Two-port formulas

All four are in ohms, and each is measured with the other port open. $z_{11}$ and $z_{22}$ are driving-point (input and output) impedances. $z_{12}$ and $z_{21}$ are transfer impedances.

All four are in siemens, and each is measured with the other port shorted. $[y]=[z{]}^{-1}$ whenever both exist. In general $y_{11}\ne 1/z_{11}$, because the two are measured under different terminations.

$h_{11}$ is the short-circuit input impedance (ohms), $h_{12}$ the open-circuit reverse voltage gain, $h_{21}$ the short-circuit forward current gain, and $h_{22}$ the open-circuit output admittance (siemens). Port 2 is shorted for $h_{11}$ and $h_{21}$; port 1 is open for $h_{12}$ and $h_{22}$.

$[g]=[h{]}^{-1}$ whenever both exist. $g_{11}$ is the open-circuit input admittance (siemens), $g_{12}$ the short-circuit reverse current gain, $g_{21}$ the open-circuit forward voltage gain, and $g_{22}$ the short-circuit output impedance (ohms).

Written out, $V_1=A{V}_2-B{I}_2$ and $I_1=C{V}_2-D{I}_2$. $A$ and $D$ are dimensionless, $B$ is in ohms and $C$ in siemens. $A$ and $C$ are measured with port 2 open, $B$ and $D$ with port 2 shorted. The minus sign is there because $-I_2$ is the current that leaves port 2 and enters whatever is connected to it.

Any one of these implies all the others (for the sets that exist). A reciprocal two-port has only three independent parameters.

These express electrical symmetry of the two ports, and they hold whether or not the network is reciprocal. For a network that is also reciprocal, only two parameters are independent (for example $z_{11}$ and $z_{12}$). Ports that can be swapped with no change at all need both $z_{11}=z_{22}$ and $z_{12}=z_{21}$.

Use ${\Delta }_T$ for a quick reciprocity check from any set. The determinant of a product is the product of the determinants, so after multiplying the ABCD matrices of reciprocal sections (built from $R$, $L$, $C$, coupled coils and ideal transformers), ${\Delta }_T$ must still equal 1. That catches arithmetic slips.

$[z]$ exists only if $C\ne 0$; $[T]$ exists only if $z_{21}\ne 0$.

$[y]$ exists only if $B\ne 0$; $[T]$ exists only if $y_{21}\ne 0$.

Note the pairs that are true reciprocals, each pair measured under the same termination: $h_{11}=1/y_{11}$ (port 2 shorted), $h_{22}=1/z_{22}$ (port 1 open), $g_{11}=1/z_{11}$ (port 2 open) and $g_{22}=1/y_{22}$ (port 1 shorted).

$g_{21}=1/A$ is the open-circuit forward voltage gain. $[g]$ exists only if $z_{11}\ne 0$ (equivalently $A\ne 0$).

$Z_a$ is the series arm at port 1, $Z_b$ the series arm at port 2, and $Z_c$ the shunt arm to the common terminal. Every reciprocal two-port that has a $[z]$ has this T equivalent, though an arm may come out negative (not realisable with positive elements). The T joins the two lower terminals, so it is equivalent only in its port voltages and currents.

$Y_a$ is the shunt arm at port 1, $Y_b$ the shunt arm at port 2, and $Y_c$ the series arm between the upper terminals. Every reciprocal two-port that has a $[y]$ has this $\pi$ equivalent, with the same caveats as the T.

Coil $L_1$ across port 1 and coil $L_2$ across port 2, mutual inductance $M$, with both dots at the upper terminals, so both port currents enter dotted terminals. Moving one dot to its lower terminal changes $M$ to $-M$ everywhere. In the $s$-domain replace $j\omega$ by $s$. The T equivalent (which may need a negative inductance) holds only if the two lower terminals may be joined.

Series arms $Z_a$ join upper terminal to upper terminal and lower terminal to lower terminal; cross arms $Z_b$ join each upper terminal to the lower terminal of the other port. Its transfer impedance changes sign when $Z_a>Z_b$.

Valid only when each sub-network still satisfies the port condition after the connection (see the validity note below).

$N_a$ is the section nearer the source; keep the order. Always valid, because the current $-I_2$ leaving $N_a$ is exactly the current entering $N_b$. That is why the minus sign is built into the ABCD definition.

Port 2 terminated in $Z_L$ (so $V_2=-I_2{Z}_L$). As $Z_L\to \infty$ this becomes $z_{11}=A/C$, the open-circuit input impedance; at $Z_L=0$ it becomes $1/y_{11}=h_{11}=B/D$, the short-circuit input impedance.

Same termination, with port 1 driven directly by $V_1$. The current delivered to the load is $-I_2$, so the load-current gain is $-I_2/I_1$. With the output open, $V_2/V_1=1/A=z_{21}/z_{11}$.

Port 1 driven by $V_s$ in series with $Z_s$. For $Z_{th}$, set $V_s$ to zero, which leaves $Z_s$ across port 1. With $V_s$ and $Z_s$ fixed, the load receives maximum power when $Z_L=Z_{th}^{\ast }$.

Each is measured at one port with the other port open (oc) or shorted (sc). For any two-port, $Z_{oc1}/Z_{sc1}=Z_{oc2}/Z_{sc2}=AD/(BC)$. So only three of the four are independent.

The image impedances are the pair for which terminating port 2 in $Z_{i2}$ makes the input impedance at port 1 equal to $Z_{i1}$, and terminating port 1 in $Z_{i1}$ makes the impedance at port 2 equal to $Z_{i2}$. For a symmetric network both equal the characteristic impedance $Z_0$: a load $Z_0$ is reproduced at the input. For a reciprocal network the image transfer constant $\theta$ satisfies $\cosh \theta =\sqrt{AD}$, which becomes $\cosh \theta =A$ when the network is also symmetric.