Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 105
Basic Concept
ОглавлениеA commonly used two‐port network is suitable to develop the concept of the S‐parameter. Even the S‐parameters of a multiport network are measured as the two‐port parameters, while other ports are terminated in the matched loads. Figure (3.9) shows the two‐port network. It is to be characterized by the S‐parameters. The port‐1 and port‐2 are terminated with the line sections of characteristic impedance Z01 and Z02, respectively. However, most of the two‐port networks have Z01 = Z02 = Z0, i.e. the identical transmission line sections at both the ports. The reference impedance Z0 is normally 50Ω. The incident voltage waves at both the ports‐ and , enter the ports and the reflected voltage waves at both the ports‐ and , come out of the ports.
The forward power, i.e. the incident power entering the port‐1, is
In equation (3.1.26b), is the RMS voltage of the voltage wave. In general for the two‐port or N‐port network, the forward power entering the ith port is written as
The incident power variable ai at the ith port is defined in a way that the power entering the port is given by the square of the power variable:
Figure 3.9 Two‐port network for evaluation of S‐parameter.
Using equations (3.1.27) and (3.1.28), the power variable ai is written in terms of the forward RMS voltage at the ith port
(3.1.29)
The forward voltage can also be written in term of the power variable as
The power variable a i is simply a normalized forward voltage wave, incident on the ith port. The normalization is done with respect to the square root of the characteristic impedance at the port. The forward power variable can also be viewed as the incident normalized current. The power entering the ith port, in terms of the incident RMS current , is given below:
(3.1.31)
The forward port current in terms of the forward power variable is
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 ith port with characteristic impedance Z0i. Figure (3.10) shows that the ith port is connected to a source with impedance Z0. For the sake of clarity, the port is taken out of the network using an interconnect line of characteristic impedance Z0 with 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 bi is
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 ith port is
(3.1.36)
where the reflection coefficient at the ith port is Γi = bi/ai. The total port voltage and the total port current in term of the power variables can be written as
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:
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 ai and bi are complex quantities. The incident and reflected power are
(3.1.39)
