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:
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:
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:
The characterization of a two-port network is made by relating the quantities present at its terminals: V1, V2, I1, I2.
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.
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):
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:
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:
Parameters Z12 und Z22 using equations (4):
Determine the Z parameters in 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).
We determine the Z parameters in Figure 7 using:
Therefore, the matrix of the impedance parameters is:
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