Two-Port Networks & ABCD Parameters

A two-port describes terminal behavior without exposing internal nodes. It is the right abstraction for a filter section, cable segment, transformer model, amplifier small-signal block, or any linear network intended to sit between a source and a load.

Chain convention #

Define $I_1$ entering port 1 and $I_2$ leaving port 2. Then

$$\begin{bmatrix}V_1\\I_1\end{bmatrix} = \begin{bmatrix}A&B\\C&D\end{bmatrix} \begin{bmatrix}V_2\\I_2\end{bmatrix}.$$

$A$ and $D$ are dimensionless, $B$ has units of impedance, and $C$ has units of admittance. If both port currents are instead defined entering the box, the equations require $-I_2$; silently mixing conventions flips signs.

The parameters follow from open- and short-circuit conditions:

$$A=\left.\frac{V_1}{V_2}\right|_{I_2=0},\quad C=\left.\frac{I_1}{V_2}\right|_{I_2=0},$$

$$B=\left.\frac{V_1}{I_2}\right|_{V_2=0},\quad D=\left.\frac{I_1}{I_2}\right|_{V_2=0}.$$

Cascades #

For stages connected in order from source to load:

$$T_{total}=T_1T_2\cdots T_n.$$

Matrix order follows physical order and is not commutative. This multiplication rule is the principal advantage of ABCD parameters over $Z$, $Y$, or hybrid parameter sets.

Reciprocity and symmetry #

For a reciprocal two-port under this convention:

$$AD-BC=1.$$

For a symmetric two-port, $A=D$. These are properties to test, not assumptions to impose on an arbitrary active or nonreciprocal network.

With load $Z_L=V_2/I_2$, the input impedance is

$$Z_{in}=\frac{AZ_L+B}{CZ_L+D}.$$

Matrix order follows physical order from input to load. Reversing the product describes the reversed cascade and generally produces a different $A$ and $D$ even though each individual section is reciprocal.