Radiometric and photometric units can be linked together using sensitivity functions of the human eye V(X), sometimes called the luminous efficiency function. In 1924, the International Commission on Illumination, CIE, introduced the concept of the sensitivity function of the human eye in photopic vision mode for point sources of radiation and an viewing angle of 2° (CIE, 1931). This function, called functions of the MKO 1931, is still the photometric standard in the USA 0.

Judd and Woe introduced in 1978 modified function V(\)(Vos, 1978; Wyszeckl, Stiles, 1982, 2000), which in this book will be called function of the ICE 1978 The changes were associated with an incompletely correct assessment of the sensitivity of the human eye in the blue and violet spectral ranges, adopted in 1931. The modified function F(A) in the spectral range of wavelengths less than 460 nm has higher values. The CIE endorsed the introduction of the 1978 V(A) function, stipulating that “the sensitivity function of the human eye for point sources of radiation can be represented as a modified Judd's V(A) function” (CIE, 1988). Moreover, in 1990, the CIE resolved that “in cases where luminance measurements in the short wavelength range consistent with the determination of color are made by an observer normal to the radiation source, it is preferable to use the modified Judd function” (CIE, 1990).

In Fig. 16.6 shows the functions V(X) CIE 1931 and 1978. The maximum sensitivity of the eye occurs at a wavelength of 555 nm, which is in the green region of the spectrum. At this wavelength, the sensitivity of the eye is equal to 1, i.e. Y(555 nm) = 1. It can be seen that the 1931 CIE function Y(A) underestimates the sensitivity of the human eye in the blue region of the spectrum (A< 460 нм). В приложении 16.П1 приведены численные значения функций У (А) 1931 г. и 1878 г.

‘) This standard is also valid in Russia.

In Fig. Figure 16.6 also shows the function Y"(A) of the sensitivity of the human eye for the scotopic vision mode. The peak sensitivity in the scotopic vision mode occurs at a wavelength of 507 nm. This value is much less than the wavelength of the maximum sensitivity in the photopic vision mode. Numerical values ​​of the function V"(\) The ICE of 1951 is given in Appendix 16.P2.

Note that, although in a number of cases the function of the U (L) CIE 1978 is preferable, it still does not belong to the category of standards, since changing standards often leads to uncertainties. However, despite this, in practice it is used quite often (WyszeckiandStiles, 2000). The 1978 CIE function U(L), shown in Fig. 16.7 can be considered the most accurate description of variations in the sensitivity of the human eye in the photopic vision mode.

To find the sensitivity function of the human eye, use minimal flash method, which is a classic way of comparing light sources by brightness and determining

Rice. 16.6. Comparison of human eye sensitivity functions V(\) CIE 1978 and 1931 for photopic vision. The eye sensitivity function is also shown here V"(\) in scotopic vision mode, which is used at low ambient light levels

Rice. 16.7. U(L) (left ordinate) and luminous efficiency measured in lumens per watt of optical power (right ordinate). The maximum sensitivity of the human eye occurs at a wavelength of 555 nm (data from the CIE, 1978)

functions Y(A). In accordance with this method, a small round light-emitting surface is alternately illuminated (with a frequency of 15 Hz) by sources of the reference and comparison colors. Since the color fusion frequency is below 15 Hz, the colors of alternating signals will be indistinguishable. However, the luminance fusion frequency of the input signals is always higher than 15 Hz, so if two color signals differ in luminance, a visible flash is observed. The researcher's goal is to adjust the color of the radiation source being tested until the observed flare is minimal.

By changing the distribution of spectral radiation power P(L), it is possible to obtain any desired color shade. One of the variants of this distribution is characterized by the maximum possible light output. The maximum luminous efficiency can be achieved by mixing radiation of a certain intensity from two monochromatic light sources (MaeAdam, 1950). In Fig. Figure 16.8 shows the maximum achievable luminous efficiency values ​​obtained using one pair of monochromatic radiation sources. Maximum luminous efficiency white light depends on color temperature. At color temperature

Rice. 16.8. Relationship between the maximum possible luminous efficiency (lm/W) and chromaticity coordinates (x,y) on the 1931 CIE color chart.

At 6500K it is ~420 lm/W, and at lower color temperatures it can exceed ~500 lm/W. The exact value of luminous efficiency is determined by the position of the shade of interest within the range white on the color chart.

The sensitivity of automatic control systems is the degree of influence of the spread of parameters and their changes during operation on the static and dynamic properties of the control system, that is, on accuracy, quality indicators, frequency properties, etc.

The parameters of the control system (transmission coefficients and time constants) are determined by the physical parameters of its constituent elements (resistors, capacitors, inductors, etc.). The values ​​of the physical parameters of the elements, firstly, have a technological spread due to tolerances for the manufacture of elements, and secondly, they are subject to operational changes over time, which is due to their aging.

Therefore, the task arises of assessing the operation of the system when the parameters of its constituent elements change and vary.

This problem is solved by quantifying the sensitivity of the system. To do this, it is necessary to describe the equation control system in normal form, i.e.

When i=1, 2, ... , n, (7.13)

where n is the order of the system;

x i - coordinates of the system state;

f i - external influences applied to the system;

a ik - equation coefficients determined by the values ​​of the physical parameters of the elements that make up the system.

The parameters of the system elements that change over time during operation and due to scatter during manufacturing will be denoted by a j (j=1, 2, ... , m).

Then the equation of system (7.13) can be written in the form

When i=1, 2, ... , n. (7.14)

The solution of equations (7.14) determines the coordinates of the system: x 1 (t), x 2 (t), ..., x n (t), forming the initial motion of the system.

Let the parameters a j change by small values ​​Da j , then we have

. . . . . . . . . .

Considering small changes in parameters a j (j=1, 2, ... , m), we obtain new equations

for i=1, 2, ... , n.

A process in the same system, but with changed parameters, determined by solving equations (7.15), i.e. , is called varied motion.

The resulting difference in the course of processes in the system due to changes in parameters

For i=1, 2, ... , n

called additional movement.

For small deviations Da j this difference can be determined as follows:

When i=1, 2, ... , n. (7.16)

Let's denote

(j=1, 2, ... , m). (7.17)

Then the additional movement will be

When i=1, 2, ... , n. (7.18)

The quantities determined by expression (7.17) are functions of the sensitivity of the i-th coordinate of the system with respect to the j-th parameter.

Thus, in order to assess the degree of influence of scatter and changes in parameters on the coordinates of the system, it is necessary to determine the sensitivity functions for each coordinate from each changing parameter.

In the case under consideration, x i (t) are the coordinates of the system state. In general, similar sensitivity characteristics are also introduced for various indicators of system quality. Then in formula (7.17) instead of x i there will be a corresponding quality indicator, and in formula (7.18) - instead of Dx i - the change in this quality indicator. The sensitivity functions for frequency characteristics will be functions not of time, but of frequency. If quality indicators are expressed not by functions, but by numbers, then u ij are called sensitivity coefficients.

If we choose external influences as changing parameters a j, then we can obtain the system sensitivity functions with respect to external influences.

The sensitivity functions are determined as follows.

Let us differentiate the original equation (7.14) according to the changing parameters a j. Then we get

Changing the order of differentiation on the left side and taking into account (7.17), we obtain the expressions

For i=1,...,n; j=1,...,m; (7.19)

which are called sensitivity equations. The solution to these equations determines the sensitivity functions.

Let's consider sensitivity functions for frequency characteristics. We write the transfer function of the open-loop system in the form

W(s) = W(s, a 1 , a 2 , ... , a m), (7.20)

where a 1, a 2, ..., a m are system parameters that have technological variation or operational changes.

Then the amplitude and phase frequency characteristics also depend on these parameters

A(w) = A(w, a 1, ..., a m);

y(w) = y(w, a 1 , ... , a m).

The sensitivity functions for the amplitude and phase frequency responses will be

J=1, 2, ... , m. (7.21)

As a result, we obtain, as a function of frequency, expressions for the deviation of frequency characteristics due to scatter and changes in system parameters:

The sensitivity functions are determined when designing systems with the least changes in quality indicators when the values ​​of the system parameters deviate from the calculated ones.

Example. Determine the sensitivity functions for the system given by the following equation (Tp+1)x(t)=kg(t), where T, k are the changing parameters.

Solution. The equation of the system in normal form has the form

Let us introduce the sensitivity functions

We obtain the sensitivity equation based on (7.19)

Having found u xk and u xT from here, we calculate the change in the process progress of the controlled variable x(t) due to changes in parameters k and T using the formula

Transfer function of the system: .

Frequency characteristics: , .

Let's find the sensitivity functions of the frequency characteristics using the parameter T

Deviations of frequency characteristics

DA(w) = u AT (w)DT, Dy(w) = u Y T (w)DT.

QUESTIONS FOR SECTION 7

1. List common methods for increasing the accuracy of control systems. Explain them.

2. Give the concept of astatic control systems. How is the degree of astatism determined?

3. What is the advantage of increasing the degree of astatism of the system using isodromic devices?

4. Which system is invariant with respect to external influences?

5. What is meant by combined control?

6. How are the transfer functions of compensating devices in combined systems determined?

7. For what purposes are non-unit principal feedbacks used?

8. Formulate the concept of sensitivity of control systems.

9. How can sensitivity equations be obtained?

10.What are sensitivity functions and sensitivity coefficients?

Analytical calculation of NI is a rather complex task and can be fully carried out using a computer.

For cascades on BT, an analytical estimate of the NI is possible for the case of small nonlinearities ( U in on the same order as φ T=25.6 mV) .

Typically, the NI level is characterized by harmonic distortion K g. The total harmonic distortion is equal to

Where K G 2 and K G 3, respectively, harmonic coefficients for the second and third harmonic components (higher order components can be neglected due to their relative smallness).

Harmonic coefficients K G 2 and K G 3, regardless of the method of switching on the BT, are determined from the following relationships:

where B is the coupling factor (loop gain).

These expressions take into account only the nonlinearity of the emitter junction and are obtained based on the Taylor series expansion of the emitter current function I e=I e 0 exp( U in/φ T).

The coupling factor depends on the method of switching on the transistor and the type feedback. For a cascade with OE and POOST we have:

Where R g- signal source resistance (or R out previous cascade); R os R os=0).

For a cascade with OE and ∥OOSN

Where R eq=R to∥R n, R os

For cascade with OK

Where R eq=R e∥R n(see subsection 2.8).

For a cascade with OB

Harmonic coefficients K G 2 and K G 3, regardless of the method of switching on the PT, are determined from the following relationships:

where A is the coefficient equal to the second term of the expansion of the expression for the nonlinear slope in the Taylor series, equal to

A=I si/U² ots,

Where I si And U ots see Figure 2.33.

The coupling factor B depends on the method of switching on the transistor and the type of feedback. For a cascade with OI and POOST we have:

B = S 0 (R os + r and),

Where R os- POOST resistance (see subsection 3.2, in the absence of POOST R os=0).

For a cascade with OI and ∥OOSN we have:

B = S 0 R g R eq/R os,

Where R eq=R with∥R n, R os- resistance ∥OOSN (see subsection 3.4).

For cascade with OS

B = S 0 (R eq + r and),

Where R eq=R with∥R n(see subsection 2.11).

For cascade with OZ

B = S 0 ((R g∥R and) + r and).

In the above expressions r and- resistance of the semiconductor body in the source circuit, r and≈1/S si, Where S si- see subsection 2.10, for low-power PTs r and=(10…200) Ohm; R and- see figure 2.38.

Given ratios for evaluation K g give good result in the case of small nonlinearities, in the mode of large nonlinearities, you should use well-known machine methods, or turn to graphical methods for estimating NI.

8.2. Calculation of control unit stability

It is convenient to assess the stability of a control unit represented by an equivalent four-port network described by Y-parameters using the definition invariant stability coefficient :

When k><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 signal source, is determined by the following relationship:

For k>1 the amplifier is unconditionally stable, for k<1 - неустойчив, k=1 соответствует границе устойчивости.

The equivalent Y-parameters of the amplifier are determined, according to the method of subsection 2.3, at specified points in the operating frequency range. The use of the invariant stability coefficient is especially convenient for machine analysis of the control system. Other methods for assessing stability are described in.

8.3. Calculation of noise characteristics of the control unit

Noise in the control unit is mainly determined by the noise of active resistances and amplifying elements located in the input stages. The greatest contribution to the noise power created by the amplifier stage is made by the amplification element. The presence of its own noise sources limits the ability to amplify weak signals.

Depending on the nature of its occurrence, the transistor’s own noise is divided into thermal, shot, current distribution noise, excess noise, etc.

Thermal noise is caused by random movements of free charge carriers in conductors and semiconductors, shot noise is caused by the discreteness of the charge of carriers (electrons and holes) and the random nature of their injection and extraction through p-n junctions. Current distribution noise is caused by fluctuations in the distribution of emitter current to collector and base currents. All of the above types of noise have a uniform spectrum.

The nature of the excess noise has not yet been fully elucidated. They are usually associated with fluctuations in the state of the surface of semiconductors. The spectral density of these noises is inversely proportional to frequency, which is why they are called 1/f noise. They are also called flicker noise, flicker noise and contact noise. 1/f type noise increases greatly with defects in the crystal lattice of the semiconductor.

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

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

Figure 8.1. Equivalent circuits of active noise impedance

The corresponding values ​​of the emf and current of these sources are as follows (see subsection 2.2):

where Δ f- operating frequency band; k=1.38·10 -23 - Boltzmann constant; T - temperature in degrees Kelvin; R w- noise resistance, G w- noise conductivity, G w=R w -1 .

For standard temperature T=290°K these formulas can be simplified:

Spectral noise densities for voltage and current are:

where , are differentials of root-mean-square voltages and noise currents as random functions of time t operating in the passband df.

Any active element can be represented as a noisy four-terminal network (Figure 8.2) and its noise characteristics can be calculated using these formulas.

Figure 8.2. Noisy quadripole

B shows expressions for the noise parameters of the BT and PT normalized spectral noise densities by voltage R w=F RU/4kT, by current G w=F RI/4kT and mutual spectral density F w, representing respectively noise resistance, noise conductivity and mutual spectral noise density.

For BT connected according to the scheme with OE:

R w = r b + 0,2I b r b 2 + 0,02I to S 0 -2 ,

G w = 0,2I b + 0,02I to g 2 S 0 -2 ,

F w = 1 + 0,02I b r b + 0,02I to gS 0 -2 ,

Where I b And I to in milliamps, g and S 0 in millisiemens. When taking into account flicker noise for frequencies f≥10Hz in these expressions, the following should be taken:

I" b = (1 + 500/f)I b,

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

For PT switched on with OP:

R w = 0,75/S 0 ,

G w = R w ω² C² zi = 40R w f² C² zi,

F w = 1 + ωC zi R w= 1 + 6.28· C zi R w.

These formulas are also applicable to other transistor circuits.

Assuming that the spectral noise densities are uniform, we can obtain an expression for the cascade noise factor:

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

By examining this expression for an extremum, we determine the optimal resistance of the signal source R g opt, at which the stage noise factor F is minimal:

In most cases, it turns out that R g opt does not match R g, optimal from the point of view of obtaining the necessary f in cascade ( R g opt>R g). The way out of this situation is to connect an anti-noise correction circuit between the first and second stages (Figure 8.3).

Figure 8.3. Simple anti-noise correction

By introducing anti-noise correction, one achieves an increase in the transmission coefficient of cascades in the HF region (by introducing attenuation at the LF and MF by the correcting circuit), thereby compensating for the decrease in gain at HF ​​due to the high-impedance R g opt.

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

Calculating the noise of cascaded quadripoles (multistage amplifier) ​​usually comes down to calculating the noise figure of the input circuit and input stage. The first stage in such an amplifier operates in low-noise mode, and the second and other stages operate in normal mode.

Calculating noise in the general case is a complex problem that can be solved using a computer. For a number of special cases, noise parameters can be calculated using the relationships given in.

8.4. Sensitivity Analysis

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

Sensitivity factor (sensitivity function or simply sensitivity ) is a quantitative assessment of the change in device parameters (including the nuclear power plant) with a given change in the parameters of its components.

The need to calculate the sensitivity function arises when it is necessary to take into account the influence of environmental factors (temperature, radiation, etc.) on the characteristics of the nuclear power plant, when calculating the required tolerances on component parameters, when determining the percentage of IC output, in optimization problems, modeling, etc.

Sensitivity function S i device parameter y to change a component parameter x i is defined as the partial derivative

This expression was obtained based on the Taylor series expansion of a function of several variables, where

Neglecting partial derivatives of the second or higher order, we obtain the relationship between the sensitivity function and the deviation of the parameter:

There are different types of sensitivity function:

◆ absolute sensitivity, absolute deviation is equal to ;

◆ relative sensitivity , the relative deviation is ;

◆ semi-relative sensitivities , .

The choice of the type of sensitivity function is determined by the type of problem being solved, for example, for a complex transmission coefficient, the relative sensitivity is equal to the relative sensitivity of the module (real part) and the semi-relative sensitivity of the phase (imaginary part):

For simple circuits The sensitivity function can be calculated by direct differentiation of the circuit function presented in analytical form. For complex circuits, obtaining an analytical expression of the circuit function is a complex task; it is possible to use direct calculation of the sensitivity function through increments. In this case, it is necessary to carry out n analyzes of the circuit, which is very irrational for complex circuits.

There is an indirect method for calculating sensitivity using transfer functions, proposed by Bykhovsky. According to this method, the sensitivity function, for example, the direct transfer coefficient, is equal to the product of the transfer functions from the input of the circuit to the element with respect to which the sensitivity is sought, and the transfer function “element - output of the circuit” (Figure 8.4a).

Figure 8.4. Indirect method for calculating sensitivity functions

Since the calculation of the sensitivity function comes down to the calculation of transfer functions, to find them it is possible to use, for example, the generalized method of nodal potentials. The indirect calculation method using transfer functions allows one to find sensitivity functions of higher orders. Figure 8.4b illustrates the determination of the second-order sensitivity function. In general, there is n! signal transmission paths, each of which contains n+1 factors.

Below we describe a method for calculating the sensitivity function, combining the direct method of differentiation and the indirect method using transfer functions, which allows one to find the sensitivity to n elements of the circuit in one analysis. Let's consider this method using examples of obtaining expressions for the absolute sensitivity of the first order S-parameters of electronic circuits described by the conductivity matrix [Y].

In the matrix representation, the characteristics of electronic circuits, including scattering parameters [S], are defined in the form of relations of algebraic additions of the matrix [Y] (see subsection 7.2). The variable parameter is included in some elements of algebraic additions. Determining the sensitivity function in this case is reduced to finding derivatives of the relations of algebraic additions (or algebraic additions and a determinant) over the elements that contain the variable parameter. In the case when the variable parameter is included in the elements of the additions of the determinant functionally, the sensitivity is defined as a complex derivative.

To determine the derivatives of algebraic complements with respect to the variable parameters of the elements included in them, we will use the theorem stating that the derivative of a determinant with respect to any element is equal to the algebraic complement of this element. The proof of the theorem is based on the Laplace expansion of the determinant

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

S ij = k ijΔ ji/Δ – δ ij.

Let us determine the sensitivity functions of the scattering parameters to a passive two-terminal network y o connected between arbitrary nodes k and l (see Figure 8.5a)

Figure 8.5. Calculation of S-parameter sensitivity

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 ijΔ j(k+l) Δ ( k+l)i /Δ² = – k ij[(Δ jk – Δ jl)(Δ ki – Δ li)]/Δ²

When obtaining this and subsequent expressions, the following matrix relations are used:

Δ ( i+j)(k+l) = Δ i(k+l) + Δ j(k+l) = (Δ ik – Δ il) + (Δ jk – Δ jl),

Δ ijΔ kl – Δ ilΔ kl = ΔΔ ij,kl.

For electronic circuits containing BTs modeled by ITUT (see subsection 2.4.1), we determine the sensitivity of the S-parameters to the conductivity of the control branch g e=1/r e and the parameter of the controlled source a included respectively between nodes k, l, and p, q (Figure 8.5b):

S S ij gе = 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 ijΔ ( 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/dα = k ij(Δ ji(k+l)(p+q) Δ – Δ ( k+l)(p+q) Δ ji)/Δ² = – k ijΔ j(p+q) Δ ( k+l)i /Δ² = – k ij[(Δ jp -Δ jq)(Δ ki –Δ li)]/Δ².

If electronic circuit contains PTs modeled by ITUN (see subsection 2.4.1), then the sensitivity of the scattering parameters to the slope S, included between nodes p, q at control nodes k, l (Figure 8.5c), is equal

S S ij S= dS ij/dS = k ij(Δ ji(k+l)(p+q) Δ – Δ ( k+l)(p+q) Δ ji)/Δ² = – k ijΔ j(k+l) Δ ( p+q)i /Δ² = – k ij[(Δ jk –Δ jl)(Δ pi –Δ qi)]/Δ².

The sensitivity of the scattering parameters to any Y-parameter of the subcircuit (Figure 8.5d), 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 ijΔ jl Δ ki /Δ².

With known sensitivity y kl to the subcircuit element parameter x (see Figure 8.5d), the sensitivity of the S-parameters of the complete circuit to this parameter, in accordance with the concept of a complex derivative, will be expressed as

S S ij x = ( dS ij/dy kl)(dy kl/dx) = S S ij ykl· S y kl x.

The last expression indicates the possibility of using the subcircuit method in analyzing the sensitivity of complex electronic circuits.

Knowing the relationship between scattering parameters and secondary parameters of electronic circuits ( K U, Z in, Z out etc.) and the sensitivity of scattering parameters to changes in circuit elements, it is possible to find sensitivity functions of secondary parameters to changes in these elements. For example, for the voltage transfer coefficient from i-th to jth node K ij=S ji/(1+S 11) sensitivity to changes in parameter x (assuming that S ij=f(x) And Sii=φ( x)) we get

S K ij x = dK ij/dx = [S S ij x(1+S ii) – S S ii x S ij]/(1 + S ii)².

Likewise for Z input(out) (Zii (jj)) we have

Zii (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 effectively used in determining the sensitivity of higher orders for all kinds of characteristics of electronic circuits. The implementation of the sensitivity calculation algorithms obtained in this way is reduced to calculating and enumerating the corresponding algebraic additions, which goes well with finding other small-signal characteristics of electronic circuits.

8.5. Machine methods for analyzing nuclear power plants

Subsection 2.3 presents the main idea of ​​the generalized method of nodal potentials, on the basis of which most of the relationships for the preliminary calculation of amplification stages were obtained. However, along with the undoubted advantages this method(ease of programming, small dimension of the resulting conductivity matrix Y, n*n, where n is the number of circuit nodes without a reference one), this method has a number of significant disadvantages. First of all, it should be noted that it is impossible to represent some ideal models of electronic circuits (short-circuited branches, voltage sources, dependent current-controlled sources, etc.) in the form of conductivity. In addition, the representation of inductance by conductivity is inconvenient in the time analysis of circuits, which is associated with the Laplace transform (Laplace operator p must be in the numerator so that the system of algebraic equations and the system of differential equations obtained as a result of the transformation have the same coefficients).

Currently, topological methods for forming a system of equations are most widespread electrical circuit, the most common of which is tabular .

In this method, all equations describing the circuit are included in common system equations containing Kirchhoff equations for currents, voltages and component equations.

Kirchhoff's equations for currents can be represented as

AI in = 0,

Where A- incidence matrix, describing the topology of the circuit, I in- branch current vector.

Kirchhoff's stress equations have the form

V in – A t V p = 0,

Where V in And V p- respectively, the vector of voltages of branches and node potentials, A t- transposed incendence matrix A.

In general, the equations describing the elements of the circuit can be presented in the following form:

Y to B to + Z to I to = W in,

Where Y in And Z in- respectively, quasi-diagonal matrices of conductivity and resistance of branches, W in- vector, which includes independent sources of voltage and current, as well as initial voltages and currents on capacitors and inductors.

Let us write the above equations in the following sequence:

V in – A t V p = 0;

Y to B to + Z to I to = W in;

AI in = 0;

and represent it in matrix form

or in general

The tabular method has mainly theoretical significance, since along with the main advantage, which is expressed in the fact that it is possible to find all currents and voltages of branches and node potentials, it has a number of significant disadvantages. First of all, it should be noted that the method is redundant, leading to a large matrix dimension T. It should further be noted that many ideal controlled sources introduce unnecessary variables. For example, the input current of voltage-controlled current and voltage sources and the input voltage of current-controlled current and voltage sources are zero, but they are treated as variables in this method.

In practical terms, a modification of the tabular method is most often used - modified node method with check .

The idea of ​​this method is to divide elements into groups; one group is formed from elements that are described using conductivities; for elements of the second group such a description is impossible. Since the voltages of the branches can be expressed through the currents of the branches of the first group, and the voltages of the branches through the nodal potentials, it is possible to exclude from the tabular equations all the voltages of the branches, and for elements of the first group also the currents of the branches. When introducing additional equations for currents in branches with elements of the second group, a check is made for the presence of previously known (zero) variables. As a result of this transformation, we obtain the equations of the modified nodal method with verification

or in general

T m X=W,

where n is the dimension of the conductivity matrix Yn 1 elements of the first group (n is the number of circuit nodes without a zero); m is the number of additional equations for elements of the second group; Jn- vector of independent current sources; I 2 - vector of currents of branches of elements of the second group; W 2 - vector, which includes independent voltage sources, as well as initial voltages and currents on capacitors and inductances, represented by elements of the second group.

To simplify programming, a matrix of coefficients of the system of equations of the modified nodal method is usually presented T m as a sum of two matrices of dimension (n+m)*(n+m)

T m = G + pC.

Into the matrix G enter all active conductivities and coefficients corresponding to frequency-independent elements, and into the matrix C- all frequency-dependent elements, and inductances are usually represented as an element of the second group, i.e. resistance. Next, a solution to this system of equations is found using Gauss-Jordan algorithms or L/U decomposition.

In frequency analysis of electronic circuits, the operator p is replaced by jω, a frequency cycle is organized, within which a system of equations is formed for each frequency point, which is solved with respect to the voltages and currents of interest.

When timing analysis of linear electronic circuits, it is possible to directly use the modified nodal form of the equations

(G+pC)X = W.

After moving to the time domain, we get

Gx + Cx" = W,

Cx" = W – Gx.

The solution to the resulting system of differential equations is found by numerical integration. One of effective methods numerical integration are methods based on linear multistep formulas , the simplest of which include Euler’s formulas (direct and inverse) and the trapezoidal formula.

By dividing the time interval into a finite number of segments h and putting t n+1 =t n +h, for each moment of time tn you can find an approximation x n to a true solution x(tn) by applying linear multi-step formulas:

x n+1 = x n + hx" n(direct Euler formula);

x n+1 = x n + hx" n+1 (inverse Euler formula);

x n+1 = xn + (h/2)(x"n + x"n+1) (trapezoid formula).

Finding x"n+1 for the (n+1)th calculation step is possible by applying Euler's direct formula.

Since the voltage on the capacitor and the current flowing through it are related by the ratio i=CdV/dt, and for inductance we have V=Ldi/dt, then the application of the inverse Euler formula is equivalent to the transition from capacitances and inductances to their equivalent circuits shown in Figure 8.6, in As a result, the circuit becomes resistive. Such models of inductance and capacitance are called grid (accompanying, discrete) models .

Figure 8.6. Grid models for Euler's inverse formula

Finding the operating point or calculating by DC is the first step in the nonlinear analysis of the control system. Analysis of the DC characteristics of circuits containing nonlinear resistances comes down to solving a system of nonlinear equations of the form f(x)=0.

Since Kirchhoff’s laws 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 discussed. The structure of the resulting tabular equations will be discussed below.

To solve a system of nonlinear equations f(x) applies Newton-Raphson method . The method involves the use of an initial approximation x 0 , carrying out the iterative procedure and, if the value |( x n +1 –x n)/x n+1 | is small enough, a statement of the fact of convergence (n is the number of iterations):

x n +1 = x n – J -1 f(x n),

Where J- Jacobian (Jacobi matrix) with dimension (m*m)

In the process of iterative processing of this system of equations, at each iteration stage the values ​​can be obtained f(x n) And J; this is equivalent to solving a linear equation of the form

J(x n +1) – x n) = –f(x n).

In other words, solving nonlinear equations can be interpreted as resolving linear equations at each step of the iterative process.

The structure of the Jacobian externally coincides with the tabulated equations of linear circuits, which are transformed taking into account the calculation for direct current - the capacitors are removed and the inductors are short-circuited.

Let the tabular equations be given in the following form:

V in – A t V p = 0;

p(V in,i in) = W;

AI in = 0;

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

The Jacobian matrix at the nth iteration will look like

Where ; Where .

To form the Jacobian, it is possible to use various modifications of the tabular method, including a modified nodal method with verification. The result of the DC circuit analysis (DC mode) can be used as an initial approximation in the timing analysis of nonlinear electronic circuits.

Nonlinear equations are easily included in circuit equations prepared by tabular or modified nodal methods. Linear elements, as before, are linear component equations. Nonlinear equations are characterized by equations in implicit form, although sometimes nonlinearities can be described in explicit form. Nonlinear capacitances and inductances are best described using additional variables - electric charges and magnetic fluxes, respectively, which must be entered into the vector of unknowns. If this is done, then the equations written using both tabular and modified nodal methods can be presented in the following form:

f(x", x, W, t) ≣ Ex" + Gx +p(x) = 0,

Where E And G- constant matrices, and all nonlinearities are reduced to a vector p(x).

The resulting system of differential equations is solved by integration using backward differentiation formulas and the Newton-Raphson algorithm, for which the Jacobian is formed. In general, the structure of the Jacobian for a linear and nonlinear circuit is identical, the difference between them is that the nonlinear capacitance (inductance) will be represented by two equations, and the charge q (flux f) will become another unknown. However, for linear capacitances and inductances, it is possible to introduce charges and magnetic fluxes as variables, which will lead to the coincidence of the Jacobian and the matrix of the system of equations. Any nonlinear conductivity will appear in the Jacobian in a similar way to linear conductivity in the matrix C modified nodal method. Thus, it becomes possible to have a unified approach to the formation and solution of equations of linear and nonlinear circuits in order to obtain their time and frequency characteristics, which is successfully implemented in modern circuit design packages.

The listed methods, as well as other issues of analysis of electronic circuits, are given in more detail in. One of the Electronics Workbench circuit design packages is described.

Sensitivity is called the reaction of various devices to changes in the parameters of its components.

Sensitivity factor (sensitivity function or simply sensitivity ) is a quantitative assessment of the change in device parameters (including the nuclear power plant) with a given change in the parameters of its components.

The need to calculate the sensitivity function arises when it is necessary to take into account the influence of environmental factors (temperature, radiation, etc.) on the characteristics of the nuclear power plant, when calculating the required tolerances on component parameters, when determining the percentage of IC output, in optimization problems, modeling, etc.

Device parameter sensitivity function y to a change in a component parameter is defined as the partial derivative

This expression was obtained based on the Taylor series expansion of a function of several variables, where

Neglecting partial derivatives of the second or higher order, we obtain the relationship between the sensitivity function and the deviation of the parameter:

.

There are different types of sensitivity function:

¨ absolute sensitivity, absolute deviation is equal to ;

¨ relative sensitivity , the relative deviation is ;

¨ semi-relative sensitivities , .

The choice of the type of sensitivity function is determined by the type of problem being solved, for example, for a complex transfer coefficient relative sensitivity is equal to the relative sensitivity of the module (real part) and the semi-relative sensitivity of the phase (imaginary part):

For simple circuits, the sensitivity function can be calculated by direct differentiation of the circuit function presented in analytical form. For complex circuits, obtaining an analytical expression of the circuit function is a complex task; it is possible to use direct calculation of the sensitivity function through increments. In this case, it is necessary to carry out n analyzes of the circuit, which is very irrational for complex circuits.

There is an indirect method for calculating sensitivity using transfer functions, proposed by Bykhovsky. According to this method, the sensitivity function, for example, the direct transfer coefficient, is equal to the product of the transfer functions from the input of the circuit to the element with respect to which the sensitivity is sought, and the transfer function “element - output of the circuit” (Figure 8.4a).

Since the calculation of the sensitivity function comes down to the calculation of transfer functions, to find them it is possible to use, for example, the generalized method of nodal potentials. The indirect calculation method using transfer functions allows one to find sensitivity functions of higher orders. Figure 8.4b illustrates the determination of the second-order sensitivity function. In general, there is n! signal transmission paths, each of which contains n+1 factors.

Below we describe a method for calculating the sensitivity function, combining the direct method of differentiation and the indirect method using transfer functions, which allows one to find the sensitivity to n elements of the circuit in one analysis. Let's consider this method using examples of obtaining expressions for the absolute sensitivity of the first order S-parameters of electronic circuits described by the conductivity matrix [Y].

In the matrix representation, the characteristics of electronic circuits, including scattering parameters [S], are defined in the form of relations of algebraic additions of the matrix [Y] (see subsection 7.2). The variable parameter is included in some elements of algebraic additions. Determining the sensitivity function in this case is reduced to finding derivatives of the relations of algebraic additions (or algebraic additions and a determinant) over the elements that contain the variable parameter. In the case when the variable parameter is included in the elements of the additions of the determinant functionally, the sensitivity is defined as a complex derivative.

To determine the derivatives of algebraic complements with respect to the variable parameters of the elements included in them, we will use the theorem stating that the derivative of a determinant with respect to any element is equal to the algebraic complement of this element. The proof of the theorem is based on the Laplace expansion of the determinant

.

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

.

Let us determine the sensitivity functions of the scattering parameters to a passive two-terminal network connected between arbitrary nodes k and l (see Figure 8.5a)

When obtaining this and subsequent expressions, the following matrix relations are used:

For electronic circuits containing BTs modeled by ITUT (see subsection 2.4.1), we determine the sensitivity of the S-parameters to the conductivity of the control branch and the parameter of the controlled source a connected, respectively, between nodes k, l, and p, q (Figure 8.5b):

If the electronic circuit contains PTs modeled by ITUN (see subsection 2.4.1), then the sensitivity of the scattering parameters to the slope S, connected between nodes p, q at control nodes k, l (Figure 8.5c), is equal to

a, A. I. Golikov a, E. V. Khoroshilova b

Annotation: The sensitivity function generated by the convex programming problem is considered, and its properties of monotonicity, subdifferentiability, and closedness are investigated. A connection is established with the Pareto-optimal set of estimates for the multicriteria convex optimization problem. Its role in systems of optimization problems is clarified. It has been established that the solution of such systems often comes down to minimizing the sensitivity function on a convex set. Numerical methods for solving such problems are proposed and their convergence is proven. Bible 20.

Keywords: sensitivity function, properties of the sensitivity function, multicriteria convex optimization problems, convergence of a numerical algorithm.

English version:
Computational Mathematics and Mathematical Physics, 2011, 51 :12, 2000-2016

Abstract databases:
Post type: Article
UDC: 519.658.4
Received by: 30.05.2011

Citation sample: A. S. Antipin, A. I. Golikov, E. V. Khoroshilova, “Sensitivity function, its properties and applications”, 51:12 (2011), 2126-2142; Comput. Math. Math. Phys. , 51 :12 (2011), 2000-2016

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This publication is cited in the following articles:

1. Yu. G. Evtushenko, M. A. Posypkin, “The method of uneven coverings for solving multicriteria optimization problems with guaranteed accuracy”, J. Comput. math. and math. physical, 53 :2 (2013), 209-224; Yu. G. Evtushenko, M. A. Posypkin, “Nonuniform covering method as applied to multicriteria optimization problems with guaranteed accuracy”, Comput. Math. Math. Phys. , 53 :2 (2013), 144-157
2. E. M. Vikhtenko, N. N. Maksimova, R. V. Namm, “Sensitivity functionals in variational inequalities of mechanics and their application to duality schemes”, Sib. magazine Comput. math., 17:1 (2014), 43-52; E. M. Vikhtenko, N. N. Maksimova, R. V. Namm, “A sensitivity functionals in varying inequalities of mechanics and their application to duality schemes”, Num. Anal. Appl. , 7 :1 (2014), 36-44
3. Yu. G. Evtushenko, M. A. Posypkin, “The method of uneven coverings for solving multicriteria optimization problems with a given accuracy”, Machine. and telemech., 2014, No. 6, 49-68; Yu. G. Evtushenko, M. A. Posypkin, “Method of non-uniform coverages to solve the multicriteria optimization problems with guaranteed accuracy”, Autom. Remote Control, 75:6 (2014), 1025-1040
4. A. V. Zhiltsov, R. V. Namm, “The Lagrange multiplier method in the problem of finite-dimensional convex programming”, Dalnevost. math. magazine, 15 :1 (2015), 53-60