Matching data lines on a printed circuit board. Load matching problem with transmission line
If a load resistance equal to the wave resistance is connected at the end of the line:
then, turning to formulas (18-23), we find that
(18-48)
i.e., the reflected wave does not occur. Such a load is called a matched load or a load without reflection.
In this case, as follows from (18-47), the reflection coefficient
From the relations written above, taking into account (18-48), we obtain:
(18-49)
This implies:
(18-51)
i.e., for any point on the line, the ratio of the complexes is equal to the wave impedance. Therefore, the operating mode of the generator feeding such a line will not change if it is cut at any section of the line and wave impedance is switched on instead of the cut part of the line. The operating mode of the remaining section of the line will also not change.
From relation (18-31) it follows that for a matched line the input impedance is equal to the wave impedance
Assuming the initial voltage phase at the end of the line equal to zero, i.e., based on (18-49) and (18-50), we write down the instantaneous values of voltage and current at any point on the line:
The resulting relationships are shown in Fig. 18-9. The intersection points of the x-axis with the voltage and current curves are shifted by a distance and, according to what was said in § 18-5, the value of g is negative. Therefore, using terms that are valid, strictly speaking, only for sinusoidal quantities, we can say that the current is ahead of the voltage in phase by an angle Voltage and current in various points the lines differ not only in amplitude, but also in phase.
The power passing through any section of the line is
(18-53)
This power decreases with distance from the beginning, since power is absorbed at each element of the line length
equal to the sum of losses in the resistance of the wires and in the conductivity of the insulation on the line element Equality of the average and right parts relations (18-54) can be shown after transformations. To do this, you should replace them in the middle part with the values from equalities (18-49), (18-50), (18-27), (18-10) and (18-11), having previously expressed them through the well-known formula
The power transmitted over a matched line is called natural or natural power. Natural power transmission mode can occur in lines if the load impedance is equal to the characteristic impedance. The average values of natural power for lines of 500, 400, 220, 110 and 35 kV, respectively, are 3 MW. This shows how the natural transmitted power increases greatly with increasing line voltage.
The power received by the line is
Transmission lines designed to channel the energy of microwave signals from the generator to the load operate best only in a certain mode– coordination mode. To analyze the optimality of energy transfer from the generator to the load, the following diagram is considered (Fig. 2.11).
Rice. 2.11. Microwave energy transfer circuit
Voltage generator with emf. (E) and internal resistance 
via a transmission line with characteristic impedance z
IN
and constant spread
connects to a load having resistance 
. IN general caseZ n
Z G
Z IN, so that reflected waves exist in the transmission line.
Elimination of reflected waves is achieved, for example, by creating additional waves reflected from the matching device. These waves must interfere, which requires ensuring equality of their amplitudes and a phase shift of 180 o. Adjusting transformers comes down to creating the conditions necessary for complete suppression of reflected waves.
Coordination of a transmission line means setting this line to traveling wave mode. Let's consider what advantages a coordinated line has over an uncoordinated one.
Maximum generator power output to the load
If the transmission line has zero length L=0 (the load is connected to the generator output), then the power released at the active resistance of the load r n , equal to
,
(2.29)
where does the maximum power output to the load come from?
.
(2.30)
Thus, with a complex internal resistance of the generator, the load must have a reactive part that is opposite in sign to the reactivity of the generator resistance. However, if the generator has a purely active resistance 
, the maximum power transfer to the load is obtained with a purely active load resistance
In what follows we will assume that the generator is matched with the transmission line, i.e. condition (2.30) is satisfied.
Let us determine what part of the power is allocated to the active resistance of the load if Z n Z IN. In this case, there is a reflected wave from the load. If the transmission line has no losses, then the active power in any section of the line, including the load, is the same. For example, at the voltage antinode it is equal to
,
(2.31)
where 
,
Where R pad– power passing through the transmission line in traveling wave mode.
Using the relations for TONE you can write down the power transferred to the load depending on SWR
.
(2.32)
Thus, if the transmission line is not matched with the load, part of the generator power is reflected and the output to the load in accordance with relations (2.31) and (2.31) is not maximum.
Maximum transmission line efficiency
Let us assume that the transmission line has losses characterized by the attenuation coefficient . The efficiency of a transmission line is defined as the ratio of the power at the end R n to power at the beginning of the line R 0
.
The power at the beginning of the line (in section 1) in Fig. 2.11 is equal to
and at load (in section 2)
Using these conditions, we obtain the dependence of the transmission line efficiency on the magnitude of the reflection coefficient modulus in the form
(2.33)
At Г=0, the efficiency is maximum and equal to
.
(2.34)
As reflection increases, efficiency decreases, especially strongly for large values. 
For the active power transmitted by the transmission line at the antinode, the voltage can be written
.
(2.35)
If the limit voltage U before(or maximum power 
) is specified in the line, it will be determined by the voltage value at the antinode 
. Therefore, from (2.35) we obtain
(2.36)
As a result, the transmitted power decreases by - 1 time.
Let's consider methods for constructing matching devices.
General principles of load matching with transmission line
Regardless of the nature and type of the matching device, as well as the frequency band within which the matching is maintained, the matching circuit looks like Fig. 2.12.
The purpose of the matching device is to eliminate the wave reflected from the load. This problem is solved by two different methods:
- by absorbing the reflected wave in a matching device. In this case, the incident wave passes through the matching device with virtually no losses.
by creating another reflected wave in the transmission line using a matching device, the amplitude of which is equal to the amplitude of the wave reflected from the load. The phases of both reflected waves differ by 180 0. As a result, the reflected waves cancel each other out.
The first matching method is based on the use of either bridge circuits or non-reciprocal devices.
The second type of matching device usually consists of reactive elements and introduces virtually no losses. It will allow you to obtain an input impedance at the junction with the line equal to the wave impedance Z input =Z IN. As a result, a traveling wave is formed in the line to the left of the junction.
Narrowband matching (NB). In the problem of narrowband matching, matching elements are built from the consideration of obtaining complete matching (G=0) at one fixed frequency. The degree of matching of the transmission line with the load is assessed by the matching characteristic, which is the dependence of the modulus of the reflection coefficient 
from frequency
.
The US band is equal to several units of percent of
0
.
From an energy point of view, matching using a non-dissipative quadripole is of greatest interest. The matching device must have the properties of an ideal transformer that converts high-frequency voltages, currents and impedances from one section to another without introducing active losses. Such transformers can be inductive, capacitive diaphragms and other inhomogeneities included in the line.
The US methodology is as follows. Load conductance is expressed through active and reactive conductivity
,
(2.37)
Where G n0, using a line segment of length l is transformed into conductivity Y 1, the active part of which is equal to the wave conductivity of the line, i.e.
.
(2.38)
R 
the reactive part of the conductivity Y 1 is compensated by parallel connection into a line of reactive conductivity equal in magnitude and opposite in sign (-iB 1). As a result, the input conductivity of the load at the terminals 11
(Fig. 2.13) becomes purely active and equal to the wave conductivity, i.e. the line is loaded with a resistance equal to its characteristic impedance, which corresponds to ideal matching. By replacing the terms conductivity with resistance everywhere, you can come to a matching circuit, where the compensating reactance (- iX) is connected to the line in series.
Let's look at the most common types of impedance transformers.
Reactive plumes. Transmission line segment with mode short circuit or idling in the load section. From the transformation formulas (2.18) and (2.19) the formulas for the reactance and conductivity of the loops follow:
,
(2.39)
.
(2.40)
Segments of short-circuited lines with a length of less than half a wave are often used as matching elements, as well as as elements of oscillatory circuits with distributed parameters. Open segments are used much less frequently. Moreover, in hollow waveguides and many other transmission lines, the idle mode is undesirable due to the intense radiation of the output hole.
Diaphragms in waveguides. A thin metal plate with a hole placed in the cross section of the waveguide is called a diaphragm. Diaphragms are used as reactive elements to match resistances.
In Fig. 2.14?, and a symmetrical diaphragm in a waveguide of rectangular cross-section is schematically depicted. The diaphragm has a rectangular cross-section with dimensions a / and b. For wave H 10, the diaphragm disturbs the magnetic field, and therefore this inhomogeneity can be represented in the form of inductance (Fig. 2.14,a). The diaphragm is called inductive. The relative value of reactance can be calculated using the following approximate formula
(2.41)
In Fig. 2.14,b shows a capacitive symmetrical diaphragm and its equivalent circuit for H 10 waves. A diaphragm of this configuration strongly disturbs the electric field of the wave. The relative value of the normalized conductivity is approximately expressed as follows:
,
(2.42)
Where Y B =1/ Z B– wave conductivity.
Long lines are widely used in radio engineering. Let's briefly look at some of them.
Long line like a transformer. Let the line be loaded with resistance. Of great interest is the property of the line to change the load resistance when it is converted to the input of the line - a property that is inherent in a conventional transformer when the load resistance is brought to the primary winding. Therefore, a long line is often called a resistance transformer.
It can be shown that:
a) a homogeneous lossless line, the length of which is equal to a quarter of a wavelength (in a more general case, an odd number of quarters of wavelength), transmits any load connected at one end to the terminals of the opposite end with a change (transformation) of this load, determined by the expression :
,For example, an oscillatory system in the form of a section of a two-wire line with copper wires, short-circuited at the end, has a quality factor of the order of several hundred. A similar oscillatory system formed by a coaxial line is characterized by a quality factor. The given figures show the advantages of oscillatory systems with distributed parameters in the VHF range compared to conventional oscillatory circuits. Calculation resonant frequencies of such oscillatory systems is produced according to formulas (7.55, 7.56, 7.61, 7.62).
Sections of long lines can also be used as filters, matching loops, etc. A short-circuited section of a line is called a loop. A more detailed presentation of these issues is given, for example, in.
Long line as a feeder. The line through which the energy of high-frequency oscillations is transferred from the generator to the load is called a feeder (from the English verb to feed– nourish).
Feeders are used in modern radio engineering devices. various types. In the meter and longer wavelength range, an open two-wire feeder is usually used to transmit energy. However, more short waves the open line begins to radiate intensely electromagnetic energy into the surrounding space, heat losses in the wires increase. As a result, the coefficient useful action of such a feeder, as the wave shortens, it drops sharply.
In the decimeter wavelength range, the coaxial transmission line is most widely used. It, unlike an open two-wire line, has practically no radiation losses, since its electromagnetic field is separated from the external space by a screen - a metal cylindrical shell. The coaxial feeder also has lower heat losses, since the conductors that form it have sufficiently large surfaces.
On centimeter waves, a waveguide is used as a feeder, which is a hollow metal pipe in which propagation electromagnetic waves. The absence of an internal conductor in the waveguide reduces energy consumption for heating and, therefore, increases the efficiency compared to the efficiency of a coaxial feeder.
When studying the features of using feeders, the issue of matching the line with the load, when maximum power is transferred to the load, is very important. This condition is equality
that is, the load resistance must be purely active and equal to the characteristic impedance of the feeder. In this case, the line has a traveling wave mode and the SWR of the line is equal to 1. There are various methods matching the line with the load. Let's look at some of them.
1. Matching a long line with a load using a quarter-wave transformer.
The principle of operation of a quarter-wave transformer is based on dependence (7.68), if we put , i.e., the product of input resistances in line sections spaced equal to each other:
select a quarter-wave transformer with the required wave impedance.
|
if necessary, require that
Based on (7.70) we have . Since the load and characteristic impedance of the line are given, the matching problem comes down to determining . As a result, when connecting a transformer with such a characteristic impedance in the cross-section, the matching condition will be satisfied
,
i.e., a traveling wave regime will take place in the line. Let us note again that if the load is active, then the transformer is connected directly to the load.
To calculate the wavelength in coaxial cable The following formula can be recommended:
Where 
;
– wavelength in air.
If the line load is complex, then the transformer cannot be connected directly to the load. Initially, you need to find the section in the line in which the resistance is active. In this case, the position is used that the input resistance of a long line under an arbitrary load in sections where there are extreme values of voltage and current is purely active in nature.
In sections where there are and ,
 |
Options for switching on a quarter-wave transformer with a complex load are shown in Fig. 7.29.
The calculation of the wave impedance of the transformer is carried out in accordance with formula (7.70). If the transformer is connected at points, i.e. we have and , then
In the section it is necessary to require that , then
If the transformer is connected at points, i.e. we have and , then
The matching condition must be satisfied in the section, then
As a result, in both cases the line was matched to the load. Matching using a quarter-wave transformer is not always convenient, since it is not always possible to select a cable with the required characteristic impedance.
More convenient from a practical point of view is the matching method developed by the Soviet scientist V.V. Tatarinov.
2. Coordination of a long line with a load using a V.V. loop. Tatarinova.
The essence of the method is as follows. There is a parallel reactive loop - a line segment (can be of variable length), short-circuited at the end with characteristic impedance (Fig. 7.30a). The input impedance of the loop is purely reactive:
It is necessary to achieve such a position that the resistance at the points is purely active (Fig. 7.30b):
Where 
|
i.e., it is necessary to require that the reactive component of this conductivity be equal to zero:
This can be achieved by selecting the required cable length, while
If the resistance at the points is not equal to the characteristic impedance of the line, then you can connect a quarter-wave transformer to the load, shown in Fig. 7.31. In this case, it is necessary to select a transformer with characteristic impedance
If it is possible to change the connection location of the loop along the line, then coordination is carried out in the following order:
– the location of the loop connection is determined;
– the length of the cable is determined.
Let the loop not be connected to the line and there is a mixed wave mode in the long line. The line always has a cross section where the active part of the input conductivity (in this case, instead of resistances, it is convenient to use conductances)
since, in accordance with formulas (7.71) and (7.72), the active component of the input conductivity of the line varies from
 |
|
|
The line is thus agreed upon. This method coordination is associated with the need to move the parallel loop along the feeder. This leads to certain design difficulties when matching coaxial lines. Therefore, devices consisting of two fixed parallel loops are used. The essence of such coordination is set out, for example, in.
Matching the transmission line with the load.
Coordination of the transmission line with the load refers to measures to ensure the transfer of the largest possible part of the power transmitted by the line from the generator to the load in a given frequency range.
Ideal matching involves transferring all power transmitted from the generator to the load. In broadband communication systems, line mismatch with load can cause distortion transmitted information and a significant increase in the noise level in the path. Typically, the reflection coefficient in such systems over the entire operating frequency band should not exceed 0.02...0.05 (VSWR from 1.04...1.1).
General principles of matching the load with the transmission line.
Matching can be carried out both with wave type conversion and without wave type conversion. Wave type conversion matching is also called excitation. Upon approval, the following conditions must be met.
1. lies in the possibility of the existence of the required type of wave in the load. To do this, you need to choose the right shape and calculate the size of the load.
2. Consists in the possible complete coincidence of the field structure in the load and the transmission line. To implement this, wave type converters are used.
3. From the point of view of circuit theory, it consists in the equality of the output resistance of the transmission line to the complex conjugate input resistance of the load. Since in the case of a traveling wave mode in the transmission line and its output resistance is purely active, it is necessary to introduce reactive elements into the transmission line to compensate for the reactive component of the load resistance.
From a theoretical point of view electromagnetic field when reflected from the load, the resulting reflected wave is compensated by the wave reflected from the reactive element introduced into the transmission line, if these waves are equal in amplitude and opposite in phase, that is, the phenomenon of wave interference is used.
As a result of the introduction of a matching element, part of the wave is reflected from it in the direction of the load, and then again to the device, and so on. At the same time, in the area between matching device and the load is formed, due to these reflections, standing wave, storing energy that is no longer supplied to the load. The amount of this stored energy also depends on the distance between the matching element and the load. The greater this distance, the more energy is stored. Therefore, the matching element should be located as close as possible to the load.
Narrowband matching.
With one matching element, when the frequency changes, the phase relationships between the wave reflected from the load and the wave reflected from the inhomogeneity are violated and the matching is disrupted. Therefore, such matching, in which reflection from the load is completely eliminated at only one frequency, is called narrowband.
The narrowband matching technique is as follows.
Load conductivity
Where, with the help of a long line segment, it is transformed into conductivity, the active part of which is equal to the wave conductivity of the line
.
To compensate for the reactive component, a reactive loop with resistance is connected to points 1-1.
As matching elements for the active components of the resistance, either a line segment long enough is used so that in glasses 1-1 the input impedance of the line segment with the load has an active component equal in value to the characteristic impedance of the line, or a quarter-wave transformer is used, which is a line segment long with the wave impedance resistance equal to
.
Pins, diaphragms, as well as short-circuited sections of lines (stubs) are used as compensating elements for reactive components.
Examples of narrowband matching
1. Matching using a short-circuited loop
It is known that the input resistance in the section of the line where the node is located 
, and in the section where the antinode is located 
A load connected to the end of a transmission line is called matched if its normalized resistance or conductivity is equal to unity: 
A traveling wave mode is established in the line. In practice, situations arise when the condition is not met and at the same time the line and load are prohibited from changing. Under these conditions, it is necessary to find a way to ensure a traveling wave mode in the line, and all the power of this wave must be absorbed in the load.
General principle, which is the basis for solving this problem, is that the load is connected to the line not directly, but through a matching transformer (Fig. 6.5, A).
A transformer is required to transform the conductivity connected to its output terminals into conductivity equal to one on its input terminals:
This is a matching condition under which a traveling wave mode is established in the line. Since there is no reflected wave, and there should be no losses in the transformer, all the power of the incident wave is absorbed in the load.
Transformers used in microwaves are made on sections of lines. Let's consider the design of a transformer that implements the so-called single-line matching (Fig. 6.5, b). A transformer is a line segment of length , at the input of which a parallel reactive discontinuity having a normalized conductivity is connected. The total conductance at the input of the transformer is the sum of two conductances connected in parallel: conductance and conductance 
This second conductance is the load conductance transformed to the input terminals by a line segment of length . So:
Substitution of (6.10) into (6.11) gives the agreement condition in the form of two equalities:
(6.13)
Condition (6.12) can be satisfied by selecting such a relative length of the transforming segment so that the active part of the conductivity is equal to unity: Condition (6.13) can be satisfied by selecting the reactive conductivity (loop conductivity): ; 
Let's give an example of a calculation.
Task. Load conductivity Wavelength in the line Calculate the length of the transforming section and the reactive conductivity at which the traveling wave mode is ensured in the main one.
Solution. Let us plot point 1 on the conductivity diagram, corresponding to (Fig. 6.6). The reading on the scale “to the generator” corresponding to this point is 0.222, and KBV = 0.23. Moving along the circle KBV = 0.23 to the generator, we certainly reach point 2, through which the circle passes. The displacement scale reading corresponding to point 2 is 0.32, and the value of the reactive conductivity is here. Thus, at a distance 
from load transformed conductivity 
To compensate for reactive conductivity, the conductivity of the loop must be chosen equal to 
So, the length of the transforming segment If the transmission line is a waveguide, then you can take a capacitive diaphragm as a loop, choosing its width according to formula (6.9).
From Fig. 6.6 it is clear that there is another version of the transformer: at point 2" on pie chart conductivity, just like at point 2, has an active part equal to unity. In this case, the reactive loop must have inductive conductivity, and the length of the segment must be greater than in the calculated case.
Work order
1. Finding the conditional ends of the line and wavelength in the waveguide. Set the oscillator frequency as specified by the teacher. Connect a plug to the end of the IL and determine the position of two adjacent conditional ends of the line. Determine the wavelength in the waveguide and compare it with the calculated value found using formula (3.19).
2. Determination of the input conductivity of the moving load. Connect a moving load to the end of the IL and install the movable wedge in some fixed position (the initial reading on the load scale should not be very large - no more than 10...15 mm). Measure the load's normalized conductance relative to its input flange. The measurement procedure is according to 4.8, but taking into account the fact that conductivity is measured, not resistance. Provide a sketch (like Fig. 1.6). Calculate the relative distances and . Using a pie chart, determine and. The report must include a drawing (like Fig. 4.7) indicating all the numerical data obtained when plotting on a pie chart. The point should be on the lower semi-axis of the conductivity diagram.
3. Calculation of single-loop matching. A further task is the single-line matching of the moving load with the conductivity 
, defined in 6.2. Using Method 6.6, calculate the minimum length of the transforming segment and the conductivity of the reactive loop. Determine which diaphragm (inductive or capacitive) needs to be included as a loop, and using (6.8) or (6.9) find the size of the diaphragm window WITH. In the report, provide a complete calculation of the matching transformer according to the problem diagram from 6.6 (an illustration like Fig. 6.6 is required).
4. Checking the quality of approval. From the available set, select an aperture with the window size you need WITH. It can only be placed in the plane of the IL output flange. Assemble the diagram of Fig. 6.7. If you leave the load scale reading equal to , then the distance between the diaphragm and the load will be zero, whereas it should be equal. Therefore, move the moved wedge away from the diaphragm. Then the reading on the load scale should be (+). Measure KBV in IL. It should be significantly closer to unity than in step 2. Ideal matching may not be achieved, since the aperture was calculated using an approximate formula. Therefore, try to increase the BEF in the IL (adjusted BEF) by small load shifts.
5. Frequency dependence of matching. Without changing anything in the “load – transformer” node, measure the dependence of the KBV in the IL on frequency. Frequency step 100...200 MHz. The number of frequency points is 3–4 above and below the operating frequency. When changing the frequency, do not forget to rebuild the IL resonator. Make a graph.
6.7. Control questions
1. How are voltage and current written as waves through reflection coefficient?
2. What is the relationship between reflection coefficient and load?
3. What is line input impedance, and what is its formula?
4. What is input line conductance, and what is its formula?
5. How does resistance (or conductivity) change along a line?
6. What is the resistance (or conductance) at the minimum and maximum voltage points and how is it related to BVV and VSWR?
7. What are the differences and similarities between pie charts of resistance and conductivity?
8. How are the concepts of normalized resistance (or conductivity) of waveguides introduced?
9. What is the longitudinal coordinate on a circular diagram of resistance (or conductivity) for waveguides?
10. How does the secondary (scattered) field reflected from the diaphragm arise?
11. How can the reactive part of the secondary field created by the diaphragm be represented? And what is its feature?
12. How is the complete field of the main type, propagating “behind” the diaphragm, represented?
13. How can we explain that an “inductive” diaphragm is precisely inductive, and a “capacitive” diaphragm is capacitive?
14. Show how to use a pie chart to find the input resistance in section 2 if it is known in section 1.
15. How, knowing the normalized resistance in a certain section of the line, find the normalized conductivity in the same section?
16. What is the principle of single-loop matching?
7. STUDY OF WAVEGUIDE QUADIPOLES
WITH TRANSVERSE INHOMOGENEITIES
Goal of the work: study of properties and measurement of elements of scattering matrices of wave multipoles.
