7. Two-Port Networks // Network Analysis // bhaswanth

For more advanced analysis (especially when we get to amplifiers, transmission lines, and S-parameter analysis) we need a more general way to describe circuit blocks. The two-port network representation is the standard.

7.1 What is a two-port?

A two-port network has two pairs of terminals: an input pair and an output pair. Examples: an amplifier has input terminals (where signal goes in) and output terminals (where the amplified signal comes out). A transmission line has input and output ports. A filter has input and output. Most analog signal-processing blocks fit this description.

plaintext

       I1 →           ← I2
      ●──────────────────●
      │                  │
  V1  │   two-port net   │  V2
      │                  │
      ●──────────────────●

We characterize the two-port by relating the four variables (input voltage $V_1$ , input current $I_1$ , output voltage $V_2$ , output current $I_2$ ) using a 2×2 matrix of parameters. The convention is that both currents flow into the network at their respective + terminals.

7.2 The four parameter sets

Different choices of independent variables give different parameter representations. There are four major sets, plus a couple of less-common ones.

Z parameters (impedance, open-circuit)

Treat input and output currents as independent. The network's response is voltage at each port:

$V_1 = z_{11} I_1 + z_{12} I_2$ $V_2 = z_{21} I_1 + z_{22} I_2$

Or in matrix form:

$\begin{pmatrix} V_1 \\ V_2 \end{pmatrix} = \begin{pmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{pmatrix} \begin{pmatrix} I_1 \\ I_2 \end{pmatrix}$

The $z_{ij}$ have units of impedance (ohms). To find each, drive one port with a current source while leaving the other port open-circuited (so the other current is zero):

$z_{11} = V_1/I_1\big|_{I_2=0}$ : input impedance with output open.
$z_{12} = V_1/I_2\big|_{I_1=0}$ : reverse transimpedance.
$z_{21} = V_2/I_1\big|_{I_2=0}$ : forward transimpedance.
$z_{22} = V_2/I_2\big|_{I_1=0}$ : output impedance with input open.

Hence "open-circuit Z parameters."

Y parameters (admittance, short-circuit)

Treat input and output voltages as independent. The network's response is current at each port:

$I_1 = y_{11} V_1 + y_{12} V_2$ $I_2 = y_{21} V_1 + y_{22} V_2$

Each $y_{ij}$ is found by applying a voltage source at one port while short-circuiting the other (so the other voltage is zero). Hence "short-circuit Y parameters."

H parameters (hybrid)

Mix one voltage and one current as independent variables. The natural choice for transistors:

$V_1 = h_{11} I_1 + h_{12} V_2$ $I_2 = h_{21} I_1 + h_{22} V_2$

H parameters are the natural representation for BJTs: $h_{11}$ is the input resistance ( $r_\pi$ ), $h_{21}$ is the current gain ( $\beta$ ), $h_{22}$ is the output conductance ( $1/r_o$ ), and $h_{12}$ models the (small) reverse voltage gain. This is why h-parameters appear so often in transistor analysis textbooks.

ABCD parameters (transmission)

Express input quantities in terms of output. Note the convention that the output current is taken out of the port for ABCD (sign reversed):

$V_1 = A V_2 + B I_2'$ $I_1 = C V_2 + D I_2'$

where $I_2' = -I_2$ is the current flowing out of port 2.

The clever thing about ABCD: when two-ports are cascaded (output of one connected to input of next), the overall ABCD matrix is the product of individual ABCD matrices. So you can analyze a chain of filters by simply multiplying their ABCD matrices. Used heavily in transmission-line analysis.

7.3 Conversion formulas between parameter sets

You often need to convert between representations. Here are some of the most useful (using $\Delta$ for the determinant of each 2×2 matrix):

Z to Y:

$Y = Z^{-1} = \frac{1}{\Delta_Z}\begin{pmatrix} z_{22} & -z_{12} \\ -z_{21} & z_{11} \end{pmatrix}$

Z to H:

$h_{11} = \frac{\Delta_Z}{z_{22}}, \quad h_{12} = \frac{z_{12}}{z_{22}}, \quad h_{21} = -\frac{z_{21}}{z_{22}}, \quad h_{22} = \frac{1}{z_{22}}$

Z to ABCD:

$A = \frac{z_{11}}{z_{21}}, \quad B = \frac{\Delta_Z}{z_{21}}, \quad C = \frac{1}{z_{21}}, \quad D = \frac{z_{22}}{z_{21}}$

There are similar formulas relating any pair, freely available in textbooks. The point is: pick the parameter set that makes your analysis cleanest, and convert at the end if needed.

7.4 T and Pi equivalents

Any reciprocal two-port (one where $z_{12} = z_{21}$ , or equivalently where applying the same source-meter swap gives the same reading, i.e., a circuit with no dependent sources) can be drawn as either a T network or a Pi network. These are the universal "lumped" representations.

plaintext

T network:                      Pi network:
 
    Z_a       Z_c                        Z_3
 ●──[Z]──●──[Z]──●          ●────[Z]─────●
         │                  │            │
        [Z_b]              [Z_1]        [Z_2]
         │                  │            │
 ●───────●───────●          ●────────────●

The T's three series-and-shunt impedances or the Pi's three shunt-and-series impedances reproduce any reciprocal two-port. For Z parameters:

T: $Z_a = z_{11} - z_{12}$ , $Z_b = z_{12}$ , $Z_c = z_{22} - z_{12}$ .
Pi: derived from Y parameters analogously.

These are workhorse representations in filter design and RF engineering. Most physical filters are some interconnection of T and Pi sections.

7.5 Worked example: Z parameters of a simple T network

Consider this T-network:

plaintext

        2 kΩ           4 kΩ
   ●───[R_a]───●───[R_c]───●
               │
              [R_b = 6 kΩ]
               │
   ●───────────●────────────●

Find $z_{11}$ : drive port 1 with current $I_1$ , leave port 2 open ( $I_2 = 0$ ). With port 2 open, no current flows through $R_c$ . So all of $I_1$ flows from the input, through $R_a$ , through $R_b$ to the bottom rail. $V_1 = I_1 (R_a + R_b) = I_1 \cdot 8000$ . Therefore $z_{11} = 8$ kΩ.

Find $z_{21}$ : same conditions, find $V_2$ . With $I_2 = 0$ and no current through $R_c$ , the voltage at port 2's top terminal equals the voltage at the middle node, which is $I_1 R_b = 6000 I_1$ . So $V_2 = I_1 \cdot 6000$ , giving $z_{21} = 6$ kΩ.

Find $z_{22}$ : drive port 2 with $I_2$ , leave port 1 open. With port 1 open, no current through $R_a$ . All of $I_2$ flows from input port 2, through $R_c$ , through $R_b$ to the bottom rail. $V_2 = I_2 (R_c + R_b) = I_2 \cdot 10000$ . So $z_{22} = 10$ kΩ.

Find $z_{12}$ : same conditions, find $V_1$ . With $I_1 = 0$ and no current through $R_a$ , the voltage at port 1's top terminal equals the middle node voltage, which is $I_2 R_b = 6000 I_2$ . So $z_{12} = 6$ kΩ.

The Z matrix:

$Z = \begin{pmatrix} 8000 & 6000 \\ 6000 & 10000 \end{pmatrix} \text{ Ω}$

Note that $z_{12} = z_{21}$ . This is a passive reciprocal network, and reciprocity guarantees the symmetry. The off-diagonal entry, $z_{12} = R_b$ , is exactly the shared shunt element of the T network. That makes sense from the topology: the only way input current "talks" to output voltage is through $R_b$ .

7.6 Series/parallel/cascade combinations

Two two-ports can be connected:

Series (input ports in series, output ports in series): Z parameters add.
Parallel (input ports in parallel, output ports in parallel): Y parameters add.
Cascade (output of one connected to input of next): ABCD matrices multiply.
Series-parallel and parallel-series: more elaborate combinations using h or g parameters.

Choose your parameter set based on how the two-ports are interconnected.

7.7 Where two-ports show up

BJT modeling with h-parameters in textbook analysis.
Transmission lines cascade as ABCD products.
Filter design chains second-order sections, each a two-port.
RF amplifier datasheets quote S-parameters (high-frequency cousins of ABCD: every microwave datasheet has S₁₁, S₂₁, S₁₂, S₂₂).
Op-amp stability analysis uses Y-parameter or ABCD blocks.
Hardware-security work: characterizing a chip's package + PCB as a two-port between silicon and supply pin, then deconvolving power traces back to internal signals. The "package as two-port" is exactly how attackers and defenders quantify the side-channel impedance budget.