Introduction To Modern Planar Transmission Lines. Anand K. Verma

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^th port is

      (3.1.36)

      where the reflection coefficient at the i^th port is Γ_i = b_i/a_i. The total port voltage and the total port 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)

      Scattering [S] Matrix

Figure (3.11) shows the N‐port network. The power entering the i^th port is given in terms of the forward voltage wave or the forward power variable a_i and the power leaving the i^th port 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^th port 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_ij relating 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_ij is defined with the help of the matched termination. The matched termination also helps to measure the matrix elements S_ij.

      Reflection Coefficient S_ii

Figure (3.12) shows that the i^th port of a multiport network is terminated in a load equal to the characteristic impedance of the port. The power wave b_i coming out of the port is incident on the load Z_L, whereas the incident power wave a_i is the reflected wave from the load. If a port is terminated in its characteristic impedance, i.e. Z_L = Z₀, then the reflection from the load at the port is zero, i.e. a_i = 0. Thus, for the excitation a_j applied at the j^th port, while all other ports are terminated in their characteristic impedances, the ports have . However, the power b_j is reflected from the j^th port.

The reflection coefficient at the j^th port is obtained from equation (3.1.40):

Figure 3.12 At the i^th port, the load is terminated in port characteristic impedance.

(3.1.41)

      Therefore, S_jj is the reflection coefficient (Γ_j) at the j^th port, provided all other ports are terminated in their characteristic impedances. However, if other ports are not terminated in their characteristic impedances, then S_jj is not a measure of the true reflection coefficient of the network or a device at the j^th port. The true reflection coefficient at the j^th port, under the unmatched load condition, is more than S_jj that is defined under the matched load condition.

      Transmission Coefficient S_ij

      If the excitation source is connected only to the j^th port and the response is seen at the i^th port, while all other ports are terminated in their characteristic impedances, it leads to . It shows that the excitation is zero at all ports, except at the j^th port. The transmitted power, i.e. the scattered power, from the j^th port is available at all