The main provisions of the theory of receiving antennas

The receiving antenna converts the energy of radio waves into the energy of high frequency currents. It is thus a low-power alternator, the load of which will serve the receiver impedance.

When considering a receiving antenna, the following questions will be of interest:  What will be the values ​​of the emf and current in the receiving antenna for a given polar-wave strength incident on the antenna;  how these quantities depend on the direction of arrival and the polarization of the wave incident on the antenna;  what is the resistance value of the receiving antenna, which plays the role of internal resistance of the generator;  What is the amount of power delivered by the receiving antenna to the receiver.

The study of the receiving antenna can be made by two different methods. The first of these consists in the direct analysis of the impact of the incoming wave on the receiving antenna (Fig. A.1.1). In particular, in the case of a wire antenna, proceed as follows. The antenna is mentally broken into elementary vibrators and find the emf induced in each elementary vibrator. Summing these EMF across all elements of the antenna, determine the EMF at the terminals of it. Knowing the EMF, you can determine the current at the input of the receiver and other parameters of the receiving antenna.

Fig. P.1.1 - Symmetric vibrator in the field of action of a flat wave

The field that creates the emf on the elementary radiator dz is determined by the relationship E z E 0 sinexpjkz cos () .

Elementary EMF d  E z d z.

In the general case, for an arbitrary antenna, this method turns out to be quite complex. Therefore, as a rule, the second method is used, based on the application of the reciprocity principle known from the theory of passive linear theory. The principle of reciprocity allows you to determine the properties and parameters of the receiving antenna, if the properties and parameters of this antenna are known when operating as a transmitting antenna. The validity of the principle of reciprocity for antennas was proved in 1927 by M. P. Sveshnikova. On the basis of this principle, M.Seiman in 1935 developed a theory of receiving antennas.

Reciprocity theorem. Consider two arbitrary and arbitrarily oriented in space antennas A1 and A2 (Fig. A.1.2). The intermediate medium is considered linear (its characteristics do not depend on the magnitude of the field strength) and isotropic. Consequently, cases of radio wave propagation in the ionosphere or some other anisotropic medium (for example, in ferrite) are excluded.

A1

A2  2 , 

1 ,  12

Fig. Item 1.2 - Mutual arrangement of transmitting and receiving antennas As it was proved by M. P. Sveshnikova, a system of two antennas can be considered as a linear passive quadrupole with clamps 1—1 and 2—12. but)

b)

Fig. P.1.3 - Models of linear quadrupoles

Consider the currents at the input and output of such a quadrupole for two cases. The first case is when A1 is a transmitting antenna; A2 - reception (fig. A.1.3, a). The input and output include additional resistances Z 1 and Z 2 , which can be considered as the internal resistance of the EMF source and the load resistance (input resistance of the receiver), respectively. Suppose that in this case with EMF  1 the current at the output of the quadrupole (current in the load) will be I21. The second case is when A2 is a transmitting antenna; A1 - receiving (fig. P.1.3, b). In this case we set the emf value equal to  2 and the current at the input terminals of the antenna will be I112.

The principle of reciprocity asserts that for the amplitudes of currents and emfs we have

II

21 12

. (A.1.1)  1  2

This ratio will be used further to find the amplitudes of the current and EMF in the receiving antenna.

Further consideration is given for antennas with linear polarization of the field. We assume that the parameters of both antennas in the transmission mode are known: F, , D, R, Z.

max  BX

Since we want to find the current in the receiving antenna depending on the size of the field incident on the antenna, it is necessary to express the EMF  1 and  2 through the parameters of the antenna when they are used to transmit and the fields generated by these antennas. We believe that the antennas are in the far zone of each other relative to each other. The distance between them is r.

The amplitude of the field excited by antenna A1 near antenna A2

F (, ).

21,111

r

 I 2 R

11 1

Since I 1 , and P  1 , we get

Z Z 2

1 BX1

rZ Z

E

1 BX1

 21 . F (, )

11 1

Similarly for the case shown in fig. 11.3, b, we get

rZ Z

2 BX2

E 12

. F (, )

22 2

In accordance with the principle of reciprocity we get

Z Z

2 BX2 R 1 D max1 F 1 (  1 ,  1 ) E12 I12

 2 

.  1

Z Z RD F (, ) EI

1 BX1 2 max22 2 2 21 21

Let's collect everything that relates to antenna A1, to the left, and to antenna A2 - to the right, then we get

R  1 D max1 F 1 ( 1 ,  1 ) E 12 R  2 D max 2 F 2 ( 2 ,  2 ) E 21

. (P.1.2)



Z Z

Z Z

1 BX1 I 12 2 BX2 I 21

The left part of the expression (A.1.2) depends only on the parameters of the first antenna. Magnitude

E 12 depends, of course, on the parameters of the second antenna, but the ratio of E 12 to the current caused by it in the first antenna I 12 depends only on the parameters of the first antenna. Thus, in (A.1.2), two independent values ​​stand on the left and on the right. The left part depends on the parameters of the first antenna, the right part - on the parameters of the second antenna. This gives grounds to conclude that each of these quantities is separately equal to the same constant, which we denote by letter N. Thus, for an arbitrary antenna

RD F (, ) E

 max

N const. (P.1.3)

Z Z BX I

The resulting relationship (A.1.3), in essence, contains the desired connections between the antenna parameters in two modes.

For the amplitude of the receiving antenna in general, we obtain

1 E RD F (, )

 max

Ip 

cos (P.1.4)

,

N

Z  ZX where  is the angle between the polarization planes of the receiving antenna and the incoming wave. Note that the plane of polarization of the receiving antenna is called the plane of polarization of the field radiated by this antenna in the transfer mode. The ratio (P.1.3) shows that the receiving antenna can be viewed as a generator EMF

 1 E RD F (, ) cos (P.1.5)

pr  max

N and internal resistance equal to the input resistance of this antenna in transmission mode.

If the internal resistance of this generator is called the antenna input resistance in the receive mode, then the ZBX in the receive and transmit modes coincide.

The directivity pattern of the receiving antenna is the dependence of the amplitude of the EMF (or current) in the receiving antenna on the angles  and , which characterize the direction of arrival of the plane electromagnetic wave, at a constant value of the field strength at the receiving point. The normalized DN of the antennas in the transmit and receive mode are the same. The phase antenna patterns in both modes also coincide. Note that the coincidence of the antenna parameters in

Transmit and receive mode takes place under the condition that the receiver and transmitter are connected to the antenna in the same way.

It can be shown that the power delivered by the receiving antenna to the load will be

E 2  2

P N  G, , (P.1.6)

240 4 where G,   D,  is the gain of the receiving antenna. As can be seen from (A.1.6), the greater the directivity of the antenna, the more power it extracts from the field. In expression (A.1.6), the first factor is the density of the stream

 2

power 5 at the point of reception. The second factor G,  has the dimension of the area and

4 is called the effective or effective area of ​​the SF antenna. Then the gain value of the receiving antenna can be determined by the formula

4S EF

G,   2 , (P.1.7)



where SF  S geom ; S geom - geometric area of ​​the antenna;  is the surface utilization factor.

The effective area is such a platform, which, multiplied by the power flux density of the incoming wave, gives the power given off by the antenna according to the load, provided that the polarization of the antenna and the incident wave coincide

The relation (A.1.7) is one of the most important antennas in the theory. It relates the effective area — a parameter convenient in reception mode to a gain.

- a parameter, the use of which is natural in transmission mode. The relation (A.1.7) is suitable for any antenna design without any limitations.

Usually, along with (A.1.7), the other relation is 4SF

D,  . (P.1.8)

 2 For most VHF antennas, the losses are small, i.e. D,  G, . At the same time, the values ​​of SF, determined in accordance with (A.1.7) and (A.1.8), practically coincide. Match the relevant instrumentation values.

Definitions

The directional coefficient of the receiving antenna in the, направлении direction is the ratio of the power input to the receiver input when receiving from the direction ,  to the average, when receiving in all directions, the value of the power input to the receiver input.

The directional ratio of the receiving antenna is called the ratio of the power received at the receiver input when receiving at this antenna to the power delivered at the receiver input when receiving at the omnidirectional antenna. In this case, it is assumed that this antenna and non-directional antenna are matched with the receiver and both antennas have a KPD equal unit.

The gain of the receiving antenna is the ratio of the power input to the receiver when it is received at a given antenna versus the power that arrives at the input when it is received at the omnidirectional antenna. It is assumed that this antenna and non-directional antenna are matched with the receiver and that the non-directional antenna has a KPD equal to one.

Appendix 2. Method-driven EMF

The method of induced electromotive forces was proposed by Brilluen and D. A. Rozhansky developed by I. G. Klyatskin, A. A. Pistolkors, V. V. Tatarinov, etc. This method can be developed in relation to slot vibrators (method induced by MDS).

The method of induced emf and the relations arising from it are widely known and described in many monographs. The essence of the induced emf method is as follows. Let have an electric vibrator to which some emf is attached. Under the influence of this EMF in the vibrator arises current. The current is distributed in such a way that the boundary conditions on the surface are vibrated, namely, the condition that the tangential component of the vector E is zero. The radiation power of the symmetric vibrator is determined by the expression



st * 

P   Re  EJ d xd y d z, 

xyz 

Consider a system of emitters (Fig. A.2.1).

Z I2

I 1 I i

l2

O l 1 l i Y

X

Fig. P.2.1 - Model of mutual placement of radiators

Boundary condition for the first radiator

E  1  Et 0.

If there are N radiating elements, the tangent component for the i-th element is determined by the formula

N

E   E,

i ik k 1

where E  ik is the tangential component near the i-th element, due to the k-th field

an item. After transformations, we obtain the following expression for the radiation power

 N  P   Re      E  i I i * d z  . (A.2.1) i1 l i 



Then the radiation resistance of the i-th element will be determined by the formula

*

Z   1 *  N  E ik I d z.

i  iIiIi k 1 l

i

The integral can be defined as some insertion resistance, then the resultant formula for the radiation resistance of the i-th element is

N I Z   k Z. (A.2.2)

 i  ik k 1 I i

The radiation resistance of the i-th element is defined as the sum of the products

relations	currents			k-th	element	to the considered			i-th	element	on	contributed by
resistance		Zik	.
Full resist				ivlen	is radiated	oi for system emit			lei and	there is a view
						N
						 Z	 iZ	.		(A.2.3)
							i 1
Consider the example of a two-element antenna array (Fig. A.2.2).

Fig. P.2.2 - System of two-active emitters: a) model of mutual placement of emitters; b) equivalent circuit

For the equivalent circuit shown in fig. P.2.2, b, the current on the first and second emitters will be determined by the expressions:

 Z  Z

1  22 2 12

I ;

1 ZZ  ZZ

11  22 12  21

 Z  Z

2 11 1  21

I 2 .

ZZ  ZZ

11  22 12  21

Consider a system of two emitters, one of which is active and the other is passive and some resistance is loaded (Fig. A.2.3)

a) b) Fig. A.2.3 - System of two emitters: a) model of reciprocal placement of emitters; b) equivalent circuit. Determine the amplitudes of the currents in the vibrators of the system of emitters shown in fig. P.2.3, but it is possible on the basis of the equivalent circuit shown in Figure P.2.3, b, where the following notation is entered:  - exciting EMF; Z

a 11

Z, Z, Z - own and mutual resistance of radiation active and

22 12 21

passive vibrators; Z í is the load resistance. In general, Z   R   jX  , and for

The antenna under study is also Z Z, Z Z with l l and Z jX. Then the relation

12 21 11 22 aní í amplitudes and the phase difference of the currents in the passive and active vibrators can be calculated by the following formulas:

R 2  X 2

12 12

q  2 2 ; (A.2.4) R  X  X  

22 22

X X X 

12 22 í

arctg arctg. (P.2.5)

 

RR

12  22 

The amplitude and phase of the current in a passive vibrator are determined by its distance from the active vibrator and tuning, which is carried out by changing the length of the vibrator arm.

The resulting values ​​for the active and reactive component of the radiation resistance of the active and passive emitters are determined by the formulas:

R R qR cosX sin;

1 11 12 12

X X qR sin X cos

 .

1 11 12 12

R R  1 R cos X sin

 ;

2 22 21 21

q

X X  1 X cos R sin

 .

2 22 21 21

q

The coefficient of directional action D can be calculated by the following expression

1201coskl 2

D . R A

To determine R  12 and X  12, you can use the graphs shown in Fig. Clause 2.4,

P.2.10.

but)

b) Fig. A.2.4 - R  12 dependence on kd and kl

but)

b) Fig. A.2.5 - Dependence X  12 on kd and kl

Appendix 3. Analysis of the two-element lattice pattern of symmetric vibrators

Consider a two-element array of symmetrical vibrators located on the OY axis, as shown in fig. P.3.1. Let the 1st and 2nd radiators are located along the axis OY. The counting system is also shown in fig. P.3.1.

Fig. P.3.1 - Layout of radiators in the array

Consider a two-element array of radiators arranged in parallel along the axis OY, then according to the multiplication theorem, the resulting radiation pattern will be determined by the expressions:  in the plane YOZ f ,   f (, ) f (, );

 el resh

 into the plane YOX f ,   f (, ) f (, );

 el resh

 in the XOZ plane f  ,   1.

If the emitters are placed parallel to the OZ axis (Fig. Section 3.1), the resulting radiation pattern of the two-element array will be determined by the expressions:  in the YOZ f ,   f (, ) f (, ) plane;

 el resh

 in plane YOX f ,   f (, );

 resh

 in the XOZ plane f  ,   fl (, ).

If the radiators are placed parallel to the OX axis (Fig. Section 3.1), the resulting radiation pattern of the two-element array will be determined by the expressions:  in the YOZ f ,   f (, ) plane;

 resh

 into the plane YOX f ,   f (, ) f (, );

 el resh

 in the XOZ plane f  ,   fl (, ). As an example, consider the case of placing emitters parallel to the axis OZ. In the projection onto the YOZ plane, the displacement schemes will look as shown in Fig. Clause 3.2.

M , , r 

j1

Currents I 1 симI 01 e and

j2

I 2 I 02 e. The radiation field of each of the symmetric vibrators shown in Fig. Clause 3.2 is characterized by a directional diagram, which is calculated by the formula coskl coscoskl

fl (, ) 

. (P.3.1)

sin



It should be noted that when placing the emitters in the YOX plane, the radiation pattern in that indicated in fig. P.3.2 the reference system of angles is determined by the formula coskl sincoskl

fl (, ) 

. (P.3.2)

cos



I 01

We introduce the notation a ;  2  1 .

I 02 If according to fig. P.3.2 r 1 r 2 , then the lattice multiplier is determined on the basis of the expression

2

sh   1a 2a cos kd sin

f (,)  . (P.3.3)

If in accordance with Fig. P.3.2 r 2 r 1 , then the lattice multiplier is determined on the basis of the expression

2

sh   1a 2a cos kd sin

f (,)  . (P.3.4)

When a 1, the expressions (A.3.3) and (A.3.4) can be given due to kd sin

fш (, ) 2 cos

. (П.3.5) 2  Рассмотрим несколько примеров для двух излучателей, размещенных, как показано на рис. П.3.2, когда на их входыподаютсяразличные по фазетоки.



Пусть a 1,   0, d  тогда множитель решетки, диаграмма

21 4

направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будет выглядеть, как показано на рис. П.3.3, а.



Пусть a 1,  2  1 0, d  тогда множитель решетки, диаграмма

2

направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будет выглядеть, как показано на рис. П.3.3, б.

Пусть a 1,  2  1 0, d  тогда множитель решетки, диаграмма направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будет выглядеть,какпоказанона рис. П.3.3, в.



f 

, град

, град

 f реш f эл 

 f  

f  

в) Рис. П.3.3 — Диаграмма направленности двухэлементной решетки симметричных вибраторов



Пусть a 1,    , d  тогда множитель решетки, диаграмма

21 2 4

направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будет выглядеть, какпоказанона рис. П.3.4, а.



Пусть a 1,  2  1  , d  тогда множитель решетки, диаграмма

22 направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будетвыглядеть, как показано на рис. П.3.4, б.



Пусть a 1,  2  1  , d  тогда множитель решетки, диаграмма

2 направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будетвыглядеть, как показано на рис. П.3.4, в.

270 270

fш  fш 

 

f эл    f эл  f  f  



a) b)

f  

в) Рис. П.3.4 — Диаграмма направленности двухэлементной решеткисимметричных вибраторов



Пусть a 1,  2  1 , d  тогда множитель решетки, диаграмма

4 направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будет выглядеть, какпоказанона рис. П.3.5, а.



Пусть a 1,   , d  тогда множитель решетки, диаграмма

21 2

направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будет выглядеть, какпоказанона рис. П.3.5, б.

Пусть a 1,  2  1 , d  тогда множитель решетки, диаграмма направленности одного элемента и результирующая диаграмма направленности в плоскости YOZ будетвыглядеть, какпоказанона рис. П.3.5, в.

90 90

180 0

180

0

, град , град



 f реш  fреш

 f эл    f эл    f   f  

a) b)

f  

в) Рис. П.3.5 — Диаграмма направленности двухэлементной решетки симметричных вибраторов



Таким образом, видно, что при d  диаграмма направленности антенной решетки

2 характеризуется наличием дифракционных максимумов. При    2  1  антенная решетка характеризуется ненаправленным излучением.Наибольшая направленность



излучения обеспечивается для случая a  1,    , d  , когда диаграмма

21 2 4

directivity of a two-element lattice has the shape of a cardioid.