The EMF of a generator of 100 ohm internal impedance is a step-function of 24 volt
amplitude. The generator drives a 20 km long section of open-circuited, lossless
transmission line of 50 ohm characteristic impedance and 2X 108 m/s propagation
velocity. A 150 ohm resistor is connected across the line 10 km away from the generator.
The step-function is launched on the line at time t = 0.
Determine:
(i) the time of arrival of the initial voltage at the 150 ohm resistor,
(ii) the amplitude of the initial voltage and,
(iii) the final steady state voltage across the resistor.

To solve this problem, we will analyze the behavior of the voltage as it propagates through the transmission line and encounters the resistor.

### Given Data:
- Internal impedance of the generator, \( Z_g = 100 \, \Omega \)
- Step-function amplitude, \( V_s = 24 \, \text{V} \)
- Transmission line characteristic impedance, \( Z_0 = 50 \, \Omega \)
- Length of the transmission line, \( L = 20 \, \text{km} \)
- Propagation velocity, \( v = 2 \times 10^8 \, \text{m/s} \)
- Resistor across the line, \( R = 150 \, \Omega \)
- Position of the resistor from the generator, \( d = 10 \, \text{km} \)

### Step 1: Time of Arrival of the Initial Voltage at the Resistor

The time \( t \) for the signal to travel from the generator to the resistor is given by:
\[
t = \frac{d}{v}
\]

Substituting the values:
\[
t = \frac{10 \times 10^3 \, \text{m}}{2 \times 10^8 \, \text{m/s}} = 5 \times 10^{-5} \, \text{s} = 50 \, \mu\text{s}
\]

So, the time of arrival of the initial voltage at the 150 ohm resistor is **50 µs**.

### Step 2: Amplitude of the Initial Voltage

The voltage amplitude launched onto the line can be determined by considering the voltage divider formed by the generator’s internal impedance and the transmission line’s characteristic impedance.

\[
V_{in} = V_s \times \frac{Z_0}{Z_g + Z_0}
\]

Substituting the given values:
\[
V_{in} = 24 \, \text{V} \times \frac{50 \, \Omega}{100 \, \Omega + 50 \, \Omega} = 24 \, \text{V} \times \frac{50}{150} = 24 \, \text{V} \times \frac{1}{3} = 8 \, \text{V}
\]

The amplitude of the initial voltage at the input to the transmission line is **8 V**.

### Step 3: Final Steady State Voltage Across the Resistor

To determine the final steady-state voltage across the resistor, we analyze the situation after multiple reflections, considering the termination at the 150 ohm resistor.

The reflection coefficient at the 150 ohm resistor is:
\[
\Gamma_R = \frac{R - Z_0}{R + Z_0} = \frac{150 \, \Omega - 50 \, \Omega}{150 \, \Omega + 50 \, \Omega} = \frac{100 \, \Omega}{200 \, \Omega} = 0.5
\]

The final voltage across the resistor is given by:
\[
V_{final} = V_{in} \times \frac{R}{R + Z_g}
\]

Substituting the values:
\[
V_{final} = 24 \, \text{V} \times \frac{150 \, \Omega}{150 \, \Omega + 100 \, \Omega} = 24 \, \text{V} \times \frac{150}{250} = 24 \, \text{V} \times 0.6 = 14.4 \, \text{V}
\]

The final steady-state voltage across the 150 ohm resistor is **14.4 V**.

### Summary of Results:
1. **Time of arrival of the initial voltage**: 50 µs.
2. **Amplitude of the initial voltage**: 8 V.
3. **Final steady state voltage across the resistor**: 14.4 V.

Would you like more details on any of these calculations, or have any questions? Here are some follow-up questions:

1. How does the reflection coefficient affect the voltage across the resistor?
2. What would happen if the resistor were located at the end of the transmission line?
3. How would the results change if the transmission line were lossy?
4. What is the significance of the characteristic impedance of the transmission line?
5. How does the propagation velocity influence the time of arrival of the signal?

**Tip**: Understanding the concept of reflection coefficients in transmission lines is crucial for analyzing voltage distributions and signal reflections in electrical circuits.

Explore the detailed analysis of voltage propagation in a transmission line with a resistor. Solve for the time of arrival, initial voltage amplitude, and final steady-state voltage across the resistor. Key concepts include transmission line theory, reflection coefficients, and voltage division. Suitable for advanced undergraduate studies.

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