You get a bonus - 1 coin for daily activity. Now you have 1 coin

Analysis of directional couplers using the symmetry method

Lecture



Directional couplers form an extensive class of enlarged basic elements used both in the construction of extensive microwave paths and in various measuring devices. Recall that a directional coupler is called a reactive eight-port network, which has two pairs of ideally matched and mutually isolated inputs. Most of the directional couplers have a plane of symmetry and therefore the selection of the nominal values ​​of their constituent elements and the analysis of the resulting scattering matrices can be performed by the method of symmetric and antisymmetric excitation

.

We introduce the numbering of the inputs of the eight-port network shown in fig. 3.7, a. In accordance with formulas (2.74) and (2.75), the scattering matrix of the eight-port network in the presence of the plane of symmetry   Analysis of directional couplers using the symmetry method should have the structure:

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method

with the second order scattering matrix

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method

refer to the partial quadrupole symmetric and antisymmetric excitation, shown in Fig. 3.7,6 and Fig. 3.7, c. These two-port networks represent the upper halves of the eight-port network, cut off by the plane of symmetry with the boundary condition Ht = 0 (symmetric excitation, “+” index) or with the boundary condition   Analysis of directional couplers using the symmetry method (antisymmetric excitation, the index "-"). The independent elements of the eight-port scattering matrix in Fig. 3.7, and are expressed as reflection coefficients as follows.   Analysis of directional couplers using the symmetry method and transfer ratios   Analysis of directional couplers using the symmetry method partial quadrupoles:

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method ;

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method ; (3.31)

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method

 

moreover, the reactivity of partial quadrupoles is equal   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method . The eight-port jet in fig. 3.7, and turns into a perfect directional coupler if the inputs are matched   Analysis of directional couplers using the symmetry method and at the same time achieved isolation of any two pairs of inputs. Depending on which inputs the decoupling is reached, the following types of direction are distinguished: 1) type 1 when decoupling pairs of inputs 1-3 and 2—4; 2) type II when decoupling pairs of inputs 1-4 and 2-3; 3) type III when decoupling pairs of inputs 1–2 and 3–4.

  Analysis of directional couplers using the symmetry method

Consider successively each of the types of orientation.

Type 1 directivity. Joint compliance of the conditions of the coupler inputs   Analysis of directional couplers using the symmetry method and once mated   Analysis of directional couplers using the symmetry method according to formulas (3.31) is equivalent to equalities

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method (3.32)

i.e., to achieve type 1 directivity, both partial quadrupoles of symmetric and antisymmetric excitation should be perfectly coordinated and differ only in the phases of transmission coefficients   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method . The phase difference of these transmission factors   Analysis of directional couplers using the symmetry method called the differential phase shift for waves passing through the coordinated partial quadrupoles of symmetric and antisymmetric excitation. The ideal scattering matrix of a directional coupler of type 1, when conditions (3.32) are fulfilled, has the structure:

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method (3.33)

Type 1 directional couplers belong to codirectional couplers, since the wave in the secondary transmission line 3-4 moves in the same direction as the wave exciting it in the primary line 1-2. In addition, type 1 directional couplers are quadrature, that is, the phase shift between elements   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method in the scattering matrix is ​​equal to   Analysis of directional couplers using the symmetry method . In the directional coupler type 1, as a rule, there is a second plane of symmetry (at least electric), passing between pairs of inputs 1-3 and 2-4. Conditions (3.32) determining the type 1 directivity can be rewritten in terms of classical transfer matrices.   Analysis of directional couplers using the symmetry method for partial quadrupoles. C using the transition formulas between the matrices A and S from table. 3.1 we get the equations   Analysis of directional couplers using the symmetry method of which, given the materiality of the elements   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method and imaginary elements   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method in a reactive quadruple should be:

  Analysis of directional couplers using the symmetry method type 1 directionality (3.34)

Differential phase shift is easily determined from the relation   Analysis of directional couplers using the symmetry method resulting in the formula

  Analysis of directional couplers using the symmetry method .

Expressions for nonzero elements of an ideal scattering matrix of a directional coupler of type 1 taking into account conditions (3.34) take the form

  Analysis of directional couplers using the symmetry method (3.35)

where the top character refers to the element   Analysis of directional couplers using the symmetry method and the bottom to the element   Analysis of directional couplers using the symmetry method .

2. Type II directivity. Joint fulfillment of junction conditions   Analysis of directional couplers using the symmetry method and matching inputs   Analysis of directional couplers using the symmetry method in accordance with formulas (3.31) is possible only if equalities hold   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method . Taking into account the canonical scattering matrix of a non-dissipative quadrupole (2.54), this leads to the expressions

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method , (3.36)

Where   Analysis of directional couplers using the symmetry method ,   Analysis of directional couplers using the symmetry method ,   Analysis of directional couplers using the symmetry method - independent real parameters that determine the scattering matrix of a reactive quadrupole.

The ideal scattering matrix of the directional type II coupler, when conditions (3.36) are fulfilled, takes the form:

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method ; (3.37)

  Analysis of directional couplers using the symmetry method

Type II directional taps refer to counter-directional taps, since the wave in the secondary transmission line 3-4 moves in the opposite direction with respect to the excitation wave in the primary transmission line 1-2. If the directional coupler of type II has a second plane of symmetry, passing between pairs of inlets 1-3 and 2-4, then   Analysis of directional couplers using the symmetry method what means   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method . Consequently, in the presence of two symmetry planes, the directional coupler of type II turns out to be quadrature. If the second plane of symmetry is not, but the selection of the parameters of the partial quadrupoles ensures equality   Analysis of directional couplers using the symmetry method , the type II directional coupler is obtained by a phase-in-phase coupler, i.e.   Analysis of directional couplers using the symmetry method .

Conditions   Analysis of directional couplers using the symmetry method ,   Analysis of directional couplers using the symmetry method Type II directivity, in terms of classical matrices, of the transfer of partial quadrupoles are as follows:

  Analysis of directional couplers using the symmetry method ;

  Analysis of directional couplers using the symmetry method

Hence, taking into account the materiality of the elements   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method and imaginary elements   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method in reactive quadrupoles the conditions follow:

  Analysis of directional couplers using the symmetry method type II directivity, (3.38)

taking into account which we obtain expressions for the calculation of the elements of the ideal scattering matrix (3.37):

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method (3.39)   Analysis of directional couplers using the symmetry method

3. Type III directionality. Joint fulfillment of junction conditions   Analysis of directional couplers using the symmetry method and matching inputs   Analysis of directional couplers using the symmetry method in accordance with formulas (3.31) is possible only if equalities hold

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method (those.   Analysis of directional couplers using the symmetry method ).   Analysis of directional couplers using the symmetry method

Taking into account the canonical scattering matrix of a reactive quadrupole, these conditions take the form:

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method (3.40)

The ideal scattering matrix of the type III directional coupler, when conditions (3.40) are met, has the structure:

  Analysis of directional couplers using the symmetry method

Conditions   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method , determining the direction of type III, in terms of classical transfer matrices of partial quadripoles give the equations

  Analysis of directional couplers using the symmetry method

  Analysis of directional couplers using the symmetry method ,

of which, by virtue of the materiality of the elements   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method and imaginary elements   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method in a reactive quadrupole the conditions follow:

  Analysis of directional couplers using the symmetry method type III directivity, (3.42)

and taking into account these conditions, we obtain the following expressions for the elements of the ideal scattering matrix (3.41):

  Analysis of directional couplers using the symmetry method ;   Analysis of directional couplers using the symmetry method . (3.43)

  Analysis of directional couplers using the symmetry method

Type 1 directional couplers can be assigned to type III directional couplers if they have a second plane.

  Analysis of directional couplers using the symmetry method

geometric symmetry   Analysis of directional couplers using the symmetry method (Fig. 3.8, a). Indeed, the rotation of the directional coupler type 1 in the plane of the pattern 90 ° clockwise and renumbering the inputs   Analysis of directional couplers using the symmetry method ,   Analysis of directional couplers using the symmetry method ,   Analysis of directional couplers using the symmetry method and   Analysis of directional couplers using the symmetry method lead to a coupler with a scattering matrix of the form (3.41), i.e., to a directional type III coupler having a plane of symmetry   Analysis of directional couplers using the symmetry method . Therefore, in directional couplers with two planes of geometric symmetry, the adjustment of partial two-port networks of symmetric and antisymmetric excitation according to conditions (3.34) and (3.42) in some cases may be equivalent and lead to identical schemes of couplers (up to renumbering inputs).

Using the formulated conditions for matching and decoupling the inputs of directional couplers in terms of the parameters of partial quadrupoles, it is possible not only to easily understand the principle of operation of one or another coupler (of course, having a plane of symmetry), but also to get the relationships necessary for its design.


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Microwave Devices and Antennas

Terms: Microwave Devices and Antennas