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
As is shown in the nodal analysis, a four-terminal network requires at least a
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.
To this end, we will set up the
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
denoted in short as
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
, we select
The differential-mode port and common-mode currents can be expressed in the
nodal currents with the aid of a base transformation matrix
where:
Equation ((241)) can now be written in the form:
Similarly, we may obtain the nodal voltages from the differential-mode port voltages from the nodal voltages as
Substitution of this result in ((243)) yields the modified network equation:
Since matrix multiplication is associative, we may write
where:
In this way we obtain
The coefficients of
: Differential-mode input admittance. : Common-mode input voltage into differential-mode input current conversion. : Differential-mode reverse transadmittance. : Differential-mode input voltage into common-mode input current conversion. : Common-mode input admittance. : Differential-mode output voltage into common-mode input current conversion. : Differential-mode transadmittance. : Differential-mode output voltage into common-mode input current conversion. : Differential-mode output admittance.
Equation ((244)) can be reduced to a two-port equation, if the
port voltages
This yields the following set of equations:
The solution of this set of equations is:
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:
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:
Z-matrix, or current-controlled representation:
Y-matrix, or voltage-controlled representation:
Hybrid-1 matrix, or current-controlled input and voltage-controlled output representation:
Hybrid-2 matrix, or voltage-controlled input and current-controlled output representation:
Transmission-1 matrix, or anti causal representation (the positive direction of the output current is opposite):
Transmission-2 matrix representation (the positive direction of the output current is opposite):
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
For design purposes, we will often use the anti-causal transmission-1 matrix
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:
The response to two excitations is equal to the sum of the responses to the individual excitations.
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
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
Nonenergic two-ports#
A two-port is called nonenergic if it has no energy storage and no power dissipation. A nonenergic two-port has:
Examples of nonenergic two-ports are ideal transformers and gyrators.