Elements of the theory of synthesis of linear frequency filters. Design (synthesis) of linear digital filters
Electric filters are four-terminal networks that, with negligible attenuation ∆A, transmit oscillations in certain frequency ranges f 0 ... f 1 (passbands) and practically do not transmit oscillations in other ranges f 2 ... f 3 (stopbands, or non-passbands).
Rice. 2.1.1. Low pass filter (LPF). Rice. 2.1.2. High pass filter (HPF).
There are many different types of electrical filter implementations: passive LC filters (circuits containing inductive and capacitive elements), passive RC filters (circuits containing resistive and capacitive elements), active filters (circuits containing 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. There is a well-developed synthesis technique for this type of filter, and the synthesis of other types of filters largely uses this
technique. Therefore in course work the focus is on synthesis
Rice. 2.1.3. Bandpass filter (PF). passive LC filters.
The synthesis task electric filter is to determine a filter circuit with the minimum possible number of elements, the frequency response of which would satisfy the specified technical requirements. Often requirements are placed on the operating attenuation characteristic. In Figures 2.1.1, 2.1.2, 2.1.3, the requirements for operating attenuation are specified by the levels of the maximum permissible attenuation in the passband A and the levels of the minimum permissible attenuation in the stopband As. The synthesis task is divided into two stages: approximation problem requirements for operating attenuation by a physically realized function and implementation task the found approximating function by an electrical circuit.
The solution to the approximation problem is to find a function of the minimum possible order, which, firstly, satisfies the given 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 electrical circuit, the frequency response of which coincides with the function found as a result of solving the approximation problem.
2.1. BASICS OF FILTERS SYNTHESIS ACCORDING TO OPERATING PARAMETERS.
Let's consider some relationships characterizing the conditions for energy transfer through an electric filter. As a rule, an electric filter is used in conditions where devices are connected from its input terminals, which can be represented in an equivalent circuit in the form of an active two-terminal network with parameters E(jω), R1, and from the output terminals devices are connected, represented in an equivalent circuit resistive resistance 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 the equivalent generator (with parameters E(jω), R1), the value of which is equal to the resistance of the generator R1. As is known, a 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.
The passage of a signal through a four-port network is characterized by the operating transfer function T(jω). Working Transmission function allows you to compare the power S 0 (jω) supplied by the generator to the load R1 (matched with 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 relationships between the emf. E(jω) and output voltage U 2 (jω). It is called the operating phase transfer constant (denoted by the Greek letter "beta"):
When transmitting 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 assessing the selective properties of electric 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 module of the working transfer function by the relations:
, (Нп); or (2.2)
, (dB). (2.3)
In the case of using formula (2.2), the operating attenuation is expressed in non-feathers, and when using formula (2.3) - in decibels.
The value is called the operating transmission constant of the quadripole network (denoted by the Greek letter “gamma”). The operating transfer function can be represented using operating attenuation and operating phase as:
In the case when the internal loss resistance 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 coefficient, defined as the ratio of the maximum power P max received from the generator by the load matched with it to the power P 2 supplied to the load R2:
A reactive quadrupole does not consume active power. Then the active power P 1 supplied by the generator is equal to the power P 2 consumed by the load:
Let us express the value of the input current module: , and substitute it into (2.5).
Using algebraic transformations, we represent (2.5) in the form:
Let's represent the numerator on the right side of the equation as:
The left side of equation (2.6) is the reciprocal of the power transfer coefficient:
The following expression represents the power reflection coefficient from the input terminals of the quadrupole:
Reflection coefficient (voltage or current) from the input terminals of the quadrupole, equal to
characterizes the matching of the filter input resistance with resistance R1.
A passive four-terminal network cannot provide power gain, that is.
Therefore, for such circuits it is advisable to use the auxiliary function defined by the expression:
Let's imagine the working attenuation in a different form, more convenient 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 the 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), we can imagine the power reflection coefficient from the input terminals of a four-terminal network:
Let's move on to writing operator images according to Laplace, taking into account that p = jω, and also that the squared 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 in the form
where , , are polynomials, for example:
From formula (2.11), taking into account (2.12), we can obtain the relationship between the polynomials:
At the stage of solving the approximation problem, the expression of the filtering function is determined, that is, the polynomials h(p), w(p) are determined; from equation (2.13) we can find the polynomial v(p).
If expression (2.8) is presented in operator form, then we can obtain the filter input resistance function in operator form:
The conditions for physical feasibility 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 an even or odd polynomial (for a low-pass filter w(p) – even, so that there is no attenuation pole at ω=0; for a high-pass filter w(p) – odd);
3. h(p) – any polynomial with real coefficients.
2.2. RATING BY RESISTANCE AND FREQUENCY.
The numerical values of the parameters of the elements L, C, R and the cutoff frequencies of real filters can take, depending on the technical conditions, very different values. Using both small and large quantities 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 relationships. The values of the coefficients are determined by the values of the parameters L, C, R filters. Therefore, normalization (changing by the same number of times) of function coefficients leads to normalization of the parameter values of the filter elements. Thus, instead of the absolute values of the resistances of the filter elements, their relative values are taken, related to the load resistance R2 (or R1).
In addition, if you normalize the frequency values relative to the cutoff frequency of the passband (this value is most often used), this will further narrow the spread of values used in the calculations and increase the accuracy of the calculations. Normalized frequency values are written in the form and are dimensionless quantities, and the normalized value of the cutoff frequency of the passband is .
For example, consider the resistance of series-connected elements L, C, R:
Standardized resistance: .
Let us introduce the normalized frequency values into the last expression: where the normalized parameters are equal to: .
The true (denormed) values of the element parameters are determined:
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 other loads from the original circuit. The introduction of normalization made it possible to create filter catalogs, which in many cases reduces the complex problem of filter synthesis to working with tables.
2.3. CONSTRUCTION OF DUAL SCHEMES.
The dual quantities, as is known, are resistance and conductivity. For each electric filter circuit, a circuit dual to it can be found. In this case, the input resistance of the first circuit will be equal to the input conductivity of the second, multiplied by the 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 transformations often turn out to be convenient, since they make it possible to reduce the number of inductive elements. As you know, inductors, compared to capacitors, are bulky and low-Q elements.
The normalized parameters of the elements of the dual circuit are determined (at =1):
2.4. APPROXIMATION OF FREQUENCY CHARACTERISTICS.
Figures 2.1.1 – 2.1.3 show graphs of the operating attenuation functions of a low-pass filter (LPF), high-pass filter (HPF), and band-pass filter (PF). The same graphs show the levels of required attenuation. In the passband f 0 ...f 1, the maximum permissible attenuation value (the so-called attenuation unevenness) ΔA is set; in the stopband f 2 ...f 3 the minimum permissible attenuation value A S is set; in the transition frequency region f 1 ... f 2 there are no requirements for attenuation.
Before starting to solve the approximation problem, the required characteristic of the operating attenuation by frequency is normalized, for example, for low-pass filters and high-pass filters:
The sought approximating function must satisfy the conditions of physical feasibility and accurately reproduce the required frequency dependence of the operating attenuation. There are various criteria for assessing the approximation error on which the Various types approximations. 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 criterion.
In the case of applying the Taylor criterion, the desired approximating function has the following form (normalized value):
where is the square of the module of the filtering function;
– order of the polynomial (takes an integer value);
ε – unevenness coefficient. Its value is related to the value of ∆A - the unevenness of attenuation in the passband (Fig. 2.4). Since on cutoff frequency passband Ω 1 =1, therefore
Filters with frequency dependences of attenuation (2.16) are called filters with extremely flat attenuation characteristics, or filters with Butterworth characteristics, who was the first to use approximation using the Taylor criterion when solving the problem of filter synthesis.
The order of the approximating function is determined based on the condition that at the cutoff frequency of the stopband Ω 2 the operating attenuation exceeds the minimum permissible value:
Where . (2.19)
Since the order of the polynomial must be an integer, the resulting value
Fig.2.4. rounded up to the nearest higher
whole value.
Let us represent expression (2.18) in operator form using the transformation jΩ→:
Let's find the roots of the polynomial: , from where
K = 1, 2, … , NB (2.20)
The roots take complex conjugate values and are located on a circle of radius . To form a 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 filtering function, according to (2.16), is equal to:
Polynomial with real coefficients; - polynomial of even order. Thus, the conditions of physical realizability are met.
2.4.2. Approximation using the Chebyshev criterion.
When using 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 circuit ).
The best slope in the transition frequency region can be obtained if, as an approximating function, we choose not a monotonic function (Fig. 2.4), but a function oscillating in the range of values 0 ... ΔA in the passband at 0<Ω<1 (рис. 2.7).
The best approximation according to the Chebyshev criterion is ensured by the use of 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 operating attenuation function (Fig. 2.7) is described by the expression
where ε is the unevenness coefficient determined by formula (2.17);
Filtering function module square;
P N (Ω) – Chebyshev polynomial of order N.
The operating attenuation in the stopband must exceed the value A S:
Substituting expression (2.22) for the values of the stopband frequencies into this inequality, we solve it with respect to the value N = NЧ - 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.
Squared modulus of the working transfer function (normalized value)
Since the zeros of the attenuation (they are also the roots of the Hurwitz polynomial) are located in the passband, expression (2.21) for the frequency values of the passband must be substituted into this expression.
Let us represent expression (2.25) in operator form using the transformation jΩ→:
The roots of a polynomial are determined by the formula:
K = 1, 2, … , NЧ, (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 recurrence formula:
Is a polynomial with real coefficients; is a polynomial of even degree. The conditions of physical realizability are met.
2.5. IMPLEMENTATION OF APPROXIMATING FUNCTION BY ELECTRIC CIRCUIT.
One of the methods for solving the implementation problem is based on the continued fraction expansion of the input resistance function
The decomposition procedure is described in the literature: , . The continued fraction decomposition can be briefly explained as follows.
A function is a ratio of polynomials. First, the numerator polynomial is divided by the denominator polynomial; then the polynomial that was the divisor becomes the dividend, and the resulting remainder becomes the divisor, and so on. The quotients obtained by division form a continued fraction. For the diagram in Figure 2.8, the continued fraction has the form (at =1):
If necessary, it is possible from the received
switch to dual schemes.
2.6. METHOD OF CONVERTING A FREQUENCY VARIABLE.
The frequency variable conversion method is used to synthesize the high-pass filter and filter filter. The conversion only applies to normalized Ω frequencies.
2.6.1. HPF synthesis. Comparing the characteristics of the low-pass filter and high-pass filter in Figures 2.9 and 2.10, you can see that they are mutually inverse. This means that if you replace the frequency variable
in the expression of the low-pass filter characteristic, then the high-pass filter characteristic is obtained. For example, for a filter with the Butterworth characteristic
Using this transformation is equivalent to replacing capacitive elements with inductive elements and vice versa:
That is
That is .
To synthesize a high-pass filter using the frequency variable conversion method, you must do the following.
Rice. 2.9. Low-pass filter with normalized Fig. 2.10. High-pass filter 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 characteristics represent the requirements for the operating attenuation of the so-called low-pass filter prototype.
3. Synthesize a low-pass filter prototype.
4. Apply formula (2.27) to move from the prototype low-pass filter to the required high-pass filter.
5. Perform denormalization of the parameters of the elements of the synthesized high-pass filter.
2.6.2. PF synthesis. In Figure 2.1.3. shows the symmetrical characteristic of the operating attenuation of a bandpass filter. This is the name of a characteristic that is geometrically symmetrical about the average frequency.
To synthesize a PF using the frequency variable transformation method, the following must be done.
1. To move from the required symmetrical 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 (op-amps), resistors and capacitors. Such schemes are designated: ARC schemes. The disadvantages of inductors are low quality factor (high losses), large dimensions, and high production costs.
2.7.1. Basic theory of ARC filters. For a linear four-port network (including a linear ARC filter), the relationship 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 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 K = N U (0) = K2 1 K2 2 ... K2 (N /2) – the value of the function H U (p) (for an even order filter) when transmitting a constant voltage (that is, at f = 0 or, in operator form, with 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 low-pass active filter (ARC). And the entire given transfer function H U (p) is a cascade connection of such four-terminal networks (Figure 2.13).
An active four-port network based on an operational amplifier has a very useful property - its input resistance is much greater than its output resistance. Connecting a very large resistance to a four-terminal network as a load (this operating mode is close to the no-load mode) does not affect the characteristics of the four-terminal network itself.
H U (p) = H1 U (p) H2 U (p) ... Hk U (p)
For example, an active low-pass filter of the 5th order can be implemented by a circuit that is a cascade connection of two quadrupoles of the second order and one quadrupole of the first order (Fig. 2.14), and a 4th order low-pass filter consists of a cascade connection of two quadripoles of the second order. Quadrupoles with a higher quality factor are connected first to the signal transmission path; The first-order four-port network (with the lowest quality factor and the lowest slope of the frequency response) is connected last.
2.7.2. ARC filter synthesis is performed using the voltage transfer function (2.29). Frequency normalization is performed relative to the cutoff frequency f c . At the cutoff frequency, the value of the voltage transfer function is less than the maximum Hmax by a factor, and the attenuation value is 3 dB
Rice. 2.14. 5th order ARC low pass filter.
Frequency characteristics are normalized relative to f c. If we solve equations (2.16) and (2.23) with respect to the cutoff frequency, we obtain the expressions
For low-pass filter with Butterworth characteristic;
With characteristics of Chebyshev.
Depending on the type of filter characteristic - Butterworth or Chebyshev - the order of the approximating function is determined using 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 through 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, acceptable values of resistive resistances (10 ... 30 kOhm) are selected. Then the real values of the capacity parameters are determined; for this purpose f c is used in expression (2.15).
The classical theory of synthesis of passive linear electrical circuits with lumped parameters involves two stages:
Finding or selecting a suitable rational function that could be a characteristic of a physically feasible circuit and at the same time be sufficiently close to a given characteristic;
Finding the structure and elements of the circuit that implements the selected function.
The first stage is called approximation of a given characteristic, the second - implementation of the circuit.
Approximation based on the use of various orthogonal functions does not pose any fundamental difficulties. The task of finding the optimal structure of a chain based on a given (physically feasible) characteristic is much more difficult. This problem does not have a unique solution. The same circuit characteristic can be implemented in many ways, differing in the circuit, the number of elements included in it and the complexity of selecting the parameters of these elements, but the sensitivity of the circuit characteristics to parameter instability, etc.
A distinction is made between circuit synthesis in the frequency domain and in the time domain. In the first case, the transfer function is specified TO(iω), and in the second - the impulse response g(t). Since these two functions are related by a pair of Fourier transforms, circuit synthesis in the time domain can be reduced to synthesis in the frequency domain and vice versa. Nevertheless, synthesis based on a given impulse response has its own characteristics, which play a large role in impulse technology in the formation of pulses with certain requirements for their parameters (front steepness, overshoot, peak shape, etc.).
This chapter discusses the synthesis of quadripoles in the frequency domain. It should be pointed out that there is currently an extensive literature on the synthesis of linear electrical circuits, and the study of the general theory of synthesis is not included in the scope of the course “Radio Engineering Circuits and Signals”. Here we consider only some particular issues of the synthesis of four-terminal networks, reflecting the features of modern radio-electronic circuits. These features primarily include:
Application of active quadripoles;
The tendency to exclude inductances from selective circuits (in microelectronic versions);
The emergence and rapid development of discrete (digital) circuit technology.
It is known that the transfer function of a four-port network TO(iω) is uniquely determined by its zeros and poles on the p-plane. Therefore, the expression “synthesis according to a given transfer function” is equivalent to the expression “synthesis according to given zeros and poles of the transfer function.” The existing theory of synthesis of four-terminal networks considers circuits whose transfer function has a finite number of zeros and poles, in other words, circuits consisting of a finite number of links with lumped parameters. This leads to the conclusion that classical methods of circuit synthesis are inapplicable to filters matched to a given signal. Indeed, the factor included in the transfer function of such a filter is e iωt 0 [see. (12.16)] is not realized by a finite number of links with lumped parameters. The material presented in this chapter is focused on quadripole networks with a small number of links. Such four-terminal networks are typical for low-pass filters, high-pass filters, stop filters, etc., widely used in radio-electronic devices.
Science refines the mind;
Learning will sharpen your memory.
Kozma Prutkov
chapter 15
ELEMENTS OF SYNTHESIS OF LINEAR STATIONARY CIRCUITS
15.1. Questions studied
WITH synthesis of analog two-terminal networks. Synthesis of stationary quadripoles according to 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 synthesizing two-terminal networks and specific ways to solve the problem according to Foster and Cauer, as well as acquire the ability to determine the possibility of implementing a particular function of the input resistance of a two-terminal network. When synthesizing electrical filters based on prototype filters, it is important to understand the advantages and disadvantages of approximating the Chebyshev and Butterworth attenuation characteristics. It is necessary to be able to quickly calculate the parameters of elements of any type of filters (low-pass filter, high-pass filter, PPF) using frequency transformation formulas.
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 a circuit that satisfies predetermined properties. In parametric synthesis, only the parameters and type of elements of a circuit whose structure is known are determined. Further we will talk only about parametric synthesis.
When synthesizing two-terminal networks, the input resistance is usually used as a source
If a function is given, then it can be implemented by a passive circuit if the following conditions are met: 1) all coefficients of the numerator and denominator polynomials are real and positive; 2) all zeros and poles are located 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 highest and lowest 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, i.e. the same input function can be implemented in several ways.
As the initial structures of synthesized two-terminal networks, Foster circuits are usually used, which are a series or parallel connection relative to the input terminals, respectively, of several complex resistances and conductivities, as well as Cauer ladder circuits.
The method of synthesis of two-terminal networks is based on the fact that a given input function is subjected to a series of successive simplifications. In this case, at each stage, an expression is identified, which is associated with a physical element of the synthesized circuit. 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 prototype low-pass filters. Possible variants of the low-pass filter prototype are shown in Fig. 15.1.
Any of the schemes can be used in the calculation, since their characteristics are identical. Designations in Fig. 15.1 have the following meaning: – inductance of a series coil or capacitance of a 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 . The transition from normalized prototype filters to another level of resistance and frequency is carried out using the following transformations of circuit elements:
;
.
Values with primes refer to the normalized prototype, while those without primes refer to the converted circuit. The initial value for synthesis is the operating power attenuation, expressed in decibels:
, dB,
– maximum power of the generator with internal resistance and emf, – output power in the load.
Typically, the frequency dependence is approximated by the most flat (Butterworth) characteristic (Fig. 15.2, A)
Where 
.
The amount of operating attenuation corresponding to the cutoff frequency is usually chosen equal to 3 dB. Wherein . Parameter n is equal to the number of active circuit elements and determines the filter order.
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
ELECTRICAL CIRCUIT
Guidelines
to course design and CDS
Publishing house Omsk State Technical University
2010
Compiled by I. V. Nikonov
The guidelines present the synthesis and analysis of an electrical circuit with important analog functional units of radio engineering: an electrical filter and an amplifier. The spectrum of the input complex periodic signal is analyzed, as well as the signal at the output of the electrical circuit (for linear operating mode).
Intended for students of specialties 210401, 210402, 090104 and directions 21030062 of full-time and part-time forms of study, studying the disciplines “Fundamentals of Circuit Theory”, “Electrical Engineering and Electronics”.
Published by decision of the editorial and publishing council
Omsk State Technical University
© GOU VPO "Omsk State
Technical University", 2010
1. Analysis of technical specifications. Main design stages 5
2. Basic principles and methods of electrical design
filters 6
2.1. Fundamentals of Filter Design 6
2.2. Methodology for synthesizing filters based on characteristic parameters 11
2.3. Methodology for synthesizing filters based on operating parameters 18
2.4. An example of the synthesis of an equivalent circuit of an electric filter 25
3. Basic principles and stages of calculating the electrical circuit of the amplifier
voltage 26
3.1.Basic principles for calculating electrical circuits of amplifiers 26
3.2. An example of calculating an electrical circuit amplifier circuit
on a bipolar transistor 28
4. Basic principles and stages of complex spectrum analysis
periodic signal 30
4.1. Principles of spectral analysis 30
4.2. Calculation formulas for spectral analysis 31
4.3. Example of input signal spectrum analysis 32
5. Analysis of the signal at the output of the electrical circuit. Recommendations
on the development of an electrical circuit diagram 33
5.1. Analysis of signal passage through an electrical circuit 33
6. Basic requirements for content, implementation, protection
coursework 35
6.1. Procedure and deadlines for issuing assignments for course design 35
6.3. Design of the graphic part of the course work (project) 36
6.4. Defense of course projects (works) 38
Bibliography 39
Applications 40
Appendix A. List of abbreviations and symbols 40
Appendix B. Options for input data for filter synthesis 41
Appendix B. Options for initial data for calculating the amplifier 42
Appendix D. Options for input data for spectrum analysis
signal 43
Appendix E. Transistor parameters for the connection circuit
OE(OI) 45
Appendix E. Task form 46
INTRODUCTION
The main tasks of electrical engineering 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, and signals. During synthesis, the inverse problem is solved - the development of analytical and graphical models (schemes) of electrical circuits and signals. If the calculations and development carried out are completed with the production of design and technological documentation, production of mock-ups or prototypes, then the term 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 ones and those with non-galvanic connections;
– analysis of complex frequency characteristics of electrical circuits;
– analysis of linear electrical circuits under complex periodic influences;
– analysis of linear electrical circuits under pulsed influences;
– theory of linear quadripoles;
– analysis of nonlinear electrical circuits;
– linear electrical filters, synthesis of electrical filters.
The listed sections are studied during classroom lessons, but course design is also an important part of the educational process. The topic of the course work (project) may correspond to one of the sections being studied, may be complex, that is, include several sections of the discipline, or may be proposed by the student.
These guidelines discuss recommendations for completing a complex course work (project), in which it is necessary to solve interrelated problems of synthesis and analysis for an analog electrical circuit.
1.
ANALYSIS OF TECHNICAL SPECIFICATIONS.
MAIN DESIGN STAGES
As a comprehensive course work (project), these guidelines propose the development of 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 a pulse generator and an analysis of the “passage” of the input signal to the output of the device. These tasks are important and practically useful, since functional units widely used in radio engineering are 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 the calculations are given in Appendices B, C, D. The numbers of task options correspond to the numbers of students in the group list, or the option number is formed in a more complex way. If necessary, students can independently set additional design requirements, for example, weight and size requirements, requirements for phase-frequency characteristics and others.
Generator
impulses
Analog Electric Filter
Analog voltage amplifier
Rice. 1
Figure 1 shows the complex effective values of the input and output electrical voltages of a harmonic form.
During course design, it is necessary to solve the following problems:
A) synthesize (develop) an equivalent electrical circuit using any method, and then a basic electrical circuit using any radio elements. Carry out calculations of attenuation and voltage transfer coefficient, illustrate the calculations with graphs;
B) develop an electrical circuit diagram of a voltage amplifier using any radio elements. Carry out calculations of the amplifier using direct current, analyze the parameters of the amplifier in the mode of small alternating signals;
D) analyze the passage of electrical voltage from the pulse generator through an electrical 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 when making various roundings, approximate analysis of the signal spectrum, and when choosing standard radio elements that are close in nominal value to the calculated values.
2.1. Basic principles of filter design
2.1.1. Basic design requirements
Electrical filters are linear or quasi-linear electrical circuits that have frequency-dependent complex total power transfer coefficients. In this case, at least one of the 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 transmission coefficients (in formula (1) the decimal logarithm is used).
The frequency range in which attenuation () approaches zero and total power transfer () approaches unity is called passband (BP). Conversely, 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 (SB). The stop band in the specialized literature on electrical filters is also called the attenuation band or attenuation band. Between the PP and the PP there is a transition frequency band. Based on the location of the passband in the frequency range, electric filters are classified into the following types:
LPF – low pass filter, the passband is at lower frequencies;
HPF – high pass filter, the passband is at high frequencies;
PF – bandpass filter, the passband is in a relatively narrow frequency range;
RF is a notch filter, the stopband is in a relatively narrow frequency range.
A real electrical filter can be made on various radio components: inductors and capacitors, selective amplification devices, selective piezoelectric and electromechanical devices, waveguides and many others. There are reference books on calculating filters on specific radio components. However, the following principle is more universal: first, an equivalent circuit is developed using ideal LC elements, and then the ideal elements are converted into any real radio components. With this recalculation, an electrical circuit diagram and a list of elements are developed, standard radio components are selected or the necessary radio components are designed independently. The simplest option for such a calculation is to develop a circuit diagram of a reactive filter with capacitors and inductors, since the circuit diagram in this case is similar to the equivalent one.
But even with such a general universal calculation, there are several different methods for synthesizing an equivalent circuit of an LC filter:
– synthesis in a coordinated mode from identical G-, T-, U-shaped units. This method is also called characteristic parameter synthesis or “k” filter synthesis. Advantages: simple calculation formulas; the calculated attenuation (attenuation unevenness) in the passband () is taken equal to zero. Flaw: This synthesis method uses various approximations, but in fact it is impossible to achieve matching over the entire passband. Therefore, filters calculated by this method may have a passband attenuation of more than three decibels;
– polynomial synthesis. In this case, the required power transfer coefficient is approximated by a polynomial, that is, the entire circuit is synthesized, and not individual links. This method is also called synthesis according to operating parameters or synthesis according to reference books of normalized low-pass filters. When using reference books, the filter order is calculated, and an equivalent low-pass filter circuit is selected that meets the requirements of the task. Advantages: the calculations take into account possible inconsistencies and deviations in the parameters of radio elements; low-pass filters are easily converted into filters of other types. Flaw: it is necessary to use reference books or special programs;
– synthesis based on 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, impulse response () is expressed in terms of transfer response () using a direct Fourier transform:
This method has found application in the synthesis of various transversal filters (filters with delays), for example digital, acoustoelectronic, for which it is easier to develop electrical circuits based on pulse characteristics than on frequency characteristics. In course work, when developing filter circuits, it is recommended to use the synthesis method based on characteristic or operating parameters.
So, in work related to the synthesis of an electric filter, it is necessary, using one of the methods, to develop an electrical equivalent circuit using ideal reactive elements, and then a basic electrical circuit using any real radio elements.
In the course design assignment regarding the synthesis of an electric filter (Appendix B), the following data can be given:
– type of synthesized filter (LPF, HPF, PF, RF);
– – active resistance of external circuits with which the filter must be fully or partially matched in the passband;
– – cut-off frequency of the filter passband;
– – cut-off frequency of the filter stop band;
– – average filter frequency (for PF and RF);
– – filter attenuation in the passband (no more);
– – attenuation of the filter in the stopband (not less);
– – PF or RF passband;
– – stop band PF or RF;
– – low-pass filter squareness coefficient, high-pass filter;
– – squareness coefficient PF, RF.
If necessary, students can independently select additional data or design requirements.
2.1.2. Normalization and frequency conversions
When synthesizing equivalent and circuit diagrams of filters, it is advisable to use normalization and frequency conversions. This allows you to reduce the number of different types of calculations and carry out synthesis using a low-pass filter as a basis. The rationing is as follows. Instead of designing for given operating frequencies and load resistance, filters are designed for normalized load resistance and normalized frequencies. Normalization of frequencies is carried out, as a rule, relative to frequency. . With this normalization, the frequency is , and the frequency is . When normalizing, an equivalent circuit with normalized elements is first developed, and then these elements are recalculated to the specified requirements using denormalizing factors:
The possibility of using normalization in the synthesis of electrical circuits follows from the fact that the type of required transfer characteristics of the electrical circuit does not change during this operation, 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 radio elements and operating frequency, and for normalized values - when normalizing factors are used.
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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 high-pass filters, filter filters, and RF filters. Thus, the recommended sequence of high-pass filter synthesis when applying frequency conversions is as follows:
– graphic requirements for the high-pass filter are normalized (the axis of normalized frequencies is introduced);
– frequency conversion of attenuation requirements is carried out due to frequency conversion:
– a low-pass filter with standardized elements is designed;
– The low-pass filter is converted into a high-pass filter with normalized elements;
– elements are denormalized in accordance with formulas (3), (4).
– graphic requirements for the PF are replaced with requirements for the low-pass filter based on the condition of equality of their bandwidths and delays;
– a low-pass filter circuit is synthesized;
– inverse frequency conversion is used to obtain a bandpass filter circuit by including additional reactive elements in the low-pass filter branches to form resonant circuits.
– graphic requirements for the RF are replaced with requirements for the high-pass filter based on the condition of equality of their bandwidths and delays;
– a high-pass filter circuit is synthesized, directly or using a prototype low-pass filter;
– the high-pass filter circuit is converted into a notch filter circuit by including additional reactive elements in the high-pass filter branches.
2.2. Filter synthesis technique
2.2.1. Basic principles of synthesis according to characteristic parameters
The rationale for the basic calculated relationships of this synthesis method is as follows.
A linear four-port network is considered, and a system of parameters is used to describe it:
where are the voltage and current at the input of the quadrupole, and are the voltage and current at the output of the quadrupole.
The transmission coefficients for an arbitrary (matched or unmatched) mode are determined:
where is the load resistance (in the general case, complex).
For an arbitrary mode, the transmission constant (), attenuation (), phase () is introduced:
. (11)
The weakening in non-feathers is given by
, (12)
and in decibels - by the expression
In an unmatched mode, the input, output and transfer characteristics of a four-port network are called operating parameters, and in a matched mode - characteristic parameters. The values of the matching input and output resistances at a given operating frequency are determined from the four-port equations (8):
In the matched mode, taking into account expressions (14), (15), the characteristic transmission constant is determined:
Taking into account the relations for hyperbolic functions
, (17)
(18)
The relationship between the characteristic parameters of the matched mode and the elements of the electrical circuit (parameters) is determined. The expressions look like
Expressions (19), (20) characterize the matched mode of an arbitrary linear two-port network. Figure 3 shows a diagram of an arbitrary
L-shaped link, the parameters of which, in accordance with expressions (8), are determined:
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With the consistent inclusion of an L-shaped link, expressions (19), (20) are transformed to the form:
, (21)
. (22)
If the longitudinal and transverse branches of the L-shaped circuit contain different types of reactive elements, then the circuit is an electrical filter.
Analysis of formulas (21), (22) for this case allows us to obtain a method for synthesizing filters based on characteristic parameters. The main provisions of this technique:
– the filter is designed from identical links connected in cascade, matched in the passband with each other and with external loads (for example, G-type links);
– attenuation in the passband () is taken equal to zero, since the filter is considered consistent throughout the entire passband;
– the required values of external active resistances () for a 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 stopband () is determined (in decibels) by the formula
; (25)
– the number of identical G-links connected in cascade is determined by the expression:
2.2.2. Low-pass filter (LPF) synthesis sequence
according to characteristic parameters
The calculation formulas are obtained from the main provisions of the synthesis methodology based on the characteristic parameters given in paragraph 2.2.1 of these guidelines. In particular, formulas (27), (28) for determining the values of link elements are obtained from expressions (23), (24). When synthesizing from characteristic parameters, the sequence of calculations for low-pass filters and high-pass filters is as follows:
A) the values of the ideal inductance and capacitance of the G-section of the filter are calculated based on the given values of the load and generator resistances and the value of the cutoff frequency of the passband:
where are the values of the load and generator resistances, and is the value of the cutoff frequency of the passband. The graph of attenuation requirements and the diagram of the L-shaped low-pass filter link are shown in Figure 4 a, b. In pictures 5 a, b the requirements for attenuation and the diagram of the L-shaped high-pass filter section are given.
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b) the link attenuation () is calculated in decibels at the cutoff frequency of the stopband () based on the specified value of the squareness coefficient (). For low pass filter:
For the high pass filter:
. (30)
In calculations using formulas (29), (30), the natural logarithm is used;
B) the number of links () is calculated for a given value of guaranteed attenuation at the stopband boundary, in accordance with formula (26):
The value is rounded to the nearest higher integer value;
D) the filter attenuation is calculated in decibels for several frequency values in the stopband (the calculated attenuation in the passband, without taking into account thermal losses, is considered equal to zero in this method). For a low pass filter:
. (31)
For the high pass filter:
; (32)
e) heat losses are analyzed (). To approximate the calculation of heat losses using a low-frequency prototype, the resistive resistances of real inductors () are first determined at frequency at independently selected values of the quality factor (). Inductors, in the future, in the electrical circuit diagram, will be introduced instead of ideal inductances (capacitors are considered to be of higher quality and their resistive losses are not taken into account). Calculation formulas:
. (34)
The filter attenuation 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 the attenuation and modulus of the voltage transmission coefficient for the low-pass filter or high-pass filter are plotted;
G) using reference books of radio elements, the standard capacitors and inductors closest in value to the ideal elements are selected for the subsequent development of an electrical circuit diagram and a list of elements of the entire electrical circuit. If there are no standard inductors of the required rating, you will need to develop them yourself. Figure 6 shows the main dimensions of a simple cylindrical coil with a single layer winding, necessary for its calculation.
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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 base of the coil.
2.2.3. Sequence of synthesis of PF (RF)
according to characteristic parameters
In pictures 7 a, b and 8 a, b graphs of attenuation requirements and the simplest L-shaped links are given, respectively, for bandpass and notch filters.
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It is recommended to synthesize PF and RF using calculations of prototype filters with the same transmission and stopband. For PF, the prototype is a low-pass filter, and for RF, a high-pass filter. The synthesis procedure is as follows:
A) at the first stage of synthesis, frequency conversion is applied, in which the graphical requirements for PF attenuation are converted into requirements for low-pass filter attenuation, and the graphical requirements for RF attenuation are converted into requirements for high-pass filter attenuation:
B) according to the previously discussed method of synthesis of low-pass filters and high-pass filters (points a–f
clause 2.2.2) an electrical circuit is being developed, equivalent to a low-pass filter for the synthesis of the PF, or a high-pass filter for the synthesis of the RF. For a low-pass filter or high-pass filter, graphs of attenuation and voltage transmission coefficient are plotted;
C) the low-pass filter circuit is converted into a bandpass filter circuit by converting longitudinal branches into serial oscillatory circuits and transverse branches into parallel oscillatory circuits by connecting additional reactive elements. The high-pass filter circuit is converted into a notch filter circuit by converting longitudinal branches into parallel oscillatory circuits and transverse branches into serial oscillatory circuits by connecting additional reactive elements. Additional reactive elements for each branch of the low-pass filter (LPF) are determined by the value of the given average frequency of the bandpass or notch filter () and the calculated values of the reactive elements of the branches of the low-pass filter (LPF), using the known expression for resonant circuits:
D) for PF or RF circuits, capacitors and inductors are developed or selected from radio element reference books using the same methodology that was discussed earlier in clause 2.2.2 (clause g) of these guidelines;
E) graphs of attenuation and voltage transfer coefficient of the low-pass filter (LPF) are converted into graphs of the PF (RF) in accordance with the relationships between the frequencies of these filters. For example, to convert low-pass filter to PF graphs:
, (41)
where are the frequencies, respectively, above and below the average frequency of the bandpass filter. Using the same formulas, the graphs of the high-pass filter are converted into graphs of the notch filter.
2.3. Methodology for synthesizing filters based on operating parameters
2.3.1. Basic principles of synthesis based on operating parameters
(polynomial synthesis)
In this synthesis method, just as in synthesis by characteristic parameters, requirements are specified for the type of filter being designed, active load resistance, attenuation or power transfer coefficient in the passband and delay band. However, it is taken into account that the input and output impedances of the filter vary 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 source data by the requirement. The method is based on a mandatory calculation for any type of filter of a low-pass filter - prototype (low-pass filter). The calculations use normalization () and frequency conversions.
The equivalent filter circuit is developed not from individual 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-standardized elements.
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The main stages of calculations on which this synthesis is based are as follows:
A) approximation - replacement of 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) transition to the operator form of recording frequency characteristics (replacing a variable with a variable in the analytical expression that approximates the power transfer coefficient);
B) transition to the expression for the input resistance of the filter, using the relationship between the power transfer coefficient, reflection coefficient and input resistance of the filter:
In expression (44), only one reflection coefficient is used, which corresponds to a stable electrical circuit (the poles of this coefficient do not have a positive real part);
D) decomposition of the analytical expression for the input resistance obtained from (44) into a sum of fractions or into a continued fraction to obtain the equivalent circuit and element values.
Polynomial synthesis in practical developments is usually carried out using filter reference books, which contain calculations for this synthesis method. The reference books provide approximating functions, equivalent circuits and normalized elements of low-pass filters. In most cases, Butterworth and Chebyshev polynomials are used as approximating functions.
The attenuation of a low-pass filter with a Butterworth approximation 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 normalized reactive elements with 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 normalized elements of the Butterworth low-pass filter of a 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 |
