Anand K. Verma - Introduction To Modern Planar Transmission Lines

(3.1.76)

The transmission parameter S 21is computed in terms of S 11. If the amplitude of the forward traveling voltage wave on the transmission line is , the total voltage on the transmission line is given by

(3.1.77)

where x is measured from the load end, as shown in Fig (3.17). The input port‐1 is located at x = − ℓ. The voltage at the port‐1 is

(3.1.78)

The port voltage V 1is obtained as a sum of the incident and reflected voltages at the port‐1:

(3.1.79)

At the port‐2, under the matched termination, Z L= nZ 0giving and . Equation (3.1.77)shows that the voltage at the port‐2, i.e. at x = 0 is

(3.1.80)

Using equation (3.1.79), the transmission coefficient, S 21of the circuit shown in Fig (3.17)is obtained as

(3.1.81)

On substituting V 1from equation (3.1.78)and V 2from equation (3.1.80)in the above equation S 21is obtained:

(3.1.82)

The present line network is symmetrical and reciprocal. It has S 11= S 22and S 21= S 12. The above expressions are checked for n = 1, i.e. for a transmission line of characteristic impedance Z 0. For this case, S 11= S 22= 0 and S 21= S 12= e −j βℓ. These are expressions of the S‐parameters for a line having characteristic impedance Z 0.

Sometimes, the conversion of one kind of network parameter to another kind is needed for the analysis of a circuit. For instance, if several circuit blocks comprising of the lumped elements and the transmission line sections are cascaded, each circuit block could be expressed by its [ABCD] matrix. It helps to get an overall [ABCD] matrix of the cascaded network. However, the final [ABCD] matrix, describing the cascaded network is further converted to the [S] matrix. Similarly, the [S] matrix of each building block of the cascaded network has to be converted to the [ABCD] matrix to get the overall [ABCD] matrix of the cascaded network. Finally, the overall [ABCD] matrix is converted to the [S] matrix of the cascaded network. The S‐parameters are measurable quantities. The performance of a network is measured in the [S] matrix using a VNA .

On several occasions, the S‐parameters of a line section or a network are known either from the simulations or from the measurements. The S‐parameters are used to get the characteristic impedance and the propagation constant of a line, or a network. However, the true S‐parameters of a network are needed for this purpose. The true S‐parameters are normally embedded in the measured or the simulated S‐parameters at the ports of measurement, or the ports of simulation. The true S‐parameters of a line or a network are extracted, i.e. de‐embedded, from the measured, or simulated, S‐parameters at the ports. This is known as the de‐embedding process [B.10]. The EM‐Simulators have provision to de‐embed the true S‐parameters from the S‐parameters obtained at the measurement or simulation ports.

This section presents the conversion of matrix parameters, de‐embedding of the S‐parameters, and extraction of the propagation characteristics.

Figure (3.18)shows a network with its known [Z] parameters. It requires conversion to the [ABCD] parameters. The [ABCD] and [Z] parameters of the network are summarized below:

(3.2.1)

(3.2.2)

Figure 3.18 Network for Z‐parameter.

The current (I 2) entering the port‐2 is taken positively. However, the Z‐parameter is defined in Fig (3.18)for the output current leaving the port. In this case, I 2is negative. Equation (3.2.2)is rearranged to get the port voltage and current V 1and I 1at the port‐1 in terms of the V 2and I 2at the port‐2:

(3.2.3)

On comparing equations (3.2.1)and (3.2.3), the following conversion expressions are obtained:

(3.2.4)

Likewise, the relations between [Y] and [ABCD] parameters are obtained:

(3.2.5)

A complete set of the conversion table of parameters is available in textbooks [B.1, B.5, B.7].

The N‐port network, having normalized reference port impedance, Z 0n= 1, is considered. The port voltage and port current in terms of the incident and reflected voltage can be written as

(3.2.6)

The above equations are written in the column matrix form:

(3.2.7)