A small town with a demand of 800 kW of electric power at 220 V is situated 15 km away from an electric plant generating power at 440 V. The resistance of the two line wires carrying power is `0.5 Omega` per km. The town gets power from the lines through a 4000-220 V step down transformer at a substation in the town.
Estimate the line power loss in the form of heat.
(b) How much power must the plant supply. assuming there is negligible power loss due to leakage?
(c) Characterize the step up transformer at the plant.

Options

A. 400 kW
B. 600 kW
C. 300 kW
D. 800 W

Explanation

To solve the problem step by step, we will break it down into parts as follows:

### Part (a): Estimate the line power loss in the form of heat.

1. **Calculate the total resistance of the transmission line**:
- The resistance of the line is given as \(0.5 \, \Omega/\text{km}\).
- The distance from the plant to the town is \(15 \, \text{km}\).
- Total resistance \(R\) can be calculated as:
\[ R = \text{Resistance per km} \times \text{Distance} = 0.5 \, \Omega/\text{km} \times 15 \, \text{km} = 7.5 \, \Omega \]

2. **Calculate the output current at the town**:
- The power demand of the town is \(800 \, \text{kW}\) at \(220 \, \text{V}\).
- The output current \(I\) can be calculated using the formula:
\[ I = \frac{P}{V} = \frac{800 \times 10^3 \, \text{W}}{220 \, \text{V}} \approx 3636.36 \, \text{A} \]

3. **Calculate the power loss in the transmission line**:
- The power loss \(P_{\text{loss}}\) in the line can be calculated using:
\[ P_{\text{loss}} = I^2 R \]
- Substituting the values:
\[ P_{\text{loss}} = (3636.36 \, \text{A})^2 \times 7.5 \, \Omega \approx 100,000,000 \, \text{W} = 100 \, \text{kW} \]

### Part (b): How much power must the plant supply?

1. **Calculate the total power supplied by the plant**:
- The total power supplied by the plant \(P_{\text{total}}\) is the sum of the power demand and the power loss:
\[ P_{\text{total}} = P_{\text{demand}} + P_{\text{loss}} = 800 \, \text{kW} + 100 \, \text{kW} = 900 \, \text{kW} \]

### Part (c): Characterize the step-up transformer at the plant.

1. **Calculate the voltage drop across the line**:
- The voltage drop \(V_{\text{drop}}\) can be calculated using:
\[ V_{\text{drop}} = I \times R = 3636.36 \, \text{A} \times 7.5 \, \Omega \approx 27,272.73 \, \text{V} \]

2. **Determine the primary and secondary voltages**:
- The primary voltage at the plant is \(440 \, \text{V}\).
- The secondary voltage after the transformer can be calculated as:
\[ V_{\text{secondary}} = V_{\text{primary}} + V_{\text{drop}} = 440 \, \text{V} + 27,272.73 \, \text{V} \approx 27,712.73 \, \text{V} \]

3. **Characterize the transformer**:
- The transformer is characterized by its input and output voltages:
- Input voltage: \(440 \, \text{V}\)
- Output voltage: \(27,712.73 \, \text{V}\)

### Summary of Results:
- (a) Line power loss: \(100 \, \text{kW}\)
- (b) Total power supplied by the plant: \(900 \, \text{kW}\)
- (c) Transformer characterization: \(440 \, \text{V} \rightarrow 27,712.73 \, \text{V}\)