8 hours · 8 marks
Two-port network theory describes linear networks with one input port and one output port using parameter matrices. Exam focus: definitions of Z, Y, ABCD, h, g; conversions (especially \(h\) from \(Z\), \(Z\) from ABCD); cascade via ABCD multiplication; reciprocity and symmetry tests.
A two-port has four terminals grouped as port 1 (\(V_1,I_1\)) and port 2 (\(V_2,I_2\)). Standard sign convention: both currents \(I_1\) and \(I_2\) enter the network through the \(+\) terminals (except for transmission/ABCD parameters, where \(I_2\) is usually defined leaving port 2 — see below).
| Parameter | Definition | Condition |
|---|---|---|
| \(Z_{11}\) | \(V_1/I_1\) | \(I_2=0\) (port 2 open) |
| \(Z_{12}\) | \(V_1/I_2\) | \(I_1=0\) |
| \(Z_{21}\) | \(V_2/I_1\) | \(I_2=0\) |
| \(Z_{22}\) | \(V_2/I_2\) | \(I_1=0\) |
\(Z_{11},Z_{22}\): driving-point impedances; \(Z_{12},Z_{21}\): transfer impedances.
Measured with the other port shorted (\(V=0\)). \(\mathbf{Y}=\mathbf{Z}^{-1}\) when both exist.
Relate sending-end quantities to receiving-end quantities. Convention: \(I_2\) leaves port 2 (so \(I_2'=-I_2\) in Z-convention):
\[ \begin{aligned} V_1 &= A V_2 + B (-I_2)\\ I_1 &= C V_2 + D (-I_2) \end{aligned} \quad\text{i.e.}\quad \begin{bmatrix}V_1\\I_1\end{bmatrix} = \begin{bmatrix}A&B\\C&D\end{bmatrix} \begin{bmatrix}V_2\\-I_2\end{bmatrix} \]| Param | Meaning | Condition |
|---|---|---|
| \(A=V_1/V_2\) | open-circuit voltage ratio | \(I_2=0\) |
| \(B=V_1/(-I_2)\) | short-circuit transfer impedance | \(V_2=0\) |
| \(C=I_1/V_2\) | open-circuit transfer admittance | \(I_2=0\) |
| \(D=I_1/(-I_2)\) | short-circuit current ratio | \(V_2=0\) |
Useful when cascading is described from load toward source. Inverse of the ABCD matrix when \(\Delta_T=AD-BC\neq 0\).
| Param | Name | Condition |
|---|---|---|
| \(h_{11}=V_1/I_1\) | input impedance | \(V_2=0\) (port 2 short) |
| \(h_{12}=V_1/V_2\) | reverse voltage gain | \(I_1=0\) |
| \(h_{21}=I_2/I_1\) | forward current gain | \(V_2=0\) |
| \(h_{22}=I_2/V_2\) | output admittance | \(I_1=0\) |
Widely used for BJT small-signal models (\(h_{ie},h_{re},h_{fe},h_{oe}\)).
\(\mathbf{g}=\mathbf{h}^{-1}\) when both exist. Useful for some FET / dual formulations.
| From → To | Key relations |
|---|---|
| Z → Y | \(\mathbf{Y}=\mathbf{Z}^{-1}\); \(Y_{11}=Z_{22}/\Delta_Z\), etc. |
| Z → ABCD | \(A=Z_{11}/Z_{21}\), \(B=\Delta_Z/Z_{21}\), \(C=1/Z_{21}\), \(D=Z_{22}/Z_{21}\) |
| Y → ABCD | \(A=-Y_{22}/Y_{21}\), \(B=-1/Y_{21}\), \(C=-\Delta_Y/Y_{21}\), \(D=-Y_{11}/Y_{21}\) |
| ABCD → h | \(h_{11}=B/D\), \(h_{12}=(AD-BC)/D\), \(h_{21}=-1/D\), \(h_{22}=C/D\) |
| Connection | Rule | When valid |
|---|---|---|
| Series (ports in series) | \(\mathbf{Z}=\mathbf{Z}_a+\mathbf{Z}_b\) | loop currents compatible; no forbidden circulating paths |
| Parallel | \(\mathbf{Y}=\mathbf{Y}_a+\mathbf{Y}_b\) | port voltages shared; Brune tests satisfied |
| Cascade (chain) | \(\begin{bmatrix}A&B\\C&D\end{bmatrix}=\mathbf{T}_1\mathbf{T}_2\) | port 2 of first feeds port 1 of second |
| Series–parallel | add \(\mathbf{h}\) | hybrid interconnection |
| Parallel–series | add \(\mathbf{g}\) | inverse hybrid interconnection |
A network of bilateral passive elements (R, L, C, mutual transformers without dependent sources / gyrators) is reciprocal:
| Parameter set | Reciprocity condition |
|---|---|
| Z | \(Z_{12}=Z_{21}\) |
| Y | \(Y_{12}=Y_{21}\) |
| ABCD | \(AD-BC=1\) |
| h | \(h_{12}=-h_{21}\) |
| g | \(g_{12}=-g_{21}\) |
Dependent sources generally destroy reciprocity.
A symmetric two-port looks the same if ports are swapped:
Ladder / T
T-network: series arms \(Z_a\) (port 1 side), \(Z_b\) (port 2 side), shunt arm \(Z_c\) from junction to common return.
Solution. With \(I_2=0\): \(Z_{11}=Z_a+Z_c\), \(Z_{21}=Z_c\).
With \(I_1=0\): \(Z_{22}=Z_b+Z_c\), \(Z_{12}=Z_c\).
\[ \mathbf{Z}=\begin{bmatrix}Z_a+Z_c & Z_c\\ Z_c & Z_b+Z_c\end{bmatrix} \]Reciprocal: \(Z_{12}=Z_{21}=Z_c\) always for this passive T.
Symmetric: iff \(Z_a=Z_b\) (then \(Z_{11}=Z_{22}\)).
Series arms \(2\,\Omega\) and \(3\,\Omega\), shunt \(6\,\Omega\). Find Z-parameters; check reciprocal/symmetric.
ABCD multiply
Cascade: series \(Z_1\) then shunt \(Y_2=1/Z_2\).
Series impedance alone:
\[ \mathbf{T}_{Z}=\begin{bmatrix}1&Z_1\\0&1\end{bmatrix} \]Shunt admittance alone:
\[ \mathbf{T}_{Y}=\begin{bmatrix}1&0\\Y_2&1\end{bmatrix} \] \[ \mathbf{T}=\mathbf{T}_{Z}\mathbf{T}_{Y}=\begin{bmatrix}1+Z_1 Y_2 & Z_1\\ Y_2 & 1\end{bmatrix} \]Then \(Z_{11}=A/C=(1+Z_1 Y_2)/Y_2=Z_2+Z_1\), \(Z_{21}=1/C=Z_2\), etc.
Network A: \(A_1=2,\,B_1=3\,\Omega,\,C_1=1\,\text{S},\,D_1=2\). Network B: \(A_2=1,\,B_2=1\,\Omega,\,C_2=1\,\text{S},\,D_2=2\). Find overall ABCD and check reciprocity.
\(\Delta=AD-BC=25-24=1\) ⇒ overall reciprocal (product of reciprocal networks).
\(A=5\neq D=5\) wait — here \(A=D=5\) so also symmetric overall. (Check: \(5\cdot5-8\cdot3=1\).)
Dependent source
Port 1: voltage \(V_1\) across input. Series resistance \(R\) then a VCVS \(V_x=\mu V_1\) feeding port 2 through the controlled source (simplified unilateral amp model: \(I_2\) depends on \(V_1\)).
Classic resistive network: left series \(1\,\Omega\), controlled current source \(2I_1\) in parallel with \(2\,\Omega\) at port 2.
Find Y or h; test reciprocity.
With dependent sources, compute parameters from definitions carefully.
Illustration with Z for a network where \(V_2=k I_1\) and input is resistive: often \(Z_{21}\neq Z_{12}\).
Conclusion pattern for exams: if \(Z_{12}\neq Z_{21}\) (or \(AD-BC\neq 1\)), the network is not reciprocal — expected when controlled sources are present.
Given \(\mathbf{Z}=\begin{bmatrix}8&6\\6&9\end{bmatrix}\). Find \(\mathbf{h}\).
\(A=5,\,B=8,\,C=3,\,D=5\). Find Z-parameters.