The T-parameters for the following cascaded network is:..

The T-parameters for the following cascaded network is:

Network description:

• Series resistor of 2 Ω
• Shunt resistor of 10 Ω to ground
• Series resistors of 4 Ω and 4 Ω
• Shunt resistor of 20 Ω to ground
• Series resistor of 8 Ω

Options:

A. \( \begin{bmatrix}1.78 & 25.84 \\ 0.195 & 3.3 \end{bmatrix} \)

B. \( \begin{bmatrix}1.25 & 45.23 \\ 1.65 & 4.58 \end{bmatrix} \)

C. \( \begin{bmatrix}1.65 & 20.23 \\ 1.95 & 6.58 \end{bmatrix} \)

D. \( \begin{bmatrix}7.2 & 16.3 \\ 2.35 & 4.3 \end{bmatrix} \)

Solution

The ladder network is treated as a cascade of series impedances and shunt admittances:

• Series: 2 Ω
• Shunt: 10 Ω → \(Y_1 = 0.1 \, S\)
• Series: 4 Ω + 4 Ω = 8 Ω
• Shunt: 20 Ω → \(Y_2 = 0.05 \, S\)
• Series: 8 Ω

For transmission (T/ABCD) matrices:

• Series \(Z\): \( \begin{bmatrix}1 & Z \\ 0 & 1\end{bmatrix} \)
• Shunt \(Y\): \( \begin{bmatrix}1 & 0 \\ Y & 1\end{bmatrix} \)

Total matrix:

\[ T = S(2)\cdot Sh(0.1)\cdot S(8)\cdot Sh(0.05)\cdot S(8) \]

Final result:

\[ T = \begin{bmatrix} 1.78 & 25.84 \\ 0.195 & 3.3 \end{bmatrix} \]

Correct Answer: Option A