Two-ports

In many engineering situations, it is convenient to model an electrical network as a two-port. In those cases, the port quantities (voltage and current) of one port are related to the port quantities of the other port, and the electrical behavior of the two-port is described with the aid of a \(2\times2\) matrix.

As is shown in the nodal analysis, a four-terminal network requires at least a \(3\times3\) matrix description, thus a two-port description can only be complete under additional constraints. These constraints are called the two-port constraints. They require that the current that flows into one port terminal equals the current flowing out of the corresponding port terminal, and a voltage placed between one of the input port terminals and one of the output port terminals does not cause a change in the port voltages and currents.

The two-port constraints are always valid if both ports are terminated with one-ports, or if the four-terminal network is a natural two-port. Ideal transformers and gyrators, as well as the controlled sources are examples of natural two-ports. A formal derivation of the two-port conditions will be given in section Two-port conditions.

With input and output port voltages and currents as port variables, there are six different combinations of dependent and independent port variables, resulting in six different two-port models. The choice of an appropriate two-port model strongly depends on the effect one wishes to study. In section Two-port representations, we will present the different representation methods. Finally, important classes of two-ports and their characteristic properties are listed in section Two-port properties.

Two-port conditions

We will now formally derive the two-port conditions.

The two-port conditions are the conditions that should apply to a four-terminal network, such that its two-port equations correctly describe its behavior, for any external network.

To this end, we will set up the \(Y\) parameter representation of the four terminal network from Fig. 585A. We will then define the differential-mode and common-mode port quantities and relate them to the nodal voltages and currents. This is illustrated in Fig. 585B. Since the two-port model only describes relations between the differential-mode quantities, we then find the conditions under which the two-port model correctly describes the behavior of the four-terminal network.

Fig. 585 Two four-terminal network representations: (A) Four-terminal network with nodal voltages and currents.

(B) Nodal voltages and currents decomposed into common-mode and differential-mode quantities. The differential-mode quantities are the two-port variables.

The four terminal network can be characterized by its \(Y\) parameters:

\[\begin{split}\left( \begin{array} [c]{c} I_{1}\\ I_{2}\\ I_{3} \end{array} \right) =\left( \begin{array} [c]{ccc} Y_{11} & Y_{12} & Y_{13}\\ Y_{21} & Y_{22} & Y_{23}\\ Y_{31} & Y_{32} & Y_{33} \end{array} \right) \cdot\left( \begin{array} [c]{c} V_{1}\\ V_{2}\\ V_{3} \end{array} \right) ,\end{split}\]

denoted in short as

(241)\[\mathbf{I}=\mathbf{Y\cdot V.} \label{eq-nodalAnalysisY}\]

In order to describe it as a two-port, we need to select the port variables. Since here the output port is connected to the ground

This is not essential: two-port conditions for a floating output port are similar to those for a floating input port.

, we select \(V_{o}=V_{3}\) and \(I_{o}=I_{3}\). Nodes 1 and 2 constitute the input port. The input port variables become the differential-mode port voltage and the differential-mode port current, \(V_{i}=V_{1}-V_{2}\) and \(I_{i} =\frac{1}{2}\left( I_{1}-I_{2}\right) \), respectively. The common-mode input port voltage and the common-mode input port current are defined as: \(V_{c}=\frac{1}{2}\left( V_{1}+V_{2}\right) \) and \(I_{c}=I_{1}+I_{2},\) respectively. These definitions are illustrated in Fig. 585B.

The differential-mode port and common-mode currents can be expressed in the nodal currents with the aid of a base transformation matrix \(\mathbf{A}\):

\[\begin{split}\left( \begin{array} [c]{c} I_{i}\\ I_{c}\\ I_{o} \end{array} \right) =\mathbf{A\cdot}\left( \begin{array} [c]{c} I_{1}\\ I_{2}\\ I_{3} \end{array} \right) ,\end{split}\]

where:

(242)\[\begin{split}\mathbf{A}=\left( \begin{array} [c]{ccc} \frac{1}{2} & -\frac{1}{2} & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{array} \right) . \label{eq-transformMatrix}\end{split}\]

Equation ((241)) can now be written in the form:

(243)\[\begin{split}\left( \begin{array} [c]{c} I_{i}\\ I_{c}\\ I_{o} \end{array} \right) =\mathbf{A\cdot Y\cdot V.} \label{eq-transformedNA}\end{split}\]

Similarly, we may obtain the nodal voltages from the differential-mode port voltages from the nodal voltages as

\[\begin{split}\mathbf{V=A}^{T}\cdot\left( \begin{array} [c]{c} V_{i}\\ V_{c}\\ V_{o} \end{array} \right) .\end{split}\]

Substitution of this result in ((243)) yields the modified network equation:

\[\begin{split}\left( \begin{array} [c]{c} I_{i}\\ I_{c}\\ I_{o} \end{array} \right) =\mathbf{A\cdot Y\cdot A}^{T}\left( \begin{array} [c]{c} V_{i}\\ V_{c}\\ V_{o} \end{array} \right) .\end{split}\]

Since matrix multiplication is associative, we may write

\[\begin{split}\left( \begin{array} [c]{c} I_{i}\\ I_{c}\\ I_{o} \end{array} \right) =\mathbf{Y}^{\prime}\cdot\left( \begin{array} [c]{c} V_{i}\\ V_{ic}\\ V_{o} \end{array} \right) ,\end{split}\]

where:

\[\mathbf{Y}^{\prime}=\mathbf{A\cdot Y\cdot A}^{T}.\]

In this way we obtain \(\mathbf{Y}^{\prime}\) as

(244)\[\begin{split}\mathbf{Y}^{\prime}=\left( \begin{array} [c]{ccc} \frac{1}{4}Y_{11}-\frac{1}{4}Y_{12}-\frac{1}{4}Y_{21}+\frac{1}{4}Y_{22} & \frac{1}{2}Y_{11}+\frac{1}{2}Y_{12}-\frac{1}{2}Y_{21}-\frac{1}{2}Y_{22} & \frac{1}{2}Y_{13}-\frac{1}{2}Y_{23}\\ \frac{1}{2}Y_{11}-\frac{1}{2}Y_{12}+\frac{1}{2}Y_{21}-\frac{1}{2}Y_{22} & Y_{11}+Y_{12}+Y_{21}+Y_{22} & Y_{13}+Y_{23}\\ \frac{1}{2}Y_{31}-\frac{1}{2}Y_{32} & Y_{31}+Y_{32} & Y_{33} \end{array} \right) . \label{eq-twoPortCMDM}\end{split}\]

The coefficients of \(\mathbf{Y}^{\prime}\)\ have the following meaning:

1. \(\mathbf{Y}_{1,1}^{\prime}\): Differential-mode input admittance.

2. \(\mathbf{Y}_{1,2}^{\prime}\): Common-mode input voltage into differential-mode input current conversion.

3. \(\mathbf{Y}_{1,3}^{\prime}\): Differential-mode reverse transadmittance.

4. \(\mathbf{Y}_{2,1}^{\prime}\): Differential-mode input voltage into common-mode input current conversion.

5. \(\mathbf{Y}_{2,2}^{\prime}\): Common-mode input admittance.

6. \(\mathbf{Y}_{2,3}^{\prime}\): Differential-mode output voltage into common-mode input current conversion.

7. \(\mathbf{Y}_{3,1}^{\prime}\): Differential-mode transadmittance.

8. \(\mathbf{Y}_{3,2}^{\prime}\): Differential-mode output voltage into common-mode input current conversion.

9. \(\mathbf{Y}_{3,3}^{\prime}\): Differential-mode output admittance.

Equation ((244)) can be reduced to a two-port equation, if the port voltages \(V_{i}\), \(V_{c}\) and \(V_{o}\) do not cause any common-mode current \(I_{c}\), and if the common-mode voltage \(V_{c}\) across the two-port does not affect the port currents \(I_{i}\) and \(I_{o}\). These conditions are satisfied if \(\mathbf{Y}_{1,2}^{\prime}\), \(\mathbf{Y}_{2,1}^{\prime}\), \(\mathbf{Y}_{2,2}^{\prime}\), \(\mathbf{Y}_{2,3}^{\prime}\) and \(\mathbf{Y} _{3,2}^{\prime}\) are zero.

This yields the following set of equations:

\[\begin{split}\frac{1}{2}Y_{11}+\frac{1}{2}Y_{12}-\frac{1}{2}Y_{21}-\frac{1}{2}Y_{22} & =0\\ \frac{1}{2}Y_{11}-\frac{1}{2}Y_{12}+\frac{1}{2}Y_{21}-\frac{1}{2}Y_{22} & =0\\ Y_{11}+Y_{12}+Y_{21}+Y_{22} & =0\\ Y_{13}+Y_{23} & =0\\ Y_{31}+Y_{32} & =0\end{split}\]

The solution of this set of equations is:

\[Y_{11}=Y_{22},~Y_{12}=Y_{21},\ Y_{21}=-Y_{22},\ Y_{13}=-Y_{23}\text{ and }Y_{31}=-Y_{32}.\]

Fig. 586 Four terminal element that satisfies the two-port conditions for arbitrary port connections. Such a circuit is referred to as a natural two-port.

Four-terminal networks that satisfy these conditions are called natural two-ports. Fig. 586 shows such a network. When it is modeled as a four-terminal network according to Fig. 585A, it satisfies the above conditions:

\[\begin{split}Y_{i} & =Y_{11}=Y_{22}=-Y_{12}=-Y_{21},\\ Y_{o} & =Y_{33},\\ Y_{r} & =-Y_{13}=Y_{23},\\ Y_{f} & =Y_{31}=-Y_{32}.\end{split}\]

Two-port representations

Linear four-terminal elements are often modeled as two-ports, one pair of terminals is considered as input port, while the other pair is considered as output port. Two-ports are then represented by matrices having only four coefficients. To do so, we select two independent variables from the four port quantities. In this way we obtain six different representation methods:

1. Z-matrix, or current-controlled representation:

   \[\begin{split}\left( \begin{array} [c]{c} V_{i}\\ V_{o} \end{array} \right) =\mathbf{Z\cdot}\left( \begin{array} [c]{c} I_{i}\\ I_{o} \end{array} \right) .\end{split}\]

2. Y-matrix, or voltage-controlled representation:

   \[\begin{split}\left( \begin{array} [c]{c} I_{i}\\ I_{o} \end{array} \right) =\mathbf{Y\cdot}\left( \begin{array} [c]{c} V_{i}\\ V_{o} \end{array} \right) .\end{split}\]

3. Hybrid-1 matrix, or current-controlled input and voltage-controlled output representation:

   \[\begin{split}\left( \begin{array} [c]{c} V_{i}\\ I_{o} \end{array} \right) =\mathbf{H\cdot}\left( \begin{array} [c]{c} I_{i}\\ V_{o} \end{array} \right) .\end{split}\]

4. Hybrid-2 matrix, or voltage-controlled input and current-controlled output representation:

   \[\begin{split}\left( \begin{array} [c]{c} I_{i}\\ V_{o} \end{array} \right) =\mathbf{H}^{\prime}\cdot\left( \begin{array} [c]{c} V_{i}\\ I_{o} \end{array} \right) .\end{split}\]

5. Transmission-1 matrix, or anti causal representation (the positive direction of the output current is opposite):

   \[\begin{split}\left( \begin{array} [c]{c} V_{i}\\ I_{i} \end{array} \right) =\mathbf{T\cdot}\left( \begin{array} [c]{c} V_{o}\\ -I_{o} \end{array} \right) .\end{split}\]

6. Transmission-2 matrix representation (the positive direction of the output current is opposite):

   \[\begin{split}\left( \begin{array} [c]{c} V_{o}\\ I_{o} \end{array} \right) =\mathbf{T}^{\prime}\cdot\left( \begin{array} [c]{c} V_{i}\\ -I_{i} \end{array} \right) .\end{split}\]

Although these six representation methods are fully equivalent, in a particular situation, one specific representation can give more insight and reduce calculations considerably. If, for example, two-ports are connected in parallel, the \(Y\) parameters of the combination are easily found by adding the individual \(Y\) matrices. If, in another situation, the inputs are connected in series and the outputs are connected in parallel, the hybrid 1 representation is convenient. In situations in which two-ports are cascaded, the transmission-1 matrix representation is convenient: the transmission-1 matrix of the cascaded two-ports is the product of the transmission-1 matrices of the individual two-ports.

Table 32 Two-port transformations: all matrices in the same row are equal. \(X_{\Delta}=x_{11}x_{22}-x_{12}x_{21}\)

	\(\mathbf{Z}\)	\(\mathbf{Y}\)	\(\mathbf{T}\)	\(\mathbf{T}^{\prime}\)	\(\mathbf{H}\)	\(\mathbf{H} ^{\prime}\)
\(\mathbf{Z}\)	\( \begin{array} {cc} z_{11} & z_{12}\\ z_{21} & z_{22} \end{array} \)	\( \begin{array} {cc} \frac{y_{22}}{Y_{\Delta}} & -\frac{y_{12}}{Y_{\Delta}}\\ -\frac{y_{21}}{Y_{\Delta}} & \frac{y_{11}}{Y_{\Delta}} \end{array} \)	\( \begin{array} {cc} \frac{t_{11}}{t_{21}} & \frac{T_{\Delta}}{t_{21}}\\ \frac{1}{t_{21}} & \frac{t_{22}}{t_{21}} \end{array} \)	\( \begin{array} {cc} \frac{t_{22}^{\prime}}{t_{21}^{\prime}} & \frac{1}{t_{21}^{\prime}}\\ \frac{T_{\Delta}^{\prime}}{t_{21}^{\prime}} & \frac{t_{11}^{\prime}}{t_{21}^{\prime}} \end{array} \)	\( \begin{array} {cc} \frac{H_{\Delta}}{h_{22}} & \frac{h_{21}}{h_{22}}\\ -\frac{h_{21}}{h_{22}} & \frac{1}{h_{22}} \end{array} \)	\( \begin{array} {cc} \frac{1}{h_{11}^{\prime}} & -\frac{h_{12}^{\prime}}{h_{11}^{\prime}}\\ \frac{h_{21}^{\prime}}{h_{11}^{\prime}} & \frac{H_{\Delta}^{\prime}}{h_{11}^{\prime}} \end{array} \)
\(\mathbf{Y}\)	\( \begin{array} {cc} \frac{z_{22}}{Z_{\Delta}} & -\frac{z_{12}}{Z_{\Delta}}\\ -\frac{z_{21}}{Z_{\Delta}} & \frac{z_{11}}{Z_{\Delta}} \end{array} \)	\( \begin{array} {cc} y_{11} & y_{12}\\ y_{21} & y_{22} \end{array} \)	\( \begin{array} {cc} \frac{t_{22}}{t_{12}} & \frac{T_{\Delta}}{t_{12}}\\ \frac{1}{t_{12}} & \frac{t_{11}}{t_{12}} \end{array} \)	\( \begin{array} {cc} \frac{t_{11}^{\prime}}{t_{12}^{\prime}} & -\frac{1}{t_{12}^{\prime}}\\ -\frac{T_{\Delta}^{\prime}}{t_{12}^{\prime}} & \frac{t_{22}^{\prime}}{t_{12}^{\prime}} \end{array} \)	\( \begin{array} {cc} \frac{1}{h_{11}} & -\frac{h_{12}}{h_{11}}\\ \frac{h_{21}}{h_{11}} & \frac{H_{\Delta}}{h_{11}} \end{array} \)	\( \begin{array} {cc} \frac{H_{\Delta}^{\prime}}{h_{22}^{\prime}} & \frac{h_{12}^{\prime}}{h_{22}^{\prime}}\\ -\frac{h_{12}^{\prime}}{h_{22}^{\prime}} & \frac{1}{h_{22}^{\prime}} \end{array} \)
\(\mathbf{T}\)	\( \begin{array} {cc} \frac{z_{11}}{z_{21}} & \frac{Z_{\Delta}}{z_{21}}\\ \frac{1}{z_{21}} & \frac{z_{22}}{z_{21}} \end{array} \)	\( \begin{array} {cc} -\frac{y_{22}}{y_{21}} & -\frac{1}{y_{21}}\\ -\frac{Y_{\Delta}}{y_{21}} & -\frac{y_{11}}{y_{21}} \end{array} \)	\( \begin{array} {cc} t_{11} & t_{12}\\ t_{21} & t_{22} \end{array} \)	\( \begin{array} {cc} \frac{t_{22}^{\prime}}{T_{\Delta}^{\prime}} & \frac{t_{12}^{\prime}}{T_{\Delta}^{\prime}}\\ \frac{t_{21}^{\prime}}{T_{\Delta}^{\prime}} & \frac{t_{11}^{\prime}}{T_{\Delta}^{\prime}} \end{array} \)	\( \begin{array} {cc} -\frac{H_{\Delta}}{h_{21}} & -\frac{h_{11}}{h_{21}}\\ -\frac{h_{22}}{h_{21}} & -\frac{1}{h_{21}} \end{array} \)	\( \begin{array} {cc} \frac{1}{h_{21}^{\prime}} & \frac{h_{22}^{\prime}}{h_{21}^{\prime}}\\ \frac{h_{11}^{\prime}}{h_{21}^{\prime}} & \frac{H_{\Delta}^{\prime}}{h_{21}^{\prime}} \end{array} \)
\(\mathbf{T}^{\prime}\)	\( \begin{array} {cc} \frac{z_{22}}{z_{12}} & \frac{Z_{\Delta}}{z_{12}}\\ \frac{1}{z_{12}} & \frac{z_{11}}{z_{12}} \end{array} \)	\( \begin{array} {cc} -\frac{y_{11}}{y_{12}} & -\frac{1}{y_{12}}\\ -\frac{Y_{\Delta}}{y_{12}} & -\frac{y_{22}}{y_{12}} \end{array} \)	\( \begin{array} {cc} \frac{t_{22}}{T_{\Delta}} & \frac{t_{12}}{T_{\Delta}}\\ \frac{t_{21}}{T_{\Delta}} & \frac{t_{11}}{T_{\Delta}} \end{array} \)	\( \begin{array} {cc} t_{11}^{\prime} & t_{12}^{\prime}\\ t_{21}^{\prime} & t_{22}^{\prime} \end{array} \)	\( \begin{array} {cc} \frac{1}{h_{12}} & \frac{h_{11}}{h_{12}}\\ -\frac{h_{22}}{h_{12}} & -\frac{H_{\Delta}}{h_{12}} \end{array} \)	\( \begin{array} {cc} -\frac{H_{\Delta}^{\prime}}{h_{12}^{\prime}} & -\frac{h_{22}^{\prime}}{h_{12}^{\prime}}\\ -\frac{h_{11}^{\prime}}{h_{12}^{\prime}} & \frac{1}{h_{12}^{\prime}} \end{array} \)
\(\mathbf{H}\)	\( \begin{array} {cc} \frac{Z_{\Delta}}{z_{22}} & \frac{z_{12}}{z_{22}}\\ -\frac{z_{21}}{z_{22}} & \frac{1}{z_{22}} \end{array} \)	\( \begin{array} {cc} \frac{1}{y_{11}} & -\frac{y_{12}}{y_{11}}\\ \frac{y_{21}}{y_{11}} & \frac{Y_{\Delta}}{y_{11}} \end{array} \)	\( \begin{array} {cc} \frac{t_{12}}{t_{22}} & \frac{T_{\Delta}}{t_{22}}\\ -\frac{1}{t_{22}} & \frac{t_{21}}{t_{22}} \end{array} \)	\( \begin{array} {cc} \frac{t_{12}^{\prime}}{t_{11}^{\prime}} & \frac{1}{t_{11}^{\prime}}\\ -\frac{T_{\Delta}^{\prime}}{t_{11}^{\prime}} & \frac{t_{21}^{\prime}}{t_{11}^{\prime}} \end{array} \)	\( \begin{array} {cc} h_{11} & h_{12}\\ h_{21} & h_{22} \end{array} \)	\( \begin{array} {cc} \frac{h_{22}^{\prime}}{H_{\Delta}^{\prime}} & \frac{h_{12}^{\prime}}{H_{\Delta}^{\prime}}\\ \frac{h_{21}^{\prime}}{H_{\Delta}^{\prime}} & \frac{h_{11}^{\prime}}{H_{\Delta}^{\prime}} \end{array} \)
\(\mathbf{H}^{\prime}\)	\( \begin{array} {cc} \frac{1}{z_{11}} & -\frac{z_{12}}{z_{11}}\\ \frac{z_{21}}{z_{11}} & \frac{Z_{\Delta}}{z_{11}} \end{array} \)	\( \begin{array} {cc} \frac{Y_{\Delta}}{y_{22}} & \frac{y_{12}}{y_{22}}\\ -\frac{y_{21}}{y_{22}} & \frac{1}{y_{22}} \end{array} \)	\( \begin{array} {cc} \frac{t_{21}}{t_{11}} & -\frac{T_{\Delta}}{t_{11}}\\ \frac{1}{t_{11}} & \frac{t_{12}}{t_{11}} \end{array} \)	\( \begin{array} {cc} \frac{t_{21}^{\prime}}{t_{22}^{\prime}} & -\frac{1}{t_{22}^{\prime}}\\ -\frac{T_{\Delta}^{\prime}}{t_{12}^{\prime}} & \frac{t_{12}^{\prime}}{t_{22}^{\prime}} \end{array} \)	\( \begin{array} {cc} \frac{h_{22}}{H_{\Delta}} & -\frac{h_{12}}{H_{\Delta}}\\ -\frac{h_{21}}{H_{\Delta}} & \frac{h_{11}}{H_{\Delta}} \end{array} \)	\( \begin{array} {cc} h_{11}^{\prime} & h_{12}^{\prime}\\ h_{21}^{\prime} & h_{22}^{\prime} \end{array} \)

For design purposes, we will often use the anti-causal transmission-1 matrix representation (\(\mathbf{T}\) representation). It will be shown that this representation is very convenient for deriving design strategies for amplifiers. As we have seen, nodal analysis uses the admittance \(\mathbf{Y}\) representation.

Table 32 gives the relations between the two-port parameters of the six representations.

Two-port properties

Linear two-ports

A two-port is linear if both the properties of superposition and homogeneity hold:

1. The response to two excitations is equal to the sum of the responses to the individual excitations.

2. The response to an enlarged excitation is equal to the equally enlarged response to the excitation.

This means that a two-port is linear if it consists of linear elements only and apart from the excitations, it has no independent sources.

Reciprocal two-ports

A reciprocal two-port is a two-port from which the input port and the output port can be interchanged without affecting the network solution. This is the case if

\[z_{12}=z_{21}\text{, }y_{12}=y_{21}\text{, }\det(\mathbf{T})=1\text{, } h_{12}=-h_{21}.\]

Any linear two-port without controlled sources is reciprocal.

An amplifier generally needs to have a large gain from input to output and a negligible gain from output to input. Hence, in amplifier design, one tries to minimize the reciprocity.

Unilateral two-ports

Unilateral behavior means that any change of the port termination at one port cannot be noticed at the other port. The input impedance of a unilateral two-port does not depend on the load impedance, and vice versa. This is often a desirable property of amplifiers: unilateral amplifiers have no reverse transfer. A unilateral two-port has

\[\det\left( \mathbf{T}\right) =0.\]

Nonenergic two-ports

A two-port is called nonenergic if it has no energy storage and no power dissipation. A nonenergic two-port has:

\[\left\vert \det\left( \mathbf{T}\right) \right\vert =1.\]

Examples of nonenergic two-ports are ideal transformers and gyrators.