Chapter 10, Problem 10.1P

A matching network consists of a length of a T-line through section in series with a capacitor. Determine the length (in wavelengths) required of the through section and the capacitor value needed (at 1.0 GHz) to match a $10-j35\text{Ω}$ load impedance to the $50-\text{Ω}$ line.

To determine

The length of section that consists of a capacitor in series and the value of capacitor for matched network.

Answer to Problem 10.1P

The length of section is $0.294\text{λ}$ and the series capacitance is $1.3\text{ pF}$ .

Explanation of Solution

Given:

The load impedance is $10-j35\Omega$ and the characteristic impedance of transmission line is $50\Omega$ . The frequency is $1\text{ GHz}$ .

Calculation:

The Smith chart for the matched network is shown in Figure 1.

  

From the Smith chart, the value of normalized load is $0.2-j0.7$ that is located on point $a$ .

The normalized load at the point $b$ is $1+j2.4$ .

The length of the section of matched network is calculated as,

  $\begin{array}{c}d=0.5\text{λ}+0.194\text{λ}-0.4\text{λ}\\ =0.294\text{λ}\end{array}$

Therefore, the length of the section is $0.294\text{λ}$ .

The value of series capacitance is calculated as,

  $\begin{array}{c}-j2.4=\frac{1}{j\text{ω}C{Z}_o}\\ C=\frac{1}{2.4\text{ω}{Z}_o}\end{array}$

Substitute $2\text{π}\times {10}^9$ for $\text{ω}$ and $50$ for $Z_o$ in above equation.

  $\begin{array}{c}C=\frac{1}{2.4\left(2\text{π}\times {10}^9\right)\left(50\right)}\\ \approx 1.3\times {10}^{-12}\text{ F}\end{array}$

Therefore, the capacitance is $1.3\text{ pF}$ .

Conclusion:

Thus, the length of section is $0.294\text{λ}$ and the series capacitance is $1.3\text{ pF}$ .