A 300 MHz generator of 50 ohm internal impedance drives a load through a section of a

transmission line of 50 ohm characteristic impedance and 3 X 108 m/s propagation

velocity. The load consists of a parallel connection of a 50 ohm resistor and a 10.7 pF

capacitor. Connected across the load is an open-circuited section of a transmission line

with electrical characteristics identical to the driving line.

For what length of the open circuited section will the load and generator be

matched?

To solve this problem, we need to determine the length of the open-circuited section of the transmission line that will result in the load and generator being matched. 

### Step 1: Calculate the Impedance of the Load

The load consists of a parallel connection of a \(50 \, \Omega\) resistor and a \(10.7 \, \text{pF}\) capacitor.

The impedance of the capacitor at a frequency \(f\) is:
\[
Z_C = \frac{1}{j\omega C} = \frac{1}{j2\pi f C}
\]
where:
- \(f = 300 \, \text{MHz} = 300 \times 10^6 \, \text{Hz}\)
- \(C = 10.7 \, \text{pF} = 10.7 \times 10^{-12} \, \text{F}\)

Substituting the values:
\[
Z_C = \frac{1}{j2\pi \times 300 \times 10^6 \times 10.7 \times 10^{-12}} = \frac{1}{j2.014 \times 10^{-2}} \approx -j49.7 \, \Omega
\]

The total load impedance \(Z_L\) is the parallel combination of the \(50 \, \Omega\) resistor and the capacitor:
\[
\frac{1}{Z_L} = \frac{1}{R} + \frac{1}{Z_C} = \frac{1}{50 \, \Omega} + \frac{1}{-j49.7 \, \Omega}
\]

Simplifying:
\[
\frac{1}{Z_L} = \frac{1}{50} + \frac{j}{49.7}
\]

To find \(Z_L\), calculate:
\[
Z_L = \left( \frac{1}{50} + \frac{j}{49.7} \right)^{-1} = \frac{1}{\frac{1 + j}{50}} = \frac{50}{1 + j} = 25 - j25 \, \Omega
\]

So, the load impedance is:
\[
Z_L = 25 - j25 \, \Omega
\]

### Step 2: Use the Open-Circuited Transmission Line to Cancel the Imaginary Part

The open-circuited section of the transmission line will create a reactive component that can be used to cancel out the reactive part of the load impedance. The impedance seen at the beginning of an open-circuited transmission line of length \(l\) is:
\[
Z_{oc}(l) = jZ_0 \tan(\beta l)
\]
where:
- \(Z_0 = 50 \, \Omega\) (characteristic impedance of the line)
- \(\beta = \frac{2\pi}{\lambda}\) is the phase constant
- \(\lambda = \frac{v}{f}\) is the wavelength

Given:
- \(v = 3 \times 10^8 \, \text{m/s}\)
- \(f = 300 \, \text{MHz}\)

Calculate the wavelength:
\[
\lambda = \frac{v}{f} = \frac{3 \times 10^8 \, \text{m/s}}{300 \times 10^6 \, \text{Hz}} = 1 \, \text{m}
\]
\[
\beta = \frac{2\pi}{\lambda} = 2\pi \, \text{rad/m}
\]

For the generator to be matched to the load, the reactive part of \(Z_L\) must be canceled out by the reactive part provided by the open-circuited transmission line:
\[
jZ_0 \tan(\beta l) = j25 \, \Omega
\]

This simplifies to:
\[
\tan(\beta l) = \frac{25}{50} = 0.5
\]

### Step 3: Determine the Length of the Transmission Line

Now, solve for \(l\):
\[
\beta l = \tan^{-1}(0.5)
\]
\[
l = \frac{1}{\beta} \tan^{-1}(0.5) = \frac{1}{2\pi} \times 0.464 \, \text{rad} \approx 0.074 \, \text{m} = 7.4 \, \text{cm}
\]

So, the length of the open-circuited section that will match the load to the generator is **7.4 cm**.

### Summary
The length of the open-circuited section of the transmission line that will result in matching the load and generator is **7.4 cm**.

Do you need further clarification or have any questions? Here are some follow-up questions:

1. How does the parallel connection of a resistor and capacitor affect the impedance of the load?
2. What would happen if the open-circuited line length were not correctly matched?
3. How does the characteristic impedance of the transmission line influence the matching condition?
4. Can this method be applied to other types of loads, such as inductive loads?
5. What are the practical implications of using open-circuited transmission lines in RF design?

**Tip**: When working with transmission lines, remember that the length of the line directly impacts the impedance it presents at the input, especially in the case of open or short-circuited lines.

Learn how to calculate the length of an open-circuited transmission line to achieve impedance matching between a generator and load. This problem involves transmission line theory, impedance matching, and parallel RLC circuits with detailed step-by-step solutions.

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