JP2643629B2 - Calculation method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to the calculation of parameters by the nonlinear least squares method and the evaluation of the importance of the obtained parameters.
It relates to the valuation method .

For example, given an equivalent circuit of an electronic component,
Measure the constants of the elements that make up the equivalent circuit at different frequencies
Estimated from the estimated impedance value, and after
It provides a way to determine the necessity.

Specifically, one variable (X) and a function
(Y) is related to several parameters (C ₀ , C ₁ ,...).
…, C _p ) derived theoretically or empirically
When expressed by a relational expression, a large number of X
Parameters matching the case from the data paired with
Meter and determine the need for its parameters
If necessary, a variable after the relational expression omitting the parameter
The function (Y) when (X) is a certain value (X = X ₁ )
Useful when estimating.

[0004]

2. Description of the Related Art In recent years, computers have developed.
As a result, CAD is also used in electronic equipment circuit design.
I'm getting up. In such CAD, an electronic circuit
Combined with the limits of the characteristics of the electronic components that make up
The electronic device operates normally under the use conditions even under the worst conditions.
It is necessary to confirm. When performing such a check,
Select the components whose electronic component characteristics are the limit
Experiment by prototyping an electronic circuit combining components
Although the operation of electronic devices has been confirmed
When it becomes complicated and a shortening of the design period is required,
It is impossible to confirm all adverse conditions by experiment,
Computer solution as an alternative to unaware experiment
The importance of analysis has increased.

At the same time , a computer circuit analysis
It has become necessary to improve the accuracy of the fruits. To improve accuracy
The equivalent circuit that approximates the characteristic values of electronic components to actual values
There has been gills considered. The simplest electronic component, a resistor
Even an equivalent circuit combines capacitors and coils in addition to resistors.
To improve accuracy, it is expressed as a combination
For this reason, the equivalent circuit is complicated and represented by many elements.

As described above, a circuit analysis by a computer
The electronic parts that make up the electronic circuit to improve the accuracy of
As a result of making the equivalent circuit of the product more complicated,
The circuit for this is much more multi-element than the actual circuit
Therefore, a computer having higher performance is required.
That is, an economic problem occurs. Therefore, equivalent
The circuit should not be unnecessarily complicated,
Unnecessary elements are desirably omitted.

[0007] The lines form the least squares method has many of the literature as "multivariate analysis" software program for analysis are commercially available. For example, Kyoritsu Shuppan Co., Ltd. 1986 1
It is shown in "Handbook of PC Statistical Analysis" edited by Kazumasa Wakimoto, Tomoyuki Tarumi, and Yutaka Tanaka published in February. Cause of n months (x _1, x _2, ......, x _n) in the case where the result y is expressed by a line format represented by the following equation (1) is applied.

[0008]

(Equation 1)

An equivalent circuit of a resistor can be represented as shown in FIG. 3, where the impedance (Z) of the resistor is
A non-line format as shown in the following equation (2).

[0010]

(Equation 2)

[0011] Instead of the measurement frequency f _i, the impedance Z _mi resistor is measured, it becomes possible to obtain the equivalent circuit constant using the nonlinear least squares method, conventional methods Sciences 1972 3 T. issued on March 10 R. It is shown in "Introduction to Numerical Computation Methods for Computers" by Mackara, translated by Isao Miura and Yoichi Tao. This includes when the given relational expression shown in equation (3), using the value y _j of y when the x = x _j, the parameters _{C 0, C 1, C 2} ,
.., And how to find C _p .

[0012]

(Equation 3)

[0013] This (Equation 3) is represented by C ₀ , C ₁ , C ₂ , ..., C _p
About C ₀₀ , C ₁₀ , C ₂₀ ,..., C _p0
And linear Taylor expansion up to the next section, and (Equation 4).

[0014]

(Equation 4)

Here, when the following (Equation 5) is set,

[0016]

(Equation 5)

(Equation 4) has the same form as (Equation 1). C _i
= C _i0 , a ₀ and a _i are known values for x = x _j . And it is possible to obtain the u _i by applying the conventional linear least squares method. Using u _i found, using the revised values of C _i0 + u _i to C _i0 is in the second calculation. Using the corrected C _i0 , a ₀ and a _i are obtained from ( _Equation 4), and u _i is obtained again. By repeating this process, it is possible to determine the parameters C _i with the necessary accuracy.

[0018]

However, the conventional method has the following problems.

(1) Generally, in order to obtain a linear expression having the form of (Equation 4) by performing a Taylor expansion of the relational expression given by (Equation 3), it is necessary to partially differentiate the relational expression with respect to parameters. It requires a fairly high level of mathematical knowledge to perform partial differentiation. In other words, only people who have considerable advanced knowledge can be used conventional non-linear least squares method, the majority of people do not take advantage. The relational expression of (Equation 3) is already given in many cases.

(2) In the conventional method, the estimated value of the parameter is used within the range where x in (Equation 3) changes, and (Equation 3)
The value of the parameter can be determined so that the difference (error) between the value y _i estimated from the above and the measured value y _i ′ is distributed almost equally. However, when the measured value greatly changes within the range where x changes, it may be better to determine the value of the parameter so that the relative error obtained by dividing the error by the estimated value is almost equally distributed. I can not cope.

(3) After obtaining the parameters, it is necessary to test the significance of the parameters. If it is not significant, the parameter may be omitted. The conventional method does not provide a test method.

The present invention solves the above-mentioned problems of the conventional method, and (1) does not require advanced mathematical knowledge.

(2) If necessary, determine parameters so that the difference (error) between the actually measured value and the estimated value is equally distributed, or the relative error obtained by dividing the error by the estimated value is equally distributed. .

(3) To provide a method for testing the significance of a parameter. It is an object.

[0025]

In order to achieve the above object, according to the present invention, a certain characteristic y has a variable x and a parameter C ₀ ,
C _1, ......, when given by the relation between the C _p, n sets of data _{(x 1, y m1),} (x 2, y m2), ......, (x n, y
In the case where the _mn) with a nonlinear least squares method obtains the values of the parameters, enter the upper limit value and the lower limit value n sets of data (x _i, y _mi) and for each parameter, the lower limit for each of the parameters from determining the five level values between the upper limit value, the characteristic value y _ei obtained from equation the measured value when the variable x to each of the five each case in combination with one another the level value of each of the parameters x _i The value obtained by multiplying the square value of the difference of y _{mi by} the weighting coefficient W _i is calculated for all the variables x _i , the sum of these values is obtained, and the minimum value of the deviation square sums for each case is smaller than the set value. If the value is also smaller, the calculation is terminated, and the level value of each parameter in this case is set to the value of the parameter to be obtained. If the minimum value is larger than the set value, the level value of each parameter that gives the minimum value To include The upper limit value and the lower limit value are limited and re-input to the input unit, and the above calculation is repeated.

[0026]

According to the present invention, since the sum of the squared deviations S is equal to or less than the initially set value, the level value of each parameter is used as the estimated value of the parameter to be obtained, so that sophisticated mathematical knowledge is not required. . Also anyone, the nonlinear <br/> least square method by the present invention it is possible to calculate the value of the parameter used. Furthermore, by utilizing the fact that the F-distribution, can be assayed the significance of the parameter C _i.

[0027]

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a block diagram showing a calculation method of an equivalent circuit constant of a resistor according to an embodiment of the present invention. FIG. 1 is a block diagram showing a method of calculating an equivalent circuit constant of a resistor according to the present embodiment. In FIG. 1, 1 is a data input unit, 2 is a parameter operation unit, 3 is a deviation square sum operation unit, 4 is an evaluation unit, 5 is a display unit, and is composed of five parts. FIG. 2 is a flowchart showing the flow of the calculation of this embodiment.

First, in the data input section 1, the measurement frequency f
Actual impedance Z of the resistor measured by changing _i
Input _mi and measurement frequency f _i (i = 1, 2, 3,...)
..., n). Although n is the number of measured values, it must be sufficiently larger than the number of parameters such as R, L ₁ , L ₂ , and C constituting the equivalent circuit. In addition, the upper limit value (K _U ) and the lower limit value (K _L ) of a region that is considered to include the true value of each parameter due to a unique technique are input for each parameter.
If this area is set sufficiently wide, it is not necessary to input the upper limit and the lower limit each time.

The parameter calculation unit 2 uses the upper limit value (K _U ) and the lower limit value (K _L ) of each parameter, such as R, L ₁ , L ₂ , and C, which constitute an equivalent circuit, to obtain five levels. Is determined as (Equation 6).

[0030]

(Equation 6)

As a method of determining the five levels, there are an isometric (Equation 7) method and an isometric (Equation 8) method.

[0032]

(Equation 7)

[0033]

(Equation 8)

I represents the parameter level, and K _i represents the parameter level value. In practice, when calculating the equivalent circuit constant of the resistor, it is appropriate to use an equivalent ratio.

FIG. 3 shows an equivalent circuit of a resistor in the deviation square sum calculation section 3. R is a resistance component and is substantially equal to a DC resistance value. L ₁ and C are an inductance component and a capacitor component caused by the resistor. L ₂ is the inductance component due to the terminal of the resistor. When the equivalent circuit of the resistor is represented in Figure 3, the impedance Z _i of the measuring frequency f _i is expressed in the form of the above equation (2).

For each case where the five levels of the equivalent circuit parameters R, L ₁ , L ₂ , and C are combined, the measurement frequency f
_The impedance Z _ei for _{i is} determined from ( _Equation 2), and the sum of squared deviations S is determined for each case from (Equation 9) from the impedance Z _ei and the actually measured impedance Z _mi when the measurement frequency is f _i .

[0037]

(Equation 9)

W _i is a weighting factor. If W _i = 1, the difference (error) between the actually measured value Z _mi of the impedance at the measured frequency f _i and the estimated value Z _ei obtained from (Equation 2) is _equally distributed. Can be calculated. If W _i = 1 / Z _ei · Z _ei , the value of the parameter can be calculated so that the value (relative error) obtained by dividing the error by the estimated value of the impedance ( Z _ei ) is _equally distributed.

The evaluation unit 4 obtains the sum of squared deviations S for each case according to (Equation 9). Let the minimum value of S be S _MIN and the maximum value be S _MAX . When each parameter of the equivalent circuit has a true value, S has a minimum value. When the degree of freedom and phi, phi but is expressed as _{φ = n-4-1, W i} = 1
When S / (φ · R ² ) and W _i = 1 / (Z _ei · Z _ei ), the square root of S / φ is a value representing a relative error. Since this relative error is determined according to the purpose of use of the equivalent circuit, α in (Equation 10) can be given first. When the minimum value S _{MIN of the} sum of squared deviations satisfies ( _Equation 10), the calculation is stopped, and the level value of each parameter in the case of providing S _MIN is set to a value to be obtained (estimated value of the true value of each parameter). .

[0040]

(Equation 10)

It should be noted, again performed when satisfying the restless (11), the same feed by changing the upper limit value K _U and the lower limit value K _L of the parameter in the parameter calculating unit 2 that the (number 10).

[0042]

[Equation 11]

Β is a value related to the calculation accuracy of the sum of squared deviations and can be initially set. When the parameter level giving S _MIN is i = 2 to 4, the new upper limit K _{1U is set} to K _1U = K _{i + 1} and the new lower limit K _{1L is set} to K _1L = K _i-1 . Also S _MIN
When the level of the parameter that gives is 1 or 5, the new upper limit K _1U and the new lower limit K _1L are respectively expressed by (Equation 12).

[0044]

(Equation 12)

In equation (12), k is k ≧ 0, and the level of the parameter giving S _MIN is increased when i = 1 or i = 5 continuously, and when i = 2 to 4, Should be small. If none of (Equation 10) and (Equation 11) are satisfied, the calculation is terminated because the equivalent circuit constant cannot be estimated.

The display unit 5 displays a value to be obtained for the equivalent circuit constant (estimated value of the true value of the parameter), a calculation time, the number of calculations, and the like.

The estimated values of the true values R ₀ , L ₁₀ , L ₂₀ , and C ₀ of the equivalent circuit constants can be obtained by the method described above. As an example, a test for the significance of the resistance R will be considered. The impedance Z _{e0i at} the measurement frequency f _i can be obtained using ( _Equation 2), which is _expressed as ( _Equation 13).

[0048]

(Equation 13)

[0049] is as if the measured value Z _mi is the measurement error epsilon _i impedance of time (several 14).

[0050]

[Equation 14]

Assuming that the measurement area is not wide, the weight coefficient W _{i is set} to W _i
= 1. The measurement error ε _i has a normal distribution N (0, σ ² ) with an average value of 0 and a standard deviation σ. Next, the resistance R is changed by ΔR, and the impedance when R = R ₀ + ΔR is Z _i.
And This is Taylor-expanded to obtain (Equation 15).

[0052]

(Equation 15)

(Equation 13), (Equation 14), (Equation 1)
Using Z _i0 , Z _mi , and Z _ei of 5), a sum of squared deviations is obtained by ( _Equation 9).

When the equivalent circuit constant is a true value, the sum of squared deviations S has a minimum value S ₀ as shown in (Equation 16). When R = R ₀ + ΔR, S _R is set as shown in (Equation 17).

[0055]

(Equation 16)

[0056]

[Equation 17]

The expected values E (S ₀ ), E of S ₀ , S _R
(S _R ) is as shown in (Equation 18).

[0058]

(Equation 18)

F ₀ given by (Equation 19) is

[0060]

[Equation 19]

When there is no effect of R, E (S) in (Equation 18)
The first term on the right side of the equation of _R ) is 0, and it is understood that F ₀ has an F distribution at this time. Therefore, the significance of R can be tested by comparing the F value obtained from the table F with F ₀ .

Further, when (Equation 20) is satisfied, "no effect of resistance R" is rejected at the risk rate α, and it is considered that R has an effect.

[0063]

(Equation 20)

F (φ, φ, α) is the degree of freedom φ of S _R and S _0.
Is the value of the upper α of the F distribution. The sum of squared deviations S _R and S ₀ are
It is calculated using an estimate of the true value of the equivalent circuit constant. The effects of other circuit constants can be tested in the same manner.

For these tests, see Akio Kudo, published by Kyoritsu Shuppan Co., Ltd. on December 10, 1983,
It is written in "Statistical Mathematics" co-authored by Hideki Uemura.

In this embodiment, 1/10 W type 1 MΩ
The impedance was measured by changing the measurement frequency to obtain an equivalent circuit constant of the carbon film resistor described above, and the data shown in Table 1 was obtained. The W _i (Equation 9) and W _i = 1, the result of obtaining the constants of the equivalent circuit of Figure 3 was as shown in (Table 2). Using the circuit constant of (Table 2), the impedance Z _{ei is} expressed by (Equation 2).
, And a relative error 10 from the measured impedance Z _{mi.
0X (Z ei -Z mi) /} Z ei When seeking (%) was as shown in (Table 3). FIG. 5 is a graph of this (Table 3). B is the measured value Z _mi, the solid line is a value obtained from the calculation.

[0067]

[Table 1]

[0068]

[Table 2]

[0069]

[Table 3]

Next, the effect of the equivalent circuit constant will be confirmed with reference to FIG. Determination of the result of the resistance R is, independently using 10% greater and low values for the estimated value of the resistance R shown in (Table 2), (number 16), S ₀ in equation (17) , S _R were calculated (Equation 19) to obtain F ₀ . 10
The F ₀ value when it is smaller by% is indicated by FR (0.9), and the F ₀ value when it is 10% larger is indicated by FR (1.1). Since the number of data is 17 and the number of estimated parameters is 4,
The degree of freedom φ is φ = 17-4-1 = 12. Risk factor α
F value when F = 5% is F (12, 12, 5%) = 2.6.
Since it is 9, both FR (1.1) and FR (0.9) are larger than F (12, 12, 5%), and it can be seen that there is an effect of the resistance R. The effect can be confirmed in the same manner for other circuit constants. The results are as shown in (Table 4), but the effect of the inductors L ₁ and L ₂ is not recognized.

[0071]

[Table 4]

Next, since the inductors L ₁ and L ₂ have no effect, an equivalent circuit constant was obtained in the same manner using the equivalent circuit of FIG. 6 where these were omitted. The equation for calculating the impedance Z _{ei in} this case is (Equation 21).

[0073]

(Equation 21)

The equivalent circuit constants are as shown in (Table 5).
Using the results of (Table 5), the impedance Z _ei and the measured impedance Z _mi obtained by (Equation 21) and the relative error are as shown in (Table 6). It can be seen that (Table 3) and (Table 6) almost coincide with each other, and it was confirmed that the inductors L ₁ and L _{2 had} no effect. In this example, (Equation 15)
ΔR = ± 0.1R ₀ was used as ΔR, but this should be determined from a specific technical point of view.

[0075]

[Table 5]

[0076]

[Table 6]

[0077] In the above example the weight coefficient W _i of (number 9) W _i
= 1, the difference between the calculated impedance value Z _ei and the actually measured value Z _mi is +3.5 except for the measurement frequency of 10 kHz.
It is dispersed within the range of 4.2 kΩ (errors tend to be large at both ends when estimating by the least squares method).

Next, the weighting coefficient W _i in (Equation 9) is calculated as W _i = 1 /
As (Z _ei · Z _ei ), R and C are calculated using the equivalent circuit of FIG.
Was determined. The estimation result was as shown in (Table 7).
Impedance Z calculated by using the result of (Table 7)
_ei and the measured values Z _mi, and the relative error _{(Z ei -Z mi) / Z} ei ×
100 (%) is shown in (Table 8). The relative errors in (Table 8) are distributed in the range of 3.5-5%. Although the impedance of the resistor decreases as the measurement frequency increases, the difference (error) between Z _i and Z _i ′ in (Table 6) is equal because W _i = 1, whereas (Table 8) Then, W _i = 1 / (Z _mi ·
Z _mi ), the relative errors between Z _ei and Z _mi are _equally distributed.

[0079]

[Table 7]

[0080]

[Table 8]

In this embodiment, the calculation of the equivalent circuit constant of the resistor is considered. However, since the capacitor and the coil are also shown in the equivalent circuit of FIG. 3 as the resistor, the equivalent circuit constant is calculated in the same manner. Can be. There are many theoretically derived or empirically created model expressions represented by (Equation 3). For example: ・ Relationship between the rotation speed of a certain drive unit and the failure rate ・ Failure rate when the applied voltage of an electric component and the ambient temperature are used as parameters ・ Discharge characteristics of capacitor (voltage change over time) ・ Temperature and catalyst are parameters And the time change of the reaction product when the reaction is performed.

[0082]

It has the following effect in the calculation method of the parameter values by non-linear least square method of the present invention, according to the present invention.

(1) In the conventional method, advanced mathematical knowledge such as partial differentiation and Taylor expansion is required. However, according to the present invention, advanced mathematical knowledge is not required. Also since it is possible to calculate the value of the parameter using a personal computer already prevalent anyone so easily be able to take advantage of the non-line-shaped <br/> least squares method, in terms of sophistication of technology It is useful. It is economical because it can be handled by a personal computer.

(2) In the equation for calculating the sum of squared deviations S, the weighting coefficient W _{i is represented} by W _i = 1 or W _i = 1 / (y _ei · y _ei ) (or W _i =
By setting 1 / (Z _ei · Z _ei )), the difference (error) between the estimated value and the actually measured value or the relative error obtained by dividing the error by the estimated value can be dispersed uniformly. When the measured value does not change much, a method of calculating a parameter in which the error is equally dispersed is required. When the measured value is largely changed, a method of calculating a parameter in which the relative error is equally dispersed is required. W _i can cope with both cases by selectively using.

(3) The effect of the parameter can be confirmed. When performing circuit analysis, it is important to omit parameters having no effect of the equivalent circuit in order to shorten the analysis time, and to perform more complicated circuit analysis when performing circuit analysis using the same computer. The method for confirming the effect according to the present invention is effective in this respect.

[Brief description of the drawings]

FIG. 1 is a block diagram of a calculation method according to an embodiment of the present invention.

Flow chart of the parameter values calculated by nonlinear top <br/> small squares method using the calculation method of an embodiment of the present invention; FIG

FIG. 3 is an equivalent circuit diagram of the resistor according to the present embodiment.

FIG. 4 is a flowchart for confirming the effect of a parameter according to the embodiment.

FIG. 5 is a diagram illustrating the relationship between the frequency characteristics of impedance obtained in the present embodiment and measured values.

FIG. 6 is an equivalent circuit diagram of the resistor according to the present embodiment, in which inductors L ₁ and L ₂ are omitted.

[Explanation of symbols]

 1 data input unit 2 parameter operation unit 3 deviation square sum operation unit 4 evaluation unit 5 display unit

 ──────────────────────────────────────────────────続 き Continuation of front page (56) References JP-A-63-75877 (JP, A) JP-A-63-282568 (JP, A) JP-A-64-84362 (JP, A)

Claims (3)

(57) [Claims] 1. A characteristic y has a variable x and a parameter C ₀ ,
C _1, ......, when given by the relation between C _m, n sets of data _{(x 1, y m1),} (x 2, y m2), ......, (x n, y
In the case where the _mn) with a nonlinear least squares method obtains the values of the parameters, enter the upper limit value and the lower limit value n sets of data (x _i, y _mi) and for each parameter, the lower limit for each of the parameters from determining the five level values between the upper limit value, the characteristic value y _ei obtained from equation the measured value when the variable x to each of the five each case in combination with one another the level value of each of the parameters x _i weighting coefficient W _i to the square value of the difference of y _mi
Is calculated for all the variables x _i , the sum of these values is calculated, and if the minimum value of the sum of squared deviations for each case is smaller than the set value, the operation is terminated and this The level value of each parameter in the case is the value of the parameter to be determined, and if the minimum value is larger than the set value, the upper limit value and the lower limit value are limited to include the level value of each parameter that gives the minimum value. Calculation method for re-inputting the value to the input unit and displaying the value of the parameter to be obtained. 2. The weighting coefficient W of a deviation square sum operation unit.
_i is the characteristic value y _ei obtained from the above relational expression and its measured value y
_When determining the parameter values so that the difference (error) between _mi is equally dispersed, W _i = 1, and the value (relative error) obtained by dividing the difference between the characteristic value y _ei and the actually measured value y _mi by y _ei is equal variance. 2. The calculation method according to claim 1, wherein when calculating the value of the parameter, W _i = 1 / (y _ei · y _ei ). 3. The value of the sum of squares deviation S ₀ calculated using the value of the parameter and the sum of squares of deviation calculated by slightly changing only the value of the parameter whose effect is to be checked from the value obtained by the above method. The effect of the parameter (confirmation of significance) is confirmed using the fact that the ratio (F ₀ = S _i / S ₀ ) to the value S _i is F-distributed. Calculation method described in 1.