Signal conversion in linear parametric circuits. Signal conversion by linear circuits with constant parameters. Signal conversion in linear circuits

Let a random process with given statistical characteristics operate at the input of a linear two-port network (Fig. 7.1) with a transfer function and impulse response; it is required to find the statistical characteristics of the process at the output of the quadrupole.

In ch. 4, the main characteristics of a random process were considered: probability distribution; correlation function; power spectral density.

Determining the last two characteristics is the simplest task. The situation is different with determining the law of distribution of a random process at the output of a linear circuit. In the general case, with an arbitrary distribution of the process at the input, finding the distribution at the output of the inertial circuit is a very difficult task.

Rice. 7.1. Linear quadripole with constant parameters

Only with a normal distribution of the input process does the problem become simpler, since for any linear operations with a Gaussian process (amplification, filtering, differentiation, integration, etc.) the distribution remains normal, only the functions change.

Therefore, if the probability density of the input process (with zero mean) is given

then the probability density at the output of the linear circuit

Dispersion is easily determined from the spectrum or from the correlation function. Thus, the analysis of the transmission of Gaussian processes through linear circuits essentially comes down to spectral (or correlation) analysis.

The next four paragraphs are devoted to transforming only the spectrum and correlation function of a random process. This consideration is valid for any probability distribution law. The question of transforming the distribution law for non-Gaussian input processes is considered in § 7.6-7.7.

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Signal conversion by linear circuits with constant parameters

1. General information

5.1 Integrating type circuits (low pass filters)

5.2 Differentiation type circuits (high pass filters)

5.3 Frequency selective circuits

Literature

1. General information

An electronic circuit is a set of elements that ensure the passage and conversion of direct and alternating currents over a wide frequency range. It includes sources of electrical energy (power supplies), its consumers and storage devices, as well as connecting wires. Circuit elements can be divided into active and passive.

In active elements it is possible to transform currents or voltages and simultaneously increase their power. These include, for example, transistors, operational amplifiers, etc.

In passive elements, the transformation of currents or voltages is not accompanied by an increase in power, but, as a rule, its decrease is observed.

Sources of electrical energy are characterized by the magnitude and direction of the electromotive force (emf) and the value of internal resistance. When analyzing electronic circuits, the concepts of ideal emf sources (generators) are used. E g (Fig. 1, a) and current I d (Fig. 1, b). They are divided into emf sources. (voltage sources) and current sources, called emf generators, respectively. (voltage generators) and current generators.

Under the emf source understand such an idealized power source, the emf of which does not depend on the current flowing through it. Internal resistance R g of this idealized power supply is zero

A current generator is an idealized power source that delivers current I g in the load, independent of the value of its resistance R n. In order for the current I g current source did not depend on load resistance R n, its internal resistance and its emf. theoretically should tend to infinity.

Real voltage sources and current sources have internal resistance R g of finite value (Fig. 2).

Passive elements of radio engineering circuits include electrical resistances (resistors), capacitors and inductors.

The resistor is an energy consumer. The main parameter of a resistor is active resistance R. Resistance is expressed in ohms (Ohms), kiloohms (kOhms) and megohms (MOhms).

Energy storage devices include a capacitor (electrical energy storage) and an inductor (magnetic energy storage).

The main parameter of a capacitor is capacitance WITH. Capacitance is measured in farads (F), microfarads (µF), nanofarads (nF), picofarads (pF).

The main parameter of an inductor is its inductance L. The inductance value is expressed in henry (H), millihenry (mH), microhenry (µH) or nanohenry (nH).

When analyzing circuits, it is usually assumed that all these elements are ideal, for which the following relationships between the voltage drop are valid: u on the element and the current flowing through it i:

If the element parameters R, L And WITH do not depend on external influences (voltage and current) and cannot increase the energy of the signal acting in the circuit, then they are called not only passive, but also linear elements. Circuits containing such elements are called passive linear circuits, linear circuits with constant parameters, or stationary circuits.

A circuit in which active resistance, capacitance and inductance are assigned to certain sections of it is called a circuit with lumped parameters. If the parameters of a circuit are distributed along it, it is considered a distributed parameter circuit.

The parameters of circuit elements can change over time according to a certain law as a result of additional influences not related to voltages or currents in the circuit. Such elements (and the chains made up of them) are called parametric:

Parametric elements include a thermistor, the resistance of which is a function of temperature, a powder carbon microphone with resistance controlled by air pressure, etc.

Elements whose parameters depend on the magnitude of currents or voltages passing through them on the elements, and the relationships between currents and voltages are described by nonlinear equations, are called nonlinear, and circuits containing such elements are called nonlinear circuits.

The processes occurring in circuits with lumped parameters are described by corresponding differential equations that connect the input and output signals through the circuit parameters.

Linear differential equation with constant coefficients a 0 ,a 1 ,a 2 …a n,b 0 ,b 1 ,..,b m characterizes a linear circuit with constant parameters

Linear differential equations with variable coefficients describe linear circuits with variable parameters.

Finally, processes occurring in nonlinear circuits are described by nonlinear differential equations.

In linear parametric systems, at least one of the parameters changes according to a given law. The result of signal conversion by such a system can be obtained by solving the corresponding differential equation with variable coefficients connecting the input and output signals.

2. Properties of linear circuits with constant parameters

As already indicated, processes occurring in linear circuits with constant lumped parameters are described by linear differential equations with constant coefficients. Let us consider the method of composing such equations using the example of a simple linear circuit consisting of series-connected elements R, L And C(Fig. 3). The circuit is excited by an ideal voltage source of arbitrary shape u(t). The task of the analysis is to determine the current flowing through the elements of the circuit.

According to Kirchhoff's second law, voltage u(t) is equal to the sum of the voltage drops across the elements R, L And C

Ri+L = u(t).

Differentiating this equation, we get

The solution of the resulting inhomogeneous linear differential equation allows us to determine the desired reaction of the circuit - i(t).

The classical method of analyzing signal conversion by linear circuits is to find a general solution to such equations, equal to the sum of the particular solution of the original inhomogeneous equation and the general solution of the homogeneous equation.

The general solution of a homogeneous differential equation does not depend on external influence (since the right side of the original equation, characterizing this influence, is taken equal to zero) and is entirely determined by the structure of the linear chain and the initial conditions. Therefore, the process described by this component of the general solution is called a free process, and the component itself is called a free component.

A particular solution to an inhomogeneous differential equation is determined by the type of exciting function u(t). Therefore, it is called the forced (forced) component, which indicates its complete dependence on external excitation.

Thus, the process occurring in the chain can be considered consisting of two overlapping processes - a forced one, which seemed to occur immediately, and a free one, which takes place only during the transition regime. Thanks to the free components, a continuous approach to the forced (stationary) mode (state) of the linear circuit is achieved in the transient process. In a steady state, the law of changes in all currents and voltages in a linear circuit, up to constant values, coincides with the law of changes in the voltage of an external source.

One of the most important properties of linear circuits, resulting from the linearity of the differential equation describing the behavior of the circuit, is the validity of the principle of independence or superposition. The essence of this principle can be formulated as follows: when several external forces act on a linear chain, the behavior of the chain can be determined by superimposing the solutions found for each of the forces separately. In other words, in a linear chain the sum of the reactions of this chain from various influences coincides with the reaction of the chain from the sum of the influences. It is assumed that the chain is free of initial energy reserves.

Another fundamental property of linear circuits follows from the theory of integration of linear differential equations with constant coefficients. For any, no matter how complex, influence in a linear circuit with constant parameters, no new frequencies arise. This means that none of the signal transformations that involve the appearance of new frequencies (i.e., frequencies not present in the spectrum of the input signal) can, in principle, be carried out using a linear circuit with constant parameters.

3. Analysis of signal conversion by linear circuits in the frequency domain

The classical method of analyzing processes in linear circuits is often associated with the need to carry out cumbersome transformations.

An alternative to the classical method is the operator (operational) method. Its essence consists in the transition through an integral transformation over the input signal from a differential equation to an auxiliary algebraic (operational) equation. Then a solution to this equation is found, from which, using an inverse transformation, a solution to the original differential equation is obtained.

The Laplace transform is most often used as an integral transform, which for a function s(t) is given by the formula:

Where p- complex variable: . Function s(t) is called the original, and the function S(p) - her image.

The reverse transition from the image to the original is carried out using the inverse Laplace transform

Having performed the Laplace transform of both sides of the equation (*), we obtain:

The ratio of the Laplace images of the output and input signals is called the transfer characteristic (operator transfer coefficient) of a linear system:

If the transfer characteristic of the system is known, then to find the output signal from a given input signal it is necessary:

· - find the Laplace image of the input signal;

· - find the Laplace image of the output signal using the formula

· - according to the image S out ( p) find the original (circuit output signal).

As an integral transformation for solving a differential equation, the Fourier transform can also be used, which is a special case of the Laplace transform when the variable p contains only the imaginary part. Note that in order for the Fourier transform to be applied to a function, it must be absolutely integrable. This limitation is removed in the case of the Laplace transform.

As is known, the direct Fourier transform of the signal s(t), given in the time domain, is the spectral density of this signal:

Having performed the Fourier transform of both sides of equation (*), we obtain:

The ratio of the Fourier images of the output and input signals, i.e. the ratio of the spectral densities of the output and input signals is called the complex transmission coefficient of a linear circuit:

If the linear system is known, then the output signal for a given input signal is found in the following sequence:

· determine the spectral density of the input signal using the direct Fourier transform;

· determine the spectral density of the output signal:

Using the inverse Fourier transform, the output signal is found as a function of time

If a Fourier transform exists for the input signal, then the complex transfer coefficient can be obtained from the transfer characteristic by replacing r on j.

Analysis of signal conversion in linear circuits using complex gain is called the frequency domain analysis method (spectral method).

In practice TO(j) are often found using circuit theory methods based on circuit diagrams, without resorting to drawing up a differential equation. These methods are based on the fact that, under harmonic influence, the complex transmission coefficient can be expressed as the ratio of the complex amplitudes of the output and input signals

linear circuit signal integrating

If the input and output signals are voltages, then K(j) is dimensionless, if current and voltage, respectively, then K(j) characterizes the frequency dependence of the resistance of a linear circuit, if voltage and current, then the frequency dependence of conductivity.

Complex transmission coefficient K(j) linear circuit connects the spectra of the input and output signals. Like any complex function, it can be represented in three forms (algebraic, exponential and trigonometric):

where is the dependence on the module frequency

Dependence of phase on frequency.

In the general case, the complex transmission coefficient can be depicted on the complex plane, plotting along the axis of real values, along the axis of imaginary values. The resulting curve is called the complex transmission coefficient hodograph.

In practice, most dependencies TO() And k() are considered separately. In this case, the function TO() is called the amplitude-frequency response (AFC), and the function k() - phase-frequency response (PFC) of the linear system. We emphasize that the connection between the spectrum of the input and output signals exists only in the complex region.

4. Analysis of signal conversion by linear circuits in the time domain

The principle of superposition can be used to determine the response, deprived of the initial energy reserves of a linear circuit, to an arbitrary input stimulus. Calculations in this case turn out to be the simplest if we proceed from the representation of the exciting signal as a sum of standard components of the same type, having first studied the reaction of the circuit to the selected standard component. A unit function (unit step) 1( t - t 0) and delta pulse (unit pulse) ( t - t 0).

The response of a linear circuit to a single step is called its transient response h(t).

The response of a linear circuit to a delta pulse is called the impulse response g(t) of that circuit.

Since a unit jump is an integral of the delta impulse, then the functions h(t) And g(t) are related to each other by the following relationships:

Any input signal of a linear circuit can be represented as a set of delta pulses multiplied by the value of the signal at times corresponding to the position of these pulses on the time axis. In this case, the relationship between the output and input signals of the linear circuit is given by the convolution integral (Duhamel integral):

The input signal can also be represented as a set of unit jumps, taken with weights corresponding to the derivative of the signal at the point of origin of the unit jump. Then

Analysis of signal conversion using impulse or step response is called by time domain analysis method (superposition integral method).

The choice of a time or spectral method for analyzing signal conversion by linear systems is dictated mainly by the convenience of obtaining initial data about the system and the ease of calculations.

The advantage of the spectral method is that it operates with signal spectra, as a result of which it is possible, at least qualitatively, to make a judgment about the change in its shape at the output of the system based on the change in the spectral density of the input signal. When using the time domain analysis method, in the general case, such a qualitative assessment is extremely difficult to make.

5. The simplest linear circuits and their characteristics

Since the analysis of linear circuits can be carried out in the frequency or time domain, the result of signal conversion by such systems can be interpreted in two ways. Time domain analysis allows you to find out the change in the shape of the input signal. In the frequency domain, this result will look like a transformation over a function of frequency, leading to a change in the spectral composition of the input signal, which ultimately determines the shape of the output signal, in the time domain - as a corresponding transformation over a function of time.

The characteristics of the simplest linear circuits are presented in Table 4.1.

5.1 Integrating type circuits (low pass filters)

Signal conversion according to the law

Where m- proportionality coefficient, - value of the output signal at the moment t= 0 is called signal integration.

The operation of integrating unipolar and bipolar rectangular pulses performed by an ideal integrator is illustrated in Fig. 4.

The complex transmission coefficient of such a device amplitude-frequency response phase-frequency response transient response h(t) = t, for t 0.

An ideal element for integrating input current i is an ideal capacitor (Fig. 5), for which

Usually the task is to integrate the output voltage. To do this, it is enough to convert the input voltage source U input into the current generator i. A result close to this can be obtained if a resistor of sufficiently high resistance is connected in series with the capacitor (Fig. 6), at which the current i = (U in - U out)/ R almost independent of voltage U exit This will be true provided U out U input Then the expression for the output voltage (at zero initial conditions U out (0) = 0)

can be replaced by the approximate expression

where is the algebraic (i.e., taking into account the sign) area under the signal expressed by a certain integral on the interval (0, t), is the result of accurate signal integration.

The degree of approximation of the real output signal to the function depends on the degree to which the inequality is satisfied U out U input or, which is almost the same thing, on the degree to which the inequality is satisfied U input . The value is inversely proportional to the value = R.C., which is called the time constant R.C.- chains. Therefore, to be able to use RC- as an integrating circuit, it is necessary that the time constant be sufficiently large.

Complex transmission coefficient R.C.-integrating type circuits

Comparing these expressions with the expressions for the ideal integrator, we find that for satisfactory integration it is necessary to satisfy the condition "1.

This inequality must be satisfied for all components of the input signal spectrum, including the smallest ones.

Step response R.C.- integrating type circuits

Thus, an integrating type RC circuit can perform signal conversion. However, very often there is a need to separate electrical oscillations of different frequencies. This problem is solved using electrical devices called filters. From the spectrum of electrical oscillations applied to the input of the filter, it selects (passes to the output) oscillations in a given frequency range (called the passband), and suppresses (weaken) all other components. According to the type of frequency response, filters are distinguished:

- low frequencies, transmitting oscillations with frequencies no higher than a certain cutoff frequency 0 (passband? = 0 0);

- treble, transmitting vibrations with frequencies above 0 (bandwidth? = 0);

- strip, which transmit vibrations in a finite frequency range 1 2 (bandwidth? = 1 2);

- rejector barriers, delaying oscillations in a given frequency band (stopband? = 1 2).

Frequency response type R.C.-integrating type circuits (Figure 4.6. b) shows that we are dealing with a circuit that effectively passes low frequencies. That's why R.C. This type of circuit can be classified as a low pass filter (LPF). With an appropriate choice of time constant, it is possible to significantly attenuate (filter) the high-frequency components of the input signal and practically isolate the constant component (if any). The cutoff frequency of such a filter is taken to be the frequency at which, i.e. the signal power transmission coefficient is reduced by 2 times. This frequency is often called cutoff frequency With (cutoff frequency 0 ). Cutoff frequency

Additional phase shift introduced R.C.-integrating type circuit at frequency c, is - /4 .

Integrating type circuits also include LR- circuit with resistance at the output (Fig. 6). Time constant of such a circuit = L/R.

5.2 Differentiation type circuits (high pass filters)

Differentiating is a circuit for which the output signal is proportional to the derivative of the input signal.

Where m- proportionality coefficient. Complex transmission coefficient of an ideal differentiating device amplitude-frequency response phase-frequency response transient response h(t) = (t).

An ideal element for converting voltage applied to it into current I, varying proportionally to the derivative is an ideal capacitor (Fig. 4.7).

To obtain a voltage proportional to the input voltage, it is enough to convert the current flowing in the circuit i into a voltage proportional to this current. To do this, just connect a resistor in series with the capacitor R(Fig. 8, b) so low resistance that the law of current change will hardly change ( i ? CdU input/ dt).

However, in reality for R.C.- the circuit shown in Fig. 4.8, A, output signal

and approximate equality U in ( t) ? RCdU input/ dt will be fair only if

Taking into account the previous expression, we get:

The fulfillment of this inequality will be facilitated by a decrease in the time constant = R.C., but at the same time the magnitude of the output signal will decrease U out, which is also proportional.

More detailed analysis of the possibility of use R.C.-circuits as a differentiating circuit can be carried out in the frequency domain.

Complex transmission coefficient for R.C.-chain of differentiating type is determined from the expression

Frequency response and phase response (Fig. 4.8, V) are given accordingly by the expressions:

Comparing the last expressions with the frequency response and phase response of an ideal differentiator, we can conclude that in order to differentiate the input signal, the inequality must be satisfied. It must be satisfied for all frequency components of the input signal spectrum.

Step response R.C.- differentiating type chains

The nature of the behavior of the frequency response R.C.-differentiation type circuit shows that such a circuit effectively passes high frequencies, so it can be classified as a high pass filter (HPF). The cutoff frequency of such a filter is taken to be the frequency at which. She is often called cutoff frequency With (cutoff frequency 0 ). Cutoff frequency

At large time constants f R.C.- differentiating-type circuits, the voltage across the resistor repeats the alternating component of the input signal, and its constant component is completely suppressed. R.C.-the chain in this case is called a dividing chain.

Has the same characteristics R.L.- circuit (Fig. 4.8, b), the time constant of which f =L/ R.

5.3 Frequency selective circuits

Frequency-selective circuits pass to the output only vibrations with frequencies lying in a relatively narrow band around the central frequency. Such circuits are often called linear bandpass filters. The simplest bandpass filters are oscillatory circuits formed by elements L, C And R, and in real circuits the resistance R(loss resistance) is usually the active resistance of reactive elements.

Oscillatory circuits, depending on the connection of their constituent elements in relation to the output terminals, are divided into serial and parallel.

The diagram of a series oscillatory circuit, when the output signal is the voltage removed from the capacitor, is shown in Fig. 9, A.

The complex transmission coefficient of such a circuit

If in a series oscillatory circuit the voltage is removed from the inductance (Fig. 4.9, b), That

At a certain frequency of input oscillations in a series oscillatory circuit, voltage resonance occurs, which is expressed in the fact that the reactances of capacitance and inductance become equal in magnitude and opposite in sign. In this case, the total resistance of the circuit becomes purely active, and the current in the circuit has a maximum value. Frequency that satisfies the condition

called resonant frequency 0:

Size:

represents the resistance module of any of the reactive elements of the oscillatory circuit at the resonant frequency and is called the characteristic (wave) impedance of the circuit.

The ratio of active resistance to characteristic resistance is called circuit attenuation:

The reciprocal d value is called the circuit quality factor:

At resonant frequency

This means that the voltage on each of the reactive elements of the circuit at resonance in Q times the voltage of the signal source.

When finding the quality factor of a real (included in any circuit) series oscillatory circuit, it is necessary to take into account the internal (output) resistance R from the input signal source (this resistance will be connected in series with the active resistance of the circuit) and the active resistance R n load (which will be connected in parallel to the output reactive element). Taking this into account, the equivalent quality factor

It follows that the resonant properties of a series oscillating circuit are best manifested with low-resistance signal sources and with high-resistance loads.

The general diagram of a parallel oscillatory circuit is shown in Fig. 10. In the above diagram, R is the active resistance of the inductance, R1 is the active resistance of the capacitor.

The input signal of such a circuit can only be a current signal, since in the case when the signal source is a voltage generator, the circuit will be shunted.

The case of greatest interest is when the resistance R 1 capacitor WITH direct current is equal to infinity. A diagram of such a circuit is shown in Fig. 4.10, b. In this case, the complex transfer coefficient

The complex transfer coefficient of a parallel oscillatory circuit (i.e., the total resistance of the circuit) is real at the resonant frequency p, satisfying the condition

where is the resonant frequency of the series oscillatory circuit.

At resonant frequency p

Note that at this frequency the currents flowing through the capacitor WITH and inductor L, shifted in phase by, equal in magnitude and in Q times the current I input of the signal source.

Due to the finiteness of internal resistance R from the signal source, the quality factor of the parallel circuit decreases:

It follows that the resonant properties of a parallel oscillatory circuit are best manifested with signal sources with a high output resistance ( R s "), i.e. current generators.

For parallel oscillatory circuits with high quality factor used in practice, the active loss resistance R significantly less inductive reactance L, therefore for the complex coefficient K(j ) we will have:

As follows from these expressions, the resonant frequency of a high-quality parallel oscillatory circuit

The impulse response of such a circuit

its transient response

For an ideal parallel oscillating circuit (lossless circuit, i.e. R = 0)

The bandwidth of the oscillatory circuits is entered similarly to the bandwidth R.C.-chains, i.e. as the frequency range within which the modulus of the complex transmission coefficient exceeds the level of the maximum (at resonance) value. With high quality factors of the circuits and small deviations (misalignments) of frequencies relative to the resonant frequency, the frequency response of the series and parallel oscillatory circuits are almost identical. This allows us to obtain, although an approximate, but quite acceptable in practice, relationship between the bandwidth and circuit parameters

Literature

Zaichik M.Yu. and others. Collection of educational and control tasks on the theory of electrical circuits. - M.: Energoizdat, 1981.

Borisov Yu.M. Electrical engineering: textbook. manual for universities / Yu.M. Borisov, D.N. Lipatov, Yu.N. Zorin. - 3rd edition, revised. and additional ; Grif MO. - Minsk: Higher. school A, 2007. - 543 s.

Grigorash O.V. Electrical engineering and electronics: textbook. for universities / O.V. Grigorash, G.A. Sultanov, D.A. Norms. - Vulture UMO. - Rostov n/d: Phoenix, 2008. - 462 s.

Lotoreychuk E.A. Theoretical foundations of electrical engineering: textbook. for students institutions prof. education / E.A. Lotoreychuk. - Grif MO. - M.: Forum: Infra-M, 2008. - 316 p.

Fedorchenko A. A. Electrical engineering with fundamentals of electronics: textbook. for students prof. schools, lyceums and students. colleges / A. A. Fedorchenko, Yu. G. Sindeev. - 2nd ed. - M.: Dashkov and K°, 2010. - 415 p.

Kataenko Yu. K. Electrical engineering: textbook. allowance / Yu. K. Kataenko. - M.: Dashkov and Co.; Rostov n/d: Akademtsentr, 2010. - 287 p.

Moskalenko V.V. Electric drive: Textbook. allowance for the environment. prof. education / V.V. Moskalenko. - M.: Masterstvo, 2000. - 366 p.

Savilov G.V. Electrical engineering and electronics: a course of lectures / G.V. Savilov. - M.: Dashkov and K°, 2009. - 322 p.

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Methods for analyzing processes in linear circuits (systems)

When analyzing processes, it is necessary to determine the response of the circuit to an input signal in the form of a signal of a given shape. From the basics of circuit theory it is known that to analyze the passage of harmonic signals through linear circuits, Kirchhoff's laws, methods of loop currents and nodal potentials, the equivalent generator method and other simple methods are used. These methods are also applicable for analysis under random influence. However, in communication theory they deal with pulsed signals, which are more diverse in shape and spectral composition and are described by a large number of parameters. These chains are also complex in structure. When analyzing the impact of signals on such circuits, spectral and operator methods and the superposition integral method are used.

Spectral method. The properties of linear circuits (four-terminal networks) can be determined using this parameter, as frequency transfer coefficient. To do this, it is necessary to consider the response of a linear two-port network to the input influence and evaluate their connection with each other.

Let us introduce the concepts of complex amplitudes of input and output harmonic voltages with angular (circular) frequency with:

The ratio of the complex amplitudes of the output and input harmonic voltages of the same frequency determines frequency gain(usually just - transmission coefficient) linear circuit (linear quadrupole):

Gain Module TO( co) = |K(co)| called amplitude-frequency response(frequency response), and the argument cp(co) - phase-frequency characteristic(PFC) of a linear quadrupole. As a rule, the frequency response has one maximum, and the phase response varies monotonically depending on the frequency (Fig. 4.2).

In the region of a certain frequency band, the response of the circuit to the input influence begins to decrease. Therefore, they use the concept bandwidth (working strip) - frequency regions where the modulus of the transmission coefficient TO(с) not less than 1/V2 = 0.707 of its maximum value. The most convenient in practice is the normalized transmission coefficient modulus K/K shks, the maximum value of which is 1. The value of 1/V2, by which the bandwidth of a linear circuit is determined, was not introduced by chance. The thing is,

Rice. 4.2.

A - frequency response; b- The phase response is that at the boundaries of the passband the modulus of the power transmission coefficient, equal to the ratio of the output and input powers, is halved. In Fig. 4.2, the passband is contained in the region from the lower co n to the upper co in frequency, and therefore its width Dso 0 = co in - co,. In practice it is often used cyclical frequency /= /(2). Then the circuit bandwidth

where / and - lower, and / in - upper boundary cyclic frequencies.

The issue of frequency transfer coefficient can be approached from another point of view. If a harmonic signal of unit amplitude is supplied to the input of a linear value, having a complex analytical model of the form uBX(t)= e J(0t , then the signal at its output will be written as uBbai(t)= TO( Substituting these expressions into formula (4.1), after simple transformations we write the frequency transmission coefficient in the form of a differential equation

According to formula (4.3), the frequency transmission coefficient of a linear circuit, in which the connection between the input and output signals is described by a differential equation with constant coefficients, is a fractional-rational function of the variable y co. In this case, the coefficients of this function coincide with the coefficients of the differential equation.

Using frequency gain TO( co) you can determine the signal at the output of a linear quadrupole. Let at the input of a linear four-port network with a frequency transfer coefficient TO( co) there is a continuous signal of arbitrary shape in the form of voltage m input (?). Applying the direct Fourier transform (2.29), we determine the spectral density of the input signal 5 in (co). Then the spectral density of the signal at the output of the linear circuit

Having carried out the inverse Fourier transform (2.30) from the spectral density (4.4), we write the output signal as

Operator method. Along with the spectral method, the operator method is used, which is based on the representation of input and output signals by Laplace transforms. The term “operator method” was introduced by O. Heaviside. He proposed a symbolic method for solving linear differential equations that describe transient processes in linear circuits. Heaviside's method is based on replacing the differentiation operator d/dt complex parameter r, which transfers signal analysis from the time domain to the domain of complex quantities. Consider a complex or real analog signal u(t), defined at t> 0 and equal to zero at time t = 0.

Laplace transform this signal is a function of a complex variable r, expressed by the integral

Analytical recording of the signal u(t) called original, and the function U(p) - his Laplace image(simpler - image). Integral

(4.6) superficially resembles the direct Fourier transform (2.29). However, there is a fundamental difference between them. The integral of the direct Fourier transform (2.29) includes the imaginary frequency acous, and the Laplace integral
(4.6) is a complex operator that can be considered as complex frequency p= a + uso (a is the real component), in this case only positive time values are considered t. Due to the multiplier e~w under the integral in formula (4.6) for U(p) The Laplace transform is also possible for non-integrable functions u(t).

The use of the concept of complex frequency in the integral transform makes it more efficient than the Fourier transform. For example, using formula (2.29) it is impossible to directly determine the spectrum of the inclusion function a(?) = 1(0- However, for the same signal directly using formula (4.6) it is easy to find its operator image:

or, because e~ a ‘°° = 0, we get

From the above example, it is obvious that the increase in the efficiency of transformation (4.6) is due to the presence of the multiplier e -a/, which ensures the convergence of this integral even for signals that do not satisfy the condition for the convergence of the integral . The presence of this multiplier allows us to interpret the Laplace transform (4.6) as a representation of the signal in the form of a “spectrum” of damped oscillations e w e,w = = e (a+уe j (in symbolic form).

The Laplace transform (4.6) has linear properties similar to the linearity property of the Fourier transform:

Among other properties, we note a simpler image transformation when differentiating and integrating a signal compared to similar Fourier transforms. Simplification is not only due to the complexity of the operator r, but also with the fact that the originals are analyzed on an infinite interval.

By analogy with the inverse Fourier transform, one introduces inverse integral Laplace transform, which is carried out using deductions:

where a, is a real variable reflected on the complex plane.

Solving differential equations using the operator method. The Laplace transform allows you to solve linear differential equations with constant coefficients. Let it be necessary to find a solution to the differential equation (4.1). Let us establish a number of assumptions:

input signal u BX (t) = 0 at t
the input signal contains only those functions for which Laplace transforms exist;
the initial conditions are zero, i.e. g/out (0) = 0.

Let us introduce correspondences between the originals of the input and output signals and their Laplace images:

Carrying out the Laplace transform of both sides of formula (4.1), we obtain

In the theory of automatic systems, the factor before U Bblx (p) in formula (4.8) are denoted by Q(p), calling own operator system, and the factor in front U nx (p) - through R(p) and call impact operator.

The operator method is based on the most important characteristic, which is the ratio of the images of the output and input signals:

and called transfer function (operator transfer coefficient) linear circuit.

Using equation (4.8), we find

Comparison of formulas (4.3) and (4.9) shows that the function K(r) reflects the result of the analytical transfer of the complex frequency transmission coefficient /((co) from the imaginary axis jeo over the entire range of complex frequencies p = a + jco.

If the transfer function is known K(r), then the output response of the circuit to a given input influence u nx (t) can be determined according to the following scheme:

record the input signal image uBX(t) -? U BX (p)
find output image 0 uyh (p) = K(p)U ux (p)
calculate output signal u ttblx(t) - 5 ? 0 out(p).

Roots of the denominator p v p 2 > ->P p in formula (4.9), i.e. roots of function

called poles transfer function K(r).

Accordingly, the roots of the numerator z v z2, z m functions K(r), those. roots of function

characterized as bullets transfer function.

In real electrical circuits p>t.

When dividing the numerator by the denominator in formula (4.9), a constant factor appears K 0, and this equation takes the so-called zero-pole transfer function representation

Actual odds values a p And b t differential equation (4.16) determines the following property of the poles and zeros of the transfer function of a linear quadripole: either all these numbers are real or form complex conjugate pairs.

Rice. 4.3.

Very often, a visual technique is used to display the zeros and poles of the transfer function on the complex plane a, . In this case, poles are usually designated by crosses, and zeros by circles. For example, in Fig. 4.3 the circle at the origin of coordinates shows zero, and the crosses 1 And 2 - poles of the transfer function of some oscillatory value. Poles 1 And 2 are negative, real and determine the difference of two damping exponentials. Complex conjugate poles 3 And 4 determine the oscillatory nature of the transfer function K(r) with the greater attenuation, the more to the left they are located, and with the greater the frequency of damped oscillations, the further they move up and down from the real axis a. The location of the poles in the left half-plane corresponds to the damped nature of the transfer function. The zeros of the transfer function can be located in both the left and right half-plane.

Dynamic representation of linear circuits. Superposition integral method. The properties of linear circuits are often easier to evaluate by the way they respond to the influence of elementary signals. Two types of dynamic representation of linear circuits have found application. According to the first of them, to analyze the response of the circuit, rectangular pulses of duration D are used as elementary signals, in the limit tending to the delta function. These pulses are directly adjacent to each other and form a sequence inscribed in or described around the curve. In the second method, the elementary signals are step functions that appear in the form of switching functions at equal time intervals A. The height of each step is equal to the signal increment over time interval D.

One of the elementary electrical signals used in analyzing the passage of various oscillations through linear circuits (four-terminal networks) is the delta function 5(?). Another elementary electrical signal in communication technology is the switching function a(?).

The delta function and the inclusion function are related analytically. The result of differentiating the inclusion function is the delta function

Respectively

Example 4.1

Let us find the derivative of the product of the exponential momentum and the inclusion function u(t) = e~ at v(t).

Solution

For function e~w at a point in time t = 0 e~ a "° = 1. Derivative As a result of the calculations, we obtain the following expression:

Pulse and transient characteristics of a linear circuit. Linearity and stationarity make it easy to find the reaction of a linear system theoretically to any input signal, knowing only one function - the system’s response to a delta function applied to the input 8 (t). This reaction is called impulse response or convolution kernel linear circuit (system) and denote h(t). Various types of real impulse responses of linear circuits h v h 2 , h 3 shown in Fig. 4.4, A.

Rice. 4.4.

A- various types of pulse; b - transitional

The response of a linear circuit to a unit function is step response g(t)(Fig. 4.4, b). Let us assume that it is necessary to determine the output signal and output (?) of a linear circuit (linear two-port network), if its impulse response is known h(t) and input signal uBX(t). Let us approximately replace the input signal curve u nx (t) a stepped line in the form of a set of fairly short rectangular pulses having the same duration At (Fig. 4.5, A).

Rice. 4.5.

A- input signal; b - pulse responses and output signal

The generation of the output signal can be explained as follows. A fairly small “piece” of the input signal with a duration of At is supplied to the input of the circuit being analyzed. If we choose the pulse duration At to be infinitesimal, then the response of a linear circuit to the first rectangular pulse will be approximately equal to the response of the same circuit to the delta function (and this will be the impulse response), multiplied by the area (and nx (0) At) of the first pulse , i.e. u nx (0)Axh(t)(Fig. 4.5, b). The response of a linear circuit to the second pulse with sufficient accuracy is the product r/in ( Ax)Axh(t - At), where and in (At) At is the area of this pulse, and the magnitude h(t- At) - impulse response of a linear circuit corresponding to a moment in time t= At. Therefore, for some arbitrary moment of time t = groin (p - the number of conditionally formed pulses per time interval ) the response of a linear circuit will be approximately expressed by the sum (dashed line in Fig. 4.5, b)

If the pulse duration At consistently approaches zero, then a small time increment At turns into dx, and the operation of summation is transformed into the operation of integration over the variable m = kAx:

For real linear circuits always h(t) = 0 at t

This fundamental relationship in linear circuit theory is imposition integral, or Duhamel integral Let us remind you that

integral (4.13) is called convolution two functions (see Chapter 2). So, a linear system convolves the input signal with its impulse response, resulting in an output signal. Formula (4.13) has a clear physical meaning: a linear stationary circuit, processing the input signal, carries out a weighted summation of all its instantaneous values that existed “in the past”.

Convolution technique. To calculate the convolution using expression (4.13), the impulse response function is reversed along its coordinate, i.e. is built in reverse time mode, and moves relative to the input signal function towards increasing values L At each current moment in time, the values of both functions are multiplied, and the product is integrated within the impulse response window. The result obtained refers to the coordinate point opposite which the value of the impulse response /?(()) is located. In the theory of electrical circuits, another, equivalent form of the Duhamel integral is used:

So, the linear system transforms with respect to the variable t functions included in formula (4.14). In this case, the input signal is converted into an output signal mout (?)> and the delta function 8(t- t) - into impulse response h(t- T). The function m in (t) does not depend on the variable t and therefore remains unchanged. The result is a formula showing that the output signal of a linear system is equal to the convolution of the input signal with its impulse response:

Let us determine the relationship between the impulse response and the frequency transmission coefficient of a linear circuit. Let's use the complex form of a harmonic signal of unit amplitude and in(?) = exp(/co?). Substituting this expression into formula (4.14) and taking it out of the integral sign, we find the response of the circuit:

The integral in parentheses is a complex function of frequency

and represents the transmission coefficient (here a formal replacement of t is made with t).

Expression (4.15) establishes an extremely important fact - the frequency transfer coefficient and the impulse response of a linear circuit are related by a direct Fourier transform. It is also obvious that there is an inverse Fourier transform for the transmission coefficient and impulse response

with which you can easily determine the impulse response of a circuit from its frequency transfer coefficient.

Since there is a simple connection between 6(7t) and a(t) according to formulas (4.10) and (4.11), all conclusions for a linear circuit made using the delta function are easily transferred to the inclusion function. Using similar reasoning and calculations, it can be shown that it is possible to simply represent input and output signals using the enable function a(t) and transient response of a linear circuit g(t). Having broken the input signal (Fig. 4.6) into elementary switching functions D mst (7) (here A And - amplitude of an elementary input voltage jump) and proceeding in the same way as when deriving relation (4.12), we obtain another form of the Duhamel integral, which allows us to determine the signal at the output of a linear circuit:

Rice. 4.6.

In the theory of linear circuits, a certain connection has been established between impulse and transient characteristics. Since the transition characteristic neiiHg(?) is a response to the unit function ст(/,), which, in turn, is an integral of the delta function 8(7) (see formula (4.11)), then between the functions h(t.) And g(t) there is an integral relation

Experimentally, the impulse response of a linear circuit can be constructed by applying a short pulse of unit area to its input and reducing the pulse duration while maintaining the area until the output signal stops changing. This will be the impulse response of the circuit.

Jean-Marie Duhamel (J. Duhamel, 1797-1872) - French mathematician.

Parametric (linear circuits with variable parameters), are called radio circuits, one or more parameters of which change over time according to a given law. It is assumed that the change (more precisely, modulation) of any parameter is carried out electronically using a control signal. In radio engineering, parametric resistance R(t), inductance L(t) and capacitance C(t) are widely used.

An example of one of the modern parametric resistances can serve as a channel of a VLG transistor, the gate of which is supplied with a control (heterodyne) alternating voltage u g (t). In this case, the slope of its drain-gate characteristic changes over time and is related to the control voltage by the functional dependence S(t)=S. If you also connect the voltage of the modulated signal u(t) to the VLG transistor, then its current will be determined by the expression:

i c (t)=i(t)=S(t)u(t)=Su(t). (5.1)

As with the class of linear circuits, the principle of superposition is applicable to parametric circuits. Indeed, if the voltage applied to the circuit is the sum of two variables

u(t)=u 1 (t)+u 2 (t), (5.2)

then, substituting (5.2) into (5.1), we obtain the output current also in the form of the sum of two components

i(t)=S(t)u 1 (t)+S(t)u 2 (t)= i 1 (t)+ i 2 (t) (5.3)

Relation (5.3) shows that the response of a parametric circuit to the sum of two signals is equal to the sum of its responses to each signal separately.

Conversion of signals in circuits with parametric resistance. Parametric resistances are most widely used to convert the frequency of signals. Note that the term “frequency conversion” is not entirely correct, since the frequency itself is unchanged. Obviously, this concept arose due to an inaccurate translation of the English word “heterodyning”. Heterodyning – it is the process of nonlinear or parametric mixing of two signals of different frequencies to produce a third frequency.

So, frequency conversion is a linear transfer (mixing, transformation, heterodyning, or transposition) of the spectrum of a modulated signal (as well as any radio signal) from the carrier frequency region to the intermediate frequency region (or from one carrier frequency to another, including a higher one) without changing type or nature of modulation.

Frequency converter(Fig. 5.1) consists of a mixer (SM) - a parametric element (for example, an MOS transistor, a varicap or a conventional diode with a quadratic characteristic), a local oscillator (G) - an auxiliary self-oscillator of harmonic oscillations with a frequency ω g, which serves for parametric control of the mixer, and an intermediate frequency filter (usually an oscillating circuit of an amplifier or UHF).

Fig.5.1. Block diagram of the frequency converter

Let's consider the operating principle of a frequency converter using the example of transferring the spectrum of a single-tone AM signal. Let us assume that under the influence of heterodyne voltage

u g (t)=U g cos ω g t (5.4)

The slope of the characteristic of the MOS transistor of the frequency converter changes over time approximately according to the law

S(t)=S o +S 1 cos ω g t (5.5)

where S o and S 1 are, respectively, the average value and the first harmonic component of the slope of the characteristic.

When an AM signal u AM (t)= U n (1+McosΩt)cosω o t arrives at the MOS transistor of the mixer, the alternating component of the output current in accordance with (5.1) and (5.5) will be determined by the expression:

i c (t)=S(t)u AM (t)=(S o +S 1 cos ω g t) U n (1+McosΩt)cosω o t=

U n (1+McosΩt) (5.6)

Let us choose as the intermediate frequency of the parametric converter

ω pch =|ω g -ω o |. (5.7)

Then, having isolated it using the IF loop from the current spectrum (5.6), we obtain a converted AM signal with the same modulation law, but a significantly lower carrier frequency

i pch (t)=0.5S 1 U n (1+McosΩt)cosω pch t (5.8)

Note that the presence of only two side components of the current spectrum (5.6) is determined by the choice of an extremely simple piecewise linear approximation of the slope of the transistor characteristic. In real mixer circuits, the current spectrum also contains components of combination frequencies

ω pc =|mω g ±nω o |, (5.9)

where m and n are any positive integers.

The corresponding time and spectral diagrams of signals with amplitude modulation at the input and output of the frequency converter are shown in Fig. 5.2.

Fig.5.2. Diagrams at the input and output of the frequency converter:

a – temporary; b – spectral

Frequency converter in analog multipliers. Modern frequency converters with parametric resistive circuits are built on a fundamentally new basis. They use analog multipliers as mixers. If two harmonic oscillations of a certain modulated signal are applied to the inputs of the analog multiplier:

u c (t)=U c (t)cosω o t (5.10)

and the local oscillator reference voltage u g (t) = U g cos ω g t, then its output voltage will contain two components

u out (t)=k a u c (t)u g (t)=0.5k a U c (t)U g (5.11)

Spectral component with difference frequency ω fc =|ω g ±ω o | is isolated by a narrow-band IF filter and is used as the intermediate frequency of the converted signal.

Frequency conversion in varicap circuits. If only heterodyne voltage (5.4) is applied to the varicap, then its capacitance will approximately change over time according to the law (see Fig. 3.2 in part I):

C(t)=C o +C 1 cosω g t, (5.12)

where C o and C 1 are the average value and the first harmonic component of the varicap capacity.

Let us assume that the varicap is affected by two signals: heterodyne and (to simplify calculations) unmodulated harmonic voltage (5.10) with amplitude U c . In this case, the charge on the varicap capacity will be determined:

q(t)=C(t)u c (t)=(C o +C 1 cosω g t)U c cosω o t=

С o U c (t)cosω o t+0.5С 1 U c cos(ω g - ω o)t+0.5С 1 U c cos(ω g + ω o)t, (5.13)

and the current flowing through it is

i(t)=dq/dt=- ω o С o U c sinω o t-0.5(ω g -ω o)С 1 U c sin(ω g -ω o)t-

0.5(ω g +ω o)С 1 U c sin(ω g +ω o)t (5.14)

By connecting an oscillatory circuit in series with a varicap, tuned to the intermediate frequency ω fc =|ω g -ω o |, the desired voltage can be selected.

With a varicap-type reactive element (for ultra-high frequencies this is varactor) you can also create a parametric oscillator, power amplifier, frequency multiplier. This possibility is based on the conversion of energy in a parametric capacitance. It is known from a physics course that the energy accumulated in a capacitor is related to its capacitance C and the charge q on it by the formula:

E = q 2 /(2C). (5.15)

Let the charge remain constant and the capacitance of the capacitor decrease. Since energy is inversely proportional to the size of the capacitance, when the latter is reduced, the energy increases. We obtain a quantitative relationship for such a connection by differentiating (5.15) with respect to the parameter C:

dE/dC= q 2 /2C 2 = -E/C (5.16)

This expression is also valid for small increments of capacity ∆C and energy ∆E, so we can write

∆E=-E (5.17)

The minus sign here shows that a decrease in capacitance of the capacitor (∆C<0) вызывает увеличение запасаемой в нем энергии (∆Э>0). The increase in energy occurs due to the external costs of performing work against the forces of the electric field when the capacitance is reduced (for example, by changing the bias voltage on the varicap).

When the parametric capacitance (or inductance) of several signal sources with different frequencies is simultaneously exposed, there will be a redistribution (exchange) of vibration energies. In practice, the oscillation energy of an external source called pump generator, is transmitted through a parametric element to the useful signal circuit.

To analyze the energy relationships in multi-circuit circuits with a varicap, let us turn to the generalized diagram (Fig. 5.3). It contains three circuits parallel to the parametric capacitance C, two of which contain sources e 1 (t) and e 2 (t), creating harmonic oscillations with frequencies ω 1 and ω 2. The sources are connected through narrow-band filters Ф 1 and Ф 2, transmitting oscillations with frequencies ω 1 and ω 2, respectively. The third circuit contains a load resistance R n and a narrow-band filter Ф 3, the so-called idle circuit, tuned to a given combination frequency

ω 3 = mω 1 +nω 2, (5.18)

where m and n are integers.

For simplicity, we will assume that the circuit uses filters without ohmic losses. If in the circuit the sources e 1 (t) and e 2 (t) supply power P 1 and P 2, then the load resistance R n consumes power R n. For a closed system, in accordance with the law of conservation of energy, we obtain the power balance condition:

P 1 + P 2 + P n = 0 (5.19)

In nonlinear electrical circuits, the connection between the input signal U In . (T) and output signal U Out . (T) described by a nonlinear functional relationship

Such a functional dependence can be considered as a mathematical model of a nonlinear chain.

Typically, a nonlinear electrical circuit is a combination of linear and nonlinear two-terminal networks. To describe the properties of nonlinear two-terminal networks, their current-voltage characteristics (CV characteristics) are often used. As a rule, the current-voltage characteristics of nonlinear elements are obtained experimentally. As a result of the experiment, the current-voltage characteristics of the nonlinear element are obtained in the form of a table. This description method is suitable for analyzing nonlinear circuits using a computer.

To study processes in circuits containing nonlinear elements, it is necessary to display the current-voltage characteristic in a mathematical form convenient for calculations. To use analytical methods of analysis, it is necessary to select an approximating function that sufficiently accurately reflects the features of the experimentally measured characteristic. The following methods of approximating the current-voltage characteristics of nonlinear two-terminal networks are most often used.

Exponential approximation. From the theory of p-n junction operation it follows that the current-voltage characteristic of a semiconductor diode at u>0 is described by the expression

. (7.3)

The exponential relationship is often used when studying nonlinear circuits containing semiconductor devices. The approximation is quite accurate for current values not exceeding a few milliamps. At high currents, the exponential characteristic smoothly turns into a straight line due to the influence of the volume resistance of the semiconductor material.

Power approximation. This method is based on the expansion of the nonlinear current-voltage characteristic into a Taylor series, converging in the vicinity of the operating point U0 :

Here are the coefficients... – some numbers that can be found from the experimentally obtained current-voltage characteristic. The number of expansion terms depends on the required accuracy of calculations.

It is not advisable to use the power-law approximation for large signal amplitudes due to a significant deterioration in accuracy.

Piecewise linear approximation It is used in cases where large signals operate in the circuit. The method is based on the approximate replacement of the real characteristic with segments of straight lines with different slopes. For example, the transfer characteristic of a real transistor can be approximated by three straight lines, as shown in Fig. 7.1.

Fig.7.1.Transfer characteristic of a bipolar transistor

The approximation is determined by three parameters: the characteristic start voltage, the slope, which has the dimension of conductivity, and the saturation voltage, at which the current stops increasing. The mathematical notation of the approximated characteristic is as follows:

(7.5)

In all cases, the task is to find the spectral composition of the current due to the effect of harmonic voltages on the nonlinear circuit. In piecewise linear approximation, circuits are analyzed using the cutoff angle method.

Let us consider, as an example, the operation of a nonlinear circuit with large signals. As a nonlinear element we use a bipolar transistor that operates with a collector current cutoff. To do this, using the initial bias voltage E The operating point is set in such a way that the transistor operates with the collector current cut off, and at the same time we supply an input harmonic signal to the base.

Fig.7.2. Illustration of current cutoff at large signals

The cutoff angle θ is half of that part of the period during which the collector current is not equal to zero, or, in other words, the part of the period from the moment the collector current reaches its maximum to the moment when the current becomes equal to zero - “cut off”.

In accordance with the designations in Fig. 7.2, the collector current for I> 0 is described by the expression

Expanding this expression into a Fourier series allows us to find the constant component I0 and amplitudes of all collector current harmonics. Harmonic frequencies are multiples of the input signal frequency, and the relative amplitudes of the harmonics depend on the cutoff angle. The analysis shows that for each harmonic number there is an optimal cutoff angle θ, At which its amplitude is maximum:

. (7.7)

Fig.7.8. Frequency multiplication circuit

Similar circuits (Fig. 7.8) are often used to multiply the frequency of a harmonic signal by an integer factor. By adjusting the oscillatory circuit included in the collector circuit of the transistor, you can select the desired harmonic of the original signal. The cutoff angle is set based on the maximum amplitude value of a given harmonic. The relative amplitude of a harmonic decreases as its number increases. Therefore, the described method is applicable for multiplication coefficients N≤ 4. Using multiple frequency multiplication, it is possible, based on one highly stable harmonic oscillator, to obtain a set of frequencies with the same relative frequency instability as that of the main generator. All these frequencies are multiples of the input signal frequency.

The property of a nonlinear circuit to enrich the spectrum, creating spectral components at the output that were initially absent at the input, is most clearly manifested if the input signal is the sum of several harmonic signals with different frequencies. Let us consider the case of the influence of the sum of two harmonic oscillations on a nonlinear circuit. We represent the current-voltage characteristic of the circuit as a polynomial of the 2nd degree:

. (7.8)

In addition to the constant component, the input voltage contains two harmonic oscillations with frequencies and , the amplitudes of which are equal to and respectively:

. (7.9)

Such a signal is called biharmonic. Substituting this signal into formula (7.8), performing transformations and grouping terms, we obtain a spectral representation of the current in a nonlinear two-terminal network:

It can be seen that the current spectrum contains terms included in the spectrum of the input signal, second harmonics of both input signal sources, as well as harmonic components with frequencies ω 1 — ω 2 and ω 1 + ω 2 . If the power-law expansion of the current-voltage characteristic is represented by a polynomial of the 3rd degree, the current spectrum will also contain frequencies. In the general case, when a nonlinear circuit is exposed to several harmonic signals with different frequencies, combination frequencies appear in the current spectrum

Where are any integers, positive and negative, including zero.

The appearance of combinational components in the spectrum of the output signal during nonlinear transformation causes a number of important effects that have to be encountered when constructing radio-electronic devices and systems. So, if one of the two input signals is amplitude modulated, then modulation is transferred from one carrier frequency to another. Sometimes, due to nonlinear interaction, an amplification or suppression of one signal by another is observed.

Based on nonlinear circuits, detection (demodulation) of amplitude-modulated (AM) signals in radio receivers is carried out. The circuit of the amplitude detector and the principle of its operation are explained in Fig. 7.9.

Fig.7.9. Amplitude detector circuit and output current shape

A nonlinear element, the current-voltage characteristic of which is approximated by a broken line, passes only one (in this case positive) half-wave of the input current. This half-wave creates voltage pulses of high (carrier) frequency on the resistor with an envelope that reproduces the shape of the amplitude-modulated signal envelope. The voltage spectrum across the resistor contains the carrier frequency, its harmonics and a low-frequency component, which is approximately half the amplitude of the voltage pulses. This component has a frequency equal to the frequency of the envelope, i.e. it represents a detected signal. The capacitor together with the resistor forms a low-pass filter. When the condition is met

(7.12)

Only the envelope frequency remains in the output voltage spectrum. In this case, the output voltage also increases due to the fact that with a positive half-wave of the input voltage, the capacitor is quickly charged through the low resistance of the open nonlinear element almost to the amplitude value of the input voltage, and with a negative half-wave, it does not have time to discharge through the high resistance of the resistor. The given description of the operation of the amplitude detector corresponds to the mode of a large input signal, in which the current-voltage characteristic of a semiconductor diode is approximated by a broken straight line.

In the small input signal mode, the initial section of the diode’s current-voltage characteristic can be approximated by a quadratic dependence. When an amplitude-modulated signal is applied to such a nonlinear element, the spectrum of which contains a carrier and side frequencies, frequencies with sum and difference frequencies arise. The difference frequency represents the detected signal, and the carrier and sum frequencies do not pass through the low-pass filter formed by the elements and .

A common technique for detecting frequency modulated (FM) waveforms is to first convert the FM waveform into an AM waveform, which is then detected in the manner described above. An oscillatory circuit detuned relative to the carrier frequency can serve as the simplest FM to AM converter. The principle of converting FM signals to AM is explained in Fig. 7.10.

Fig.7.10. Converting FM to AM

In the absence of modulation, the operating point is on the slope of the circuit's resonance curve. When the frequency changes, the amplitude of the current in the circuit changes, i.e., the FM is converted to AM.

The circuit of the FM to AM converter is shown in Fig. 7.11.

Fig.7.11. FM to AM converter

The disadvantage of such a detector is the distortion of the detected signal, which arises due to the nonlinearity of the resonant curve of the oscillatory circuit. Therefore, in practice, symmetrical circuits with better characteristics are used. An example of such a circuit is shown in Fig. 7.12.

Fig.7.12. FM signal detector

Two circuits are tuned to extreme frequency values, i.e., to frequencies AND. Each of the circuits converts FM to AM, as described above. AM oscillations are detected by appropriate amplitude detectors. Low-frequency voltages are opposite in sign, and their difference is removed from the output of the circuit. The detector response, i.e. the output voltage versus frequency, is obtained by subtracting the two resonance curves and is more linear. Such detectors are called discriminators.

Signal conversion in linear parametric circuits. Signal conversion by linear circuits with constant parameters. Signal conversion in linear circuits

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