The network. It is a T-network: a 4 Ω resistor in series with port 1, a 2 Ω resistor shunting from the middle node down to the common rail, and a 1 Ω resistor in series with port 2.

Definition of the z-parameters. The impedance (open-circuit) parameters are defined by

\(V_1 = Z_{11}I_1 + Z_{12}I_2, \qquad V_2 = Z_{21}I_1 + Z_{22}I_2\)

so each parameter is measured with one port left open:

\(Z_{11}=\left.\dfrac{V_1}{I_1}\right|_{I_2=0},\ Z_{12}=\left.\dfrac{V_1}{I_2}\right|_{I_1=0},\ Z_{21}=\left.\dfrac{V_2}{I_1}\right|_{I_2=0},\ Z_{22}=\left.\dfrac{V_2}{I_2}\right|_{I_1=0}\)

Z₁₁ — drive port 1, port 2 open. With I₂ = 0 no current flows in the 1 Ω arm, so it drops no voltage and can be ignored. The current I₁ flows through the 4 Ω series arm and then through the 2 Ω shunt arm:

\(Z_{11} = 4 + 2 = 6\ \Omega\)

Z₂₂ — drive port 2, port 1 open. Now the 4 Ω arm carries no current, and I₂ flows through the 1 Ω series arm and the 2 Ω shunt arm:

\(Z_{22} = 1 + 2 = 3\ \Omega\)

Z₂₁ — transfer impedance. With port 2 open, all of I₁ flows through the shunt 2 Ω resistor, and the open-circuit voltage at port 2 is the voltage across that shunt element (the 1 Ω arm drops nothing):

\(V_2 = 2 I_1 \Rightarrow Z_{21} = 2\ \Omega\)

Z₁₂. The network contains only resistors, so it is reciprocal and \(Z_{12} = Z_{21} = 2\ \Omega\). Physically, the 2 Ω shunt arm is the branch common to both loops, and for a T-network the mutual term is always the shunt element.

Assemble the matrix.

\([Z] = \begin{bmatrix} 6 & 2 \\ 2 & 3 \end{bmatrix}\Omega\)

Quick check on the options. A reciprocal network must give a symmetric matrix (Z₁₂ = Z₂₁), which immediately rules out options 2, 3 and 4; only option 1 is symmetric, and its diagonal entries 6 Ω and 3 Ω match the series-plus-shunt sums computed above.

Shortcut worth memorising. For a T-network with series arms Z_A (port 1), Z_B (port 2) and shunt arm Z_C: \(Z_{11}=Z_A+Z_C\), \(Z_{22}=Z_B+Z_C\), \(Z_{12}=Z_{21}=Z_C\).

Hence, the z-parameter matrix is [[6, 2], [2, 3]] Ω.