S.V. Schmidt, student, D.Yu. Belova, student
Scientific Director: B.Z. Kaliev, to. T. N., Professor
Innovative Eurasian University
G. Pavlodar, Kazakhstan

This work was carried out in accordance with scientific program Enhance the efficiency of the use of resources of Kazakhstan by developing a mathematical model and optimal control algorithms for electric power systems, as defined as the strategic task of the Republic of Kazakhstan in the President of the President of the country to the people of Kazakhstan "Kazakhstan 2030". The same program is based on the development of a global energy-level long-term strategy prepared on the basis of studies of scientists of Russia and Kazakhstan noted in the fundamental work of Nursultan Abishevich Nazarbayev "Strategy of the Radical Update of the Global Community and a Partnership of Civilization". The purpose of this scientific article is to increase the efficiency of quality management of generated electricity by improving the mathematical model of stationary regimes. An analysis of the substitution schemes makes it possible to identify patterns whose application will improve high-quality electricity indicators, the efficiency of operation and design of the system itself on the basis of improving the mathematical model of its stationary regimes.
Optimization of the state of the electrical system is a thin and time-consuming task solved on the basis of the analysis and synthesis of T E. Operating modes. In industrial conditions, due to a number of reasons (change in temperature, equipment wear, reducing the activity of the catalyst, a decrease in thermal conductivity, etc.) the parameters of the control system are gradually changed, and their valid values \u200b\u200bare always different from the calculated. The problem of controlling the quality of electricity, taking into account the influence of existing regulatory devices, is currently solved on the basis of multiple calculations, the method of consistent approximation. In market conditions, it is difficult to agree with a similar approach to the calculation and optimization of the power supply system.
In this paper, a solution was obtained above the marked problems by improving mathematical models using sensitivity functions so that the desired parameters of the mode are determined directly on the independent parameters for replacing the transmission system and the distribution of electricity.
The practical value is that the use of sensitivity functions allows you to change the methodology for maintaining the regime, the meaning of which is to ensure, first of all, consumers of high-quality electricity, taking into account the reliable and economic indicators of the power supply system of the power supply system, reducing unjustified labor costs.
The sensitivity function is one of the most important quality indicators of the frequency-selective chains. Sensitivity information is used for various purposes:
1. The sensitivity function is a criterion for a comparative assessment. different configurations electronic chains.
2. Sensitivity analysis results are used to determine the tolerances on the parameters of the chain elements.
3. The absolute sensitivity function is used when optimizing the characteristics of electronic circuits to calculate the gradient of the target function. 4. Sensitivity allows you to understand how the variations of any parameter on the chain characteristics affect.
When designing management and regulation systems, it is important to know how the change in the parameters of the elements affects the chain characteristics. This effect is evaluated using sensitivity functions. The relative sensitivity function H (JW) to variations AI is determined by the formula:
Let [Y] H [V] be functions of the parameter A I, and the vector of the right part does not depend on this parameter. Differentiating (1.2) by a i, we get:
Formula (1.3) allows you to determine the sensitivity of all elements of the vector [V] to variations of the parameter A i.
But in practice, it is usually necessary to determine the sensitivity of any one function of the chain, i.e. It is necessary to find the sensitivity of one variable V i to variations of several parameters A i. To find the sensitivity V i, multiply the left and right parts of equality (1.3) per unit vector:
When considering sensitivity functions in the time domain, independent sources may have an arbitrary current and voltage form. The choice of analysis time may be arbitrary, including from the very beginning of transient processes that occur in the chain when sources are turned on. Therefore, private derivatives according to the parameters of the elements will be determined from the values \u200b\u200b(currents and stresses) presented as time functions. Let the response at the output of the chain, the voltage U is (T). We will search for private derivatives:
Current through the same jet element in the attached scheme (Fig. 1 b)
The result obtained by the method of the attached scheme can be confirmed by direct differentiation of the chain reaction:
Replace the DC element equivalent to it the DIC current source (Fig. 2 b).
At the output of the circuit, you can observe the response to the effect of the source of the perturbation di c. If you divide the amount of impact on the DC constant, the response will change to the same value. Thus, the response at the output of the chain will be numerically equal to the DUI / DC derivative (Fig. 2. B).
Output:
The actual values \u200b\u200bof the parameters of the control of electric power systems are almost always different from the calculated one. These parameter changes can lead to a change in the static and dynamic properties of the system. This circumstance is desirable to take into account in advance in the process of designing and setting up the system, which can be feasible to the use of sensitivity functions, directly the method of attached schemes.
In this paper, a method was identified to optimize the state of the electrical system by improving mathematical models using sensitivity functions so that the desired parameters of the mode are determined directly by independent parameters of the scheme for replacing the transmission system and distribution of electricity, which has an important promising theoretical and practical value. When solving the optimization problem, electrical networks of the power system, taking into account the probabilistic nature of the initial data, there is a need to allocate the most significant factors. When approaching the limit throughput Modes The accuracy of setting the parameters of the substitution scheme parameters has the greatest effect on the calculation accuracy.
This article has great importance For circuit design electrical schemes and their optimization, to determine the degree of influence of the parameters of the components of the circuit on its output parameters, as well as to predict the scatter of the output parameters.
Bibliography:
1. Akhmetbaev D.S. Simulation of stationary modes of the transmission and distribution of electricity. - Almaty. 2010. - P. 28-30.
2. Kaliev B.Z. Materials of the International Scientific and Practical Conference "Industrial and Innovative Development At the Modern Stage: Status and Prospects." - Pavlodar. 2009. - P. 18-20.

Bibliographic link to the article:
S.V. Schmidt, D.Yu. Belova, B.Z. Kaliev Application of sensitivity functions to energy tasks // Online electrician: electric power industry. New technologies, 2012..php? ID \u003d 30 (date of handling: 12/20/2019)

Analytical calculation is not a rather complicated task and can be fully carried out using a computer.

For cascades, an analytical assessment is possible for BT for the case of small non-linearities ( U Vh One order S. φ T.\u003d 25.6 mV).

Usually the level is neither characterized by the harmonic coefficient K g. The total harmonic coefficient is equal

where K. g.2 I. K. g.3, respectively, the coefficients of harmonics in the second and third harmonic components (components of a higher order can be neglected due to their relative smallness).

Harmonic coefficients K. g.2 I. K. g.3, regardless of the method of inclusion of BT, are determined from the following relations:

where b is a communication factor (loop strengthening).

These expressions take into account only the nonlinearity of the emitter transition and obtained on the basis of decomposition in the Taylor series of the Emitter current I E.=I E. 0 EXP ( U Vh/φ T.).

The communication factor depends on the method of incorporating the transistor and the type of feedback. For a cascade with OE and POSTA we have:

where R G. - resistance of the source of the signal (or R out previous cascade); R OS. R OS.=0).

For cascade with OE and ∥OSN

where R eq.=R K.∥R N., R OS.

For cascade with ok

where R eq.=R E.∥R N.(See subsection 2.8).

For a cascade with about

Harmonic coefficients K. g.2 I. K. g.3, regardless of the mode of inclusion of PTs, are determined from the following relations:

where A is a coefficient equal to the second member of the expression expression for nonlinear steepness in a series of Taylor equal

A.=I SI./U.² UC,

where I SI. and U Ots. See Figure 2.33.

Communication factor B depends on the method of incorporating the transistor and the type of OOS. For Cascade with OI and Posy, we have:

B. = S. 0 (R OS. + r I.),

where R OS. - Resistance to the POST (see subsection 3.2, in the absence of a fighter R OS.=0).

For a cascade with OI and ∥OSN we have:

B. = S. 0 R g r eq/R OS.,

where R eq.=R S.∥R N., R OS. - Resistance ∥OSN (see subsection 3.4).

For cascade with

B. = S. 0 (R eq. + r I.),

where R eq.=R S.∥R N. (See subsection 2.11).

For cascade with oz

B. = S. 0 ((R G.∥R I.) + r I.).

In the above expressions r I. - body resistance of the semiconductor in the source chain, r I.≈1/S SIwhere S SI - See subsection 2.10, for low-power Fri r I.\u003d (10 ... 200) Ohm; R I. - See Figure 2.38.

Preparations for evaluation K g They give a good result in the case of small nonlinearities, in the mode of large nonlinearities, you should use well-known machine methods, or refer to graphic assessment methods.

8.2. Calculation of stability UU.

Evaluation of the stability of the UU represented by the equivalent quadrupole, described by Y-parameters, is convenient to carry out by definition invariant stability coefficient :

For k\u003e<1 - потенциально неустойчив, т.е. существуют такие сочетания полных проводимостей нагрузки и источника сигнала, при которых возможно возникновение генерации.

The stability of the amplifier, taking into account the conductivity of the load and the source of the signal, is determined by the following ratio:

When k\u003e 1, the amplifier is certainly stable when<1 - неустойчив, k=1 соответствует границе устойчивости.

Equivalent Y-parameters of the amplifier are determined according to the procedure of subsection 2.3, at the specified points of the operating frequency range. The use of an invariant sustainability coefficient is particularly convenient for machine analysis by UU. Other resistance assessment methods are described in.

8.3. Calculation of noise characteristics уу

Noises in UU are mainly determined by the noise of active resistance and amplifying elements located in the input cascades. The greatest contribution to the noise power generated by an amplifying cascade makes an amplifying element. The presence of own sources of noise limits the possibility of enhancing weak signals.

Depending on the nature of the occurrence, the intrinsic noises of the transistor are divided into heat, fractional, shock distribution noises, excess, etc.

The heat noises are caused by disorderly movements of free charge carriers in conductors and semiconductors, fractional - discreteness of the charge of carriers (electrons and "holes") and a random nature of injection and extraction them through P-N-transitions. Talking noise is caused by fluctuations of the distribution of the emitter current on the currents of the collector and the base. All of the above types of noise have a uniform spectrum.

The nature of excessive noise is not fully found out. They are usually associated with fluctuations of the state of the surface of semiconductors. The spectral density of these noise is inversely proportional to the frequency, which served as a reason for the name of their noises of type 1 / f. They are also called flicker noises, flicker noises and contact noises. The noise of type 1 / F is strongly increasing when defects in the semiconductor crystal lattice.

The most significant contribution to the power of the noise of amplifying elements is made by thermal noises.

The noises of the active elements can be represented as a voltage source (Figure 8.1a) or the current source (Figure 8.1b).

Figure 8.1. Equivalent schemes of active noise resistance

The corresponding EMF values \u200b\u200band current of these sources are as follows (see subsection 2.2):

where Δ. f. - streak of working frequencies; k.\u003d 1.38 · 10 -23 - Boltzmann's constant; T - temperature in degrees of Kelvin; R Sh - noise resistance, G Sh - noise conductivity, G Sh=R Sh -1 .

For standard temperature T \u003d 290 ° K, these formulas can be simplified:

Spectral density of voltage noise and current make up:

where, - differentials from the standard voltages and noise currents as random functions of T, acting in the DF bandwidth.

Any active element can be represented by a noisy four-pole (Figure 8.2) and according to the formulas, calculate its noise characteristics.

Figure 8.2. Noisy quadrupole

The expressions for the noise parameters of BT and PT of the normalized spectral densities of noise noise R Sh=F ru./4kt.in the current G sh=F RI/4kt. and mutual spectral density F S., respectively, noise resistance, noise conductivity and mutual spectral noise density.

For BT, included in the Scheme with OE:

R Sh = r B. + 0,2I b r b 2 + 0,02I to S. 0 -2 ,

G Sh = 0,2I B. + 0,02I to G. 2 S 0 -2,

F S. = 1 + 0,02I b r b + 0,02I to gs. 0 -2 ,

where I B. and I K. in milliamperes, G and S. 0 in millisimeters. When taking into account the flicker noise for frequencies of F≥10Hz in these expressions, it should be accepted:

I "B. = (1 + 500/f.)I B.,

I "K. = (1 + 500/f.)I K..

For PT included with OI:

R Sh = 0,75/S. 0 ,

G Sh = R w Ω.² C² Zi. = 40R W F.² C.² Z

F S. = 1 + ωc z r sh \u003d 1 + 6,28 · C ZI R W.

These formulas are applicable for other transistor inclusion schemes.

Believing uniform spectral density of noise, according to can be obtained for the coefficient of Cascade Noise:

F. = (R G. + R Sh + G w r g + 2F w r g)/R G..

Exploring this expression on the extremum, determine the optimal resistance of the signal source Rg Opt.in which the noise coefficient of the Cascade F is minimal:

At the same time, in most cases it turns out that Rg Opt. S. does not coincide R G., optimal in terms of obtaining necessary f B. Cascade ( Rg Opt.>R G.). The output from this situation is the inclusion between the first and second cascades of the cornet correction chain (Figure 8.3).

Figure 8.3. Simple anti-free correction

The introduction of anti-net correction is to increase the coefficient of transmission of cascades in the region of the RF (by making the adjusting attenuation chain on the LF and SC), thereby compensating for the recession of the increase in the RF due to the high-alone Rg Opt..

Approximately the parameters of the anti-net correction can be determined from the equality of its time constant RC time τ B. uncorrected cascade.

Calculation of noise cascade connected four-pole (multi-stage amplifier) \u200b\u200bis usually reduced to the calculation of the noise coefficient of the input chain and the input cascade. The first cascade in such an amplifier works in a low-noise mode, and the second and other cascades are in normal mode.

The calculation of the noise in the general case is a complex task solved by computer. For a number of special cases, noise parameters may be calculated by the ratios given in.

8.4. Sensitivity analysis

Sensitivity called reaction various devices To change the parameters of its component.

Sensitivity coefficient (sensitivity function or simply sensitivity ) It is a quantitative assessment of the change in the parameters of the device (including and AEU) for a given change in the parameters of its component.

The need for calculating the sensitivity function occurs if you need to take into account the effect on the characteristics of the environmental factors (temperature, radiation, etc.), when calculating the required tolerances on the component parameters, when determining the percentage of the EMR exit, in the optimization, modeling tasks, etc.

Sensitivity function S I. Device parameter y. To change the component parameter x I. Determined as a private derivative

This expression obtained on the basis of decomposition in a series of Taylor function of several variables, where

Needed by private derivatives of the second or more order, we obtain the connection of the sensitivity and deflection function of the parameter:

There are varieties of sensitivity function:

◆ absolute sensitivity, absolute deviation at the same time ;

◆ relative sensitivity , relative deviation is equal ;

◆ Semi-retinuous sensitivity , .

Selecting the type of sensitivity function is determined by the type of problem being solved, for example, for the complex transmission coefficient, the relative sensitivity is equal to the relative sensitivity of the module (valid part) and the sequencing sensitivity of the phase (imaginary part):

For simple schemes Calculation of sensitivity functions can be carried out by direct differentiation of the circuit function represented in analytical form. For complex schemes, obtaining an analytical expression of a circuit function is a complex task, it is possible to apply the direct calculation of the sensitivity function through increments. In this case, it is necessary to conduct N analyzes of the scheme, which is very irrational for complex schemes.

There is an indirect method for calculating sensitivity by transmission functions proposed by Bykhovsky. According to this method, the sensitivity function, for example, the direct transmission coefficient is equal to the product of the transmission functions from the login of the circuit to the element relative to which the sensitivity, and the gear ratio "element - the output of the circuit" (Figure 8.4a) is searched.

Figure 8.4. Indirect method for calculating sensitivity functions

Since the calculation of the sensitivity functions is reduced to the calculation of the transfer functions, then it is possible to use, for example, a generalized method of nodal potentials. The indirect method of calculating the gear ratios allows you to find the sensitivity functions of higher orders. Figure 8.4b illustrates the found function of the sensitivity of the second order. In general, there is n! Signal transmission paths, each of which contains N + 1 womb.

The following describes the method of calculating the sensitivity function, which combines the direct differentiation method and indirectly by gentlemen, allowing the sensitivity to n elements in one analysis. Consider this method On examples of obtaining expressions for the absolute sensitivity of the first order of S-parameters of the electronic circuits described by the conductivity matrix [Y].

In the matrix representation, the characteristics of the electronic circuits, including the scattering parameters [S], are determined in the form of the relationship of algebraic additions of the matrix [Y] (see subsection 7.2). The variable parameter enters into some elements of algebraic additions. The definition of the sensitivity function is reduced in this case to finding derivatives of algebraic add-ons (or algebraic additions and determinants) by elements containing a variable parameter. In the case when the variable parameter enters the elements of the additions of the determinant is functionally, the sensitivity is defined as a complex derivative.

To determine derivatives of algebraic additions according to the variable parameters of the elements included in them, we use the theorem that approvers that the derivative of the determinant according to any element is equal to the algebraic addition of this element. The proof of the theorem is based on the decomposition of the determinant in Laplas

The overall expression for S-parameters through algebraic supplements has the form (see subsection 7.2)

S ij. = k ij.Δ Ji./Δ – Δ IJ..

Determine the functions of the sensitivity of scattering parameters to the passive two-general y O. included between arbitrary nodes k and l (see Figure 8.5A)

Figure 8.5. Calculation of sensitivity of s-parameters

S s ij. y.0 = dS IJ./dY. 0 = k ij.(Δ ji.(k.+l.)(k.+l.) Δ – Δ ( k.+l.)(k.+l.) Δ Ji.)/Δ² = – k ij.Δ j.(k.+l.) Δ ( k.+l.)i. /Δ² = – k ij.[(Δ JK. – Δ jl)(Δ Ki. – Δ LI)]/Δ²

Upon receipt of this and subsequent expressions, the following matrix ratios are used:

Δ ( i + J.)(k + L.) = Δ i.(k + L.) + Δ j.(k + L.) = (Δ IK – Δ il) + (Δ JK. – Δ jl),

Δ IJ.Δ KL. – Δ ilΔ KL. = ΔΔ IJ, KL..

For electronic circuits containing BT, simulated items (see subsection 2.4.1), we define the sensitivity of S-parameters to the conductivity of the control branch g E.=1/r E. and the parameter of the controlled source A turned on, respectively, between the nodes K, L, and P, Q (Figure 8.5b):

S s ij ge = dS IJ./dG E. = k ij.[(Δ ji.(k.+l.)(k.+l.) Δ + αΔ iJ.(k.+l.)(p.+q.))Δ – (Δ ( k.+l.)(k.+l.) Δ+αΔ ( k.+l.)(p.+q.) Δ IJ.])/Δ² = – k ij.Δ ( k.+l.)i. (Δ j.(k.+l.) + αΔ j.(p.+q.))/Δ² = – k ij.(Δ ki -Δ lI)[(Δ jk -Δ jl)+ α(Δ jP. - Δ JQ.)/Δ²,

S s ij. α = dS IJ./d.α = k ij.(Δ ji.(k.+l.)(p.+q.) Δ – Δ ( k.+l.)(p.+q.) Δ Ji.)/Δ² = – k ij.Δ j.(p.+q.) Δ ( k.+l.)i. /Δ² = – k ij.[(Δ jp -Δ jQ.)(Δ ki -Δ lI)]/Δ².

If a electronic circuit It contains PTs, simulated by ITUN (see subsection 2.4.1), then the sensitivity of scattering parameters to the steepness S, included between the nodes P, q with the control nodes K, L (Figure 8.5V), is equal to

S s ij. S \u003d. dS IJ./ DS \u003d. k ij.(Δ ji.(k.+l.)(p.+q.) Δ – Δ ( k.+l.)(p.+q.) Δ Ji.)/Δ² = – k ij.Δ j.(k.+l.) Δ ( p.+q.)i. /Δ² = – k ij.[(Δ jk -Δ jl)(Δ PI -Δ qI)]/Δ².

The sensitivity of scattering parameters to any Y-parameter of the subcursion (Figure 8.5g), for example, y KL.will be equal

S s ij ykl = dS IJ./dy kl. = k ij.(Δ ji, KL. Δ – Δ kL. Δ IJ.)/Δ² = – k ij.Δ jl Δ ki. /Δ².

With known sensitivity y KL. to the parameter of the element of the depths X (see Figure 8.5g) The sensitivity of the s-parameters of the full scheme to this parameter, in accordance with the concept of a complex derivative, will express as

S s ij. x \u003d ( dS IJ./dy kl.)(dy kl./dX.) = S s ij ykl· S y kl x.

The latter expression indicates the possibility of applying the method of refside when analyzing the sensitivity of complex electronic circuits.

Knowing the connection of scattering parameters with secondary parameters of electronic circuits ( K U., Z Vh, Z out etc.) and the sensitivity of scattering parameters to change the elements of the scheme, it is possible to find the sensitivity functions of secondary parameters to the change of these elements. For example, for the transmission coefficient on the voltage from the i-go j-th knot K ij.=S ji./(1+S. 11) sensitivity to the change of parameter x (believing that S ij.=f.(x.) I. S II.=φ( x.)) Receive

S k ij x = dK IJ./dX. = [S s ij x(1 + s II.) – S S II X S ij] / (1 + s II.)².

Similar to Z. vK(out) (Z II. (jJ.)) Have

Z II. (jJ.) = Z g (n.) · (1 + s II. (jJ.)) / (1 - s II. (jJ.));

S Z I. i.(jJ.) x. = dZ. iI.(jJ.) /dX. = –2Z g (n.) · S S I. i.(jJ.) x. · S. II. (jJ.) / (1 - s II. (jJ.))².

This method can be equally efficiently used in determining the sensitivity of higher orders for all sorts of electronic circuit characteristics. The implementation of the sensitivity calculation thus obtained in this way is reduced to the calculation and the mischief of the corresponding algebraic additions, which is well combined with the finding of other uninimony characteristics of the electronic circuits.

8.5. Machine Methods Analysis AEU

Subsection 2.3 shows the basic idea of \u200b\u200bthe generalized method of nodal potentials, on the basis of which most relations were obtained for the sketching calculation of amplifying cascades. However, along with undoubted advantages this method (simplicity of programming, low dimension of the obtained conduction matrix Y., N * n, where N is the number of circuit nodes without reference), this method has a number of significant flaws. First of all, it should be noted the impossibility of representing the conductivity of some ideal models of electronic circuits (short-circuited branches, voltage sources, dependent sources, driven by current, etc.). In addition, the presentation of inductance is inconvenient with temporary analysis of schemes, which is associated with the Laplace transformation (Laplace operator p. It should be in a numerator in order for the system of algebraic equations and the system of differential equations resulting from the conversion of the differential equations).

Currently, topological methods of formation of the system of equations obtained the greatest distribution. electrical chain, most common of which is tabular .

In this method, all equations describing the chain are included in general System equations containing the Kirchhoff equations for currents, voltages and component equations.

Kirchoff equations for currents can be represented as

AI B. = 0,

where A.- Inspection matrix describing the topology of the chain, I B. - Vector of current branches.

Kirchhoff equations for stresses are viewed

V B. – A t v n = 0,

where V B. and V P. - respectively, vectors of branches and nodal potentials, A T. - Transposed Communction Matrix A..

In general, the equations describing the elements of the chain can be represented in the following form:

Y in b in + Z in i in = W B.,

where Y B. and Z B. - respectively, quasidiagonal conductivity matrices and resistance of the branches, W B. - Vector where independent voltage and current sources are included, as well as initial stresses and currents on capacitors and inductors.

We write the same equations in the following sequence:

V B. – A t v n = 0;

Y in b in + Z in i in = W B.;

AI B. = 0;

and imagine in matrix form

or in general

The tabular method is mainly theoretical value, since along with the main advantage, which is, it is possible to find all currents and stresses of branches and nodal potentials, has a number of significant flaws. First of all, it should be noted the redundancy of the method leading to the large dimension of the matrix T.. This should be noted that many ideal controlled sources lead to the appearance of unnecessary variables. For example, the input current controlled by voltage sources of current and voltage, as well as the input voltage of the current current and voltage sources is zero, but in this method they are considered as variables.

In practical terms, the modification of the table method is most often used - modified nodal method with verification .

The idea of \u200b\u200bthis method lies in the separation of elements into groups; One group is formed from the elements that are described by the use of conductors, for the elements of the second group, this description is impossible. Since the branches of the branches of the first group can be expressed by the branches of the branches, and the branches of the branches through the nodal potentials, then all the branches voltage from the table equations can be excluded, and for the elements of the first group of branches. With the introduction of additional equations for currents in branches with elements of the second group, an inspection is made for the presence of pre-known (zero) variables. As a result of such a transformation, we obtain the equation of a modified node method with verification

or in general

T M x \u003d W,

where n is the dimension of the conduction matrix Y N. 1 elements of the first group (n - the number of circuit nodes without zero); M is the number of additional equations for the elements of the second group; J N. - vector of independent current sources; I. 2 - vector of currents of branches of the elements of the second group; W. 2 - vector where independent voltage sources are included, as well as initial voltages and currents on capacitors and inductances represented by elements of the second group.

To simplify programming, usually represent the matrix of the coefficients of the system of equations of the modified node method T M. in the form of the sum of two matrices dimension (n + m) * (n + m)

T M \u003d G + PC.

In the matrix G. All active conductivity and coefficients corresponding to the frequency-independent elements are made, and in the matrix C. - all frequency-dependent elements, and inductance usually represent the element of the second group, i.e. resistance. Further find the solution of this system of equations using the Gauss-Jordan algorithms or L / U-decomposition.

With frequency analysis of electronic circuits operator p. Replaced by jω., A cycle is organized in frequency, inside of which a system of equations is formed for each frequency point, which is solved relatively interesting stresses and currents.

With a temporary analysis of linear electronic circuits, it is possible to directly use the modified integral form of equations

(G + PC.)X \u003d W..

After the transition to the temporary region we get

GX + CX "\u003d W,

CX "\u003d W - GX.

The solution of the obtained system of differential equations is located by numerical integration. Alone effective methods numerical integration are methods based on linear multi-shaped formulas The simplest of which includes the Euler formulas (direct and inverse) and the formula of the trapezium.

After breaking the time interval to a finite number of segments H and putting t. n.+1 \u003d T n + Hfor each moment of time t N. You can find approximation x N. to a true decision x.(t N.) By applying linear multi-step formulas:

x. n.+1 \u003d x n + hx "n (direct formula Euler);

x. n.+1 \u003d X n + HX " n.+1 (reverse formula Euler);

x. n.+1 \u003d X N. + (h./2)(x "N. + x "N. +1) (Formula of the trapezium).

Finding x "N. +1 For (n + 1), the step of calculations is possible by applying the direct formula of the Euler.

Since the voltage on the condenser and the current flowing through it is associated with the ratio I \u003d CDV / DT, and for inductance we have V \u003d LDI / DT, the use of reverse formula of the Euler is equivalent to the transition from tanks and inductances to their equivalent circuits shown in Figure 8.6, in The result of which the chain becomes resistive. Such inductance and containers are called grid (accompanying, discrete) models .

Figure 8.6. Network models for reverse formula Euler

Introducing the working point or calculation of a DC is the first step with a nonlinear HU analysis. Analysis of the characteristics of the DC schemes containing nonlinear resistance is reduced to solving the system of nonlinear equations of the form f (x) \u003d 0.

Since the laws of Kirchhoff apply not only to linear, but also to nonlinear elements, to form a system of equations f (x) It is possible to use the table methods already considered. The structure of the resulting tabular equations will be discussed below.

To solve the system of nonlinear equations f (x)applied newton Rafson Method . The method provides for the use of the initial approximation x. 0, Iterative procedure and, if the value | ( x N. +1 –x N.)/x N. +1 | It is enough small, the establishment of the fact of convergence (N- Number of iterations):

x N. +1 = x N. – J. -1 f.(x N.),

where J - Jacobian (Matrix Jacobi) dimension (M * M)

In the process of iterative processing of this system of equations at each stage of iterations, values \u200b\u200bcan be obtained f.(x N.) I. J.; This is equivalent to solving a linear equation in the form

J.(x N. +1) – x N.) = –f.(x N.).

In other words, the solution of nonlinear equations can be interpreted as a re-solution of linear equations at each stage of the iterative process.

The structure of Jacobian externally coincides with the table equations of linear circuits, which are converted with regard to the calculation of the DC, are removed capacitors and the coils of inductance are cleaned.

Let the tabular equations are specified in the following form:

V B. – A t v n = 0;

p.(V B.,i B.) = W.;

AI B. = 0;

System of equations p.(V B.,i B.) = W. Determines the relationship between currents and branches in implicit form, some of these dependencies can be linear.

Matrix Jacoba by nth iteration will have kind

where ; where.

For the formation of Jacobian, it is possible to use various modifications of the table method, including a modified nodal with a check. The result of an analysis of the DC scheme (DC mode) can be used as an initial approximation with a temporary analysis of nonlinear electronic circuits.

Nonlinear equations are easily included in the chain equations compiled by a table or modified nodal method. Linear elements, as before, linear component equations. For nonlinear equations, the equations in an implicit form are characteristic, although sometimes nonlinearity can be described in explicit form. Nonlinear containers and inductances are best described using additional variables - electrical charges and magnetic streams, respectively, which must be introduced into the vector of unknown. If this is done, the equations recorded both tabular and modified nodal methods can be represented as follows:

f.(x ", x., W., t.) ≣ Ex " + GX. +p.(x.) = 0,

where E. and G.- Permanent matrices, and all non-linearities are reduced to the vector p (X).

The resulting system of differential equations is solved by integrating using differentiation formulas back And the Newton-Rafson algorithm, for which Jacobian is formed. In general, the jacobian structure for a linear and nonlinear circuit is identical, the difference between them is that the nonlinear container (inductance) will be represented by two equations, and the charge Q (stream F) will become another unknown. However, for linear containers and inductances, it is possible to introduce charges and magnetic streams as variables, which will lead to the coincidence of the Jacobian and the matrix of the equation system. Any nonlinear conductivity will appear in Jacobian in the same way as linear conductivity in the matrix C. modified node method. Thus, it becomes possible to form a single approach to the formation and solving equations of linear and nonlinear circuits in order to obtain their time and frequency characteristics, which is successfully implemented in modern schemes of circuit design.

Listed methods, as well as other issues of analyzing electronic circuits, are given in. The in describes one of the packets of the ELECTRONICS WORKBENCH schematic design.

Sensitivity the reaction of various devices to change the parameters of its component is called.

Sensitivity coefficient (sensitivity function or simply sensitivity ) It is a quantitative assessment of the change in the parameters of the device (including and AEU) for a given change in the parameters of its component.

The need for calculating the sensitivity function occurs if you need to take into account the effect on the characteristics of the environmental factors (temperature, radiation, etc.), when calculating the required tolerances on the component parameters, when determining the percentage of the EMR exit, in the optimization, modeling tasks, etc.

Device sensitivity function y. A change in the parameter of the component is defined as a private derivative

This expression was obtained on the basis of decomposition in a series of Taylor function of several variables, where

Needed by private derivatives of the second or more order, we obtain the connection of the sensitivity and deflection function of the parameter:

.

There are varieties of sensitivity function:

¨ absolute sensitivity, absolute deviation at the same time ;

¨ relative sensitivity , relative deviation is equal ;

¨ Celebration sensitivity , .

Selection of the type of sensitivity function is determined by the type of problem being solved, for example, for a complex transmission coefficient Relative sensitivity is equal to the relative sensitivity of the module (the actual part) and the sequity sensitivity of the phase (imaginary part):

For simple schemes, the calculation of the sensitivity function can be carried out by direct differentiation of the circuit function represented in the analytical form. For complex schemes, obtaining an analytical expression of a circuit function is a complex task, it is possible to apply the direct calculation of the sensitivity function through increments. In this case, it is necessary to conduct N analyzes of the scheme, which is very irrational for complex schemes.

There is an indirect method for calculating sensitivity by transmission functions proposed by Bykhovsky. According to this method, the sensitivity function, for example, the direct transmission coefficient is equal to the product of the transmission functions from the login of the circuit to the element relative to which the sensitivity, and the gear ratio "element - the output of the circuit" (Figure 8.4a) is searched.

Since the calculation of the sensitivity functions is reduced to the calculation of the transfer functions, then it is possible to use, for example, a generalized method of nodal potentials. The indirect method of calculating the gear ratios allows you to find the sensitivity functions of higher orders. Figure 8.4b illustrates the found function of the sensitivity of the second order. In general, there is n! Signal transmission paths, each of which contains N + 1 womb.

The following describes the method of calculating the sensitivity function, which combines the direct differentiation method and indirectly by gentlemen, allowing the sensitivity to n elements in one analysis. Consider this method on examples of obtaining expressions for the absolute sensitivity of the first order of S-parameters of the electronic circuits described by the conductivity matrix [Y].

In the matrix representation, the characteristics of the electronic circuits, including the scattering parameters [S], are determined in the form of the relationship of algebraic additions of the matrix [Y] (see subsection 7.2). The variable parameter enters into some elements of algebraic additions. The definition of the sensitivity function is reduced in this case to finding derivatives of algebraic add-ons (or algebraic additions and determinants) by elements containing a variable parameter. In the case when the variable parameter enters the elements of the additions of the determinant is functionally, the sensitivity is defined as a complex derivative.

To determine derivatives of algebraic additions according to the variable parameters of the elements included in them, we use the theorem that approvers that the derivative of the determinant according to any element is equal to the algebraic addition of this element. The proof of the theorem is based on the decomposition of the determinant in Laplas

.

The overall expression for S-parameters through algebraic supplements has the form (see subsection 7.2)

.

We define the sensitivity functions of the scattering parameters to the passive two-tip-compartment included between arbitrary nodes K and L (see Figure 8.5A)

Upon receipt of this and subsequent expressions, the following matrix ratios are used:

For electronic circuits containing BT, simulated items (see subsection 2.4.1), we define the sensitivity of the S-parameters to the conductivity of the control branch and the parameter of the controlled source A on, respectively, between the nodes K, L, and P, Q (Figure 8.5b):

If the electronic circuit comprises a PT, simulated by ITUN (see subsection 2.4.1), then the sensitivity of scattering parameters to the steepness S, included between the nodes P, q with the control nodes K, L (Figure 8.5V), is equal to

In section. 2.4 The main provisions of this computational method were indicated, which makes it possible to obtain private derivatives (the coefficients of the influence of parameters) according to the respective system parameters. These derivatives can be determined simultaneously with the solution of the initial differential equation.

The application range based on the study of sensitivity (influence) of parameters is wider than parameter estimation methods. Maceinger leads the following list of possible applications:

a) Prediction of solutions in the neighborhood of a well-known solution by linear extrapolation.

b) Definition tolerances for parameters using linear prediction, selection of critical parameters.

c) applications to statistical studies: Evaluation of the influence of random parameters of the system or initial conditions, extrapolation of the results obtained at random input signals.

d) optimization of the parameters of the system with gradient methods in accordance with a certain quality criterion.

e) analysis of the sensitivity of the decision to errors.

e) identification of the boundaries of the system of system stability.

g) changing the constant time of various processes; Changing the rise time, sediment time.

h) Decision of the boundary value problem for ordinary differential equations.

We restrict ourselves to the discussion of the application of this method to evaluate the object parameters.

Methods based on the study of the influence (sensitivity) of parameters

We now highlight the main positions of the method that uses the function of the influence of parameters. Consider the following inhomogeneous linear differential equation

with initial conditions

It is required to obtain a solution at specific values \u200b\u200bof the parameters Consider while only one parameter is for clarity; Then it will be the function of two variables, for example, by the curve of the solution obtained by the value of the parameter by extrapolation, you can find a close curve corresponding to

The number of members in this expansion necessary for satisfactory approximation depends on the magnitude and behavior of the decision and its private derivatives of the area you are interested in. Here will be considered only approximation with an accuracy of first-order members.

A private derivative is a function of a function of influence or a function of the sensitivity of the first order parameter. Other coefficients of influence relating to equation (9.67) are

The last two members characterize sensitivity to changes in the initial conditions. Differentiating (9.67) by and considering that and depend on

Changing the procedure for differentiation and using the designation coming to a differential equation for

with initial conditions

as follows from the fact that the initial values \u200b\u200bare constant and do not depend on equation (9.70) known as the system sensitivity equation relative to the parameter with small changes from this equation you can get information about the approximate gradient value. This equation is not difficult to modify, replacing private derivatives with full:

(approximate sensitivity equation). The reason that this equation is just approaching

it is that the ratio between private and complete having

Consequently, equation (9.71) is a good approximation if changes in time parameters are sufficiently small.

Similarly, you can derive approximate sensitivity equations relative to the four parameters under consideration

Each of these equations can be modeled using a separate sensitivity model (see the block diagram in FIG. 9.8). In the linear case under consideration, all approximate sensitivity equations are equal, except for differences in the right parts. This means that the sensitivity functions can be sequentially determined on the same model using the appropriate "binding member" or. Further simplifications are obtained if we consider that, according to formulas (9.73a), (9.736),

according to formulas (9.73B), (9.73g),

a comparison of formula (9.67) with (9.73B) and (9.73g) gives

Thus, it is sufficient to modify equation (9.736) and take advantage of the ratios (9.74) - (9.76) to simultaneously obtain the sensitivity functions of all four parameters (Fig. 9.9, b). Such a practical implementation scheme requires significantly smaller costs than the scheme corresponding to FIG. 9.8.

If the initial conditions are also the parameters of interest, it is easy to see that in the respective sensitivity equations, the "binding member" is generally absent. When we get a homogeneous differential equation

with initial conditions

This equation is solved simply by reuse The main model is identically equal to zero managing function and, accordingly, changed initial conditions.

Applications of the method of influence of parameters are not limited to linear syrttember. As an example of a nonlinear system, we consider the equation

The sensitivity equations are

Again the equations differ only in the "binding members". Therefore, it can be sequentially used to use the same model with control functions The considered task can be generalized on the system of differential equations with parameters

The sensitivity equations relative to which derivatives are determined in the form

The initial conditions are zero, unless the initial conditions of the initial differential equation are considered as parameters. The above wording is valid for both linear and nonlinear systems. To study the effect of a separate parameter, it is necessary to simulate (or program) the entire system of sensitivity equations (9.81), even if this parameter explicitly enters one equation of the source system (9.80). If, for example, a "binding member" appears in the sensitivity equation, then, at nevertheless, all other sensitivity equations contain in an implicit form in the form of members and are associated with the equation.

Another area of \u200b\u200bapplications is detected in the study of the exclusion effect of derivatives

high order of differential equation. Suppose that the equation is studied

It is necessary to figure out the influence of a third-order member

Sensitivity equations relatively and have

Therefore, and from the sensitivity model, you can get the value of the influence of this parameter in the surrounding area.

Until now, this section considered the absolute functions of the sensitivity of the parameters, for example, it is sometimes possible to use relative sensitivity functions, for example

Method using sensitivity points

In the previous section, it was found that for the simultaneous definition of several sensitivity functions, in addition to the model of the object, another number of additional sensitivity models are needed. This is due to the complication of the analog computing circuit or with an increase in the machine time required to solve such tasks.

On the other hand, in section. 9.1 It was shown that when using a generalized model of additional sensitivity models, it is not necessary - sensitivity functions can be measured directly. This is explained by the linearity of the generalized model regarding the parameters.

Given the desirability of the highest possible simplification of the modeling and cutting scheme

time, it makes sense to study the types of models that make it possible to find the wisest number of sensitivity functions (from among those subject to definition). For this purpose, the so-called method of sensitivity points is used.

Its main idea can be explained as follows. Consider the linear object with the transmission function depending on the parameters, the laptose conversion from the input signal is then the output signal is determined by the formula

The output of the corresponding model is

Considering the differentiability of the production by parameters, we obtain

(absolute) parameter sensitivity functions

relative parameter sensitivity functions

The following example helps to illustrate this idea (Fig. 9.10, a, b). For the model, the ratio is valid

Hence the relative sensitivity functions get

As a result, we arrive at the scheme of FIG. 9.10, b. Called sensitivity points. With analogue

FIG. 9.10. (see scan)

modeling both sensitivity functions can be measured simultaneously, with digital calculations, both functions are determined by the same program.

This idea can be extended to multi-system systems with feedback (Fig. 9.11). It is assumed here that in each of the elementary blocks there is only one parameter for which it is necessary to calculate the sensitivity function. As before, it is not difficult to show that it is a point of sensitivity for a parameter from the block remains to consider the issue

(Click to view the scan)

how the parameter enters into the transfer function is solved by the introduction of an additional gear ratio

This is a logarithmic gear ratio of sensitivity introduced earlier than a bode. The input serves a signal, removed from the sensitivity point to the output -

Some special cases:

In this case, the signal C is a sensitivity function and there is no need to add any elements into a sensitivity model (Fig. 9.9, b and 9.10, b).

b) if i.e. gear ratio, is the product of two gear ratios, from which only one contains the parameter representing for us,

i.e. coincides with the transfer function of the part of the model that contains

These ideas can also be distributed to the functions of the highest order sensitivity, for example

which are obviously obtained from the first-order sensitivity functions. It turns out that in this case another sensitivity model is necessary.

Of course, the sensitivity analysis was also used to describe objects in the time domain. An overview of the appropriate literature can be found in the work. Many interesting articles contain two collections of reports of symposiums Iifak on sensitivity.

Continuous custom models

The scheme considered here is shown in FIG. 9.12. The error is defined as

where some functionality. It is necessary to minimize the criterion that can be written as functionality from even functions.

The model setting is carried out by changing the parameters in accordance with the value of the gradient.

The components of the gradient vector are determined by differentiation:

moreover, it is a ratio of the effect of the parameter. Now you can define the following

operator:

where do you get

As indicated in the previous section, a set of operators depending on the parameter A and acting on the signal and, allows you to obtain all the sensitivity functions of the parameters.

Example. We use the results of work. Object and model are described by equations according to

The sensitivity equation is obtained as a result of differentiation of the equation of the model:

where and is considered constant. Apply as a criterion a minimum condition

and we will use the method of the Great Desk to configure

since only depends

The behavior of the model setup scheme is described by formulas (9.98) - (9.102). Due to the limitation requiring constancy A B (9.102), these formulas allow you to pour to approximately describe changes A, when these changes occur quite slowly. The work has examined convergence issues for cases when the input and is a step or sinusoidal signal. In the first case, you can prove the stability of the equilibrium point

The second case leads to the Mathieu equations that may have both (asymptotically) sustainable and periodic and unstable solutions.

When studying stability, the second Lyapunov method was used: see, as well as the works cited in the previous section.

Note that the sensitivity functions of the parameters play the role of auxiliary variables by analogy with the above in ch. 6 and 7 for the case of discrete signals.

Examples of modeling, practical implementation and applications

Although the work is not directly related to the rating of parameters, it can be mentioned as another example of using the effects of the parameters. The system under study is shown in FIG. 9.13. The object parameters (for example, changing the angular velocity of the aircraft along the pitch axis from the deviation of control surfaces) change. These changes are compensated

setting parameters and in the feedback loop. The desired indicators of the "Object + Feedback Chain" system are set by the reference model, which is a fixed analog scheme. The purpose of the setting is to minimize some even functional from an error that means that.

This result is obtained by generating the coefficients of the effect of the parameters of the reference model instead of the corresponding coefficients covered by the feedback of the object. If fixed, this approach has the advantage that the generated effects of the influence of parameters are the required private derivatives. (This is not true for the model setup scheme under the above.)

Intermittent models setting

As noted in section. 9.2, for continuous setup schemes, it is difficult to identify the properties of convergence. This is primarily due to the complexity of determining the gradient when changing the parameters of the model. We now consider the schemes in which the parameters of the model remain constant when determining the gradient. After the measurement interval, the model parameters are configured, then the measurement period begins again and so on.