Two-Port Circuits – Parameters – Examples

A two-port network is an electrical network with two different ports for input and output.

A port is a pair of terminals through which electrical current can enter and exit. As an example of a one-port circuit are the passive elements of a network: resistors, inductors and capacitors. A one-port network is represented by diagram of Figure 1:

Figure 1

On the other hand, a four-terminal circuit, such as those made up of operational amplifiers, transistors or transformers, is considered a two-port network, which can be represented by the following diagram in Figure 2:

Figure 2

The study of two-port networks is justified because it allows treating or modeling complex circuits as a “black box”, that is, as a box where we do not know in detail what is inside. A signal feeds this “box” through one of its ports (input port); the signal is processed by the linear network of Figure 2 and is then delivered to a load by the other port (output port), as exemplified in Figure 3:

Figure 3

The characterization of a two-port network is made by relating the quantities present at its terminals: V₁, V₂, I₁, I₂.

Restrictions.

The complex network model as a two port network has certain restrictions:

• There can be no energy stored within the circuit.
• There can be no independent sources within the circuit; Dependent sources, however, are allowed.
• The current entering the port (input or output) must be equal to the current leaving the port (input or output).

The equations that relate the quantities V₁, V₂, I₁, I₂ present at the input and output ports of a two-port network are called parameters.

Impedance parameters.

To derive the impedance parameters, we supply the two-port network with a voltage source (which can be the Thevenin voltage supplied by the circuit connected at the input port) or by a current source (which can be the Norton provided by the circuit connected at the input port) as shown in Figure 4 a) and b):

Figure 4

From either of these two configurations, we can express the relationships between voltages and currents as:

Equations (1) allow to represent the model for a network of two ports, the “black box”, in matrix form:

The Z terms are called impedance parameters. To evaluate these parameters, we run the following tests. The value of the parameters can be evaluated by setting I₁=0 A (input port in open circuit), or I₂=0 A (output port in open circuit). In summary:

According to the table of equations (2), we can evaluate Z₁₁ and Z₁₂ by connecting a voltage source V₁ (or a current source I₁) to port 1 with port 2 in open circuit, as in Figure 5:

Figure 5

Then, from the circuit of Figure 5, by means of circuit analysis, we determine the value of I₁ und V₂, and then obtain the parameters Z₁₁ und Z₂₁ using equations (3):

Similarly, parameters Z₁₂ und Z₂₂ are obtained by the following experiment, as in Figure 6:

Figure 6

 Parameters Z₁₂ und Z₂₂ using equations (4):

Example.

Determine the Z parameters in Figure 7:

Figure 7

To determine Z₁₁ und Z₂₁, a voltage source V₁ is applied to the input port and the output port is left open, as in Figure 8a). To determine Z₁₂ and Z₂₂, a voltage source V₂ is applied to the input port and the output port is left open, as in Figure 8 b).

Figure 7

 We determine the Z parameters in Figure 7 using:

Therefore, the matrix of the impedance parameters is:

Sources:

1. Introduccion-al-analisis-de-circuitos-robert-l-boylestad,
2. Análisis de Redes – Van Valkenburg,
3. Fundamentos_de_circuitos_electricos_5ta
4. Fundamentos_de_Señales_y_Sistemas_usando la Web y Matlab

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