Analysis of quadrupole cascade structure using transfer matrices

Many four-pole and two-pole microwave devices have a cascade structure, characterized by the fact that the output of the preceding four-port network is the input of the next four-port network, etc. The analysis of such a connection is much simpler if you characterize the four-terminal networks, as well as their combination with special matrices - transmission matrices. A feature of the transfer matrices is that a pair of electrical quantities is used as an impact on the quadrupole, which determine the mode of one input (usually the second), and the response is a corresponding pair of quantities that determines the mode of another input (usually the first).

In determining the classical transfer matrix (we will also call it matrix A), the relation between the action and the reaction is

(3.1)

or in algebraic form

(3.1, a)   

To determine the physical meaning of the elements of the matrix A, we should consider the mental experiments of idling and short circuit at the inputs of the quadrupole. Then follows:

(3.2)

Therefore, the element is the voltage transfer ratio when the output is open, the element is the current transfer ratio when there is a short circuit at the output. Elements and, respectively, represent the normalized mutual resistance in the event of a short circuit and mutual conductivity when the circuit is idled at the output.

Using (3.1), it is easy to show that the classical transfer matrix of the composite quadrupole in Fig. 3.1 is equal to the product of the transfer matrix of the partial quadrupole:

t. e. the main property of transfer matrices takes place:

A = A'A. " (3.3)

Of course, this property of the transfer matrix extends to any number of cascade-connected two-port networks. The product of two matrices does not obey in the general case the translational law, and therefore it is necessary to multiply the transfer matrices of the quadrupoles during cascading in the sequence. in which they are included in the path.

Along with the classical transfer matrix A, when analyzing cascade-connected microwave quadrupoles, the transmission matrix T is also used, which links the modes of the first and second inputs in terms of incident and reflected waves:

(3.4)

Due to the change in the order of the incident and reflected waves in the excitation column related to the second input, the transmission wave matrix for the cascade connection of two quadrupoles is determined by a rule similar to (3.3), i.e.

. (3.5)

Thus, when using any transfer matrices — classical or wave — the cascade connection matrix of N different quadrupoles is equal to the product of N transfer matrices of individual cascades. It should be noted that the elements of the transmission wave matrix T do not have a clear physical meaning, except for the element that is equal to the reciprocal of the element of the scattering matrix (i.e. ).

Let us briefly analyze the two-terminal cascade structure (Fig. 3.2). Such bipolar networks are a cascade connection of a number of quadrupoles, the last of which is closed to the terminal load with resistance . The input impedance of the composite two-pole can be found by dividing the upper equation (3.1 a) by the bottom:

(3.6)

Thus, the analysis of a two-terminal cascade structure is reduced to finding the transfer matrix followed by using formula (3.6).

If the terminal load is a short circuit or idling, then formula (3.6) is simplified and takes one of the following forms:

at at (3.7)

The analysis of cascade bipolar networks can be carried out in terms of the transfer matrix . In this case, the division of the second equation (3.4) into the first with the subsequent substitution of the reflection coefficient of the terminal load allows us to find the input reflection coefficient of the composite two-pole:

(3.8)

The input reflection coefficient can be found using the usual formula by substituting the value from (3.6) into it.