The classical method of analyzing processes in linear circuits often turns out to be associated with the need for cumbersome transformations.
An alternative to the classical method is the operator (operational) method. Its essence consists in the transition by means of 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 the inverse transformation, a solution to the original differential equation is obtained.
As an integral transform, the Laplace transform is most often used, which for the 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
After completing the Laplace transform of both sides of the equation (*), we get:
The ratio of the Laplace images of the output and input signals is called the transfer characteristic (operator transfer ratio) of the linear system:
If the transfer characteristic of the system is known, then to find the output signal for a given input signal, it is necessary:
· - find the Laplace image of the input signal;
- find the Laplace image of the output signal by the formula
- according to the image S out ( p) find the original (circuit output).
The Fourier transform, which is a special case of the Laplace transform, when the variable p contains only the imaginary part. Note that in order to apply the Fourier transform to a function, it must be absolutely integrable. This limitation is lifted in the case of the Laplace transform.
As you know, 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 the equation (*), we get:
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 the linear circuit:
If the complex gain of a 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 direct Fourier transform;
Determine the spectral density of the output signal:
Using the inverse Fourier transform, find the output signal as a function of time
If there is a Fourier transform for the input signal, then the complex gain can be obtained from the gain by replacing R on the j.
Analysis of the transformation of signals in linear circuits using a complex gain is called a frequency domain (spectral) analysis method.
On practice TO(j) are often found by the methods of circuit theory on the basis of schematic diagrams, without resorting to drawing up a differential equation. These methods are based on the fact that under harmonic action, 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, respectively, by current and voltage, then K(j) characterizes the frequency dependence of the resistance of a linear circuit, if by voltage and current, then - the frequency dependence of conductivity.
Complex transmission ratio K(j) of a 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
Phase versus 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 of the addiction TO() and k() are considered separately. In this case, the function TO() is called the amplitude-frequency characteristic (AFC), and the function k() - phase-frequency characteristic (PFC) of the linear system. We emphasize that the relationship between the spectrum of the input and output signals exists only in the complex domain.
Parametric (linear circuits with variable parameters), are called radio circuits, one or more parameters of which change in time according to a given law. It is assumed that the change (more precisely, the modulation) of a parameter is carried out electronically using a control signal. In radio engineering, parametric resistances R (t), inductance L (t) and capacitance C (t) are widely used.
An example of one of the modern parametric resistances the channel of the VLG transistor can serve, the gate of which is supplied with a control (heterodyne) alternating voltage u g (t). In this case, the steepness of its drain-gate characteristic changes over time and is associated with the control voltage by the functional dependence S (t) = S. If the voltage of the modulated signal u (t) is also connected 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 to the class of linear ones, we apply the principle of superposition 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.
Converting signals in a circuit with parametric resistance. The most widely used parametric resistances are 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 from an inaccurate translation of the English word "heterodyning". Heterodyne - it is the process of non-linear or parametric mixing of two signals of different frequencies to obtain 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 to another, including a higher one) without changing type or nature of modulation.
Frequency converter(Figure 5.1) consists of a mixer (CM) - a parametric element (for example, an MOS transistor, varicap or a conventional diode with a square-law characteristic), a local oscillator (G) - an auxiliary oscillator of harmonic oscillations with a frequency of ω g, which serves for parametric control of the mixer, and an intermediate frequency filter (usually an UHF or UHF oscillatory circuit).
Figure 5.1. Block diagram of the frequency converter
Let us consider the principle of operation of a frequency converter using the example of transferring the spectrum of a single-tone AM signal. Suppose that under the influence of a heterodyne voltage
u g (t) = U g cos ω g t (5.4)
the slope of the characteristic of the MIS transistor of the frequency converter varies in time approximately according to the law
S (t) = S o + S 1 cos ω g t (5.5)
where S o and S 1 - respectively the average value and the first harmonic component of the slope of the characteristic.
When the AM signal u AM (t) = U n (1 + McosΩt) cosω o t arrives at the MIS transistor of the mixer, the AC 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) cos ω o t =
U n (1 + McosΩt) (5.6)
Let as the intermediate frequency of the parametric converter be chosen
ω psc = | ω г -ω о |. (5.7)
Then, isolating it with the help of the IF amplifier circuit from the current spectrum (5.6), we obtain a converted AM signal with the same modulation law, but a significantly lower carrier frequency
i psc (t) = 0.5S 1 U n (1 + McosΩt) cosω psc 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 transistor characteristic slope. In real mixer circuits, the current spectrum also contains the components of the combination frequencies
ω psc = | mω г ± nω о |, (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.
Figure 5.2. Frequency converter input and output diagrams:
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 a modulated signal is applied to the inputs of the analog multiplier two harmonic oscillations:
u с (t) = U c (t) cosω o t (5.10)
and the reference voltage of the local oscillator 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)
The spectral component with the difference frequency ω psc = | ω g ± ω o | selected by a narrow-band IF filter and used as the intermediate frequency of the converted signal.
Frequency conversion in a circuit with varicap... If only a heterodyne voltage (5.4) is applied to the varicap, then its capacitance will approximately vary in time according to the law (see Figure 3.2 in Part I):
C (t) = C o + C 1 cosω г t, (5.12)
where C about and C 1 is the average value and the first harmonic component of the varicap capacitance.
Suppose that two signals act on the varicap: a heterodyne and (to simplify calculations) an unmodulated harmonic voltage (5.10) with an amplitude U c. In this case, the charge on the varicap capacitance will be determined by:
q (t) = C (t) u c (t) = (С о + С 1 cosω g t) U c cosω o t =
С о 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
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 in series with the varicap an oscillatory circuit tuned to the intermediate frequency ω psc = | ω g - ω o |, it is possible to select the desired voltage.
With a reactive element of the varicap type (for ultrahigh frequencies, this is varactor) you can also create a parametric generator, power amplifier, frequency multiplier. This possibility is based on the conversion of energy into a parametric capacitance. It is known from the physics course that the energy accumulated in a capacitor is related to its capacity C and the charge on it q by the formula:
E = q 2 / (2C). (5.15)
Let the charge remain constant and the capacitance of the capacitor decreases. Since energy is inversely proportional to the value of the capacitance, then the decrease in the latter increases the energy. 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 capacitance ∆С and energy ∆E, therefore it is possible to write
∆E = -E (5.17)
The minus sign here shows that a decrease in the capacitance of the capacitor (∆С<0) вызывает увеличение запасаемой в нем энергии (∆Э>0). The increase in energy occurs due to external costs for performing work against the forces of the electric field with a decrease in capacitance (for example, by changing the bias voltage on the varicap).
With the simultaneous action on the parametric capacitance (or inductance) of several signal sources with different frequencies, between them will occur redistribution (exchange) of vibrational energies. In practice, the vibration energy of an external source, called pump generator, through the parametric element is transmitted to the useful signal circuit.
To analyze the energy ratios in multi-circuit circuits with a varicap, we turn to the generalized scheme (Figure 5.3). In it, parallel to the parametric capacitance C, three circuits are connected, two of which contain sources e 1 (t) and e 2 (t), which create harmonic oscillations with frequencies ω 1 and ω 2. The sources are connected through narrow-band filters Ф 1 and Ф 2, which transmit vibrations 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) give off the power P 1 and P 2, then the load resistance R n consumes the power P n. For a closed-loop system, in accordance with the energy conservation law, we obtain the power balance condition:
P 1 + P 2 + P n = 0 (5.19)
In order to transform the input signal into a form convenient for storage, playback and management, it is necessary to justify the requirements for the parameters of signal conversion systems. To do this, it is necessary to mathematically describe the relationship between the signals at the input, output of the system and the parameters of the system.
In the general case, the signal conversion system is nonlinear: when a harmonic signal enters it, harmonics of other frequencies appear at the output of the system. The parameters of the nonlinear conversion system depend on the parameters of the input signal. There is no general theory of nonlinearity. One way to describe the relationship between the input E in ( t) and the weekend E out ( t) signals and parameter K the nonlinearity of the conversion system is as follows:
| (1.19) |
where t and t 1 - arguments in the space of the output and input signals, respectively.
The nonlinearity of the transformation system is determined by the type of function K.
To simplify the analysis of the signal transformation process, the assumption about the linearity of the transformation systems is used. This assumption is applicable to nonlinear systems if the signal has a small amplitude of harmonics, or when the system can be considered as a combination of linear and nonlinear links. An example of such a nonlinear system is light-sensitive materials (a detailed analysis of their transforming properties will be done below).
Consider signal conversion in linear systems. The system is called linear if its reaction to the simultaneous action of several signals is equal to the sum of the reactions caused by each signal acting separately, that is, the principle of superposition is fulfilled:
where t, t 1 - arguments in the space of the output and input signals, respectively;
E 0 (t, t 1) - impulse response of the system.
Impulse response system the output signal is called if a signal described by the Dirac delta function is applied to the input. This function δ ( x) are determined by three conditions:
| δ( t) = 0 for t ≠ 0; | (1.22) | |
| (1.23) | ||
| δ( t) = δ(– t). | (1.24) |
Geometrically, it coincides with the positive part of the vertical coordinate axis, that is, it looks like a ray going up from the origin. Physical implementation of the Dirac delta function in space there is a point with infinite brightness, in time - an infinitely short pulse of infinitely high intensity, in spectral space - an infinitely strong monochromatic radiation.
The Dirac delta function has the following properties:
 | (1.25) | |
 | (1.26) |
If the impulse occurs not at the zero sample, but at the value of the argument t 1, then such "shifted" by t 1 delta function can be described as δ ( t–t 1).
To simplify expression (1.21) connecting the output and input signals of a linear system, the assumption is made that the linear system is insensitive (invariance) to a shift. The linear system is called shear insensitive if, when the impulse is shifted, the impulse response only changes its position, but does not change its shape, that is, it satisfies the equality:
| E 0 (t, t 1) = E 0 (t – t 1). | (1.27) |
Rice. 1.6. Insensitivity of impulse response of systems
or filters to shift
Optical systems, being linear, are shear-sensitive (not invariant): the distribution, illumination and size of the "circle" (in the general case, not a circle) of scattering depend on the coordinate in the image plane. As a rule, in the center of the field of view, the diameter of the "circle" is smaller, and the maximum value of the impulse response is greater than at the edges (Fig. 1.7).
Rice. 1.7. Shear Sensitivity of Impulse Response
For shift-insensitive linear systems, expression (1.21) connecting the input and output signals takes on a simpler form:
From the definition of convolution, it follows that expression (1.28) can be presented in a slightly different form:
which for the considered transformations gives
| (1.32) |
Thus, knowing the signal at the input of a linear and shear-invariant system, as well as the impulse response of the system (its response to a unit impulse), using formulas (1.28) and (1.30), one can mathematically determine the signal at the output of the system without physically realizing the system itself.
Unfortunately, from these expressions it is impossible to directly find one of the integrands E in ( t) or E 0 (t) on the second and known output signal.
If a linear, shear-insensitive system consists of several filter units that pass the signal in sequence, then the impulse response of the system is a convolution of impulse responses of the constituent filters, which can be abbreviated as
which corresponds to the preservation of a constant value of the constant component of the signal during filtering (this will become obvious when analyzing filtering in the frequency domain).
Example... Let us consider the conversion of an optical signal when receiving targets with a cosine intensity distribution on a photosensitive material. The world is called a lattice or its image, consisting of a group of stripes of a certain width. The luminance distribution in the grating is usually rectangular or cosine. The worlds are necessary for the experimental study of the properties of optical signal filters.
A diagram of a device for recording a cosine target is shown in Fig. 1.8.
Rice. 1.8. Diagram of the device for obtaining the worlds
with cosine intensity distribution
Moving evenly at speed v photographic film 1 is illuminated through a slit 2 of width A. The change in illumination over time is performed according to the cosine law. This is achieved by passing the light beam through the lighting system 3 and two polaroid filters 4 and 5. The polaroid filter 4 rotates uniformly, the filter 5 is stationary. The rotation of the axis of the movable polarizer relative to the fixed one provides a cosine change in the intensity of the transmitted light beam. Illumination change equation E(t) in the plane of the slot has the form:
The filters in the system under consideration are a slit and a photographic film. Since a detailed analysis of the properties of light-sensitive materials will be given below, we will only analyze the filtering effect of slit 2. Impulse response E 0 (X) slit 2 wide A can be represented as:
 | (1.41) |
then the final form of the equation for the signal at the output of the slit is as follows:
Comparison E out ( x) and E in ( x) shows that they differ only in the presence of a factor in the variable part. The graph of a sinc function is shown in Fig. 1.5. It is characterized by an oscillating decay with a constant period from 1 to 0.
Consequently, with an increase in the value of the argument of this function, i.e., with an increase in the product w 1 A and decreasing v, the amplitude of the variable component of the signal at the output decreases.
Moreover, this amplitude will vanish when
This is the case when
Where n= ± 1, ± 2 ...
In this case, instead of the world on the film, you will get a uniform blackening.
Changes in the constant component of the signal a 0 did not occur, since the impulse response of the gap was here normalized in accordance with condition (1.37).
Thus, by adjusting the recording parameters of the world v, A, w 1, it is possible to choose the amplitude of the variable component of illumination that is optimal for a given light-sensitive material, equal to the product a sinc ((w 1 A)/(2v)), and prevent marriage.
When analyzing the passage of a stationary LB through linear electrical circuits (Fig. 1), we will assume that the circuit mode is steady, ie. After the signal is applied to the circuit input, all turn-on transients have ended. Then the output SP will also be stationary. The problem under consideration will be to determine from a given correlation function of the input signal or its spectral power density B(t) or G(w) output signal.
Let's first consider the solution to this problem in the frequency domain. The input SP is given by its spectral power density GX(
). Output power spectral density G y (w) is determined by the formula) = GX( )K 2 ( ), (1)where K 2 (
) is the square of the modulus of the complex transfer function of the chain. The squaring of the modulus is based on the fact that the desired characteristic is a real function of the frequency and energy characteristic of the output process.To determine the relationship between the correlation functions, it is necessary to apply the inverse Fourier transform to both sides of equality (1):
Bx(
) = F -1 [G x( )]; F -1 [K 2 ( )] = Bh( )Correlation function of the impulse response of the investigated circuit:
Bh(
)= h(t)h(t- )dt.Thus, the correlation function of the output SP is
) =B x( ) B h() = Bx ( t)B h(t-t) dt.EXAMPLE 1 of a stationary random wideband signal passing through RC-circuit (low-pass filter), represented by the diagram in fig. 2.
Broadband is understood in such a way that the energy bandwidth of the input SP is much larger than the bandwidth of the circuit (Fig. 3). With such a ratio between the form K 2 (
) and G x() it is possible not to consider the course of the characteristic G x() in the high frequency range.Considering that in the frequency band where K 2 (w) significantly differs from zero, the spectral power density of the input signal is uniform, it is possible to approximate the input signal with white noise without a significant error, i.e. put G x(
) = G 0 = const. This assumption greatly simplifies the analysis. Then G y( ) = G 0 K 2 ( )For a given chain
) = 1 /, then G y( ) = G 0 /.Let us determine the energy width of the output signal spectrum. Output SP power
P y = s y 2 = (2p) - 1 G y(
)d = G 0 /(2RC), then e = (G0) -1 Gy( )d= p / (2RC).In fig. 4 shows the correlation function of the output SP and its spectral power density.
The power spectral density is shaped like the square of the modulus of the complex transfer function of the circuit. Maximum value G y(
) equals G 0. The maximum value of the correlation function of the output SP (its variance) is equal to G 0 /(2RC). It is not difficult to determine the area limited by the correlation function. It is equal to the value of the spectral power density at zero frequency, i.e. G 0:.
Linear-parametric circuits - radio engineering circuits, one or several parameters of which change in time according to a given law, are called parametric (linear circuits with variable parameters). It is assumed that the change of any parameter is carried out electronically using a control signal. In a linear-parametric circuit, the parameters of the elements do not depend on the signal level, but can independently change over time. In reality, a parametric element is obtained from a non-linear element, the input of which is the sum of two independent signals. One of them carries information and has a small amplitude, so that in the area of its changes, the parameters of the circuit are practically constant. The second is a control signal of large amplitude, which changes the position of the operating point of the nonlinear element, and, consequently, its parameter.
In radio engineering, parametric resistance R (t), parametric inductance L (t), and parametric capacitance C (t) are widely used.
For parametric resistance R (t), the controlled parameter is the differential slope
An example of a parametric resistance is the channel of an MOS transistor, to the gate of which a control (heterodyne) alternating voltage is applied u Г (t). In this case, the slope of its drain-gate characteristic changes with time and is related to the control voltage by the dependence S (t) = S. If the voltage of the modulated signal is also connected to the MOS transistor u (t), then its current is determined by the expression
The most widely used parametric resistances are used to convert the frequency of signals. Heterodyning is a process of nonlinear or parametric mixing of two signals of different frequencies to obtain oscillations of the third frequency, as a result of which the spectrum of the original signal is shifted.
Rice. 24. Block diagram of the frequency converter
The frequency converter (Fig. 24) consists of a mixer (CM) - a parametric element (for example, an MIS transistor, varicap, etc.), a local oscillator (G) - an auxiliary harmonic oscillator with a frequency ωg, which serves for parametric control of the mixer, and an intermediate frequency filter (IFF) - a bandpass filter
Let us consider the principle of operation of a frequency converter using the example of transferring the spectrum of a single-tone AM signal. Suppose that under the influence of a heterodyne voltage
the slope of the MOS transistor characteristic varies approximately according to the law
where S 0 and S 1 - respectively the average value and the first harmonic component of the slope of the characteristic. When AM signal arrives at the converting MIS transistor of the mixer
the alternating component of the output current will be determined by the expression:
Let the frequency be selected as the intermediate frequency of the parametric converter
