3 Microwave filters, attenuators, phase shifters. Branched microwave paths.

The microwave path of any radio system consists of a large number of different microwave devices. These include segments of transmission lines, connectors, bends and twists, matching devices, phase shifters, microwave filters, microwave power dividers, non-reciprocal microwave devices using ferrites, switching devices, etc. Common to these and similar devices is that they belong to the class of devices with distributed parameters. The geometrical dimensions of these devices are comparable with the wavelength of electromagnetic rings *** ***. This determines all the specifics of the calculation and design of microwave devices, since the processes occurring in them have a wave character. The theory of UHF devices is closely related to electrodynamics and includes two large sections: the analysis of microwave devices and the synthesis of microwave devices. The task of the analysis is to study the external characteristics of the microwave, as well as determine these external characteristics by solving the corresponding internal problem using applied electrodynamics methods or from an experiment.

The task of the synthesis of microwave devices is to determine the structure and geometrical dimensions of the element of the microwave device according to its specified characteristics.

The study of the external characteristics of microwave devices can be done without specifying their internal structure. This allows us to consider the microwave device as a kind of “black box”, having a certain number of microwave transmission lines emerging from it. Each of these transmission lines is also a device with distributed parameters, for which the wave nature of electromagnetic processes is non-transient. This leads to the need to fix the longitudinal coordinates of the cross sections of transmission lines or, as they say, to fix the terminal plane. With respect to these terminal planes, the phases, and in some cases, the amplitudes of the incident and reflected waves are counted. The displacement of the terminal planes along the input transmission lines leads to a change in the external characteristics of microwave devices. In most cases, in the input transmission lines of the microwave devices, the single propagating wave is the main type wave. Waves of other types are in a supercritical mode. The terminal plane of the microwave device tend to be arranged so that the amplitudes of supercritical waves in them can be neglected.

In the following, we will consider passive linear microwave devices. The microwave device is called passive if it does not include active transforming or amplifying elements, for example, transistors, electronic microwave devices, etc. The linearity of microwave devices means the independence of its characteristics from the input to

25

him power. External characteristics of passive linear microwave devices are interconnected by systems of linear algebraic equations. Therefore, in the theory of microwave devices, the mathematical apparatus of matrix theory is widely used.

Methods for describing inhomogeneities of the waveguide path In the microwave technology, it is customary for each microwave device to put in correspondence some multi-terminal network. In this case, each propagating wave in the input transmission lines of the microwave device is assigned a pair of terminals of this equivalent multipole. In the future, we assume that only the main types of waves propagate in the input transmission lines. Then the number of terminal pairs of an equivalent multipole coincides with the number of input transmission lines of the RF device. The inputs of the microwave device are cross sections of the input transmission lines.

On each pair of clams equivalent multipole can be determined

complex voltages u n and currents i n . Methods for setting equivalent voltages and currents may be different. We will consider linear passive multipoles. The currents and voltage at the terminals of a multipole can be connected by a ratio system,

U ZJ ZJ ...

  ZJ

1 11 1 12 2 1nn

 U ZJ ZJ ...

 ZJ

2 21 1 22 2 2nn





 ... 

...

 U ZJ ZJ ...

 ZJ

nn11 n22 nn n

which is in a matrix form

 ZZ ... Z  J

U 

1 1112 1n 1

 

UZ Z ... ZJ

2 2122 2n  

 2 ...  ...  *  ... (1)

 

... ... ...

  

UZ Z ... ZJ

n nn 

n1 n2  n

U i

where z ik  - mutual resistances of the i –th and k – th input at the idle mode at J k all inputs except k –th; U

i is the intrinsic resistance of the i – th input, is determined in idle mode on

z ii  J i all inputs except the i –th. Use the normalization of the element matrix of resistance [Z] 1 111

** **

P JJW i ii

i  ii iii  UU i ii

2W i 2 22

U U / W, i  Wj

iii ii

W k

Z Z

ik ik

W i

[]  1  [].

Yz

A multipole is reactive if the elements of its resistance matrix are purely reactive.

When analyzing the operation of microwave devices, there is always a formal possibility of transition from electric and magnetic fields, characterized by their strengths to their equivalent voltages, “applied” to the device terminals, and equivalent currents, “current”, at these terminals. At the same time, the functional features of any waveguide element, which determine its interaction with various nodes of the path, are described with the help of some coefficients combined into a matrix. The matrix of any element is defined if the structure of the electromagnetic field is known.

The total field in the transmission line can be represented as an estimate of the sum of the incident and reflected waves (or falling from the other arm) with the normalized amplitudes Upad and Uotr. This representation is convenient to use when describing microwave elements, and the resulting wave matrices combine the coupling coefficients between the values ​​of the incident and reflected waves in the arms of this element.

For example, consider a triple-arm waveguide element (see Figure 3.1).

U  pa1

U  pad2

U  Ref2

U  Ref1

U  pad3

U  Ref3

 

U SU SU SU;

Ref 1 11 pad1 12 pad2 13 pad3

U  SU  SU  SU  ; (3.1)

 ot2 21 pad1 22 pad2 23 pad3 

 

 U neg3  S 31 U pad1  S 32 U pad2  S 33 U pad3 ,



where S ij - complex coefficients characterizing the waveguide node.

Let the source be included in the i-th arm U pad i 0, and to the remaining arms of the waveguide

element is connected to the agreed load, i.e. U j 0. Then when j i coefficient

pad

S ii is the wave reflection coefficient in the i-th arm.



U

den

S ii   , (3.2)

U

padi

and for i j, the coefficient S ij is the transmission coefficient from the jth arm of the i-th one:



U

den

S ij   . (3.3)

U

pad j

In cases when U const system (5.1) can be written in matrix form:

padi



U U 

Ref 1 SS S  pad1

1112 13

  U  neg 2   S 21 S 22 S 23   U  pad 2 , (3.4)  

 

U SSS U

 ot3  3132 33  pad3 

 

where the matrix  is the scattering matrix.

S

Multipole scattering matrix. Consider a multipole (Fig. 3.2).

The scattering matrix for him has the form: SS ... S

 1112 1N 



SS ... S

2122 2N

S .

 ... ... ... ... 



SS ... S

 N1 N 2 NN 

For symmetric multipolar S S. Number of matrix elements

ij ji

NN 1

which should define is.

2 We use the property of the unitarity of the matrix when N 1, j k;

*

 S ij S k 

i  1 0, j k.

All energy supplied to arm j is distributed over all shoulders, which must be matched with its loads. The non-excitation of the neighboring shoulder is a condition for the independent excitation of the shoulders.

Waveguide H-tee scattering matrix. H-tee - a device for branching energy along the narrow wall of the waveguide (Fig. 3.3). The division of power is carried out in phase. In order to obtain total power from shoulder 3 from shoulders 1 and 2, it is necessary to excite it in phase.

The H-plane waveguide tee scattering matrix, when internally matched from the side of the shoulder 3, and the shoulders 1 and 2 are loaded on matched loads, has the form:

0	one	one
one	0	one
one	one	0





one

 

 

S



,

2

where the numbering of the rows of the slavers corresponds to the shoulder numbering.

The scattering matrix of the waveguide E-tee. E-tee - a device for branching energy along the wide wall of the waveguide (Fig. 3.4).

The scattering matrix of the E-tee, when internally matched from the side of the shoulder 3, and the shoulders 1 and 2 are loaded with matched loads, has the form:



01 1



one

 

 

one

S



ten



.

2

one



ten

The division of power is carried out in antiphase. In order to obtain total power in shoulder 3 from shoulders 1 and 2, the latter must be excitement out of phase, and to subtract the power of shoulders 1 and 2, excite them in phase.

The scattering matrix of a double waveguide tee (Fig. 3.5). It is assumed that at the inputs of the double tee there are waves of the main type H 10 . In such a multipole there are no non-reciprocal devices. Such a multipole is mutual.

Let the multi-pole is triggered by the combination 3. The rest of the inputs are consistent. In this case, it is polarized at the outputs 1 and 2 (see Fig. 3.6, a) in a pertriphase phase: S S.

13 23

Excitation to input 4 will not pass: S 43 0.

Imagine that input 4 is excited, the others are consistent. The principle of power distribution is the same as in the H-tee: S 14 S 24 . The energy in the shoulder 3 does not fall: S 34 0.

2 

2 

2 

2 1;

S 11 S 12

S 13

S 14 2 

2 

2 

2 1;

S 31 S 32

S 33

S 34 2 

2 

2 

2 1.

S 41 S 42

S 43

S 44

From here

212 

.

S 41

S 31

2

1 i 1 i

S 

e; S 31 

e 41 2 2

S 31 S 32 .

The full scattering matrix is

i i

00ee   i i

10 0 ee





S

.

i  i

2 e e 00   i i 

ee 00  



For the case of a consistent scattering matrix, when the phases of the elements of the matrix are 0, then: 00 11 

10 0 11

 

S

.

2 1 10 0 

11 00



The matrix gives the relationship between the amplitudes of the incident and reflected waves at the inputs of the multipole:

U  U  

Ref1 Pad1

 

Uu

 ot2  pad2 

S

   .

 

Uu

Neg3 Pad3

   U  neg4   U  pad4 



We define U  re i :

 1

 U  ref 1  U  U  ;

pad3 pad4

2

  U  ref 2  1 U   U  ;

pad3 pad4

2



one

 

 2 U U ;

U Ref3 

pad1 pad2



  1  U   U   .

U 

 ot4

pad1 pad2

 2

Microwave bridge devices. Bridge devices are eight-port networks with the following properties: when one arm is excited, the signal on the other arm does not pass (the decoupling property). The signal at the remaining 2 inputs is divided in half.

Examples of bridge devices. Ring bridge, waveguide slotted bridge, double waveguide tee.

Ring bridge runs on coaxial, waveguide, strip transmission lines (see. Fig. 3.7).

a) b) 1 - transitions; 2 - housing; 3 - microstrip transmission line. 3.7 - Ring bridge: a) scheme; b) design

Suppose that input 2 is excited and the rest are loaded on matched loads. In an annular coaxial line, two waves of opposite direction are excited. Passing the same path, the fields of these waves are folded in section a (voltage antinodes)



a-section is equivalent to idling and lags behind input 4 on. Idling through

44 is recalculated into a short circuit. The signal at input 4 will not pass, but is divided in half between inputs 1 and 3. Similarly, we can consider the excitation from any other input. Determine the wave resistance of the ring  K , if the wave resistances are known

inputs P.

Consider the case of common-mode signals at inputs 1 and 3 - the ring bridge is made up of H-tees. Resistance  LP is recalculated from input 1 to input 2, as a kind of resistance Z 1 and from input 3 is converted to Z 2 to input 2. These resistances at input 2 are parallel.

Z

Z  2 ; Z  2;

VH2 LP 2 LP

2  Z  2 ;  

LP 2 QC LP In the case when the ring bridge is composed of E-tees, the wave impedance of the ring and transmission lines should be related by

 LP

.

K 2 Wave-gap bridge. Consider the waveguide-gap bridge presented in Fig. 3.8.

We assume that in waveguides at the inputs 1, 2, 3, 4, H 10 waves are excited. Let the bridge be excited from the side of the shoulders 1 and 2 (Fig. 3.9). In the case shown in fig. 3.9, and in the field of the excitation wave H 10 is excited.

a) b) Fig. 3.9 - Description of the principle of operation of the waveguide-slot bridge

Determine the wavelength in the region of the slit



.

 1  

 2  2

1 1 

  

 4a 

 KR 

In the case of antiphase excitation (see Fig. 3.9, b), an H 20 wave is excited in the region of the slit



.

 2 



 2  2

1  1 

2a 

 KR 

Waves of these types propagating in the region of the gap will receive a phase delay of 2 по 2

 l;  l.

H10 H 20  1  2

At inputs 3 and 4, the fields of these waves are folded with a phase difference of 2 2

 .

   l

 12  

In order for the phase difference to be selected, the length of the slit

2

1  1 2

l .

4  2  1

Consider the signal at input 3. The signal at inputs 3 and 4 is divided in half. In this case, the signal at input 3 is ahead of the signal at input 4. Waves, propagating in the region of the gap at inputs 3 and 4, encounter an inhomogeneity. As a result, reflected waves appear. To compensate for them (to adjust the bridge) enter the adjustable pin. You can adjust either transient attenuation or directivity (isolation at input 2 and the level of signals at input 3 and 4). You can also consider excitation from other inputs.

Directional couplers. Directed couplers (fig. 3.10) are 8– pole cells, one input of which is isolated. This device is used for tapping part of the power to some secondary transmission line. This is a completely mutual device.

S S S S  A;

1122 33 44



S S S S B;

12 2134 43



S S S S C;

 133124 42

S S S S D.

 14 412332

Then the scattering matrix can be written as

ABCD



B ADC

S .

CDAB



DC BA



In practice, the directional coupler is used in two modes of operation - see fig. 3.11.

one

2 1

2

4 4

a) b) Fig. 3.11 - Modes of operation of the directional coupler: a) codirectional; b) opposing

For the codirectional coupler, the scattering matrix is

2

 0 j 1D 0 D 

 2 

j1D 0 D 0 



S

.

 2 

0 D 0 j1D



  D 0 j 1D 2 0  



Directed couplers have characteristics: 1) transient attenuation

P 1

A 10lg 20lg D, dB

P 4

2) working attenuation

P12

L 10lg 10lg1D, dB P 2

3) orientation

P

D

M 10lg 4 20lg

db

P 3

C

4) standing ratio 1

S 11

KSV

. 1

S 11

See also