Anand K. Verma - Introduction To Modern Planar Transmission Lines

The forward port current in terms of the forward power variable is

(3.1.32)

The multiplication of the voltage and current of equations (3.1.30)and (3.1.32), again provides the forward power, . Thus, the definitions of the power variable both as the normalized voltage wave and as the normalized current wave are consistent. However, one must be careful about the presence of the square root of the characteristic impedance in the numerator and denominator for two definitions.

Consider the reflected power wave at the i thport with characteristic impedance Z 0i. Figure (3.10)shows that the i thport is connected to a source with impedance Z 0. For the sake of clarity, the port is taken out of the network using an interconnect line of characteristic impedance Z 0with zero length, ℓ = 0. The total power available from the source does not enter the network. A part of it gets reflected. The reflected power in terms of the reflected power variable b iis

Figure 3.10 A section of the multiport network. Port is shown extended with length ℓ = 0.

(3.1.33)

The reflected power variable is related to the reflected port voltage and the reflected port current as follows:

(3.1.34)

(3.1.35)

The power entering the i thport is

(3.1.36)

where the reflection coefficient at the i thport is Γ i= b i/a i. The total port voltage and the total por t current in term of the power variables can be written as

(3.1.37)

The reflected port current is negative, such that the reflected power travels in the opposite direction, i.e. it travels away from the port. Using equation (3.1.37), the power variables can be written in terms of the total port voltage and the total port current:

(3.1.38)

The definition of the power wave variables given by the equations are valid for the special cases of the forward wave and reflected waves. The definition, given in equation (3.1.38), is valid for the general case. It is applicable at any port for any kind of termination. The power‐variables a i and b i are complex quantities . The incident and reflected power are

(3.1.39)

Figure (3.11)shows the N‐port network. The power entering the i thport is given in terms of the forward voltage wave or the forward power variable a iand the power leaving the i thport is given in terms of the reflected (backward) voltage wave or the reflected (backward) power variable b i. A part of the microwave power is reflected at the i thport itself and remaining power entering the network comes out of all other ports as ( ) or as ( ). The outcoming power from any port is a linear combination of the transmitted power from all other ports. Using either the voltage variables or the power variables, the incident power and reflected power at various ports are correlated as follows:

(3.1.40)

The incident power at the port is treated as the excitation , and reflected/transmitted power at the port is considered as the response . The network is characterized by the S‐parameters.

Therefore, the matrix elements S ijrelating the excitation ( or a i) to the response ( or b i) are described as follows:

Figure 3.11 N‐port network showing power variables (a i, b i) in terms of voltage variables .

The S ijis defined with the help of the matched termination. The matched termination also helps to measure the matrix elements S ij.

Figure (3.12)shows that the i thport of a multiport network is terminated in a load equal to the characteristic impedance of the port. The power wave b icoming out of the port is incident on the load Z L, whereas the incident power wave a iis the reflected wave from the load. If a port is terminated in its characteristic impedance, i.e. Z L= Z 0, then the reflection from the load at the port is zero, i.e. a i= 0. Thus, for the excitation a japplied at the j thport, while all other ports are terminated in their characteristic impedances, the ports have . However, the power b jis reflected from the j thport.