In electronics, a two-port network (a kind of four-terminal network or quadripole) is an electrical network (i.e. a circuit) or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the current entering one terminal must equal the current emerging from the other terminal on the same port.Gray, §3.2, p. 172Jaeger, §10.5 §13.5 §13.8 The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port and port 2 is considered the output port.
It is commonly used in mathematical circuit analysis.
Application The two-port network model is used in mathematical circuit analysis techniques to isolate portions of larger circuits. A two-port network is regarded as a "black box" with its properties specified by a matrix of numbers. This allows the response of the network to signals applied to the ports to be calculated easily, without solving for all the internal voltages and currents in the network. It also allows similar circuits or devices to be compared easily. For example, transistors are often regarded as two-ports, characterized by their -parameters (see below) which are listed by the manufacturer. Any linear circuit with four terminals can be regarded as a two-port network provided that it does not contain an independent source and satisfies the port conditions.
Examples of circuits analyzed as two-ports are filters, matching networks, transmission lines, transformers, and small-signal models for transistors (such as the hybrid-pi model). The analysis of passive two-port networks is an outgrowth of reciprocity theorems first derived by Lorentz.
In two-port mathematical models, the network is described by a 2 by 2 square matrix of complex numbers. The common models that are used are referred to as -parameters, -parameters, -parameters, -parameters, and -parameters, each described individually below. These are all limited to linear networks since an underlying assumption of their derivation is that any given circuit condition is a linear superposition of various short-circuit and open circuit conditions. They are usually expressed in matrix notation, and they establish relations between the variables , voltage across port 1 , current into port 1 , voltage across port 2 , current into port 2
which are shown in figure 1. The difference between the various models lies in which of these variables are regarded as the independent variables. These current and voltage variables are most useful at low-to-moderate frequencies. At high frequencies (e.g., microwave frequencies), the use of power and energy variables is more appropriate, and the two-port current–voltage approach is replaced by an approach based upon scattering parameters.
General properties There are certain properties of two-ports that frequently occur in practical networks and can be used to greatly simplify the analysis. These include:
Reciprocal networks A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port 1 is the same as the voltage appearing at port 1 when the same current is applied to port 2. Exchanging voltage and current results in an equivalent definition of reciprocity. A network that consists entirely of linear passive components (that is, resistors, capacitors and inductors) is usually reciprocal, a notable exception being passive circulators and isolators that contain magnetized materials. In general, it will not be reciprocal if it contains active components such as generators or transistors.Nahvi, p. 311. Symmetrical networks A network is symmetrical if its input impedance is equal to its output impedance. Most often, but not necessarily, symmetrical networks are also physically symmetrical. Sometimes also antimetrical networks are of interest. These are networks where the input and output impedances are the duals of each other.Matthaei et al, pp. 70–72. Lossless network A lossless network is one which contains no resistors or other dissipative elements.Matthaei et al, p. 27.
Impedance parameters (z-parameters)
where
All the -parameters have dimensions of ohms.
For reciprocal networks . For symmetrical networks . For reciprocal lossless networks all the are purely imaginary.Matthaei et al, p. 29.
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 . That is, the resistors 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 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 is represented by its emitter resistance :
a simplification made possible because the dependent current source in the hybrid-pi model for draws the same current as a resistor connected across . The second transistor 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 The double vertical bar denotes a parallel connection of the resistors: .
The negative feedback introduced by resistors can be seen in these parameters. For example, when used as an active load in a differential amplifier, , making the output impedance of the mirror approximately
compared to only without feedback (that is with = 0Ω). At the same time, the impedance on the reference side of the mirror is approximately
only a moderate value, but still larger than 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.
Admittance parameters (y-parameters)
where
All the Y-parameters have dimensions of siemens.
For reciprocal networks . For symmetrical networks . For reciprocal lossless networks all the are purely imaginary.
Hybrid parameters (h-parameters)
where
This circuit is often selected when a current amplifier is desired at the output. The resistors shown in the diagram can be general impedances instead.
Off-diagonal -parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another.
For reciprocal networks . For symmetrical networks . For reciprocal lossless networks and are real, while and are purely imaginary.
Example: common-base amplifier
Note: Tabulated formulas in Table 2 make the -equivalent circuit of the transistor from Figure 6 agree with its small-signal low-frequency hybrid-pi model in Figure 7. Notation: is base resistance of transistor, is output resistance, and is mutual transconductance. The negative sign for reflects the convention that are positive when directed into the two-port. A non-zero value for means the output voltage affects the input voltage, that is, this amplifier is bilateral. If , the amplifier is unilateral.
+ Table 2 Expression Approximation
History The -parameters were initially called series-parallel parameters. The term hybrid to describe these parameters was coined by D. A. Alsberg in 1953 in "Transistor metrology".56 IRE 28.S2, p. 1543 In 1954 a joint committee of the IRE and the AIEE adopted the term -parameters and recommended that these become the standard method of testing and characterising transistors because they were "peculiarly adaptable to the physical characteristics of transistors".AIEE-IRE committee report, p. 725 In 1956, the recommendation became an issued standard; 56 IRE 28.S2. Following the merge of these two organisations as the IEEE, the standard became Std 218-1956 and was reaffirmed in 1980, but has now been withdrawn.IEEE Std 218-1956
Inverse hybrid parameters (g-parameters)
where
Often this circuit is selected when a voltage amplifier is wanted at the output. 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 -equivalent circuit of the transistor from Figure 8 agree with its small-signal low-frequency hybrid-pi model in Figure 9. Notation: is base resistance of transistor, is output resistance, and is mutual transconductance. The negative sign for reflects the convention that are positive when directed into the two-port. A non-zero value for means the output current affects the input current, that is, this amplifier is bilateral. If , the amplifier is unilateral.
+ Table 3 Expression Approximation
ABCD-parameters The -parameters are known variously as chain, cascade, or transmission parameters. There are a number of definitions given for parameters, the most common is,Matthaei et al, p. 26.Ghosh, p. 353.
Note: Some authors chose to reverse the indicated direction of I2 and suppress the negative sign on I2.
where
For reciprocal networks . For symmetrical networks . For networks which are reciprocal and lossless, and are purely real while and are purely imaginary.
This representation is preferred because when the parameters are used to represent a cascade of two-ports, the matrices are written in the same order that a network diagram would be drawn, that is, left to right. However, a variant definition is also in use,A. Chakrabarti, p. 581, , Dhanpat Rai & Co pvt. ltd.
where
The negative sign of arises to make the output current of one cascaded stage (as it appears in the matrix) equal to the input current of the next. Without the minus sign the two currents would have opposite senses because the positive direction of current, by convention, is taken as the current entering the port. Consequently, the input voltage/current matrix vector can be directly replaced with the matrix equation of the preceding cascaded stage to form a combined matrix.
The terminology of representing the parameters as a matrix of elements designated etc. as adopted by some authorsFarago, p. 102. and the inverse parameters as a matrix of elements designated etc. is used here for both brevity and to avoid confusion with circuit elements.
Table of transmission parameters The table below lists and inverse parameters for some simple network elements.
Element matrix matrix Remarks Series impedance , impedance Shunt admittance , admittance Series inductor , inductance , complex angular frequency Shunt inductor , inductance , complex angular frequency Series capacitor , capacitance, complex angular frequency Shunt capacitor , capacitance, complex angular frequency Transmission line Clayton, p. 271. , characteristic impedance, propagation constant (), length of transmission line ()Impedance Matching Network, Source Impedance , Load Impedance , Phase Shift
, real part of , imaginary part of
Scattering parameters (S-parameters)
The previous parameters are all defined in terms of voltages and currents at ports. -parameters are different, and are defined in terms of incident and reflected waves at ports. -parameters are used primarily at UHF and microwave frequencies where it becomes difficult to measure voltages and currents directly. On the other hand, incident and reflected power are easy to measure using directional couplers. The definition is,Vasileska & Goodnick, p. 137