A high-voltage direct-current generating station delivers power to a city. The generator output voltage is V0=400 kV, the city is represented by resistance RL=3542 ohms, and each wire of the transmission line round-trip between the station and city is represented by resistance RTL=195 ohms. What is the voltage VL when the electricity reaches the city (in kilo volt)?
The Correct Answer and Explanation is:
To find the voltage $V_L$ when the electricity reaches the city, we can use principles of voltage division in a series circuit, since the transmission line and city load resistances are in series.
Given:
- Generator voltage: $V_0 = 400 \, \text{kV} = 400,000 \, \text{V}$
- Load resistance (city): $R_L = 3542 \, \Omega$
- Total transmission line resistance (round-trip): $R_{TL} = 195 \, \Omega$
Step 1: Total resistance in the circuit
$$
R_{\text{total}} = R_{L} + R_{TL} = 3542 \, \Omega + 195 \, \Omega = 3737 \, \Omega
$$
Step 2: Current in the circuit
Using Ohm’s Law:
$$
I = \frac{V_0}{R_{\text{total}}} = \frac{400000 \, \text{V}}{3737 \, \Omega} \approx 107.09 \, \text{A}
$$
Step 3: Voltage across the city load
$$
V_L = I \cdot R_L = 107.09 \, \text{A} \times 3542 \, \Omega \approx 379400 \, \text{V}
$$
Convert to kilovolts:
$$
V_L \approx \frac{379400}{1000} = \boxed{379.4 \, \text{kV}}
$$
✅ Final Answer: $\boxed{379.4 \, \text{kV}}$
Explanation (300+ words):
This problem illustrates the concept of voltage drop in a series circuit due to resistive losses along a transmission line. In long-distance power transmission, especially with high-voltage direct current (HVDC) systems, minimizing energy loss is crucial. Here, the generator outputs 400 kV, but due to resistance in the wires, not all of this voltage reaches the city.
The entire setup is modeled as a simple series circuit: the transmission line resistance ($R_{TL}$) and the city’s electrical resistance load ($R_L$). Because current is the same through all elements in a series circuit, we can calculate the total resistance to find the current. Using Ohm’s Law $(V = IR)$, the total current $I$ drawn from the generator is calculated by dividing the total voltage by the total resistance.
Once we know the current, we find how much of the generator’s voltage actually drops across the city’s load. Since $V = IR$, multiplying the current by $R_L$ gives us the voltage reaching the city. In this case, the city receives approximately 379.4 kV, which means about 20.6 kV is lost due to resistance in the wires.
This emphasizes the importance of using high voltage in power transmission: for the same power, higher voltages mean lower currents, and since resistive losses are proportional to the square of the current $(P = I^2R)$, they are significantly reduced. This is a fundamental principle behind HVDC systems, which are used for efficient long-distance energy transfer
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