Likewise, the expressions for Y22 and Y12 could be computed by short‐circuiting the port‐1, V1 = 0. Final [Y] matrix of the T‐network is
(3.1.15)
The above matrix is a reciprocal of the [Z] matrix, given in of length ℓ equation (3.1.7).
Example 3.4
Determine the [Y] parameter of a section of the transmission line of length ℓ shown in Fig (3.3).
Solution
The incident voltage
At the port‐1, the incident current Iinc enters the port, so it is positive, whereas at the port‐1, the reflected current Iref leaves the port, so it is negative. At the port‐2, the incident current Iinc enters the port‐2 from the port‐1 side and leaves the port‐2, so it is negative, whereas at the port‐2, the reflected current Iref from the terminated load, enters the port‐2, so it is positive. The total voltage and the total current at the port‐2 are
The line section is symmetrical and reciprocal giving the [Y] parameter:
(3.1.16)
3.1.3 Transmission [ABCD] Parameter
On many occasions, two or more circuit elements or circuit blocks are interconnected in such a way that the output voltage and current of the first circuit block become the input to the next circuit block. To facilitate such combination or cascading, the circuit elements and blocks are characterized using the transmission parameters, i.e. the [ABCD] matrix, instead of [Z] or [Y] matrix. The great strength of the transmission parameter, i.e. the [ABCD] parameter, is due to its ability to provide [ABCD] matrix of the complete cascaded network, as a multiplication of the [ABCD] matrices of the individual circuit element or circuit block. The [ABCD] parameter, different from the T‐matrix, is applicable to a two‐port network only.
To obtain the transmission matrix description of a two‐port network, the output voltage and current are treated as the independent variables. The following expressions relate to the input and output voltage and current of the two‐port network shown in Fig (3.4):
Figure 3.4 Two‐port network for transmission parameter.
These expressions can be written in the matrix form,
In the case of the [Z] and [Y] parameters, the positive current I2 enters the port, while in the above network defining the [ABCD] parameter in Fig (3.4), the output current I2 leaving the port is taken as positive [B.1, B.3]. It is an input to the next circuit block, as shown in Fig (3.5). However, like defining the [Z] and [Y] parameters, to define the [ABCD] parameter current, I2 could be taken as the current entering the output port. In this case, I2 in equation (3.1.18) is replaced by (−I2) [B.1, B.4].
The matrix elements A, B, C, D can be determined from the open and short circuit conditions at the output port. When the output is open‐circuited, I2 = 0. Equation (3.1.17) provides the parameter‐A and C:
(3.1.19)
The parameter A is the voltage ratio that is a reciprocal of the voltage gain. The parameter C is the trans‐admittance of a network. It relates the output voltage of a network to its input current source.
When the output is short‐circuited, V2 = 0. Equation (3.1.17) again provides the parameters‐B and D:
(3.1.20)
The parameter B is the trans‐impedance of a network. It provides the output current when the input of a network is excited by the voltage source. The parameter D is the current ratio giving a reciprocal of the current gain of a network.
Fig (3.5) demonstrates the usefulness of the transmission parameters to obtain an equivalent [ABCD] parameter of the cascaded networks. The [ABCD] parameters for the first and the second network are written as
At the junction of two networks, I2 = I3 and V2 = V3. Therefore, from the above equations, the following expression is obtained:
(3.1.21)
