Definitions

# Two-port network

A Two-Port Network (or four-terminal network or quadripole) is an electrical circuit or device with two pairs of terminals (i.e., the circuit connects two dipoles). Two terminals constitute a port if they satisfy the essential requirement known as the port condition: the same current must enter and leave a port. Examples include small-signal models for transistors (such as the hybrid-pi model), filters and matching networks. The analysis of passive two-port networks is an outgrowth of reciprocity theorems first derived by Lorentz. See review by Jasper J. Goedbloed, Reciprocity and EMC measurements.

A two-port network makes possible the isolation of either a complete circuit or part of it and replacing it by its characteristic parameters. Once this is done, the isolated part of the circuit becomes a "black box" with a set of distinctive properties, enabling us to abstract away its specific physical buildup, thus simplifying analysis. Any linear circuit with four terminals can be transformed into a two-port network provided that it does not contain an independent source and satisfies the port conditions.

The parameters used in order to describe a two-port network are the following: z, y, h, g, T. They are usually expressed in matrix notation and they establish relations between the following parameters (see Figure 1):

$\left\{V_1\right\}$ = Input voltage
$\left\{V_2\right\}$ = Output voltage
$\left\{I_1\right\}$ = Input current
$\left\{I_2\right\}$ = Output current
These variables are most useful at low to moderate frequencies. At high frequencies (for example, microwave frequencies) power and energy are more useful variables, and the two-port approach based on current and voltages that is discussed here is replaced by an approach based upon scattering parameters.

Though some authors use the terms two-port network and four-terminal network interchangeably, the latter represents a more general concept. Not all four-terminal networks are two-port networks. A pair of terminals can be called a port only if the current entering one is equal to the current leaving the other (the port condition). Only those four-terminal networks in which the four terminals can be paired into two ports can be called two-ports.

## Impedance parameters (z-parameters)

$left\left[begin\left\{array\right\}\left\{c\right\} V_1 V_2 end\left\{array\right\} right\right] = left\left[begin\left\{array\right\}\left\{cc\right\} z_\left\{11\right\} & z_\left\{12\right\} z_\left\{21\right\} & z_\left\{22\right\} end\left\{array\right\} right\right] left\left[begin\left\{array\right\}\left\{c\right\}I_1 I_2 end\left\{array\right\} right\right]$.

$z_\left\{11\right\} = \left\{V_1 over I_1 \right\} bigg|_\left\{I_2 = 0\right\} qquad z_\left\{12\right\} = \left\{V_1 over I_2 \right\} bigg|_\left\{I_1 = 0\right\}$

$z_\left\{21\right\} = \left\{V_2 over I_1 \right\} bigg|_\left\{I_2 = 0\right\} qquad z_\left\{22\right\} = \left\{V_2 over I_2 \right\} bigg|_\left\{I_1 = 0\right\}$

Notice that all the z-parameters have dimensions of ohms.

### Example: bipolar current mirror with emitter degeneration

Figure 3 shows a bipolar current mirror with emitter resistors to increase its output resistance. Transistor Q1 is diode connected, which is to say its collector-base voltage is zero. Figure 4 shows the small-signal circuit equivalent to Figure 3. Transistor Q1 is represented by its emitter resistance rEVT / IE (VT = thermal voltage, IE = Q-point emitter current), a simplification made possible because the dependent current source in the hybrid-pi model for Q1 draws the same current as a resistor 1 / gm connected across rπ. The second transistor Q2 is represented by its hybrid-pi model. Table 1 below shows the z-parameter expressions that make the z-equivalent circuit of Figure 2 electrically equivalent to the small-signal circuit of Figure 4.

Table 1 Expression Approximation
R_{21} = begin{matrix} {V_mathrm{2} over I_mathrm{1} }end{matrix} Big>_{I_{2}=0} $- \left( beta r_O - R_E \right)$ $begin\left\{matrix\right\} frac \left\{r_E +R_E \right\}\left\{r_\left\{ pi\right\}+r_E +2R_E\right\} end\left\{matrix\right\}$ $- beta r_o$$begin\left\{matrix\right\} frac \left\{r_E+R_E \right\}\left\{r_\left\{ pi\right\} +2R_E\right\}end\left\{matrix\right\}$
R_{11}= begin{matrix} frac{V_{1}}{I_{1}}end{matrix} Big>_{I_{2}=0} $\left(r_E + R_E\right)$ $//$ $\left(r_\left\{ pi\right\} +R_E\right)$ 
R_{22} = begin{matrix} frac{V_{2}}{I_{2}}end{matrix} Big>_{I_{1}=0} $\left($$1 + beta$ $begin\left\{matrix\right\} frac \left\{R_E\right\} \left\{r_\left\{ pi\right\} +r_E+2R_E \right\} end\left\{matrix\right\} \right)$ $r_O$ $+ begin\left\{matrix\right\} frac \left\{ r_\left\{ pi\right\}+r_E +R_E \right\}\left\{r_\left\{ pi\right\}+r_E +2R_E \right\} end\left\{matrix\right\}$$R_E$ $\left($$1 + beta$$begin\left\{matrix\right\} frac \left\{R_E\right\} \left\{r_\left\{ pi\right\}+2R_E \right\} end\left\{matrix\right\} \right)$ $r_O$
R_{12} = begin{matrix} {V_mathrm{1} over I_mathrm{2} }end{matrix} Big>_{I_{1}=0} $R_E$ $begin\left\{matrix\right\} frac \left\{r_E+R_E\right\} \left\{r_\left\{ pi\right\} +r_E +2R_E\right\} end\left\{matrix\right\}$ $R_E$ $begin\left\{matrix\right\} frac \left\{r_E+R_E\right\} \left\{r_\left\{ pi\right\} +2R_E\right\} end\left\{matrix\right\}$

The negative feedback introduced by resistors RE can be seen in these parameters. For example, when used as an active load in a differential amplifier, I1 ≈ -I2, making the output impedance of the mirror approximately R22 -R21 ≈ 2 β rORE /(rπ+2RE ) compared to only rO without feedback (that is with RE = 0 Ω) . At the same time, the impedance on the reference side of the mirror is approximately R11 -R12$begin\left\{matrix\right\} frac \left\{r_\left\{pi\right\}\right\} \left\{r_\left\{pi\right\}+2R_E\right\} end\left\{matrix\right\}$ $\left(r_E+R_E\right)$, only a moderate value, but still larger than rE with no feedback. In the differential amplifier application, a large output resistance increases the difference-mode gain, a good thing, and a small mirror input resistance is desirable to avoid Miller effect.

$left\left[begin\left\{array\right\}\left\{c\right\} I_1 I_2 end\left\{array\right\} right\right] = left\left[begin\left\{array\right\}\left\{cc\right\} y_\left\{11\right\} & y_\left\{12\right\} y_\left\{21\right\} & y_\left\{22\right\} end\left\{array\right\} right\right] left\left[begin\left\{array\right\}\left\{c\right\}V_1 V_2 end\left\{array\right\} right\right]$.

where

$y_\left\{11\right\} = \left\{I_1 over V_1 \right\} bigg|_\left\{V_2 = 0\right\} qquad y_\left\{12\right\} = \left\{I_1 over V_2 \right\} bigg|_\left\{V_1 = 0\right\}$

$y_\left\{21\right\} = \left\{I_2 over V_1 \right\} bigg|_\left\{V_2 = 0\right\} qquad y_\left\{22\right\} = \left\{I_2 over V_2 \right\} bigg|_\left\{V_1 = 0\right\}$

The network is said to be reciprocal if $y_\left\{12\right\} = y_\left\{21\right\}$. Notice that all the Y-parameters have dimensions of siemens.

## Hybrid parameters (h-parameters)

$\left\{V_1 choose I_2\right\} = begin\left\{pmatrix\right\} h_\left\{11\right\} & h_\left\{12\right\} h_\left\{21\right\} & h_\left\{22\right\} end\left\{pmatrix\right\}\left\{I_1 choose V_2\right\}$.

where

$h_\left\{11\right\} = \left\{V_1 over I_1 \right\} bigg|_\left\{V_2 = 0\right\} qquad h_\left\{12\right\} = \left\{V_1 over V_2 \right\} bigg|_\left\{I_1 = 0\right\}$

$h_\left\{21\right\} = \left\{I_2 over I_1 \right\} bigg|_\left\{V_2 = 0\right\} qquad h_\left\{22\right\} = \left\{I_2 over V_2 \right\} bigg|_\left\{I_1 = 0\right\}$

Often this circuit is selected when a current amplifier is wanted at the output. The resistors shown in the diagram can be general impedances instead.

Notice that off-diagonal h-parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another.

### Example: common base amplifier

Note: Tabulated formulas in Table 2 make the h-equivalent circuit of the transistor from Figure 6 agree with its small-signal low-frequency hybrid-pi model in Figure 7. Notation: rπ = base resistance of transistor, rO = output resistance, and gm = transconductance. The negative sign for h21 reflects the convention that I1, I2 are positive when directed into the two-port. A non-zero value for h12 means the output voltage affects the input voltage, that is, this amplifier is bilateral. If h12 = 0, the amplifier is unilateral.

Table 2 Expression Approximation
h_{21} = begin{matrix} {I_mathrm{2} over I_mathrm{1} }end{matrix} Big>_{V_{2}=0} $begin\left\{matrix\right\} - frac \left\{frac \left\{beta \right\}\left\{beta+1\right\}r_O +r_E\right\} \left\{r_O+r_E\right\} end\left\{matrix\right\}$ $begin\left\{matrix\right\} - frac \left\{beta \right\}\left\{beta+1\right\}end\left\{matrix\right\}$
h_{11}= begin{matrix} frac{V_{1}}{I_{1}}end{matrix} Big>_{V_{2}=0} $r_E//r_O$ $r_E$
h_{22} = begin{matrix} frac{I_{2}}{V_{2}}end{matrix} Big>_{I_{1}=0} $begin\left\{matrix\right\} frac \left\{1\right\} \left\{\left(beta +1\right) \left(r_O +r_E\right)\right\} end\left\{matrix\right\}$ $begin\left\{matrix\right\} frac \left\{1\right\} \left\{\left(beta +1\right) r_O \right\} end\left\{matrix\right\}$
h_{12} = begin{matrix} {V_mathrm{1} over V_mathrm{2} }end{matrix} Big>_{I_{1}=0} $begin\left\{matrix\right\} frac \left\{r_E\right\} \left\{r_E+r_O\right\} end\left\{matrix\right\}$ $begin\left\{matrix\right\} frac \left\{r_E\right\} \left\{r_O\right\} end\left\{matrix\right\}$ << 1

## Inverse hybrid parameters (g-parameters)

$\left\{I_1 choose V_2\right\} = begin\left\{pmatrix\right\} g_\left\{11\right\} & g_\left\{12\right\} g_\left\{21\right\} & g_\left\{22\right\} end\left\{pmatrix\right\}\left\{V_1 choose I_2\right\}$.

where

$g_\left\{11\right\} = \left\{I_1 over V_1 \right\} bigg|_\left\{I_2 = 0\right\} qquad g_\left\{12\right\} = \left\{I_1 over I_2 \right\} bigg|_\left\{V_1 = 0\right\}$

$g_\left\{21\right\} = \left\{V_2 over V_1 \right\} bigg|_\left\{I_2 = 0\right\} qquad g_\left\{22\right\} = \left\{V_2 over I_2 \right\} bigg|_\left\{V_1 = 0\right\}$

Often this circuit is selected when a voltage amplifier is wanted at the output. Notice that off-diagonal g-parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another. The resistors shown in the diagram can be general impedances instead.

### Example: common base amplifier

Note: Tabulated formulas in Table 3 make the g-equivalent circuit of the transistor from Figure 8 agree with its small-signal low-frequency hybrid-pi model in Figure 9. Notation: rπ = base resistance of transistor, rO = output resistance, and gm = transconductance. The negative sign for g12 reflects the convention that I1, I2 are positive when directed into the two-port. A non-zero value for g12 means the output current affects the input current, that is, this amplifier is bilateral. If g12 = 0, the amplifier is unilateral.

Table 3 Expression Approximation
g_{21} = begin{matrix} {V_mathrm{2} over V_mathrm{1} }end{matrix} Big>_{I_{2}=0} $begin\left\{matrix\right\} frac \left\{r_o\right\}\left\{r_\left\{ pi\right\}\right\}end\left\{matrix\right\}$$+ g_m r_O + 1$ $g_m r_O$
g_{11}= begin{matrix} frac{I_{1}}{V_{1}}end{matrix} Big>_{I_{2}=0} $begin\left\{matrix\right\} frac \left\{1\right\} \left\{r_\left\{pi\right\}\right\} end\left\{matrix\right\}$ $begin\left\{matrix\right\} frac \left\{1\right\} \left\{r_\left\{pi\right\}\right\} end\left\{matrix\right\}$
g_{22} = begin{matrix} frac{V_{2}}{I_{2}} Big>_{V_{1}=0} end{matrix} $r_O$ $r_O$
g_{12} = begin{matrix} {I_mathrm{1} over I_mathrm{2} }end{matrix} Big>_{V_{1}=0} $-$$begin\left\{matrix\right\} frac\left\{ beta + 1\right\}\left\{ beta\right\} end\left\{matrix\right\}$ $-1$

## ABCD-parameters

The ABCD-parameters are known variously as chain, cascade, or transmission parameters.

$\left\{V_2 choose I_2\right\} = begin\left\{pmatrix\right\} A & B C & D end\left\{pmatrix\right\}\left\{V_1 choose I_1\right\}$.

where

$A = \left\{V_2 over V_1 \right\} bigg|_\left\{I_1 = 0\right\} qquad B = \left\{V_2 over I_1 \right\} bigg|_\left\{V_1 = 0\right\}$

$C = -\left\{I_2 over V_1 \right\} bigg|_\left\{I_1 = 0\right\} qquad D = -\left\{I_2 over I_1 \right\} bigg|_\left\{V_1 = 0\right\}$

Note that we have inserted negative signs in front of the fractions in the definitions of parameters C and D. The reason for adpoting this convention (as opposed to the convention adopted above for the other sets of parameters) is that it allows us to represent the transmission matrix of cascades of two or more two-port networks as simple matrix multiplications of the matrices of the individual networks. This convention is equivalent to reversing the direction of I2 so that it points in the same direction as the input current to the next stage in the cascaded network.

An ABCD matrix has been defined for Telephony four-wire Transmission Systems by P K Webb in British Post Office Research Department Report 630 in 1977.

### Table of transmission parameters

The table below lists transmission parameters for some simple network elements.

Element Matrix Remarks
Series resistor $begin\left\{pmatrix\right\} 1 & -R 0 & 1 end\left\{pmatrix\right\}$ R = resistance
Shunt resistor $begin\left\{pmatrix\right\} 1 & 0 -1/R & 1 end\left\{pmatrix\right\}$ R = resistance
Series conductor $begin\left\{pmatrix\right\} 1 & -1/G 0 & 1 end\left\{pmatrix\right\}$ G = conductance
Shunt conductor $begin\left\{pmatrix\right\} 1 & 0 -G & 1 end\left\{pmatrix\right\}$ G = conductance
Series inductor $begin\left\{pmatrix\right\} 1 & -Ls 0 & 1 end\left\{pmatrix\right\}$ L = inductance
s = complex angular frequency
Shunt capacitor $begin\left\{pmatrix\right\} 1 & 0 -Cs & 1 end\left\{pmatrix\right\}$ C = capacitance
s = complex angular frequency

## Combinations of two-port networks

Series connection of two 2-port networks: Z = Z1 + Z2
Parallel connection of two 2-port networks: Y = Y1 + Y2

Suppose we have a two-port network consisting of a series resistor R followed by a shunt capacitor C. We can model the entire network as a cascade of two simpler networks:

$mathbf\left\{T\right\}_1 = begin\left\{pmatrix\right\} 1 & -R 0 & 1 end\left\{pmatrix\right\}$

$mathbf\left\{T\right\}_2 = begin\left\{pmatrix\right\} 1 & 0 -Cs & 1 end\left\{pmatrix\right\}$

The transmission matrix for the entire network T is simply the matrix multiplication of the transmission matrices for the two network elements:

$mathbf\left\{T\right\} = mathbf\left\{T\right\}_2 cdot mathbf\left\{T\right\}_1$

$= begin\left\{pmatrix\right\} 1 & 0 -Cs & 1 end\left\{pmatrix\right\} cdot begin\left\{pmatrix\right\} 1 & -R 0 & 1 end\left\{pmatrix\right\}$

$= begin\left\{pmatrix\right\} 1 & -R -Cs & 1+RCs end\left\{pmatrix\right\}$

Thus:

$begin\left\{pmatrix\right\} V_2 I_2 end\left\{pmatrix\right\} = begin\left\{pmatrix\right\} 1 & -R -Cs & 1+RCs end\left\{pmatrix\right\} begin\left\{pmatrix\right\} V_1 I_1 end\left\{pmatrix\right\}$

### Notes regarding definition of transmission parameters

1) It should be noted that all these examples are specific to the definition of transmission parameters given here. Other definitions exist in the literature, such as:

$\left\{V_1 choose I_1\right\} = begin\left\{pmatrix\right\} A & B C & D end\left\{pmatrix\right\}\left\{V_2 choose -I_2\right\}$

2) The format used above for cascading (ABCD) examples cause the "components" to be used backwards compared to standard electronics schematic conventions. This can be fixed by taking the transpose of the above formulas, or by making the $V_1, I_1$ the left hand side (dependent variables). Another advantage of the $V_1, I_1$ form is that the output can be terminated (via a transfer matrix representation of the load) and then $I_2$ can be set to zero; allowing the voltage transfer function, 1/A to be read directly.

3) In all cases the ABCD matrix terms and current definitions should allow cascading. 4

## Networks with more than 2 ports

While two port networks are very common (e.g. amplifiers and filters), other electrical networks such as directional couplers and isolators have more than 2 ports. The following representations can be extended to networks with an arbitrary number of ports:

They are extended by adding appropriate terms to the matrix representing the other ports. So 3 port impedance parameters result in the following relationship:

$left\left[begin\left\{array\right\}\left\{c\right\} V_1 V_2 V_3 end\left\{array\right\} right\right] = left\left[begin\left\{array\right\}\left\{ccc\right\} Z_\left\{11\right\} & Z_\left\{12\right\} & Z_\left\{13\right\} Z_\left\{21\right\} & Z_\left\{22\right\} &Z_\left\{23\right\} Z_\left\{31\right\} & Z_\left\{32\right\} & Z_\left\{33\right\} end\left\{array\right\} right\right] left\left[begin\left\{array\right\}\left\{c\right\}I_1 I_2 I_3 end\left\{array\right\} right\right]$.

It should be noted that the following representations cannot be extended to more than two ports:

• Hybrid (h) parameters
• Inverse hybrid (g) parameters
• Transmission (ABCD) parameters
• Scattering Transmission (T) parameters