \text{ V}
\]

### Summary

- **Phase Voltage (\(V_{\text{ph}}\))**: 444 V
- **Line Voltage (\(V_{\text{L}}\))**: 769.3 V

These values are approximate and based on ideal assumptions (such as no losses and perfectly balanced windings). In practical scenarios, there might be additional factors to consider such as winding resistance and leakage flux.

Answer 2

To determine the phase and line voltages of a 3-phase alternator, we can follow these steps. Given the alternator's specifications:

- Frequency (\( f \)) = 50 Hz
- Number of Poles (\( P \)) = 8
- Number of Slots (\( S \)) = 120
- Conductors per Slot (\( C \)) = 8
- Flux per Pole (\( \Phi \)) = 0.05 Wb

### Step 1: Calculate the Synchronous Speed

The synchronous speed \( N_s \) of the alternator can be found using the formula:

\[ N_s = \frac{120 \cdot f}{P} \]

Substituting the given values:

\[ N_s = \frac{120 \cdot 50}{8} = 750 \text{ RPM} \]

### Step 2: Calculate the Number of Phases and Slots per Phase

In a star-connected alternator with 120 slots, the number of slots per phase can be calculated as:

\[ \text{Slots per Phase} = \frac{S}{\text{Number of Phases}} \]

Since this is a 3-phase alternator:

\[ \text{Slots per Phase} = \frac{120}{3} = 40 \text{ slots per phase} \]

### Step 3: Calculate the Phase Voltage

The phase voltage (\( V_{\text{ph}} \)) can be calculated using the following formula for a sinusoidally distributed flux:

\[ V_{\text{ph}} = \frac{4.44 \cdot f \cdot \Phi \cdot N_{\text{ph}}}{\text{Number of Phases}} \]

where \( N_{\text{ph}} \) is the number of turns per phase. For a star-connected alternator, the number of turns per phase is given by the total number of conductors per phase divided by the number of conductors in series per phase.

Let's calculate the number of turns per phase:

\[ N_{\text{ph}} = \text{Conductors per Slot} \times \text{Slots per Phase} = 8 \times 40 = 320 \]

Now, substituting the values:

\[ V_{\text{ph}} = 4.44 \cdot 50 \cdot 0.05 \cdot \frac{320}{3} \]

\[ V_{\text{ph}} = 4.44 \cdot 50 \cdot 0.05 \cdot 106.67 \]

\[ V_{\text{ph}} = 4.44 \cdot 50 \cdot 5.333 \]

\[ V_{\text{ph}} = 1186.67 \text{ Volts} \]

### Step 4: Calculate the Line Voltage

For a star-connected alternator, the line voltage (\( V_L \)) is related to the phase voltage by:

\[ V_L = \sqrt{3} \cdot V_{\text{ph}} \]

Substituting the calculated phase voltage:

\[ V_L = \sqrt{3} \cdot 1186.67 \]

\[ V_L \approx 1.732 \cdot 1186.67 \]

\[ V_L \approx 2050 \text{ Volts} \]

### Summary

- **Phase Voltage (\( V_{\text{ph}} \))**: Approximately 1186.67 Volts
- **Line Voltage (\( V_L \))**: Approximately 2050 Volts