Elements of the theory of synthesis of linear frequency filters. Design (synthesis) of linear digital filters. Filter power characteristics
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 methodological instructions present the synthesis and analysis of an electrical circuit with important analog functional units of radio engineering: an electrical 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 methodological instructions, the development of electrical equivalent and schematic diagrams an electrical circuit containing an electrical filter and an amplifier, as well as analyzing the spectrum of the input signal of the pulse generator and analyzing 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 for 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 consistent mode of an arbitrary linear quadrupole... 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, connected in cascade, matched in the passband with each other and with external loads of the 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 frequencies in the stopband (the calculated attenuation in the passband, 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 calculation stages 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 |
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.
The classical theory of synthesis of passive linear electric circuits with lumped parameters provides for two stages:
Finding or selecting a suitable rational function that could be a characteristic of a physically feasible chain and, at the same time, be sufficiently close 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 is the implementation of the circuit.
The approximation based on the use of various orthogonal functions does not cause fundamental difficulties. The task of finding the optimal structure of a chain for a given (physically feasible) characteristic is much more difficult. This problem does not have an unambiguous solution. One and the same characteristic of the circuit can be implemented in many ways, differing in the circuit, in the number of elements included in it and the complexity of the selection of the parameters of these elements, but the sensitivity of the characteristics of the circuit to instability of parameters, etc.
Distinguish between the synthesis of circuits in the frequency domain and in the time domain. In the first case, it is given 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, the synthesis of the circuit in the time domain can be reduced to synthesis in the frequency domain and vice versa. Still, synthesis for a given impulse response has its own characteristics that play a large role in impulse technology when forming pulses with certain requirements for their parameters (front steepness, overshoot, peak shape, etc.).
This chapter deals with the synthesis of quadripoles in the frequency domain. It should be pointed out that currently there is an extensive literature on the synthesis of linear electrical circuits, and the study of the general theory of synthesis is not included in the task of the course "Radio engineering circuits and signals". Here, only some particular issues of the synthesis of two-port networks are considered, reflecting the features of modern radioelectronic circuits. These features primarily include:
The use of active four-port networks;
The tendency to exclude inductances from selective circuits (in microelectronic design);
The emergence and rapid development of discrete (digital) circuit technology.
It is known that the transfer function of a two-port network TO(iω) is uniquely determined by its zeros and poles on the p-plane. Therefore, the expression "synthesis by the given transfer function" is equivalent to the expression "synthesis by the given zeros and poles of the transfer function". The existing theory of synthesis of two-port networks considers circuits, the transfer function of which 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 matched to a given signal. Indeed, the factor e iωt 0 entering the transfer function of such a filter [see. (12.16)] is not realized by a finite number of links with lumped parameters. The material presented in this chapter is focused on four-port networks with a small number of links. Such quadripoles are typical for low-pass filters, high-pass filters, suppression filters, etc., which are widely used in electronic devices.
Electric filters are four-port networks, which, with negligible attenuation ∆A, pass oscillations in certain frequency ranges f 0 ... f 1 (passbands) and practically do not pass oscillations in other ranges f 2 ... f 3 (stop bands, or non-transmission bands).
Rice. 2.1.1. Low pass filter (LPF). Rice. 2.1.2. High Pass Filter (HPF).
There are many different types of implementation of electrical filters: passive LC filters (circuits contain inductive and capacitive elements), passive RC filters (circuits contain resistive and capacitive elements), active filters (circuits contain operational amplifiers, resistive and capacitive elements), waveguide, digital filters and others. Among all types of filters, LC filters occupy a special position, as they are widely used in telecommunications equipment in various frequency ranges. A well-developed synthesis technique exists for this type of filter, and the synthesis of other types of filters makes much of this.
methodology. Therefore, the course work focuses on the synthesis
Rice. 2.1.3. Band pass filter (PF). passive LC filters.
The task of synthesis an electrical filter is to define a filter circuit with the smallest possible number of elements, the frequency response of which would meet the specified specifications. Requirements are often made on the characteristic of the working attenuation. In Figures 2.1.1, 2.1.2, 2.1.3, the requirements for the operating attenuation are set by the levels of the maximum allowable attenuation in the passband A and the levels of the minimum allowable attenuation in the passband As. The synthesis task is divided into two stages: approximation problem requirements for the working weakening of a physically realizable function and implementation task found approximating function by the electric circuit.
The solution to the approximation problem consists in finding such a function of the minimum possible order, which, firstly, satisfies the specified technical requirements for the frequency response of the filter, and, secondly, satisfies the conditions of physical realizability.
The solution to the implementation problem is to determine the electrical circuit, the frequency response of which coincides with the function found as a result of solving the approximation problem.
2.1. FUNDAMENTALS OF FILTER SYNTHESIS BY OPERATING PARAMETERS.
Let us consider some relations characterizing the conditions for the transfer of energy through an electric filter. As a rule, an electrical filter is used in conditions when devices are connected from the side of its input terminals, which in the equivalent circuit can be represented in the form of an active two-port network with parameters E (jω), R1, and from the side of the output terminals, devices represented in the equivalent circuit are connected resistor R2. The electrical filter connection diagram is shown in Figure 2.2.1.
Figure 2.2.2 shows a diagram in which, instead of a filter and resistance R2, a load resistance is connected to an equivalent generator (with parameters E (jω), R1), the value of which is equal to the resistance of the generator R1. As you know, the generator delivers maximum power to a resistive load if the load resistance is equal to the resistance of the internal losses of the generator R1.
Signal passage through a four-port network is characterized by an operating transfer function T (jω). The working transfer function makes it possible to compare the power S 0 (jω) given by the generator to the load R1 (matched to its own parameters) with the power S 2 (jω) supplied to the load R2 after passing through the filter:
The argument of the working transfer function arg (T (jω)) characterizes the phase relations between the emf E (jω) and output voltage U 2 (jω). It is called the working phase constant transmission (denoted Greek letter"beta"):
When transferring energy through a four-port network, changes in power, voltage and current in absolute value are characterized by the modulus of the working transfer function. When evaluating the selective properties of electrical filters, a measure determined by a logarithmic function is used. This measure is the working attenuation (denoted by the Greek letter "alpha"), which is related to the working transfer function module by the ratios:
, (Нп); or (2.2)
, (dB). (2.3)
In the case of using formula (2.2), the working attenuation is expressed in nepers, and when using formula (2.3) - in decibels.
The value is called the working constant of the four-port transmission (denoted by the Greek letter "gamma"). The working transfer function can be represented using the working attenuation and the working phase as:
In the case when the resistance of the internal losses of the generator R1 and the load resistance R2 are resistive, the powers S 0 (jω) and S 2 (jω) are active. It is convenient to characterize the passage of power through the filter using the power transfer factor, defined as the ratio of the maximum power P max received from the generator by the load matched to it to the power P 2 supplied to the load R2:
A reactive four-port network does not consume active power. Then the active power P 1 given by the generator is equal to the power P 2 consumed by the load:
We express the value of the input current modulus:, and substitute it in (2.5).
Using algebraic transformations, we represent (2.5) in the form:
We represent the numerator of the right side of the equation in the form:
The left side of equation (2.6) is the reciprocal of the power transfer factor:
The following expression represents the reflectance of the power from the input terminals of a four-port network:
Reflection coefficient (voltage or current) from the input terminals of the four-port network, equal to
characterizes the matching of the input resistance of the filter with the resistance R1.
A passive four-port network cannot provide power amplification, that is.
Therefore, for such circuits, it is advisable to use an auxiliary function defined by the expression:
Let us represent the working attenuation in a different, more convenient form for solving the problem of filter synthesis:
Obviously, the nature of the frequency dependence of the operating attenuation is associated with the frequency dependence of a function called the filtering function: the zeros and poles of the filtering function coincide with the zeros and poles of the attenuation.
Based on formulas (2.7) and (2.9), it is possible to represent the power reflection coefficient from the input terminals of a four-port network:
Let's move on to recording operator images according to Laplace, taking into account that p = jω, and also that the square of the modulus of a complex quantity is expressed, for example. Expression (2.10) in operator form has the form
Operator expressions,, are rational functions of the complex variable "p", and therefore they can be written as
where,, - are polynomials, for example:
From formula (2.11), taking into account (2.12), one can obtain the relation between the polynomials:
At the stage of solving the approximation problem, the expression of the filtration function is determined, that is, the polynomials h (p), w (p) are determined; from equation (2.13) one can find the polynomial v (p).
If expression (2.8) is presented in operator form, then we can obtain the function of the input resistance of the filter in operator form:
The conditions for physical realizability are as follows:
1. v (p) - must be a Hurwitz polynomial, that is, its roots are located in the left half of the plane of the complex variable p = α + j · Ω (chain stability requirement);
2. w (p) - must be either even or odd polynomial (for low-pass filter w (p) - even, so that there is no attenuation pole at ω = 0; for high-pass filter w (p) - odd);
3. h (p) is any polynomial with real coefficients.
2.2. REGULATION ON RESISTANCE AND FREQUENCY.
The numerical values of the parameters of the elements L, C, R and the cutoff frequencies of real filters can take on a variety of values, depending on the technical conditions. The use of both small and large values in calculations leads to a significant calculation error.
It is known that the nature of the frequency dependences of the filter does not depend on the absolute values of the coefficients of the functions describing these dependences, but is determined only by their ratios. The values of the coefficients are determined by the values of the parameters L, C, R filters. Therefore, the normalization (change by the same number of times) of the coefficients of the functions leads to the normalization of the values of the parameters of the filter elements. Thus, instead of the absolute values of the resistances of the filter elements, their relative values are taken, referred to the load resistance R2 (or R1).
In addition, if the frequency values are normalized relative to the cutoff frequency of the passband (this value is most often used), then this will further narrow the spread of the values used in the calculations and increase the accuracy of the calculations. The normalized frequency values are written as and are dimensionless values, and the normalized value is the cutoff frequency of the passband.
For example, consider the resistance of the series-connected elements L, C, R:
Normalized resistance:.
Let us introduce the normalized frequency values into the last expression: where the normalized parameters are equal to:.
The true (denormalized) values of the parameters of the elements are determined by:
By changing the values of f 1 and R2, it is possible to obtain new circuits of devices operating in other frequency ranges and under different loads from the original circuit. The introduction of standardization made it possible to create catalogs of filters, which in many cases reduces the complex problem of filter synthesis to working with tables.
2.3. CONSTRUCTION OF DUAL CIRCUITS.
As you know, the dual quantities are resistance and conductivity. A dual circuit can be found for each electrical filter circuit. In this case, the input impedance of the first circuit will be equal to the input conductivity of the second, multiplied by a coefficient. It is important to note that the operating transfer function T (p) for both schemes will be the same. An example of constructing a dual circuit is shown in Figure 2.3.
Such conversions are often convenient, since they can reduce the number of inductive elements. As you know, inductors, in comparison with capacitors, are bulky and low-Q elements.
The normalized parameters of the elements of the dual circuit are determined (at = 1):
2.4. FREQUENCY CHARACTERISTICS APPROXIMATION.
Figures 2.1.1 - 2.1.3 show the graphs of the functions of the operating attenuation of the low-pass filter (LPF), high-pass filter (HPF), band-pass filter (BPF). The same graphs show the levels of the required attenuation. In the passband f 0 ... f 1, the maximum allowable attenuation value (the so-called attenuation unevenness) ΔA is set; in the non-transmission band f 2 ... f 3 the minimum allowable value of the attenuation A S is set; in the transition region of frequencies f 1 ... f 2 requirements for attenuation are not imposed.
Before proceeding with the solution of the approximation problem, the required characteristics of the operating attenuation in frequency are normalized, for example, for a low-pass filter and a high-pass filter:
The sought-for approximating function must satisfy the conditions of physical feasibility and sufficiently accurately reproduce the required frequency dependence of the operating attenuation. There are various criteria for evaluating the approximation error, which are based on different types approximation. In problems of approximation of amplitude-frequency characteristics, the optimality criteria of Taylor and Chebyshev are most often used.
2.4.1. Approximation by Taylor's criterion.
In the case of the application of the Taylor criterion, the sought approximating function has the following form (normalized value):
where is the square of the modulus of the filtering function;
- the order of the polynomial (takes an integer value);
ε - coefficient of unevenness. Its value is related to the value of ∆А - non-uniformity of attenuation in the passband (Fig. 2.4). Since at the cutoff frequency of the passband Ω 1 = 1, therefore
Filters with frequency dependences of attenuation (2.16) are called filters with maximally flat attenuation characteristics, or filters with characteristics of Butterworth, who first applied the Taylor criterion approximation when solving the filter synthesis problem.
The order of the approximating function is determined based on the condition that at the cutoff frequency Ω 2 the operating attenuation exceeds the minimum allowable value:
Where . (2.19)
Since the order of the polynomial must be an integer, the resulting value is
Figure 2.4. rounded to the nearest higher
integer value.
Expression (2.18) can be represented in operator form using the transformation jΩ →:
Find the roots of the polynomial:, whence
K = 1, 2, ..., NB (2.20)
The roots take complex conjugate values and are located on a circle of radius. To form the Hurwitz polynomial, you need to use only those roots that are located in the left half of the complex plane:
Figure 2.5 shows an example of placing the roots of a 9th order polynomial with a negative real component in the complex plane. Module square
Rice. 2.5. the filtration function, according to (2.16), is equal to:
Polynomial with real coefficients; is an even order polynomial. Thus, the conditions of physical realizability are fulfilled.
2.4.2. Approximation by the Chebyshev criterion.
When using the power polynomials Ω 2 N B for Taylor approximation, a good approximation to the ideal function near the point Ω = 0 is obtained, but in order to ensure a sufficient steepness of the approximating function for Ω> 1, it is necessary to increase the order of the polynomial (and, consequently, the order of the scheme ).
The best slope in the transition frequency range can be obtained if, as an approximating one, we choose not a monotonic function (Fig. 2.4), but a function that fluctuates in the range of values 0 ... ΔA in the passband at 0<Ω<1 (рис. 2.7).
The best approximation by the Chebyshev criterion is provided by using the Chebyshev polynomials P N (x) (Fig. 2.6). In the interval -1< x < 1 отклонения аппроксимирующих функций от нулевого уровня равны ±1 и чередуются по знаку.
In the interval -1< x < 1 полином Чебышёва порядка N описывается выражением
P N (x) = cos (N arccos (x)), (2.21)
for N = 1 P 1 (x) = cos (arccos (x)) = x,
for N = 2 P 2 (x) = cos (2 arccos (x)) = 2 cos 2 (arccos (x)) - 1 = 2 x 2 - 1,
for N≥3, the polynomial P N (x) can be calculated using the recurrence formula
P N +1 (x) = 2 x P N (x) - P N -1 (x).
For x> 1, the values of the Chebyshev polynomials increase monotonically and are described by the expression
P N (x) = ch (N Arch (x)). (2.22)
The function of the working weakening (Fig.2.7) is described by the expression
where ε is the coefficient of unevenness, determined by the formula (2.17);
Filtering function module square;
P N (Ω) is a Chebyshev polynomial of order N.
The operating attenuation in the stopband must exceed the value of A S:
Substituting expression (2.22) for the values of the frequencies of the band of non-transmission into this inequality, we solve it with respect to the value N = Np - the order of the Chebyshev polynomial:
The order of the polynomial must be an integer, so the resulting value must be rounded to the nearest higher integer value.
The square of the modulus of the operating transfer function (standardized value)
Since the attenuation zeros (they are also the roots of the Hurwitz polynomial) are located in the passband, expression (2.21) for the values of the passband frequencies must be substituted into this expression.
Expression (2.25) can be represented in operator form using the transformation jΩ →:
The roots of the polynomial are determined by the formula:
K = 1, 2, ..., NCH, (2.26)
Complex conjugate roots in the complex plane are located on an ellipse. The Hurwitz polynomial is formed only by roots with a negative real component:
Filtering function module square; therefore, we find the polynomial using the recurrent formula:
It is a polynomial with real coefficients; is a polynomial of even degree. The conditions of physical realizability are fulfilled.
2.5. IMPLEMENTATION OF THE APPROXIMATING FUNCTION BY THE ELECTRIC CIRCUIT.
One of the methods for solving the implementation problem is based on expanding the input resistance function into a continued fraction
The decomposition procedure is described in the literature:,. The continued fraction expansion can be briefly explained as follows.
The function is a ratio of polynomials. First, the numerator polynomial is divided by the denominator polynomial; then the polynomial that was the divisor becomes divisible, and the resulting remainder becomes the divisor, and so on. The quotients obtained by division form a continued fraction. For the circuit in Figure 2.8, the continued fraction has the form (for = 1):
If necessary, you can from the received
schemes go to dual.
2.6. FREQUENCY VARIABLE CONVERSION METHOD.
The frequency variable conversion method is used to synthesize the high-pass filter and the high-frequency filter. The conversion applies only to normalized Ω frequencies.
2.6.1. HPF synthesis... Comparing the characteristics of the LPF and HPF in Figures 2.9 and 2.10, you can see that they are mutually inverse. This means that if we change the frequency variable
in the expression of the characteristics of the low-pass filter, then the characteristic of the high-pass filter will be obtained. For example, for a filter with a Butterworth characteristic
Using this transformation is equivalent to replacing capacitive elements with inductive ones and vice versa:
That is
That is .
To synthesize a high-pass filter using the frequency-variable conversion method, you need to do the following.
Rice. 2.9. LPF with normalized Fig. 2.10. HPF with normalized
characteristic. characteristic.
1. Perform normalization of the frequency variable.
2. Apply formula (2.27) to transform the frequency variable
The recalculated requirements for the operating attenuation characteristic represent the requirements for the operating attenuation of the so-called LPF prototype.
3. Synthesize a low-pass filter prototype.
4. Apply the formula (2.27) for the transition from the low-pass filter prototype to the required high-pass filter.
5. Denormalize the parameters of the elements of the synthesized HPF.
2.6.2. PF synthesis... Figure 2.1.3. depicts the symmetrical characteristic of the operating attenuation of the bandpass filter. This is the name of the characteristic that is geometrically symmetrical about the center frequency.
To synthesize the TF using the frequency variable transform method, you need to do the following.
1. To switch from the required symmetric characteristic of the PF to the normalized characteristic of the low-pass filter prototype (and use the already known synthesis technique), it is necessary to replace the frequency variable (Figure 2.11)
2.7. ACTIVE FILTERS.
Active filters are characterized by the absence of inductors, since the properties of inductive elements can be reproduced using active circuits containing active elements (operational amplifiers), resistors and capacitors. Such schemes are designated: ARC schemes. The disadvantages of inductors are low Q-factor (high losses), large dimensions, high production cost.
2.7.1. Fundamentals of ARC Filter Theory... For a linear four-port network (including a linear ARC filter), the ratio between the input and output voltage (in operator form) is expressed by the voltage transfer function:
where w (p) is an even (K p 0 for a low-pass filter) or an odd (for a high-pass filter) polynomial,
v (p) is a Hurwitz polynomial of order N.
For a low-pass filter, the transfer function (normalized value) can be represented as a product of factors
where К = Н U (0) = К2 1 К2 2 ... operator form, for p = 0);
the factors in the denominator are formed by the product of complex conjugate roots
in the case of an odd-order filter, there is one factor formed using the root of the Hurwitz polynomial with a real value.
Each transfer function factor can be implemented by a second or first order active low pass filter (ARC). And the entire given transfer function H U (p) is a cascade connection of such four-port networks (Figure 2.13).
An active four-port network based on an operational amplifier has a very useful property - its input impedance is much greater than its output impedance. Connecting to a four-terminal network as a load of a very large resistance (this operating mode is close to the idle mode) does not affect the characteristics of the four-terminal network itself.
Н U (р) = Н1 U (p) H2 U (p) ... Hk U (p)
For example, a 5th-order active low-pass filter can be implemented by a circuit that is a cascade connection of two second-order four-port networks and one first-order four-port network (Fig.2.14), and a 4th-order low-pass filter consists of a cascade connection of two second-order four-port networks. Quadrupoles with a higher Q-factor are connected first to the signal transmission path; a first-order four-port network (with the lowest Q factor and the lowest frequency response steepness) is connected last.
2.7.2. ARC filter synthesis produced using the voltage transfer function (2.29). Frequency normalization is performed relative to the cutoff frequency f c. At the cutoff frequency, the voltage transfer function value is less than the maximum Hmax by times, and the attenuation value is 3 dB
Rice. 2.14. ARC 5th order low pass filter.
The frequency characteristics are normalized relative to f c. If we solve equations (2.16) and (2.23) with respect to the cutoff frequency, then we obtain the expressions
For LPF with Butterworth characteristic;
With the characteristic of Chebyshev.
Depending on the type of filter characteristic - Butterworth or Chebyshev, - the order of the approximating function is determined by formulas (2.19) or (2.26).
The roots of the Hurwitz polynomial are determined by formulas (2.20) or (2.26). The voltage transfer function for a second-order four-port network can be formed using a pair of complex-conjugate roots, and, in addition, can be expressed in terms of the parameters of the circuit elements (Fig. 2.14). The analysis of the circuit and the derivation of expression (2.31) are not given. Expression (2.32) for a first-order four-port network is written in a similar way.
Since the value of the load resistance does not affect the characteristics of the active filter, denormalization is performed based on the following. First, the acceptable values of the resistive resistances are selected (10 ... 30 kOhm). Then the real values of the capacitance parameters are determined; for this, f c is used in expression (2.15).
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