Theoretical foundations of filter synthesis. "Synthesis of linear filters. Transfer function of a two-port network
Science sharpens the mind;
Learning will revive the memory.
Kozma Prutkov
chapter 15
ELEMENTS OF SYNTHESIS OF LINEAR STATIONARY CIRCUITS
15.1. Studied issues
WITH Intez analog two-terminal devices. Synthesis of stationary four-port networks for a given frequency response. Butterworth and Chebyshev filters.
Directions. When studying the issues, it is necessary to clearly understand the ambiguity of solving the problem of synthesis of two-terminal devices and specific ways of solving the problem according to Foster and Cauer, as well as to acquire the ability to determine the possibility of implementing one or another function of the input resistance of a two-terminal network. When synthesizing electric filters based on prototype filters, it is important to understand the advantages and disadvantages of approximating the attenuation characteristics according to Chebyshev and Butterworth. It is necessary to be able to quickly calculate the parameters of elements of any types of filters (LPF, HPF, PPF) using the formulas of frequency transformations.
15.2. Brief theoretical information
In circuit theory, it is customary to talk about structural and parametric synthesis. The main task of structural synthesis is the choice of the structure (topology) of the circuit that satisfies the predetermined properties. In parametric synthesis, only the parameters and the type of circuit elements are determined, the structure of which is known. In what follows, we will only talk about parametric synthesis.
The input impedance is usually used as a starting point in the synthesis of two-port networks.
If a function is given, then it can be implemented by a passive circuit under the following conditions: 1) all the coefficients of the polynomials of the numerator and denominator are real and positive; 2) all zeros and poles are either in the left half-plane or on the imaginary axis, and the poles and zeros on the imaginary axis are simple; these points are always either real or form complex conjugate pairs; 3) the higher and lower degrees of the polynomials of the numerator and denominator differ by no more than one. It should also be noted that the synthesis procedure is not unambiguous, that is, the same input function can be implemented in several ways.
Foster circuits are usually used as the initial structures of the synthesized two-terminal networks, which are a series or parallel connection with respect to the input terminals, respectively, of several complex resistances and conductances, as well as Cauer ladder circuits.
The method of synthesis of two-port networks is based on the fact that a given input function or is subjected to a number of successive simplifications. At the same time, at each stage, an expression is highlighted, which is associated with a physical element of the synthesized chain. If all components of the selected structure are identified with physical elements, then the synthesis problem is solved.
The synthesis of four-port networks is based on the theory of low-pass filter prototypes. Possible options LPF prototype are shown in Fig. 15.1.
Any of the schemes can be used in the calculation, since their characteristics are identical. Fig. 15.1 have the following meaning: - the inductance of the series coil or the capacitance of the parallel capacitor; - generator resistance, if, or generator conductivity, if; - load resistance, if or load conductivity, if.
The values of the prototype elements are normalized so that the cutoff frequency is also. The transition from normalized prototype filters to a different level of resistances and frequencies is carried out using the following transformations of the circuit elements:
;
.
The dashed values refer to the normalized prototype, and those without the dash to the transformed circuit. The initial value for synthesis is the operating power attenuation, expressed in decibels:
, dB,
- the maximum power of the generator with internal resistance and emf, - the output power in the load.
Usually, the frequency dependence is approximated by the maximally flat (Butterworth) characteristic (Fig.15.2, a)
where 
.
The value of the operating attenuation corresponding to the cutoff frequency is usually chosen equal to 3 dB. Wherein . Parameter n is equal to the number of active elements in the circuit and determines the order of the filter.
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Lecture number 15.
Design (synthesis) of linear digital filters.
The design (synthesis) of a digital filter is understood as the choice of such coefficients of the system (transfer) function, at which the characteristics of the resulting filter satisfy the specified requirements. Strictly speaking, the design problem also includes the choice of a suitable filter structure (see lecture 14), taking into account the finite accuracy of the calculations. This is especially important when implementing filters in hardware form (in the form of specialized LSIs or digital signal processors). Therefore, in general, the design of a digital filter consists of the following stages:
- Solving an approximation problem to determine filter coefficients and a system function that meets specific requirements.
- The choice of the filter construction scheme, that is, the transformation of the system function into a specific block diagram filter.
- Evaluation of the effects of quantization, that is, the effects associated with the finite precision of the representation of numbers in digital systems with a finite bit depth.
- Checking by simulation methods whether the obtained filter meets the specified requirements.
Methods for synthesizing digital filters can be classified according to various criteria:
- by the type of filter received:
- finite impulse response filter synthesis methods;
- 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 compute the output signal with limited input in convolution that does not require impulse response truncation. Second, finite impulse response filters can have a strictly linear phase response in the passband, which makes it possible to design filters with an amplitude response that does not distort the input signals. Third, 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 by non-recursive schemes, but also using recursive forms.
Let's note the disadvantages of FIR filters:
- An impulse response with a large number of samples is required to approximate filters whose frequency response is sharp. Therefore, when using normal convolution, it is necessary to perform a large amount of computation. Only the development of fast convolution methods based on a highly efficient FFT algorithm has allowed FIR filters to successfully compete with IIR filters that have sharp cuts in the frequency response.
- The delay in linear phase FIR filters is not always an integer number of sample bins. In some applications, such a multiple delay can cause certain difficulties.
One of the design options for digital filters is associated with a given sequence of impulse response samples, which are used to obtain and analyze its frequency response (frequency gain).
Let us obtain a condition under which a non-recursive filter has a strictly linear phase response. The system function of such a filter is:
, (15.1)
where the filter coefficients are the 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 have the form:
(15.2)
where is the constant phase delay, expressed in terms of the number of sampling intervals. Let's write the frequency response as follows:
(15.3)
Equating the real and imaginary parts, we get:
, (15.4)
. (15.5)
Where:
. (15.6)
There are two possible solutions to equation (15.6). One (at) is not of interest, the other is appropriate for the case. Crosswise multiplying the terms of equation (15.6), we get:
(15.7)
Since equation (15.7) has the form of a Fourier series, the solution to the equation must satisfy the following conditions:
, (15.8)
and (15.9)
From condition (15.8) it follows that for each there is only one phase delay at which strict linearity of the filter phase response can be achieved. From (15.9) it follows that for a given, satisfying condition (15.8), the impulse response must have a well-defined symmetry.
It is advisable to consider the use of conditions (15.8) and (15.9) separately for the even and odd cases. 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 the number is even, then it is a fractional number, and the delay in the filter is equal to a non-integer number of sampling intervals. For example, for we obtain, and the center of symmetry of the impulse response lies in the middle between two samples.
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 symmetric 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 proceeds from the required (or desired) frequency response, followed by the calculation of the filter coefficients. There are several methods for calculating such filters:method of designing with the help of windows, method of frequency sampling, method of calculating the optimal (according to Chebyshev) filter.Consider a windowing design idea using a low-pass FIR filter as an example.
First of all, the desired frequency response of the designed filter is set. For example, take the ideal continuous frequency response of a low-pass filter with a gain equal to one per low frequencies ah and equal to zero at frequencies exceeding some cutoff frequency ... A discrete representation of an ideal low-pass filter is a periodic characteristic that can be specified by samples at a periodicity interval equal to the sampling frequency. Determination of the low-pass filter coefficients by the inverse DFT methods (either analytically or using a program that implements the inverse DFT) gives an infinite in both directions sequence of impulse response samples, which has the form of a classical function.
To obtain a realizable non-recursive filter of a 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 the use ofrectangular windowgiven by special function Due to the truncation of the samples, the initially specified 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, sidelobe ripple occurs in the passband of the frequency response.
To weaken the listed effects and, above all, to reduce the level of the lobes in the stop band, the truncated impulse response is multiplied by the weighting function (window), smoothly falling towards the edges. Thus, the windowed FIR filter design method is a method of reducing window gaps by using non-rectangular windows. In this case, the weighting 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 frequency response of the window should decrease rapidly when approaching.
As weight functions, the windows of Hamming, Kaiser, Blackman, Chebyshev, etc. are used.
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
Methodical instructions
to course design and CPC
Publishing house OmSTU
2010
Compiled by I. V. Nikonov
The guidelines provide synthesis and analysis electrical circuit with important analog functional units of radio engineering: an electric filter and an amplifier. An 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 electrical circuit (for a linear mode of operation).
Are intended for students of specialties 210401, 210402, 090104 and directions 21030062 full-time and part-time forms of study, studying disciplines "Fundamentals of circuit theory", "Electrical engineering and electronics".
Reprinted by the decision of the Editorial and Publishing Council
Omsk State Technical University
© GOU VPO "Omsk State
Technical University ", 2010
1. Analysis of technical specifications. Main design stages 5
2. Basic principles and methods of designing electrical
filters 6
2.1. Fundamental Filter Design Principles 6
2.2. Technique for the synthesis of filters by characteristic parameters 11
2.3. Technique for the synthesis of filters by operating parameters 18
2.4. An example of 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 of calculating electrical circuits of amplifiers 26
3.2. An example of calculating an electrical circuit amplifier
bipolar transistor 28
4. Basic principles and stages of complex spectrum analysis
periodic signal 30
4.1. Spectral Analysis Principles 30
4.2. Calculation formulas for spectral analysis 31
4.3. Example for analyzing the spectrum of an input signal 32
5. Analysis of the signal at the output of the electrical circuit. Recommendations
on the development of an electrical schematic diagram 33
5.1. Analysis of Signal Flow Through an Electrical Circuit 33
6. Basic requirements for content, performance, protection
term paper 35
6.1. The procedure and timing for issuing an assignment for course design 35
6.3. Registration of the graphic part of the course work (project) 36
6.4. Protection course projects(works) 38
Bibliography 39
Appendices 40
Appendix A. List of abbreviations and symbols 40
Appendix B. Variants of initial data for filter synthesis 41
Appendix B. Variants of the initial data for calculating the amplifier 42
Appendix D. Options for input data for spectrum analysis
signal 43
Appendix D. Parameters of transistors for the switching circuit
OE (OI) 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 for known models, circuits, devices, signals. In the synthesis, the inverse problem is solved - the development of analytical and graphic models (diagrams) of electrical circuits and signals. If the calculations and development are completed with the manufacture of design and technological documentation, the manufacture of models or prototypes, then the term is used design.
The first disciplines of radio engineering specialties of higher educational institutions, in which various problems of analysis and synthesis are considered, are the disciplines "Fundamentals of the theory of electrical circuits" and "Electrical engineering and electronics". The main sections of these disciplines:
- steady-state analysis of linear resistive electrical circuits, linear reactive electrical circuits, including resonant and non-galvanic circuits;
- analysis of complex frequency characteristics of electrical circuits;
- analysis of linear electrical circuits with complex periodic influences;
- analysis of linear electrical circuits under impulse influences;
- theory of linear four-port networks;
- analysis of nonlinear electrical circuits;
- linear electric filters, synthesis of electric filters.
The listed sections are studied during classroom sessions, 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 studied sections, it may be complex, that is, it may include several sections of the discipline, it 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 the interrelated problems of synthesis and analysis for an analog electrical circuit.
1.
ANALYSIS OF THE TECHNICAL REFERENCE.
MAIN STAGES OF DESIGN
As a complex course work (project) in these guidelines, it is proposed to develop electrical equivalent and schematic diagrams of an electrical circuit containing an electrical filter and an amplifier, as well as an analysis of the spectrum of the input signal of the pulse generator and 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. Options for tasks for individual sections of calculations are given in Appendices B, C, D. The numbers of options for tasks 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 Electric Filter
Analog voltage amplifier
Rice. one
Figure 1 shows the complex effective values of the input and output electric voltages of the harmonic form.
When designing coursework, it is necessary to solve the following tasks:
A) synthesize (develop) by any method an electric equivalent circuit, and then - an electric circuit diagram on any radioelements. Calculate attenuation and voltage transmission coefficient, illustrate the calculations with graphs;
B) develop an electrical schematic diagram of a voltage amplifier on any radioelements. Carry out calculations of the amplifier for direct current, analyze the parameters of the amplifier in the mode of small variable signals;
D) analyze the passage of electric 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 in various rounding, approximate analysis of the signal spectrum, when choosing standard radioelements that are close in nominal value to the calculated values.
2.1. Basic principles of filter design
2.1.1. Basic design requirements
Electrical filters are linear or quasi-linear electrical circuits with frequency-dependent complex apparent power transmission coefficients. In this case, at least one of two transmission coefficients is also frequency-dependent: voltage or current. Instead of dimensionless transmission coefficients, attenuation (), measured in decibels, is widely used in the analysis and synthesis of filters:
, (1)
where,, are the modules of the transfer coefficients (in the formula (1), the decimal logarithm is used).
The frequency range in which the attenuation () approaches zero and the apparent power gain () approaches unity is called the bandwidth (BW). And vice versa, in the frequency range, where the power transfer coefficient is close to zero, and the attenuation is several tens of decibels, there is a stopband (FB). The stopband is also called the stopband or stopband in the electrical filter literature. There is a transition frequency band between the SP and the PS. 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 the lower frequencies;
HPF - high-pass filter, the passband is at the high frequencies;
PF - bandpass filter, the passband is in a relatively narrow frequency range;
RF - notch filter, stopband is in a relatively narrow frequency range.
A real electrical filter can be implemented on various radio components: inductors and capacitors, selective amplifying devices, selective piezoelectric and electromechanical devices, waveguides, and many others. There are handbooks for calculating filters on well-defined radio components. However, the following principle is more universal: first, an equivalent circuit is developed based on ideal LC-elements, and then the ideal elements are recalculated into any real radio components. With such a recalculation, an electrical schematic diagram, a list of elements is developed, standard radio components are selected or the necessary radio components are independently designed. The simplest version of such a calculation is the development of a schematic diagram of a reactive filter with capacitors and inductors, since the schematic diagram in this case is similar to an 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 technique is also referred to as characterization or “k” filter synthesis. Dignity: simple calculation formulas; the calculated attenuation (attenuation unevenness) in the passband () is taken to be zero. Flaw: This synthesis method uses different approximations, but in fact, matching across the entire bandwidth is not possible. Therefore, filters calculated by this method can have attenuation in the passband of more than three decibels;
- polynomial synthesis. In this case, the required power transfer factor is approximated by a polynomial, that is, the entire circuit is synthesized, and not individual links. This method is also called synthesis according to operating parameters or synthesis according to reference books of normalized low-pass filters. When using reference books, the order of the filter is calculated, an equivalent low-pass filter is selected that meets the requirements of the task. Dignity: the calculations take into account possible inconsistencies and deviations of the parameters of radioelements, low-pass filters are easily converted into filters of other types. Flaw: it is necessary to use reference books or special programs;
- synthesis by impulse or transient characteristics... 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 response () using direct conversion Fourier:
This method has found application in the synthesis of various transverse filters (filters with delays), for example, digital, acoustoelectronic, for which it is easier to develop electrical circuits in terms of impulse than in frequency characteristics. V term paper When designing filter circuits, it is recommended to apply the synthesis method according to characteristic or operating parameters.
So, in the work concerning the synthesis of an electric filter, it is necessary, by one of the methods, to develop an electric equivalent circuit on ideal reactive elements, and then an electric circuit diagram on any real radioelements.
In the assignment for course design in the part concerning the synthesis of an electric filter (Appendix B), the following data can be given:
- the type of the synthesized filter (LPF, HPF, PF, RF);
- - active resistances of external circuits, with which the filter must be matched in full or in part in the passband;
- - cutoff frequency of the filter passband;
- is the cutoff frequency of the filter stop band;
- - average filter frequency (for PF and RF);
- - attenuation of the filter in the passband (no more);
- - attenuation of the filter in the stop band (not less);
- - bandwidth of the PF or RF;
- - band of retention PF or RF;
- - 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. Rationing and frequency conversions
When synthesizing equivalent and basic filter circuits, it is advisable to apply normalization and frequency transformations. This allows you to reduce the number of different types of calculations and carry out synthesis, taking as a basis a low-pass filter. Rationing is as follows. Instead of designing for given operating frequencies and load resistances, filters are designed for normalized load resistance and normalized frequencies. Frequency normalization is carried out, as a rule, relative to frequency. ... With this normalization, the frequency, and the frequency. When normalizing, an equivalent circuit with normalized elements is first developed, and then these elements are recalculated to the specified requirements using denorming 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 at normalized values - when using normalizing factors.
Rice. 2 
Without rationing:
, (5)
with standardization:
. (6)
In expression (6), in the general case, the normalizing factors can be arbitrary real numbers.
The additional use of frequency transformations makes it possible to significantly simplify the synthesis of HPF, PF, RF. So, the recommended sequence of HPF synthesis, when using frequency transformations, is as follows:
- graphic requirements for HPF are normalized (the axis of normalized frequencies is introduced);
- frequency conversion of attenuation requirements due to frequency conversion is performed:
- a low-pass filter with normalized elements is being designed;
- LPF is converted to HPF with normalized elements;
- elements are denormalized in accordance with formulas (3), (4).
- the graphic requirements for the PF are replaced with the requirements for the LPF from the condition that their bandwidth and delay are equal;
- a low-pass filter circuit is synthesized;
- an inverse frequency conversion is applied to obtain a bandpass filter circuit by including additional reactive elements in the LPF branches to form resonant circuits.
- the graphic requirements for the RF are replaced with the requirements for the high-pass filter on the condition that their bandwidth and delay are equal;
- a high-pass filter circuit is synthesized, either 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. Basic principles of synthesis by characteristic parameters
The substantiation of the main calculated relations of this synthesis method is as follows.
A linear two-port network is considered; a system of parameters is used to describe it:
where are the voltage and current at the input of the four-port device, are the voltage and current at the output of the four-terminal device.
The transmission coefficients for an arbitrary (matched or unmatched) mode are determined:
where is the load resistance (in the general case, complex).
For arbitrary mode, the transmission constant (), attenuation (), phase () are introduced:
. (11)
Attenuation in nepers is determined by the expression
, (12)
and in decibels - by the expression
In inconsistent mode, input, output, and transfer characteristics four-port networks are called operating parameters, and in the agreed mode - characteristic. The values of the matching input and output resistances at a given operating frequency are determined from the equations of the four-port network (8):
In a consistent mode, taking into account expressions (14), (15), the characteristic constant of the transmission 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. Expressions are of the form
Expressions (19), (20) characterize the coordinated mode of an arbitrary linear four-port 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 transformed 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 electrical filter.
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 the same, cascaded, matched in the passband with each other and with external loads of links (for example, G-type links);
- attenuation in the passband () is taken to be 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 resistances of the "branches" of the L-shaped link according to the approximate formula
- the cutoff frequency of the passband () is determined from the condition
- link attenuation () at the cutoff frequency of the stop band () is determined (in decibels) by the formula
; (25)
- the number of identical G-links included in cascade is determined by the expression:
2.2.2. LPF synthesis sequence (HPF)
by characteristic parameters
The design formulas are obtained from the main provisions of the synthesis methodology for the characteristic parameters given in paragraph 2.2.1 of the data 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 nominal values of the ideal inductance and capacitance of the G-link of the filter are calculated according to the given values of the load resistances, 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 diagram of the attenuation requirements and the diagram of the L-shaped link of the low-pass filter are shown in Figures 4 a, b... Figures 5 a, b the requirements for the attenuation and the diagram of the L-shaped HPF link are given.
Rice. 4
Rice. 5 
b) the link attenuation () is calculated in decibels at the cutoff frequency of the stop band () according to the given value of the squareness coefficient (). For LPF:
For the high pass filter:
. (30)
In calculations using formulas (29), (30), the natural logarithm is used;
C) the number of links () is calculated according to a given value of guaranteed attenuation at the stop band boundary, in accordance with formula (26):
The value is rounded to the nearest higher integer value;
D) the attenuation of the filter in decibels is calculated for several values of the frequencies in the stop band (the calculated attenuation in the pass band, excluding heat losses, in this method is considered equal to zero). For a low pass filter:
. (31)
For the high pass filter:
; (32)
e) heat losses are analyzed (). For an approximate calculation of heat losses for a low-frequency prototype, the resistive resistances of real inductors () are first determined at a frequency at independently selected values of the quality factor (). Inductors, in the future, in the electrical schematic diagram, will be introduced instead of ideal inductors (capacitors are considered higher Q and their resistive losses are not taken into account). Calculation formulas:
. (34)
The attenuation of the filter in decibels, taking into account heat losses, is determined by:
and the modulus of the voltage transfer coefficient () is determined from the relation connecting it with the attenuation of the filter:
E) based on the results of calculations using formulas (35), (36), graphs of attenuation and modulus of the voltage transfer coefficient for a low-pass filter or a high-pass filter are built;
G) according to the reference books of radioelements, standard capacitors and inductors that are closest to the ideal elements are selected for the subsequent development of an electrical schematic diagram and a list of elements of the entire electrical circuit. In the absence of standard inductance coils of the required rating, you must develop them yourself. Figure 6 shows the basic dimensions of a simple cylindrical single-layer coil required 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 the synthesis of PF (RF)
by characteristic parameters
Figures 7 a, b and 8 a, b the graphs of the requirements for attenuation and the simplest L-shaped links, respectively, for the bandpass and notch filters are shown.
Rice. 7 
Rice. eight 
It is recommended to synthesize PF and RF using the 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 technique is as follows:
A) at the first stage of the synthesis, frequency conversion is applied, in which the graphic requirements for the attenuation of the PF are recalculated into the requirements for the weakening of the low-pass filter, and the graphic requirements for the weakening of the RF are recalculated into the requirements for the weakening of the high-pass filter:
B) according to the previously considered method for the synthesis of LPF and HPF (items a – f
p. 2.2.2) an electric circuit is being developed that is equivalent to a low-pass filter for the synthesis of a PF, or a high-pass filter - for a synthesis of the RF. For a low-pass filter or high-pass filter, 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 successive 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 series oscillatory circuits by connecting additional reactive elements. Additional reactive elements for each LPF branch (HPF) are determined by the value of the given average frequency of the band-pass or notch filter () and the calculated values of the reactive elements of the LPF branches (HPF) using the well-known expression for the resonant circuits:
D) for PF or RF circuits, capacitors and inductors are developed or selected according to the reference books of radioelements according to the same methodology that was considered earlier in paragraph 2.2.2 (point g) of these guidelines;
E) the graphs of the attenuation and the voltage transfer coefficient of the LPF (HPF) are recalculated into the PF (RF) graphs in accordance with the ratios between the frequencies of these filters. For example, to convert LPF to PF graphs:
, (41)
where are the frequencies, respectively, above and below the center frequency of the bandpass filter. The same formulas are used to recalculate the high-pass filter graphs into the notch filter graphs.
2.3. Technique for the synthesis of filters by operating parameters
2.3.1. Basic principles of synthesis by operating parameters
(polynomial synthesis)
In this synthesis method, as in the synthesis by characteristic parameters, requirements are set for the type of the designed filter, active load resistance, attenuation or power transfer coefficient in the passband and stopband. However, it is taken into account that the input and output impedances of the filter change in the passband. In this regard, the filter is synthesized in an inconsistent mode, that is, according to operating parameters, which is reflected in the initial data by the requirement. The method is based on compulsory calculation for any type of low-pass filter - prototype (low-pass filter). The calculations use normalization () and frequency transformations.
An equivalent filter 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 non-normalized elements.
Rice. 9 
Rice. 10 
The main stages of calculations on which this synthesis is based are as follows:
A) approximation - replacement of the graphical requirements for the power transfer coefficient with an analytical expression, for example, the ratio of polynomials in powers, which corresponds to the formulas for the frequency characteristics of real reactive filters;
B) the transition to the operator form of recording the 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 applied, 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 the sum of fractions or in a continued fraction to obtain the equivalent circuit and the values of the elements.
In practical developments, polynomial synthesis is usually carried out using filter reference books, in which calculations for a given synthesis method are performed. 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 the low-pass filter with the Butterworth approximating function is described by the expression:
where is the order of the filter (a positive integer numerically equal to the number of reactive elements in the equivalent filter circuit).
The filter order is determined by the expression
Tables 1, 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 |
Target: Mastering the technique of synthesis of linear filters (low pass, high pass and band pass) based on maximally flat and Chebyshev approximations.
Brief theoretical information: To perform this work, you need to be able to analyze various types of linear circuits and find their main characteristics. (frequency transmission ratio, transfer function and its poles); knowledge of the principles of synthesis of linear low-pass filters based on the maximum-flat and Chebyshev approximations and the principles of transition from known low-pass filter schemes to high-pass filter and band-pass filter circuits.
LPFs are designed for transmission with minimal attenuation of oscillations, the frequencies of which do not exceed a certain cutoff frequency, which is called cutoff frequency, in this case, oscillations with frequencies higher than the cutoff frequency should be significantly attenuated.
Properties of the transfer function of a two-port network :
The poles of the transfer function of the two-port network should be located in the left half-plane of the complex frequency p. They can be real or form complex conjugate pairs.
The number of poles of the transfer function must always exceed the number of zeros.
Unlike the poles, the zeros of the transfer function can be located in any half-plane, i.e., over the entire plane of the complex frequency p.
Filter synthesis steps :
Formulation of technical requirements for filter characteristics depending on the specified bandwidth. In this case, no restrictions are imposed on the filter structure. This approach is called synthesis for a given frequency response... As a rule, the ideal characteristic is not realizable in practice.
An approximation of an ideal characteristic using a function that can belong to a physically realizable circuit.
Implementation of the selected approximated function and obtaining a filter circuit diagram with the nominal values of the elements included in it.
The most widespread are two types of approximation: maximally flat and Chebyshev.
Maximum flat approximation based on the use of the frequency power transmission factor function, given in the form:
where 
- dimensionless normalized frequency.
A filter whose frequency response satisfies this function is called a filter with a maximum-flat characteristic or a Butterworth filter.
The synthesis procedure begins with determining the poles of the filter transfer function, for which it is necessary to go to the normalized complex frequency R n and determine the poles of the frequency power transfer coefficient function of the filter:
;
In the general case, the roots of this equation can be determined using the Moivre formula (calculating the roots n-th power of a complex number). In this case, it is necessary to take into account the value of the phase of the complex number z= - 1 ( = ).
When finding the roots of this equation for any filter order n the following should be done general
pattern:
all poles are located at the same angular distance from each other and this distance is always equal to 
; if n- odd, then the first pole is always 1, if n- even, then the first pole 
.
Using the property of the quadrant symmetry of the location of the poles of the frequency power transfer coefficient function and the conditions of stability and physical realizability of the two-port networks, for the filter transfer function it is necessary to select only those poles that are located in the left half-plane of the complex frequency and write for them zero-pole representation transfer function.
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