Question

Calculate the distribution factor for 36 slots, 4 pole, single layer three phase winding.

Answer 1

To calculate the distribution factor (k_d) for a three-phase winding with 36 slots, 4 poles, and a single layer, we can use the following formula:

\[
k_d = \frac{\sin\left(\frac{m \cdot \pi}{P}\right)}{m \cdot \sin\left(\frac{\pi}{P}\right)}
\]

Where:
- \( m \) is the number of slots per pole per phase.
- \( P \) is the number of poles.

### Step 1: Calculate \( m \)

First, we need to determine the number of slots per pole per phase. This can be calculated using:

\[
m = \frac{\text{Total Slots}}{\text{Number of Phases} \times \text{Number of Poles}} = \frac{36}{3 \times 4} = \frac{36}{12} = 3
\]

### Step 2: Substitute Values into the Distribution Factor Formula

Now, substituting the values into the distribution factor formula:

- \( P = 4 \)
- \( m = 3 \)

\[
k_d = \frac{\sin\left(\frac{3 \cdot \pi}{4}\right)}{3 \cdot \sin\left(\frac{\pi}{4}\right)}
\]

### Step 3: Calculate Sine Values

- \(\sin\left(\frac{3\pi}{4}\right) = \sin\left(135^\circ\right) = \frac{\sqrt{2}}{2}\)
- \(\sin\left(\frac{\pi}{4}\right) = \sin\left(45^\circ\right) = \frac{\sqrt{2}}{2}\)

### Step 4: Plug in the Sine Values

\[
k_d = \frac{\frac{\sqrt{2}}{2}}{3 \cdot \frac{\sqrt{2}}{2}} = \frac{\frac{\sqrt{2}}{2}}{\frac{3\sqrt{2}}{2}} = \frac{1}{3}
\]

### Conclusion

The distribution factor \( k_d \) for a three-phase winding with 36 slots and 4 poles is:

\[
\boxed{\frac{1}{3}}
\]

Answer 2

To calculate the distribution factor (Kd) for a three-phase winding, you need the number of slots and the number of poles. The distribution factor is given by:

\[ K_d = \frac{\sin(\frac{m \beta}{2})}{m \sin(\frac{\beta}{2})} \]

where:
- \( m \) = number of slots per pole per phase
- \( \beta \) = slot angle

First, find the number of slots per pole per phase \( m \):

\[ m = \frac{\text{Number of Slots}}{\text{Number of Poles} \times \text{Number of Phases}} \]

For your case:
\[ m = \frac{36}{4 \times 3} = 3 \]

The slot angle \( \beta \) is given by:

\[ \beta = \frac{360^\circ}{\text{Number of Slots}} \]

For 36 slots:
\[ \beta = \frac{360^\circ}{36} = 10^\circ \]

Now plug these into the formula:

\[ K_d = \frac{\sin(\frac{3 \times 10^\circ}{2})}{3 \sin(\frac{10^\circ}{2})} = \frac{\sin(15^\circ)}{3 \sin(5^\circ)} \]

Using standard values:
- \( \sin(15^\circ) \approx 0.2588 \)
- \( \sin(5^\circ) \approx 0.0872 \)

Thus:
\[ K_d \approx \frac{0.2588}{3 \times 0.0872} \approx 0.99 \]

The distribution factor \( K_d \) is approximately 0.99.