For the decomposition of most quadrupoles of the microwave cascade structure, only six basic elements are enough, which we will also call elementary quadrupoles. The equivalent circuits of these quadrupoles, as well as the corresponding classical and wave transfer matrices, are summarized in Table. 3.2. The transition to other matrixes of parameters can be easily performed by the transition formulas table. 3.1. Note that any elementary quadrupole in Table. 3.2 is characterized by only one parameter (complex or real).
Let us explain how the classical transfer matrices of each elementary quadrupole can be compiled. The first quadrupole in the table. 3.2 is a segment of a regular lossless transmission line with a fixed electrical length . In accordance with (3.2), the element of the classical transfer matrix of this quadrupole describes the voltage distribution in a regular transmission line open at the end, and therefore equal to . Similarly, the element in the matrix A describes the current distribution in a short-circuited regular transmission line and is . The elements and in the matrix A of the segment of the transmission line are:
;
where denoted by the input resistance and input conductance of the transmission line during a short circuit and idling at the output.
Similarly, the values of the elements of the classical transfer matrix for the second quadrupole in Table 2 are set. 3.2, representing a segment of a regular transmission line with losses.
The third quadrupole in table. 3.2 represents the junction of two transmission lines, differing in the values of wave impedances and . In the junction plane, the equality of complete unnormalized voltages and currents, that is, i . The minus sign takes into account that the currents at each input flow into the quadrupole.
Table 3.2. Elementary quadruple circuits
Passing with the help of the relation (1.18) in these equalities to the normalized voltages and currents, and also adding for completeness the zero terms, we obtain the system of equations:
from which you can determine all the elements of the classical transfer matrix of the interface. Note that the joint is characterized in fact by one material parameter , called a wave impedance.
The fourth quadrupole in table. 3.2 is a lumped resistance z, connected in series to the gap between two identical transmission lines. In accordance with Ohm’s law, the normalized voltage at the first input of such a quadrupole is equal and, moreover, equality takes place . From these two conditions, and follow the values of the elements of the transfer matrix And in the fourth row of the table. 3.2.
For the fifth quadrupole in table. 3.2, which is a concentrated conductance that shunts a regular transmission line, there are equalities
From these equalities the values of the elements of the classical transfer matrix in the fifth row of the table follow. 3.2.
The sixth elementary quadripole in table. 3.2 is a non-reciprocal phase shifter.
We emphasize that all elementary quadrupoles in the table. 3.2, except the first two, have zero electrical length and, therefore, are extremely simplified mathematical models of the corresponding real elements. The inevitable delay in the propagation of an electromagnetic wave in real elements of the path, to which the equivalent circuit 4-6 in the table. 3.2, can easily be taken into account by cascade connection of the segments of transmission lines at the input and output of each element.
Example. As the simplest example of decomposing a complex device into a series of cascade-connected elementary two-port networks, consider a transmission line in which a wave-like change of wave impedance is twice made, with the distance between the joints equal to l (Fig. 3.3). Choosing the position of the reference planes in the places where the impedance of the transmission lines with the impedances and impedances changes , we obtain a quadrupole cascade structure with the following classical transmission matrix:
(3.19)
When formula (3.13) takes the form
(3.20)
where is the normalized characteristic impedance of the average segment of the transmission line included in the path with a unit normalized characteristic impedance. If in real microwave devices cascaded segments with differing wave impedances are used
Fig. 3.3. Transforming line segment
transmission in the microwave path
each segment with a classical transfer matrix can be considered as an enlarged base element, characterized by two parameters — wave resistance and length.
Let us analyze the behavior of the input reflection coefficient for the quadrupole shown in Fig. 3.3. Using the transition formula from the transfer matrix in the form (3.19) to the scattering matrix, we obtain:
(3.21)It is easy to check that the reflection coefficient can go to zero in the following cases:
1) for any , if , which corresponds to a regular transmission line;
2) for any , if and , which corresponds to a half-wave transformer; moreover, the half-wave transformer is a symmetrical four-port network ( );
3) with , if that corresponds to a quarter-wave transformer; moreover, the quarter-wave transformer is an antimetric quadrupole ( ).
By decomposing the trigonometric functions into Taylor series in relative frequency detuning in the vicinity of the points and , simple approximate formulas can be obtained for estimating the error at the inputs of transformers in the frequency band.
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Microwave Devices and Antennas
Terms: Microwave Devices and Antennas