R. G
The classical theory of synthesis of passive linear electrical circuits with lumped parameters involves two stages:
Finding or selecting a suitable rational function that could be a characteristic of a physically feasible circuit and at the same time be close enough to a given characteristic;
Finding the structure and elements of the circuit that implements the selected function.
The first stage is called the approximation of a given characteristic, the second - the implementation of the chain.
Approximation based on the use of various orthogonal functions does not cause fundamental difficulties. Much more difficult is the problem of finding the optimal structure of a chain according to a given (physically feasible) characteristic. This problem does not have a unique solution. The same circuit characteristic can be implemented in many ways, differing in the scheme, the number of elements included in it and the complexity of selecting the parameters of these elements, but the sensitivity of the circuit characteristics to parameter instability, etc.
A distinction is made between circuit synthesis in the frequency domain and in the time domain. In the first case, set Transmission function To(iω), and in the second - the impulse response g(t). Since these two functions are linked by a pair of Fourier transforms, circuit synthesis in the time domain can be reduced to synthesis in the frequency domain and vice versa. Nevertheless, synthesis according to a given impulse response has its own characteristics, which play an important role in impulse technology in the formation of impulses with certain requirements for their parameters (front steepness, surge, peak shape, etc.).
This chapter deals with the synthesis of quadripoles in the frequency domain. It should be pointed out that there is currently an extensive literature on the synthesis of linear electrical circuits, and the study general theory synthesis is not included in the task of the course "Radio circuits and signals". Here, only some particular issues of the synthesis of quadripoles are considered, reflecting the features of modern radio-electronic circuits. These features primarily include:
The use of active quadripoles;
Tendency to exclude inductors from selective circuits (in microelectronic design);
The emergence and rapid development of discrete (digital) circuit technology.
It is known that the transfer function of the quadripole To(iω) is uniquely determined by its zeros and poles on the p-plane. Therefore, the expression "synthesis by a given transfer function" is equivalent to the expression "synthesis by given zeros and poles of the transfer function". The existing theory of quadripole synthesis considers circuits whose transfer function has a finite number of zeros and poles, in other words, circuits consisting of a finite number of links with lumped parameters. This leads to the conclusion that the classical methods of circuit synthesis are inapplicable to filters that are compatible with given signal. Indeed, the factor e iωt 0 included in the transfer function of such a filter [see Fig. (12.16)] is not realized by a finite number of links with lumped parameters. The material presented in this chapter is focused on quadripoles with a small number of links. Such quadripoles are typical for low-pass filters, high-pass filters, barrier filters, etc., which are widely used in electronic devices.
Circuit theory is usually divided into two broad areas, closely related to each other - analysis and synthesis. The task of the analysis is to find the external and internal characteristics of an electrical circuit, the structure of which is predetermined, for example, in the form of a circuit diagram. The task of chain synthesis is diametrically opposed - external characteristic, such as frequency voltage gain, input or output impedance, etc., is assumed to be known. It is required to find a circuit structure that implements this characteristic.
Unlike analysis, chain synthesis is generally an ambiguous procedure. Therefore, among the set of structures with the same properties, it is necessary to find the one that is optimal in a certain sense. So, it is always desirable that the synthesized circuit contains the minimum possible number of elements. In many cases, it is necessary that the circuit be insensitive to the choice of the values of the elements included in it.
Circuit synthesis is a developed area of modern theoretical radio engineering. A number of synthesis methods, sometimes very complex, have been developed, which the reader can get acquainted with on his own. Circuit synthesis methods have acquired exceptionally great importance in connection with the introduction of computer-aided design systems for radio engineering devices.
In this chapter, we will study the simplest problem of synthesizing frequency filters, which are linear stationary quadripoles formed by elements L, C and R. The initial data for synthesis in all cases will be given by amplitude-frequency characteristics.
13.1. Frequency characteristics of quadripoles
Quadripoles are called electrical circuits that look like a "black box" with two pairs of available clamps. One pair is the input, the other is the output of the signal. In operating mode, a signal source is connected to the input, and the output terminals are loaded with load resistances
It is assumed that the reader is familiar with the methods of analysis of quadripoles, which are presented in the circuit theory course. The material of this section sheds light on certain aspects essential for the synthesis of quadripoles.
Matrix description.
The most important property of a linear stationary quadripole is that four complex amplitudes at any frequency of external action are related by two linear algebraic equations. Two arbitrarily chosen complex amplitudes can be taken as independent quantities, while the other two must be determined in terms of them. This serves as the basis for the matrix description of linear quadripoles. So, a transfer matrix (-matrix) is often used, assuming that the voltage and current at the output are independent variables. Wherein
Coefficients A, B, C and D have different physical dimensions and can be determined from open circuit and short circuit tests. Transfer matrices are especially convenient for describing the cascade connection of quadripoles, since the resulting matrix is the product of matrices of individual links.
If the four-port matrix and load resistance are given, then the so-called circuit functions can be calculated, which include, for example:
a) input impedance
b) transfer resistance
c) frequency voltage transfer coefficient
The functions of the circuit depend on general case from frequency. Any function of the circuit is expressed through the elements of the quadripole matrix and through the load resistance. So, dividing the left and right sides of equation (13.1) into each other, we find that the input impedance
Similarly, the frequency voltage transfer coefficient
Let us pay attention to the fact that the function depends on the direction of energy transfer in the system. If the source and load are reversed, then the frequency gain is introduced in the reverse direction (load on the left):
Transfer function of a quadripole.
In the future, not only the variable but also the complex frequency will be used as an argument of the frequency transfer coefficient, i.e., along with the function, more general characteristics- Transmission function . The transfer function of a quadripole has all the properties of the transfer functions of linear stationary systems considered in Chap. eight.
Thus, a linear quadripole with constant parameters corresponds to the function
where is a constant value. If the circuit is stable, then the poles should be located in the left half-plane, forming complex conjugate pairs.
Usually, an additional condition is introduced - the number of poles of the function must exceed the number of zeros, i.e., at an infinitely distant point, there must be not a pole, but a zero of the transfer function. Then the impulse response of the circuit
turns out to be limited, because with an infinitely large radius of the integration contour C, the exponential factor of the integrand can "extinguish" the integral along the arc.
The location of the zeros of the transfer function.
In contrast to the poles, the zeros of the function of a stable linear quadripole can be located both in the left and in the right half-plane of the variable . Indeed, if then this only means that at some the image of the output voltage vanishes. This does not contradict the properties of stable systems.
Quadripoles that do not have zeros of the transfer function in the right half-plane are called minimum-phase circuits. If there are zeros in the right half-plane, then such quadripoles are called non-minimum-phase circuits.
This terminology is associated with the following circumstances. Consider the plane of the complex frequency, on which some points are indicated in the left and right half-planes. Let these points be the zeros of the transfer function of the four-terminal network. If the circuit is under harmonic external influence, so that these points correspond to two vectors on the complex plane: which correspond to the corresponding factors in the numerator of formula (13.5). Both vectors rotate and change their length as the frequency changes. The difference between them is that the vector with frequency change from to increases the phase angle of the frequency gain by radians, while the vector under the same conditions decreases the phase by the same amount. The transfer coefficient of the quadripole is a fractional rational function, the change in the argument of which
Therefore, with the same number of zeros and poles, the non-minimum-phase circuit provides a greater absolute value of the change in the phase of the transmission coefficient compared to the minimum-phase circuit.
The location of the zeros of the function is related to the topological structure of the circuit. In circuit theory, it is shown that any four-terminal network with the following property will be minimum-phase: signal transmission from input to output can be completely stopped by breaking a single branch. In particular, the minimum-phase circuits will be any quadripoles of the ladder structure.
Non-minimum-phase quadripoles have, as a rule, the structure of bridge (crossed) circuits, in which the output signal passes through two or more channels. The simplest non-minimum-phase circuit is a symmetrical bridge quadripole formed by elements. Here, as is easy to see, the voltage transfer function
This function has a single zero which is in the right half-plane.
However, the bridge structure does not automatically guarantee that the circuit belongs to the non-minimum-phase class. In each individual case, the presence or absence of zeros of the transfer function in the right half-plane should be checked.
Relationship between the frequency response and phase response of a minimum-phase four-terminal network.
The transfer function of any stable quadrupole in the right half-plane of the variable is an analytic function. If, moreover, this four-terminal network belongs to the number of circuits of the minimum-phase type, then its transfer function in the right half-plane does not have zeros either. This means that the function is analytic
In accordance with the material of Ch. 5 the boundary values of the real and imaginary parts of the function on the imaginary axis, i.e., at are interconnected by a pair of Hilbert transformations:
Thus, realizing the given AFC of a minimum-phase type four-terminal network, it is impossible to obtain any PFC in this case.
Based on the properties of the Hilbert transforms, it can be argued, for example, that if the frequency response of a minimum-phase two-terminal network reaches a maximum at some frequency, then the PFC in the vicinity of this frequency passes through zero.
If the quadripole belongs to the class of non-minimal phase circuits, then the frequency response and phase response are independent of each other. Among non-minimum-phase circuits, a particularly important role is played by the so-called all-transmitting quadripoles, in which the modulus of the transmission coefficient is constant and does not depend on frequency. An example is a symmetrical bridge-four-terminal network, for which, in accordance with equality (13.6)
Similar quadripoles are used for phase correction of signals. They make it possible to partially compensate for distortions in the shape of signals that have passed through radio engineering devices.
n1.docx
Ministry of Education and Science of the Russian FederationState educational institution
higher professional education
"Omsk State Technical University"
ANALYSIS AND SYNTHESIS OF THE SCHEME
ELECTRIC CIRCUIT
Guidelines
to course design and CDS
Publishing house OmSTU
2010
Compiler I. V. Nikonov
The guidelines present the synthesis and analysis of an electrical circuit with important analog functional units of radio engineering: an electric filter and an amplifier. The analysis of the spectrum of the input complex periodic signal is carried out, as well as the analysis of the signal at the output of the electric circuit (for the linear mode of operation).
Designed for students of specialties 210401, 210402, 090104 and direction 21030062 of full-time and part-time education, studying the disciplines "Fundamentals of Circuit Theory", "Electrical Engineering and Electronics".
Published by decision of the editorial and publishing council
Omsk State Technical University
© GOU VPO "Omsk State
Technical University", 2010
1. Analysis of the terms of reference. Main design steps 5
2. Basic principles and methods for designing electrical
filters 6
2.1. Basic filter design principles 6
2.2. Technique for synthesizing filters by characteristic parameters 11
2.3. Technique for synthesizing filters according to operating parameters 18
2.4. An example of the synthesis of the equivalent circuit of an electric filter 25
3. Basic principles and stages of calculation electrical circuit amplifier
voltage 26
3.1.Basic principles for calculating the electrical circuits of amplifiers 26
3.2. An example of calculating the circuit of an electrical principle amplifier
bipolar transistor 28
4. Basic principles and stages of complex spectrum analysis
periodic signal 30
4.1. Principles spectral analysis 30
4.2. Calculation formulas for spectral analysis 31
4.3. Input Spectrum Analysis Example 32
5. Analysis of the signal at the output of the electrical circuit. Recommendations
for the development of an electrical circuit diagram 33
5.1. Analysis of the signal flow through an electrical circuit 33
6. Basic requirements for content, performance, protection
term paper 35
6.1. The procedure and terms for issuing assignments for course design 35
6.3. Registration of the graphic part of the course work (project) 36
6.4. Defense of course projects (works) 38
References 39
Apps 40
Appendix A. List of abbreviations and symbols 40
Appendix B. Variants of initial data for filter synthesis 41
Appendix B. Variants of initial data for calculating the amplifier 42
Appendix D. Variants of initial data for spectrum analysis
signal 43
Appendix E. Transistor parameters for the switching circuit
OE(OE) 45
Appendix E. Task Form 46
INTRODUCTION
The main tasks of electrical and radio engineering disciplines are the analysis and synthesis of electrical circuits and signals. In the first case, currents, voltages, transmission coefficients, spectra are analyzed with known models, circuits, devices, signals. During synthesis, the inverse problem is solved - the development of analytical and graphical models (schemes) of electrical circuits and signals. If the calculations, development are completed with the production of design and technological documentation, the production of mock-ups or prototypes, then the term design.
The first disciplines of radio engineering specialties of higher educational institutions, in which various tasks analysis and synthesis, are the disciplines "Fundamentals of the theory of electrical circuits" and "Electrical engineering and electronics". The main sections of these disciplines:
- analysis in the steady state of linear resistive electrical circuits, linear reactive electrical circuits, including resonant and non-galvanic connections;
– analysis of the complex frequency characteristics of electrical circuits;
– analysis of linear electrical circuits under complex periodic effects;
– analysis of linear electric circuits under impulse influences;
– theory of linear quadripoles;
– analysis of non-linear electrical circuits;
– linear electric filters, synthesis of electric filters.
The listed sections are studied during the classroom, however, course design is also an important part of the educational process. The topic of the course work (project) may correspond to one of the sections studied, may be complex, that is, include several sections of the discipline, may be proposed by the student.
In these guidelines, recommendations are considered for the implementation of a comprehensive course work (project), in which it is necessary to solve interrelated problems of synthesis and analysis for an analog electrical circuit.
1.
ANALYSIS OF TECHNICAL REQUIREMENTS.
MAIN STAGES OF DESIGN
As a comprehensive course work (project) in these guidelines, the development of electrical equivalent and circuit diagrams an electric circuit containing an electric filter and an amplifier, as well as an analysis of the spectrum of the input signal of the pulse generator and an analysis of the “passage” of the input signal to the output of the device. These tasks are important, practically useful, since functional units widely used in radio engineering are being developed and analyzed.
The electrical structural diagram of the entire device for which it is necessary to carry out calculations is shown in Figure 1. Task options for individual sections of calculations are given in Appendixes B, C, D. The numbers of task options correspond to the numbers of students in the group list, or the option number is formed in a more complex way. If necessary, students can independently set additional design requirements, for example, weight and size requirements, requirements for phase-frequency characteristics, and others.
Generator
impulses
Analog electrical filter
Analog voltage amplifier
Rice. one
Figure 1 shows the complex effective values of the input and output electrical voltages of harmonic form.
During course design it is necessary to solve the following tasks:
A) to synthesize (develop) by any method an electrical equivalent circuit, and then - an electrical circuit diagram on any radio elements. Calculate the attenuation and voltage transfer coefficient, illustrate the calculations with graphs;
B) develop an electrical circuit diagram of a voltage amplifier on any radioelements. Carry out amplifier calculations according to direct current, analyze the parameters of the amplifier in the mode of small variable signals;
D) analyze the passage of electrical voltage from the pulse generator through an electric filter and amplifier, illustrate the analysis with graphs of the amplitude and phase spectrum of the output signal.
In this sequence, it is recommended to carry out the necessary calculations, and then arrange them in the form of sections of an explanatory note. Calculations must be performed with an accuracy of at least 5%. This should be taken into account for various roundings, an approximate analysis of the signal spectrum, and when choosing standard radio elements that are close in nominal value to the calculated values.
2.1. Basic filter design principles
2.1.1. Basic design requirements
Electrical filters are linear or quasi-linear electrical circuits with frequency-dependent complex total power transfer coefficients. At the same time, at least one of the two transmission coefficients is also frequency-dependent: voltage or current. Instead of dimensionless transfer coefficients, attenuation (), measured in decibels, is widely used in the analysis and synthesis of filters:
, (1)
where , , are the moduli of the transfer coefficients (in formula (1) the decimal logarithm is used).
The frequency range in which the attenuation () approaches zero, and the total power transfer ratio () approaches unity, is called the bandwidth (BW). Conversely, in frequency range, where the power transfer coefficient is close to zero, and the attenuation is several tens of decibels, the stopband (TB) is found. The stop band is also called the attenuation band or the attenuation band in the literature on electrical filters. Between the PP and PZ is a transitional frequency band. According to the location of the passband in the frequency range, electrical filters are classified into the following types:
LPF - low-pass filter, the passband is at lower frequencies;
HPF - high-pass filter, the passband is at high frequencies;
PF - bandpass filter, the passband is in a relatively narrow frequency range;
RF is a notch filter, the stopband is in a relatively narrow frequency range.
A real electric filter can be made on various radio components: inductors and capacitors, selective amplifying devices, selective piezoelectric and electromechanical devices, waveguides and many others. There are reference books on the calculation of filters on well-defined radio components. However, the following principle is more universal: first, an equivalent circuit is developed on ideal LC elements, and then the ideal elements are recalculated into any real radio components. With such a recalculation, an electrical circuit diagram is developed, a list of elements, standard ones are selected or the necessary radio components are designed independently. Most simple option Such a calculation is the development of a circuit diagram of a reactive filter with capacitors and inductors, since the circuit diagram in this case is similar to the equivalent one.
But even with such a general universal calculation, there are several different methods for synthesizing the equivalent circuit of an LC filter:
- synthesis in a coordinated mode from the same G-, T-, U-shaped links. This method is also called characteristic-parameter synthesis or k-type filter synthesis. Advantages: simple calculation formulas; the calculated attenuation (attenuation ripple) in the passband () is assumed to be zero. Flaw: This synthesis method uses various approximations, but in fact, it is impossible to obtain an agreement over the entire bandwidth. Therefore, filters calculated by this method can have passband attenuation greater than three decibels;
– polynomial synthesis. In this case, the required power transfer coefficient is approximated by a polynomial, that is, the entire circuit is synthesized, and not individual links. This method is also called synthesis by operating parameters or synthesis by reference books of normalized low-pass filters. When using directories, the filter order is calculated, an equivalent low-pass filter circuit is selected that meets the requirements of the task. Advantages: possible inconsistencies and deviations of the parameters of radio elements are taken into account in the calculations, low-pass filters are easily converted into filters of other types. Flaw: it is necessary to use reference books or special programs;
– synthesis by pulse or transient response. It is based on the relationship between the time and frequency characteristics of electrical circuits through various integral transformations (Fourier, Laplace, Carson, etc.). For example, the impulse response () is expressed in terms of the transfer characteristic () using the direct Fourier transform:
This method has found application in the synthesis of various transversal filters (filters with delays), for example, digital, acoustoelectronic, for which it is easier to develop electrical circuits based on impulse responses than on frequency responses. AT term paper when developing filter circuits, it is recommended to use the synthesis method according to characteristic or operating parameters.
So, in the work related to the synthesis of an electric filter, it is necessary to use one of the methods to develop an electrical equivalent circuit on ideal reactive elements, and then a circuit electrical circuit on any real radio elements.
In the task for the course design in the part relating to the synthesis of an electric filter (Appendix B), the following data can be given:
– type of synthesized filter (LPF, HPF, PF, RF);
- - active resistances of external circuits, with which the filter in the passband must be fully or partially matched;
– – cutoff frequency of the filter passband;
– – cutoff frequency of the filter stopband;
– – average filter frequency (for PF and RF);
– – filter attenuation in the passband (no more);
– – filter attenuation in the stopband (not less than);
– – PF or RF bandwidth;
– – PF or RF stopband;
– – coefficient of squareness of LPF, HPF;
– – coefficient of squareness PF, RF.
If necessary, students can independently select additional data or design requirements.
2.1.2. Normalization and frequency transformations
When synthesizing equivalent and circuit diagrams of filters, it is advisable to apply normalization and frequency transformations. This makes it possible to reduce the number of different types of calculations and carry out synthesis based on a low-pass filter. The normalization is as follows. Instead of designing for given operating frequencies and load resistances, filters are designed for normalized load resistance and normalized frequencies. Normalization of frequencies is carried out, as a rule, relative to the frequency. . With this normalization, the frequency is , and the frequency is . When normalizing, an equivalent circuit with normalized elements is first developed, and then these elements are recalculated to the specified requirements using denormalizing factors:
The possibility of applying normalization in the synthesis of electrical circuits follows from the fact that the form of the required transfer characteristics of the electrical circuit during this operation does not change, they are only transferred to other (normalized) frequencies.
For example, for the voltage divider circuit shown in Figure 2, the voltage transfer coefficient is similar both for given radioelements and operating frequency, and for normalized values - when normalizing multipliers are used.
Rice. 2 
Without normalization:
, (5)
with normalization:
. (6)
In expression (6), in the general case, the normalizing factors can be arbitrary real numbers.
Additional application of frequency transformations allows to significantly simplify the synthesis of HPF, PF, RF. So, the recommended sequence of HPF synthesis, when applying frequency transformations, is as follows:
– graphical requirements for HPF are normalized (the axis of normalized frequencies is introduced);
– the attenuation requirements are frequency converted by frequency conversion:
– LPF is designed with normalized elements;
– LPF is converted to HPF with normalized elements;
– elements are denormalized in accordance with formulas (3), (4).
– graphical requirements for the PF are replaced by requirements for the LPF from the condition of equality of their bandwidths and delays;
– a low-pass filter circuit is synthesized;
- an inverse frequency conversion is used to obtain a band-pass filter circuit by including additional reactive elements in the low-pass filter branches to form resonant circuits.
– graphic requirements for RF are replaced by requirements for HPF from the condition of equality of their bandwidths and delays;
– a high-pass filter circuit is synthesized, directly or using a prototype – a low-pass filter;
– the HPF circuit is converted into a notch filter circuit by including additional reactive elements in the HPF branches.
2.2. Filter synthesis technique
2.2.1. The main provisions of the synthesis by characteristic parameters
The substantiation of the main calculated relations of this synthesis method is as follows.
A linear four-terminal network is considered; a system of -parameters is used to describe it:
where are the voltage and current at the input of the quadripole, are the voltage and current at the output of the quadripole.
The transmission coefficients for an arbitrary (consistent or inconsistent) mode are determined:
where is the load resistance (generally complex).
For arbitrary mode, the transmission constant (), attenuation (), phase () is introduced:
. (11)
The attenuation in neperes is given by
, (12)
and in decibels - the expression
In inconsistent mode, input, output, and transfer characteristics quadripole are called operating parameters, and in a coordinated mode - characteristic. The values of the matching input and output impedances at a given operating frequency are determined from the equations of the two-terminal network (8):
In the coordinated mode, taking into account expressions (14), (15), the characteristic transfer constant is determined:
Taking into account the relations for hyperbolic functions
, (17)
(18)
the relationship between the characteristic parameters of the matched mode and the elements of the electrical circuit (-parameters) is determined. The expressions look like
Expressions (19), (20) characterize the matched mode of an arbitrary linear two-terminal network. Figure 3 shows a diagram of an arbitrary
L-shaped link, the parameters of which, in accordance with expressions (8), are determined:
Rice. 3 
With the coordinated inclusion of the L-shaped link, expressions (19), (20) are converted to the form:
, (21)
. (22)
If there are different types of reactive elements in the longitudinal and transverse branches of the L-shaped circuit, then the circuit is an electric filter.
An analysis of formulas (21), (22) for this case allows one to obtain a method for synthesizing filters by characteristic parameters. The main provisions of this technique:
– the filter is designed from identical, cascaded, matched in the passband with each other and with external loads of links (for example, L-type links);
– attenuation in the passband () is taken equal to zero, since the filter is considered matched over the entire passband;
- the required values of external active resistances () for the matched mode are determined through the resistance of the "branches" of the L-shaped link according to the approximate formula
– cutoff frequency of the passband () is determined from the condition
– link attenuation () at the cutoff frequency of the stopband () is determined (in decibels) by the formula
; (25)
- the number of identical G-links connected in cascade is determined by the expression:
2.2.2. Sequence of synthesis of LPF (HPF)
by characteristic parameters
The calculation formulas are derived from the main provisions of the synthesis methodology for the characteristic parameters given in paragraph 2.2.1 of these guidelines. In particular, formulas (27), (28) for determining the values of the link elements are obtained from expressions (23), (24). When synthesizing by characteristic parameters, the sequence of calculations for LPF and HPF is as follows:
A) the values of the ideal inductance and capacitance of the G-link of the filter are calculated according to the given values \u200b\u200bof the resistance of the load, the generator and the value of the cutoff frequency of the passband:
where are the values of the load and generator resistances, is the value of the cutoff frequency of the passband. The schedule of attenuation requirements and the scheme of the L-shaped link of the low-pass filter are shown in Figures 4 a, b. In figures 5 a, b the requirements for attenuation and the scheme of the L-shaped HPF link are given.
Rice. four
Rice. 5 
b) the attenuation of the link () in decibels at the cutoff frequency of the stopband () is calculated according to the given value of the squareness coefficient (). For LPF:
For the high pass filter:
. (30)
In calculations by formulas (29), (30), the natural logarithm is used;
C) the number of links () is calculated according to the given value of guaranteed attenuation at the stopband boundary, in accordance with formula (26):
The value is rounded up to the nearest higher integer value;
D) Calculate filter attenuation in decibels for several frequencies in the stopband (calculated attenuation in the passband, without taking into account thermal losses, is considered equal to zero in this method). For the low pass filter:
. (31)
For the high pass filter:
; (32)
e) heat losses are analyzed (). For an approximate calculation of heat losses according to a low-frequency prototype, the resistive resistances of real inductors () are first determined at a frequency with self-selected Q-factors (). Inductors, in the future, in the electrical circuit diagram, will be introduced instead of ideal inductances (capacitors are considered to be of higher quality and their resistive losses are not taken into account). Calculation formulas:
. (34)
The attenuation of the filter in decibels, taking into account thermal losses, is determined by:
and the module of the voltage transfer coefficient () is determined from the relationship relating it to the attenuation of the filter:
E) based on the results of calculations using formulas (35), (36), graphs of the attenuation and the module of the voltage transfer coefficient for the low-pass filter or high-pass filter are plotted;
G) according to the directories of radio elements, standard capacitors and inductors closest in value to ideal elements are selected for the subsequent development of an electrical circuit diagram and a list of elements of the entire electrical circuit. In the absence of standard inductors of the required rating, it is necessary to develop them yourself. Figure 6 shows the main dimensions of a simple cylindrical coil with a single-layer winding, necessary for its calculation.
Rice. 6 
The number of turns of such a coil with a ferromagnetic core (ferrite, carbonyl iron) is determined from the expression
where is the number of turns, is the absolute magnetic permeability, is the relative magnetic permeability of the core material,
is the length of the coil, where is the radius of the coil base.
2.2.3. Sequence of synthesis of PF (RF)
by characteristic parameters
Figures 7 a, b and 8 a, b graphs of attenuation requirements and the simplest L-shaped links, respectively, for band-pass and notch filters are shown.
Rice. 7 
Rice. eight 
Synthesis of PF and RF is recommended to be carried out using calculations of prototype filters with the same bandwidth and delay. For PF, the prototype is a low-pass filter, and for RF, a high-pass filter. The synthesis procedure is as follows:
A) at the first stage of synthesis, a frequency transformation is applied, in which the graphical requirements for the attenuation of the PF are recalculated into the requirements for the attenuation of the LPF, and the graphical requirements for the attenuation of the RF are recalculated into the requirements for the attenuation of the HPF:
B) according to the previously considered method for the synthesis of low-pass filters and high-pass filters (points a-f
2.2.2), an electrical circuit equivalent to a low-pass filter is developed for the synthesis of the PF, or the HPF - for the synthesis of the RF. For LPF or HPF, graphs of attenuation and voltage transfer coefficient are plotted;
C) the low-pass filter circuit is converted into a band-pass filter circuit by converting the longitudinal branches into serial ones oscillatory circuits and transverse branches into parallel oscillatory circuits by connecting additional reactive elements. The HPF circuit is converted into a notch filter circuit by converting the longitudinal branches into parallel oscillatory circuits and the transverse branches into serial oscillatory circuits by connecting additional reactive elements. Additional reactive elements for each branch of the low-pass filter (HPF) are determined by the value of the specified average frequency of the band-pass or notch filter () and the calculated values of the reactive elements of the low-pass filter (HPF) branches, using the well-known expression for resonant circuits:
D) for PF or RF circuits, capacitors and inductors are developed or selected from reference books of radio elements according to the same methodology that was considered earlier in clause 2.2.2 (clause g) of these guidelines;
E) graphs of attenuation and voltage transfer coefficient of LPF (HPF) are recalculated into graphs of PF (RF) in accordance with the ratios between the frequencies of these filters. For example, to convert LPF to PF charts:
, (41)
where are the frequencies, respectively, above and below the average frequency of the band pass filter. The same formulas are used to recalculate the high-pass filter graphs into notch filter graphs.
2.3. Technique for Synthesizing Filters Based on Operating Parameters
2.3.1. Basic principles of synthesis by operating parameters
(polynomial synthesis)
In this synthesis method, as well as in the synthesis by characteristic parameters, the requirements are set for the type of the designed filter, active load resistance, attenuation or power transfer coefficient in the pass and delay bands. However, it is taken into account that the input and output resistances of the filter vary in the passband. In this regard, the filter is synthesized in an inconsistent mode, that is, according to the operating parameters, which is reflected in the initial data by the requirement . The method is based on the obligatory calculation for any types of filters of a low-pass filter - a prototype (low-pass filter). The calculations use normalization () and frequency transformations.
The filter equivalent circuit is not developed from separate identical links, but completely at once, usually in the form of a chain structure circuit. Figure 9 shows a view of a U-shaped chain circuit of a low-pass filter, and Figure 10 shows a view of a T-shaped circuit of the same filter with unnormalized elements.
Rice. 9 
Rice. ten 
The main stages of calculations on which this synthesis is based are as follows:
A) approximation - replacing the graphical requirements for the power transfer coefficient with an analytical expression, for example, the ratio of polynomials in degrees, which corresponds to the formulas for the frequency characteristics of real reactive filters;
B) transition to the operator form of recording frequency characteristics (replacement of a variable by a variable in an analytical expression approximating the power transfer coefficient);
C) transition to the expression for the input impedance of the filter, using the relationship between the power transfer coefficient, the reflection coefficient and the input impedance of the filter:
In expression (44), only one reflection coefficient is used, which corresponds to a stable electrical circuit (the poles of this coefficient do not have a positive real part);
D) expansion of the analytical expression for the input resistance obtained from (44) into a sum of fractions or into a continued fraction to obtain an equivalent circuit and element values.
Polynomial synthesis in practical developments is usually carried out using filter reference books, in which calculations are made for this method synthesis. The reference books contain approximating functions, equivalent circuits and normalized elements of low-pass filters. In most cases, Butterworth and Chebyshev polynomials are used as approximating functions.
The attenuation of a low-pass filter with an approximating Butterworth function is described by the expression:
where is the filter order (a positive integer numerically equal to the number of reactive elements in the equivalent filter circuit).
The order of the filter is determined by the expression
Tables 1 and 2 show the values of the normalized reactive elements in the Butterworth approximation, calculated for different orders of the low-pass filter (for circuits similar to those in Figures 9, 10).
Table 1
Values of the normalized elements of the Butterworth LPF of the U-shaped circuit
| | | | | | | | |
| 1 | 2 | ||||||
| 2 | 1,414 | 1,414 | |||||
| 3 | 1 | 2 | 1 | ||||
| 4 | 0,765 | 1,848 | 1,848 | 0,765 | |||
| 5 | 0,618 | 1,618 | 2 | 1,618 | 0,618 | ||
| 6 | 0,518 | 1,414 | 1,932 | 1,932 |
Lecture number 15.
Design (synthesis) of linear digital filters.
Under design (synthesis) digital filter understand the choice of such coefficients of the system (transfer) function, in which the characteristics of the resulting filter satisfy the specified requirements. Strictly speaking, the design task also includes the choice of an appropriate filter structure (see Lecture 14), taking into account the finite accuracy of calculations. This is especially true when implementing filters in hardware form (in the form of specialized LSI or digital signal processors). Therefore, in general, the design of a digital filter consists of the following steps:
- Solving an approximation problem in order to determine the filter coefficients and a system function that meets specific requirements.
- The choice of filter construction scheme, that is, the transformation of a system function into a specific filter block diagram.
- Evaluation of the effects of quantization, that is, the effects associated with the finite precision of the representation of numbers in digital systems, which have a finite capacity.
- Checking by simulation methods whether the resulting filter satisfies the specified requirements.
Methods for synthesizing digital filters can be classified according to various criteria:
- by filter type:
- methods for synthesizing filters with a finite impulse response;
- methods for synthesizing filters with infinite impulse response;
- by the presence of an analog prototype:
- synthesis methods using an analog prototype;
- direct synthesis methods (without using an analog prototype).
In practice, FIR filters are often preferred for the following reasons. First, FIR filters provide the ability to accurately calculate the output signal with limited input over convolution that does not require impulse response truncation. Second, filters with a finite impulse response can have a strictly linear phase response in the passband, which allows you to design filters with an amplitude response that does not distort the input signals. Thirdly, FIR filters are always stable and, with the introduction of an appropriate finite delay, are physically realizable. In addition, FIR filters can be implemented not only in non-recursive schemes, but also using recursive forms.
Note the disadvantages of FIR filters:
- To approximate filters whose frequency responses have sharp cutoffs, an impulse response with a large number of samples is required. Therefore, when using conventional convolution, it is necessary to perform a large amount of calculations. Only the development of fast convolution methods based on the high-performance FFT algorithm allowed FIR filters to successfully compete with IIR filters that have sharp cutoffs in the frequency response.
- The delay in FIR filters with a linear phase response is not always equal to an integer number of sampling intervals. In some applications, this multiple delay can be problematic.
One of the options for designing digital filters is associated with a given sequence of samples of the impulse response, which are used to obtain and analyze its frequency response (frequency gain).
We obtain the condition under which the non-recursive filter has a strictly linear phase response. System function such a filter looks like:
, (15.1)
where the filter coefficients are impulse response samples. The Fourier transform of is the frequency response of the filter, periodic in frequency with a period. We represent it for a real sequence in the form: We obtain the conditions under which the impulse response of the filter will ensure the strict linearity of its phase response. The latter means that the phase characteristic should look like:
(15.2)
where is the constant phase delay expressed in terms of the number of sampling intervals. We write the frequency response in the form:
(15.3)
Equating the real and imaginary parts, we get:
, (15.4)
. (15.5)
Where:
. (15.6)
There are two possible solutions equations (15.6). One (when) is of no interest, the other corresponds to the case. Cross-multiplying the terms of equation (15.6), we get:
(15.7)
Since equation (15.7) has the form of a Fourier series, the solution of the equation must satisfy the following conditions:
, (15.8)
and (15.9)
It follows from condition (15.8) that for each there is only one phase delay, under which strict linearity of the filter phase characteristic can be achieved. From (15.9) it follows that for a given one that satisfies condition (15.8), the impulse response must have a well-defined symmetry.
It is expedient to consider the use of conditions (15.8) and (15.9) separately for the cases of even and odd. If an odd number, then an integer, that is, the delay in the filter is equal to an integer number of sampling intervals. In this case, the center of symmetry falls on the reference. If it's an even number, then fractional number, and the delay in the filter is equal to a non-integer number of sampling intervals. For example, for we get, and the center of symmetry of the impulse response lies in the middle between two readings.
The values of the impulse response coefficients are used to calculate the frequency response of the FIR filters. It can be shown that for a symmetrical impulse response with an odd number of samples, the expression for a real function that takes positive and negative values is:
, (15.10)
where
Most often, when designing an FIR filter, one starts from the required (or desired) frequency response and then calculates the filter coefficients. There are several methods for calculating such filters:windowed design method, frequency sampling method, method for calculating the optimal (according to Chebyshev) filter.Consider the idea of windowed design using the FIR low-pass filter as an example.
First of all, the desired frequency response of the designed filter is set. For example, let's take an ideal continuous frequency response of a low-pass filter with a gain equal to unity at low frequencies and equal to zero at frequencies exceeding some cutoff frequency . discrete representation The ideal low-pass filter is a periodic characteristic, which can be set by readings on a periodicity interval equal to the sampling frequency. Determining the low-pass filter coefficients using inverse DFT methods (either analytically or using an inverse DFT program) yields a sequence of impulse response samples that is infinite in both directions, which has the form of a classical function.
To get an implementable non-recursive filter given order this sequence is truncated the central fragment of the required length is selected from it. Simple truncation of impulse response samples is consistent with usingrectangular window, given by a special function Due to the truncation of samples, the initially given frequency response is distorted, since it is a convolution in the frequency domain of the discrete frequency response and the DFT of the window function:
, (15.11)
where DFT As a result, ripple occurs in the passband of the frequency response due to the side lobes.
To mitigate the above effects and, above all, to reduce the level of lobes in the stopband, the truncated impulse response is multiplied by a weight function (window) that gradually decreases towards the edges. Thus, the method of designing FIR filters with windows is a method of reducing window gaps by using non-rectangular windows. In this case, the weight function (window) must have the following properties:
- the width of the main lobe of the frequency response of the window containing as much of the total energy as possible should be small;
- the energy in the side lobes of the window's frequency response should decrease rapidly as k is approached.
Hamming, Kaiser, Blackman, Chebyshev, etc. windows are used as weight functions.
The general theory of the synthesis of linear electrical circuits is not included in the task of the course "Radio circuits and signals".
This chapter discusses only some particular, specific questions for the synthesis of radio circuits:
synthesis of active quadripoles in the form cascading connection elementary non-interacting (decoupled) links of the first or second order;
construction of selective circuits that do not contain inductors (integrated circuits);
elements of synthesis of discrete (digital) circuits and the relationship between the frequency response and phase response of digital filters.
The synthesis of analog circuits in this chapter is carried out only in the frequency domain, i.e., according to a given transfer function; for digital circuits, synthesis is also considered for a given impulse response (briefly).
It is known that the transfer function of a linear quadripole is uniquely determined by its zeros and poles on the -plane (analogue circuits) or on the z-plane (digital circuits). Therefore, the expression "synthesis by a given transfer function" is equivalent to the expression "synthesis by given zeros and poles of the transfer function". The existing theory of quadripole synthesis considers circuits whose transfer function has a finite number of zeros and poles, in other words, circuits consisting of a finite number of links with lumped parameters. The material presented below is focused on quadripoles with a small number of links, which are typical for low-pass filters, high-pass filters, barrier filters, etc., which are widely used in electronic devices.
