A transmission line 90 meter has a characteristic impedance of 75 ohms. The line is connected through a 15 V dc source and 75 ohms source resistance at time t = 0 The voltage at the input end changes to 10 V, with the same polarity as before, at time t = 1.5mu*s What is the resistance that terminates the lines?


To find the resistance that terminates the line, we can use the theory of transmission lines and reflections. Let's break down the problem into steps.

### Given dataTo find the resistance that terminates the line, we can use the theory of transmission lines and reflections. Let's break down the problem into steps.

### Given 
- Length of the transmission line = 90 meters
- Characteristic impedance of the transmission line (\(Z_0\)) = 75 ohms
- Source voltage at \(t = 0\) is 15 V DC.
- Source resistance = 75 ohms (same as \(Z_0\)).
- At \(t = 1.5 \ \mu s\), the voltage at the input end changes to 10 V.

### Step 1: Calculate the time delay (\(\tau\)) of the transmission line.
The propagation delay (\(\tau\)) of a signal through a transmission line is given by:

\[
\tau = \frac{\text{length of the transmission line}}{\text{signal velocity}}
\]

Assuming the transmission line is air-filled (or vacuum-like), the signal velocity \(v\) is approximately equal to the speed of light \(c = 3 \times 10^8 \ \text{m/s}\).

\[
\tau = \frac{90}{3 \times 10^8} = 3 \times 10^{-7} \ \text{seconds} = 0.3 \ \mu s
\]

This means that it takes 0.3 microseconds for the signal to travel from one end of the transmission line to the other.

### Step 2: Voltage reflection coefficient and behavior at \( t = 1.5 \ \mu s \).

At time \( t = 1.5 \ \mu s \), the voltage at the input changes from 15 V to 10 V. The change in voltage is due to the reflection from the load (termination). To understand this, we can analyze the reflection using the reflection coefficient \(\Gamma\).

The reflection coefficient at the load is given by:

\[
\Gamma = \frac{R_L - Z_0}{R_L + Z_0}
\]

Where:
- \( R_L \) is the load resistance (which we need to find),
- \( Z_0 \) is the characteristic impedance of the line (75 ohms).

We know that the initial voltage at the source was 15 V and changed to 10 V due to reflection at time \( t = 1.5 \ \mu s \). The voltage change implies that the reflected voltage must have been \( 15 \ \text{V} - 10 \ \text{V} = 5 \ \text{V} \).

The reflected voltage \( V_r \) is related to the incident voltage \( V_i \) by:

\[
V_r = \Gamma V_i
\]

So, the reflection coefficient \(\Gamma\) is:

\[
\Gamma = \frac{V_r}{V_i} = \frac{5}{15} = \frac{1}{3}
\]

### Step 3: Solve for \(R_L\).

Now, we can use the reflection coefficient to solve for the terminating resistance \(R_L\):

\[
\frac{R_L - 75}{R_L + 75} = \frac{1}{3}
\]

Multiply both sides by \( (R_L + 75) \) to clear the denominator:

\[
R_L - 75 = \frac{1}{3}(R_L + 75)
\]

Now, multiply through and simplify:

\[
R_L - 75 = \frac{1}{3}R_L + 25
\]

Multiply everything by 3 to clear the fraction:

\[
3R_L - 225 = R_L + 75
\]

Simplifying:

\[
3R_L - R_L = 225 + 75
\]

\[
2R_L = 300
\]

\[
R_L = 150 \ \text{ohms}
\]

### Conclusion:
The resistance that terminates the line is \( \boxed{150 \ \text{ohms}} \).

---

Would you like more details or have any questions about the steps? Here are 5 related questions to explore further:

1. How does the signal velocity in a transmission line depend on the medium inside the line?
2. What happens when a transmission line is terminated with a short circuit or an open circuit?
3. How does the reflection coefficient affect the power transmitted through the line?
4. What are the consequences of mismatching the load resistance with the characteristic impedance?
5. How do we calculate the input impedance of a transmission line for different lengths?

### Tip:
To avoid signal reflections, always match the load impedance to the characteristic impedance of the transmission line!

A transmission line 90 meter has a characteristic impedance of 75 ohms. The line is connected through a 15 V dc source and 75 ohms source resistance at time t = 0. The voltage at the input end changes to 10 V, with the same polarity as before, at time t = 1.5μs. What is the resistance that terminates the lines?

Explore how to calculate the termination resistance of a 90-meter transmission line with a characteristic impedance of 75 ohms, connected to a 15 V DC source. This problem involves key concepts such as the reflection coefficient, voltage changes, and transmission line theory, suitable for undergraduate electrical engineering students.

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