Two-port network


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.cite book
author=P.R. Gray, P.J. Hurst, S.H. Lewis, and R.G. Meyer
title=Analysis and Design of Analog Integrated Circuits
year= 2001
edition=Fourth Edition
publisher=Wiley
location=New York
isbn=0471321680
pages=§3.2, p. 172
url=http://worldcat.org/isbn/0471321680
] cite book
author=R. C. Jaeger and T. N. Blalock
title=Microelectronic Circuit Design
year= 2006
edition=Third Edition
publisher=McGraw-Hill
location=Boston
isbn=9780073191638
pages=§10.5 §13.5 §13.8
url=http://worldcat.org/isbn/9780073191638
] 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 [http://www.ieee.org/organizations/pubs/newsletters/emcs/summer03/jasper.pdf 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)::{V_1} = Input voltage :{V_2} = Output voltage :{I_1} = Input current :{I_2} = 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.cite book
author=P.R. Gray, P.J. Hurst, S.H. Lewis, and R.G. Meyer
title=Analysis and Design of Analog Integrated Circuits
year= 2001
edition=Fourth Edition
publisher=Wiley
location=New York
isbn=0471321680
pages=§3.2, p. 172
url=http://worldcat.org/isbn/0471321680
] cite book
author=R. C. Jaeger and T. N. Blalock
title=Microelectronic Circuit Design
year= 2006
edition=Third Edition
publisher=McGraw-Hill
location=Boston
isbn=9780073191638
pages=§10.5 §13.5 §13.8
url=http://worldcat.org/isbn/9780073191638
]

Impedance parameters (z-parameters)

: left [ egin{array}{c} V_1 \ V_2 end{array} ight] = left [ egin{array}{cc} z_{11} & z_{12} \ z_{21} & z_{22} end{array} ight] left [ egin{array}{c}I_1 \ I_2 end{array} ight] .

:z_{11} = {V_1 over I_1 } igg|_{I_2 = 0} qquad z_{12} = {V_1 over I_2 } igg|_{I_1 = 0}

:z_{21} = {V_2 over I_1 } igg|_{I_2 = 0} qquad z_{22} = {V_2 over I_2 } igg|_{I_1 = 0}

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. [The emitter-leg resistors counteract any current increase by decreasing the transistor "VBE". That is, the resistors "RE" cause negative feedback that opposes change in current. In particular, any change in output voltage results in less change in current than without this feedback, which means the output resistance of the mirror has increased.] 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 "rE" ≈ "VT / 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.

ABCD-parameters

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

: {V_2 choose I_2} = egin{pmatrix} A & B \ C & D end{pmatrix}{V_1 choose I_1} .

where

:A = {V_2 over V_1 } igg|_{I_1 = 0} qquad B = {V_2 over I_1 } igg|_{V_1 = 0}

:C = -{I_2 over V_1 } igg|_{I_1 = 0} qquad D = -{I_2 over I_1 } igg|_{V_1 = 0}

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 "I"2 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.

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

Example: Cascading two networks

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{T}_1 = egin{pmatrix} 1 & -R \ 0 & 1 end{pmatrix}

: mathbf{T}_2 = egin{pmatrix} 1 & 0 \ -Cs & 1 end{pmatrix}

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

: mathbf{T} = mathbf{T}_2 cdot mathbf{T}_1

::: = egin{pmatrix} 1 & 0 \ -Cs & 1 end{pmatrix} cdot egin{pmatrix} 1 & -R \ 0 & 1 end{pmatrix}

::: = egin{pmatrix} 1 & -R \ -Cs & 1+RCs end{pmatrix}

Thus:

: egin{pmatrix} V_2 \ I_2 end{pmatrix} = egin{pmatrix} 1 & -R \ -Cs & 1+RCs end{pmatrix} egin{pmatrix} V_1 \ I_1 end{pmatrix}

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:

: {V_1 choose I_1} = egin{pmatrix} A & B \ C & D end{pmatrix}{V_2 choose -I_2}

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:

*Admittance (Y) Parameters
*Impedance (Z) Parameters
*Scattering (S) Parameters

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

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

ee also

*Admittance parameters
*Impedance parameters
*Scattering parameters
*Ray transfer matrix
*Quadrupole — An abstract charge configuration.

References


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