Anand K. Verma - Introduction To Modern Planar Transmission Lines

For the lossless transmission line, γ = j β, α = 0 and [Z] is

(3.1.8)

Figure 3.3 A transmission line section.

To define the [Y] parameters, the voltage is taken as an independent variable and current as of the dependent one for a two‐port network shown in Fig (3.1). In this case, the voltage is a source of excitation, and current at the port is the response . Thus, for a linear network, the total port current is a superposition of currents due to the voltages applied at both the ports:

(3.1.9)

where [V] and [I] are the voltage and current column matrices. The admittance matrix of the two‐port network is

(3.1.10)

The Y‐parameters are defined as the short‐circuited parameters . For the short‐circuited port‐2, V 2 =0, and Y 11and Y 21are defined from equation (3.1.9):

(3.1.11)

Likewise, for the short‐circuited port‐1, the Y‐parameters are

(3.1.12)

The [Y] parameters are extended to a multiport network by defining its matrix elements as follows:

(3.1.13)

Equation (3.1.13)shows that to get Y ii, i.e. the diagonal elements of the [Y] matrix, all the ports are short‐circuited, except the i thport. The current is evaluated at the i thport for the voltage applied at the i thport itself. To get Y ij, i.e. the off‐diagonal elements of the [Y] matrix, the voltage is applied at the j thport. Y ijis the mutual admittance describing the coupling between the j thport and the i thport. The current at the i thport is evaluated or measured, while all other ports are short‐circuited. The admittance element Y ijis evaluated as

(3.1.14)

Fig (3.2)shows the T‐network. Determine the [Y] parameter of the network.

The loop equations for the circuit are written as

For the short‐circuited port‐2, V 2= 0: .

From the above equations:

Likewise, the expressions for Y 22and Y 12could be computed by short‐circuiting the port‐1, V 1= 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).

Determine the [Y] parameter of a section of the transmission line of length ℓ shown in Fig (3.3).

The incident voltage excites the port‐1, and it reaches the port‐2 as The port‐2 is short‐circuited to determine the [Y] parameter. Under the short‐circuit condition, the reflected voltage at the port‐2 is such that the total voltage at the port‐2 is zero. The reflected voltage at the port‐1 is . The total voltage and the total current at the port‐1 are

At the port‐1, the incident current I incenters the port, so it is positive, whereas at the port‐1, the reflected current I refleaves the port, so it is negative. At the port‐2, the incident current I incenters 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 I reffrom 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)

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.