JP2011145944A - Method and device for calculating numerical value for solving electric circuit - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a technology for solving an electric circuit including a CPE within a time area with respect to an arbitrary circuit configuration and an arbitrary input voltage. <P>SOLUTION: A computer executes: an acquisition step for acquiring input voltage applied to the electric circuit at least as a boundary condition; a first calculation step for calculating a branch current flowing into the CPE based on the boundary condition; a second calculation step for calculating a branch current flowing into a circuit element other than the CPE based on the boundary condition; and a step for determining an unknown variable in a time area from the boundary condition and calculation results obtained by the first and second calculation steps by using Kirchhoff's current law and Kirchhoff's voltage law. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a numerical calculation method for solving an electric circuit including a constant phase element (CPE) in the time domain, a numerical calculation device, a program, and a recording medium storing the program.

In recent years, when designing an electrically driven or controlled device such as an electrical product, the electrical behavior of circuits, modules, materials, etc. incorporated in the device is predicted by numerical calculation (simulation). Efficiency is being improved.
In order to perform numerical calculations, it is first necessary to accurately grasp the electrical characteristics of an object to be analyzed. As a method for investigating electrical characteristics, for example, AC impedance measurement is known. AC impedance measurement is a method of measuring the impedance to be analyzed by applying an AC voltage while changing the frequency, and from the measurement result, identification of an equivalent circuit and determination of physical property values of circuit elements can be performed.

In AC impedance measurement, fitting is performed using a number of circuit elements to identify an equivalent circuit, and a well-known circuit element is Constant Phase Element (CPE). CPE is a circuit element having an intermediate property between resistance and capacitance, and its impedance is expressed by the following equation (1).

Z _CPE = 1 / {(iω) ^α T} (1)

Here, Z _CPE is the impedance of CPE, i is an imaginary unit, T and α are fitting parameters and physical property values of the substance expressed by CPE.

  When the equivalent circuit to be numerically calculated includes CPE, the following problems arise.

When calculating the numerical value by expanding the frequency of the input waveform, the CPE can be solved in the form of Equation (1). However, frequency expansion is possible only within a linear range, and physical properties cannot be applied to actual materials because they often exhibit nonlinear behavior such as electric field dependence. When numerically calculating a substance exhibiting non-linear behavior, the numerical calculation is generally performed in such a manner that the time evolution of the time evolution is reduced to behave piecewise linearly. Therefore, it is necessary to rewrite the equation in the time domain.
Also, in an electric circuit or the like, a rectangular wave is often used as an input voltage, and infinite frequency components are required when the frequency is expanded, and the calculation cost is very high. Therefore, in order to cope with an arbitrary input waveform, it is necessary to rewrite the equation in the time domain instead of the frequency.

ここでｄ^α／ｄｔ^αは分数階微分を示す記号であり、ΔＶはＣＰＥの両端にかかる電位差である。また、非特許文献１では、ＣＰＥもしくは抵抗が２つ直列に並んだ回路に対して
特定の種類の入力電圧を印加した場合に、回路素子間の電位がMittag-Leffler’s functionを用いて書くことができると述べている。しかしながら、非特許文献１で述べられている数値計算方法は、特定の回路構成と特定の入力電圧に限定されており、この方法を任意の回路構成、任意の入力電圧の場合に汎用的に適用することは難しい。 According to Non-Patent Document 1, the current I flowing through the CPE can be written as in Equation (2) using fractional differentiation.

Here, d ^α / dt ^α is a symbol indicating fractional differentiation, and ΔV is a potential difference applied to both ends of the CPE. In Non-Patent Document 1, when a specific type of input voltage is applied to a circuit in which two CPEs or resistors are arranged in series, the potential between circuit elements is written using Mittag-Leffler's function. You can do it. However, the numerical calculation method described in Non-Patent Document 1 is limited to a specific circuit configuration and a specific input voltage, and this method is generally applied to an arbitrary circuit configuration and an arbitrary input voltage. Difficult to do.

回路の方程式を導く場合、この関係はいかなる時刻によっても成り立つと考える。そのため電流は式（４）のようになる。

Ｃが時間に依存しないならば式（５）のようになる。

以上の事柄とαが１の時にＣＰＥはコンデンサと一致する必要があることとの両者から推察するとＣＰＥの電流の表式（２）は物性値Ｔが時間に依存するときには適切ではなく、このままでは物質の電流電圧特性を正確に再現することができない。 Moreover, physical properties often have electric field dependence in actual materials. That is, the physical property value T is a function of the electric field, and when the electric field changes with time, the physical property value T depends on time.
Now, in an ordinary capacitor, the capacitance C is defined as shown in Equation (3).

When deriving circuit equations, this relationship is considered to hold at any time. Therefore, the current is as shown in Equation (4).

If C does not depend on time, equation (5) is obtained.

From the above facts and the fact that when α is 1, CPE needs to coincide with the capacitor, the expression (2) of the CPE current is not appropriate when the physical property value T depends on time. The current-voltage characteristics of a substance cannot be accurately reproduced.

  In addition, nonpatent literatures 2-4 are mention | raise | lifted as a reference literature regarding fractional differentiation. Non-Patent Document 2 solves a fractional differential equation in the time domain for a homogeneous viscoelastic body. Non-Patent Document 3 shows the definition of fractional differential, etc., and Non-Patent Document 4 introduces some discretization techniques when performing numerical calculation of fractional differential. Further, non-patent document 5 describes in detail numerical calculation (simulation) of a general electric circuit.

S. Westerlund, “Capacitor Theory”, IEEE Transactions on Dielectrics and Electrical Insulation 1 5 (1994) 826-839. L. B. Eldred, W. P. Baker, A. N. Palazotto, “Numerical Application of Fractional Derivative Model Constitutive Relations for Viscoelastic Matericals”, Computer & Structures 60 6 (1996) 875-882. I. Podlubny, “Fractional Differential Equations”, Academic Press, (1999) ISBN0-12-558840-2. K. B. Oldham, J. Spanier, “The Fractional Calculus”, Dover Publications Inc. (2006) ISBN0-486-45001-5. Hideaki Asai and Takayuki Watanabe, "Electronic Circuit Simulation Technique", Science and Technology Publishing (2002) ISBN4-87653-200-1.

The present invention has been made in view of the above circumstances, and an object of the present invention is to be able to solve an electric circuit including a CPE in an arbitrary circuit configuration and an arbitrary input voltage in a time domain. Is to provide.

  A first aspect of the present invention is a numerical calculation method for solving an electric circuit including a CPE (Constant Phase Element) as a circuit element in a time domain, wherein a computer inputs at least the electric circuit as a boundary condition An acquisition step of acquiring a voltage; a first calculation step in which a computer calculates a branch current flowing in the CPE based on the boundary condition; and a branch current flowing in a circuit element other than the CPE based on the boundary condition. And the computer calculates an unknown variable in the time domain using the Kirchhoff's current law and Kirchhoff's voltage law from the boundary condition and the calculation results in the first and second calculation steps. A numerical calculation method characterized by comprising:

  A second aspect of the present invention is a numerical calculation apparatus for solving an electric circuit including a CPE (Constant Phase Element) as a circuit element in a time domain, and acquires at least an input voltage applied to the electric circuit as a boundary condition Obtaining means, first calculating means for calculating a branch current flowing in the CPE based on the boundary condition, and second calculating means for calculating a branch current flowing in a circuit element other than the CPE based on the boundary condition; And means for obtaining an unknown variable in the time domain using Kirchhoff's current law and Kirchhoff's voltage law from the boundary condition and the calculation results of the first and second calculation means. It is a numerical calculation device.

  According to a third aspect of the present invention, there is provided a program that causes a computer to execute each step of the numerical calculation method.

  According to a fourth aspect of the present invention, there is provided a computer-readable recording medium on which the program is recorded.

  According to the present invention, an electric circuit including a CPE can be solved in a time domain for an arbitrary circuit configuration and an arbitrary input voltage.

The flowchart of the numerical calculation method which concerns on embodiment of this invention. The structural example of the numerical calculation apparatus which concerns on embodiment of this invention. Circuits to be calculated in the first, fifth, and sixth embodiments. The calculation result in Example 1. FIG. A circuit to be calculated in the second embodiment. The calculation result in Example 2. A circuit to be calculated in the third embodiment. Calculation results in Example 3 (when α = 1 and when α = 0.5). A circuit to be calculated in the fourth embodiment. Calculation results in Example 4 (when α = 1 and when α = 0.5). A circuit to be calculated in the fifth embodiment. The graph which shows the applied voltage in Example 5 and 6. FIG. The calculation result in Example 5. The calculation result in Example 6 (when T is not time-dependent and time-dependent).

An embodiment of the present invention provides a numerical calculation method for solving an electric circuit including a CPE (Constant Phase Element) as a circuit element in a time domain. As a circuit element, CP
In addition to E, there are resistors, capacitors, coils, etc., but this method can be applied to any circuit configuration including any circuit element. At this time, various physical property values (CPE physical property values α and T, resistance value, capacitance, inductance, etc.) may be different for each circuit element. Moreover, this method is not limited to the analysis object itself being an electric circuit. When the electrical behavior of an object can be approximated by an equivalent electric circuit (equivalent circuit), the present method can be preferably applied to numerical calculation of such an equivalent circuit.

  FIG. 1 shows a flowchart of a numerical calculation method according to this embodiment. This method roughly includes the step of acquiring data necessary for calculation (S100), the step of calculating a branch current flowing through the CPE (S101), the step of calculating a branch current flowing through circuit elements other than the CPE (S102), unknown The step includes a step of obtaining a variable (S103). As will be described later, these steps are executed by a computer. Hereinafter, each step will be described.

(1) Step S100 (acquisition step)
In step S100, various data necessary for numerical calculation are acquired. For example, data such as circuit configuration such as physical property values of circuit elements and connection relations, boundary conditions such as input voltage applied to an electric circuit, calculation conditions such as calculation end time and time increment, and the like are acquired. These data may be input by the user each time, or the data stored in the storage medium may be read.

(2) Step S101 (first calculation step)
In step S101, the branch current flowing through the CPE is calculated based on the boundary condition obtained in step S100.

As described above, the impedance Z _CPE of _CPE is represented by the following formula (1).

Z _CPE = 1 / {(iω) ^α T} (1)

Here, α and T are physical property values of CPE, i is an imaginary unit, and ω is an angular frequency. However, as a matter of course, this expression is a notation within a linear range. For example, when T has an electric field dependency, Expression (1) cannot be used unless a minute alternating current that can maintain linearity is used. In order to obtain the electric field dependence value of T, it is necessary to apply a DC bias.
When a CPE having the above characteristics exists between a terminal 0 having a potential V ₀ and a terminal 1 having a potential V ₁ , the branch current I of the CPE at time t is a fractional differential _a D _t ^α Is obtained as in the following formula (6) or the following formula (7). Which formula is used is different in each embodiment, and will be described in each embodiment.

I = T _a D _t ^α (V ₀ −V ₁ ) (6)

I = _a D _t ^α {T (V ₀ −V ₁ )} (7)

Here, a is the calculation start time, and the direction from terminal 0 to terminal 1 is positive.

但しαは実数、ａは計算開始時刻、ｔは現在の時間、ｍΔｔ＝ｔ−ａであり、ｍは整数で、Δｔは時間刻み幅である。また式(９）の定義を用いている。

上記の定義を元に離散化を行い、数値計算を行った。具体的にはGrunwald-Letnikovの
分数階微分の定義で極限をとらずにそのまま計算した。つまり以下の実施例では式（１０）のように数値計算を行った。

There are several types of definitions of fractional differentials. Here, calculation is performed using the G1 algorithm shown in Non-Patent Document 4 based on the definition of Grunwald-Letnikov. Details are shown below.
In the definition of Grunwald-Letnikov's fractional differential, the α-order differential acting on the function f is defined as equation (8).

Where α is a real number, a is a calculation start time, t is a current time, mΔt = ta, m is an integer, and Δt is a time step. Also, the definition of equation (9) is used.

Discretization was performed based on the above definition, and numerical calculations were performed. Specifically, Grunwald-Letnikov's definition of fractional derivative was used as it was without taking the limit. That is, in the following examples, numerical calculation was performed as shown in Expression (10).

  As described above, in this embodiment, the branch current of the CPE is calculated in the time domain using the fractional differential, so that an accurate calculation result can be obtained with a small calculation cost for an input voltage having an arbitrary waveform. it can. Note that various calculation methods other than those described above can be considered as the discretization method for fractional differentiation (see, for example, Non-Patent Document 4), but any method may be used for the calculation in step S101.

但し端子０から端子１に流れる向きを正とする。
静電容量Ｃを持つコンデンサが、電位がＶ０である端子０と電位がＶ１である端子１との間に存在するとした場合、時刻ｔにおけるコンデンサに流れる枝電流Ｉは、下記式（１２）のように求まる。静電容量が時間依存性を持つことを想定しているので、静電容量は時間の関数である。

但し端子０から端子１に流れる向きを正とする。 (3) Step S102 (second calculation step)
In step S102, the branch current flowing in the circuit elements other than the CPE is calculated based on the boundary condition obtained in step S100. Methods for calculating branch currents flowing in circuit elements (resistors, capacitors, coils, etc.) other than CPE are known (see, for example, Non-Patent Document 5), and these calculation methods may be used in step S102.
Hereinafter, a method of calculating the branch current flowing through the resistor and the capacitor will be exemplified.
When a resistor having a resistance value R exists between a terminal 0 having a potential V0 and a terminal 1 having a potential V1, the branch current I flowing through the resistor at time t is expressed by the following equation (11). I want to.

However, the direction from terminal 0 to terminal 1 is positive.
When a capacitor having a capacitance C exists between a terminal 0 having a potential V0 and a terminal 1 having a potential V1, the branch current I flowing through the capacitor at time t is expressed by the following equation (12). It is obtained as follows. Since it is assumed that the capacitance is time-dependent, the capacitance is a function of time.

However, the direction from terminal 0 to terminal 1 is positive.

(4) Step S103
In step S103, an unknown variable is obtained in the time domain using Kirchhoff's current law and Kirchhoff's voltage law from the boundary conditions obtained in step S100 and the calculation results in steps S101 and S102. Although the unknown variable varies depending on the circuit configuration and the purpose of numerical calculation, for example, the potential at each node of the circuit, the branch current flowing through the entire circuit, and the like are unknown variables.

(Device configuration)
FIG. 2 shows an example of the configuration of a numerical calculation apparatus that performs the numerical calculation described above. The numerical calculation device is configured by a computer including a CPU 202, an input device 203, a primary storage device 204, an output device 205, a communication device 206, a secondary storage device 207, and the like. The CPU 202 is a processing device responsible for calculation and control. The input device 203 is an information input device such as a mouse, a keyboard, or a scanner. This is used for inputting boundary conditions and calculation conditions in step S100. The primary storage device 204 is a temporary storage device such as a memory (RAM). The output device 205 is an information output device such as a display or a printer. Used for displaying and printing out the results of numerical calculations. The communication device 206 is a LAN board or the like, and is a device that enables wired or wireless communication with other computers and network devices. The secondary storage device 207 indicates a device for storing data, such as a hard disk, a flexible disk, a CD-ROM, a DVD-ROM. Reference numeral 208 denotes a bus line, which has a role of transmitting information between device groups in the computer. Reference numeral 209 denotes a LAN cable, which is connected to a communication device between different computers and transmits information.

  A program for realizing the above-described numerical calculation method is stored in the secondary storage device group 207. The program is read into the primary storage device group 204 and executed by the CPU 202, thereby realizing the above-described processing of steps S100 to S103. When this program is provided to the user, the program may be recorded on a computer-readable recording medium such as a flash memory, a CD-ROM, a DVD-ROM, or provided through a network. May be.

[Example 1]
As Example 1, a numerical calculation example of a circuit in which a resistor and a CPE are arranged in parallel as shown in FIG. 3 will be described. In this embodiment, the formula (6) is used. It is assumed that a potential of V _A sin (2πft) [V] is applied to the terminal 301. Let V _A = 100. f is a frequency, and in this embodiment, in order to create a Cole-Cole Plot (Cole-Cole plot), it is calculated by shaking in the range of about 30 [Hz] to 1 × 10 ⁸ [Hz]. Cole-Cole Plot is a graph in which real and imaginary components of impedance at each frequency are plotted on a complex plane. The potential of the terminal 302 is fixed at 0 [V]. Reference numeral 303 denotes a resistance, and a resistance value R = 1 [Ω]. Reference numeral 304 denotes CPE, where T = 1.59 × 10 ⁻³ [F / sec ^0.5 ] and α = 0.5. The direction in which the current flows from the terminal 301 to the terminal 302 is positive.

  In this embodiment, since all potentials are given as boundary conditions, it is only necessary to obtain the circuit branch current (that is, the sum of the branch current flowing through the resistor 303 and the branch current flowing through the CPE 304) in step S103. .

図４に示すように理論予測と計算結果はよい一致をしており、本方法を用いて図３に示す回路が適切に計算できていることがわかる。 FIG. 4 shows the result of calculating the branch current of the circuit shown in FIG. 3 by using the above-described numerical calculation method, obtaining the impedance, and obtaining the Cole-Cole Plot. In calculating Cole-Cole Plot, the current was the sum of branch currents flowing through 303 and 304. The calculation result of this method is shown by the plot (black circle) in FIG.
Here, in order to verify the calculation results, a prediction was made using a theoretical formula of impedance. Since the impedance of the circuit in which the resistor and the CPE are arranged in parallel can be written as the following equation (13), the theoretical value was calculated using this.

As shown in FIG. 4, the theoretical prediction and the calculation result are in good agreement, and it can be seen that the circuit shown in FIG. 3 can be calculated appropriately using this method.

[Example 2]
As Example 2, a numerical calculation example of a circuit in which two circuits of FIG. 3 as shown in FIG. 5 are arranged in series is shown. In this embodiment, the formula (6) is used. _A potential of V _A sin (2πft) [V] is applied to the terminal 501. Let V _A = 100. f is a frequency, and in this embodiment, it is assumed that the Cole-Cole Plot is swung in the range of about 1 [Hz] to 1 × 10 ⁸ [Hz] and calculated. The potential of the terminal 503 is fixed at 0 [V]. Reference numerals 504 and 506 both denote resistances, and both are set to a resistance value R = 1 [Ω]. Reference numerals 505 and 507 both indicate CPE, CPE 505 has T = 1.59 × 10 ⁻³ [F] and α = 1.0, and CPE 507 has T = 1.59 × 10 ⁻³ [F / sec ^{0. 5} ], α = 0.5. The direction in which the current flows from the terminal 501 to the terminal 503 is positive.
In step S103, the potential of the node 502 is first obtained as a variable, and then the branch current of the entire circuit is obtained.

  FIG. 6 shows the Cole-Cole Plot obtained by calculating the branch current of the circuit shown in FIG. The voltage is the difference between the terminals 501 and 503, and the current is the sum of the branch currents flowing through the resistor 504 and the CPE 505 to obtain the impedance. The calculation result of this method is shown by the plot (black circle) in FIG. Moreover, the prediction result by a theoretical formula is shown by the continuous line of FIG. 6 for verification of a calculation result. The combined impedance when two circuits are arranged in series as in the circuit of FIG. 5 is the sum of the impedances of the two circuits.

  As shown in FIG. 6, the theoretical prediction and the calculation result are in good agreement, and it can be seen that the circuit shown in FIG. 5 can be calculated appropriately using this method. That is, by using this method, it was shown that a plurality of CPEs having different values of α and T exist and a circuit including elements other than the CPE such as resistors can be solved.

[Example 3]
Next, as Example 3, an example in which it is easy to understand that the problem is solved in the time domain is shown. In this embodiment, the formula (6) is used. The circuit to be solved is composed of one CPE as shown in FIG. Terminal 701 is applied with a potential of 0 [V] at time 0 [sec] ≦ t <3 [sec] and a potential of 100 [V] at time 3 [sec] ≦ t ≦ 10 [sec]. And The potential of the terminal 702 is fixed at 0 [V]. Reference numeral 703 indicates CPE. The direction in which the current flows from the terminal 701 to the terminal 702 is positive.
In this embodiment, since all potentials are given as boundary conditions, it is only necessary to obtain the branch current of the circuit in step S103.
The start time is t = 0 [sec], and the end time is t = 10 [sec]. The time increment is Δt = 1.0 × 10 ⁻² [sec].

本方法の計算結果をプロット（黒丸）で示し、式（１４）を用いた理論式による予測結果を実線で示す。 FIG. 8A shows an example of numerical calculation when CPE 703 is T = 0.318 [F] and α = 1.0. The horizontal axis of FIG. 8A is time [sec], and the vertical axis is current [A] (sum of displacement current and conduction current).
FIG. 8B shows an example of numerical calculation when CPE 703 is T = 0.318 [F / sec ^0.5 ] and α = 0.5. The horizontal and vertical axes in FIG. 8B are the same as those in FIG. 8A.
Here, in order to verify the calculation results, predictions were made using theoretical formulas. When f (t) is a snake site function H (t), it can be written as equation (14), which is used. Where Γ represents a gamma function.

The calculation result of this method is shown by a plot (black circle), and the prediction result by the theoretical formula using the formula (14) is shown by a solid line.

When α is 1, the CPE matches a normal capacitor. Therefore, when an input potential is applied as in this embodiment, the current becomes a delta function. In the numerical calculation, since the time step is discretized, it does not exactly match the delta function. However, the current shown in FIG. 8A has a shape similar to the delta function, and is calculated correctly. Recognize.
On the other hand, when α is 0.5, the current is not a delta function due to fractional differentiation, and a tail is expected. In FIG. 8B, as expected, a calculation result in which the current has a tail is obtained. Moreover, the theoretical prediction and the calculation result are in good agreement, and it can be seen that the calculation is correct.
By using this method in this way, a simple solution can be obtained even when the input waveform behaves like a heavisite, that is, when a very large number of frequency components are required when solving in the frequency domain. It was shown that

[Example 4]
In Example 3, there was no need to obtain the potential, but Example 4 shows a case in which the potential needs to be obtained. In this embodiment, the formula (6) is used. The circuit to be solved is a series circuit of a CPE and a resistor as shown in FIG. Terminal 1001 is applied with a potential of 0 [V] at time 0 [sec] ≦ t <3 [sec] and a potential of 100 [V] at time 3 [sec] ≦ t ≦ 10 [sec]. And The potential of the terminal 1003 is fixed at 0 [V]. Reference numeral 1004 denotes a resistance, and a resistance value R = 1 [Ω]. 1005 indicates the CPE. The direction in which current flows from the terminal 1001 to the terminal 1003 is positive.
In step S103, the potential of the node 1002 is first obtained as a variable, and then the branch current of the circuit is obtained.
The start time is t = 0 [sec], and the end time is t = 10 [sec]. The time increment is Δt = 1.0 × 10 ⁻² [sec].

  FIG. 10A shows an example of numerical calculation when CPE 1005 is T = 0.318 [F] and α = 1.0. The horizontal axis of FIG. 10A is time [sec], and the vertical axis is current [A] (sum of displacement current and conductive current). The calculation result of this method is shown by a plot (black circle), and the prediction result by the theoretical formula is shown by a solid line. When α is 1, the CPE matches a normal capacitor. It is simply a series circuit of a resistor and a capacitor, and exhibits a behavior with a time constant τ = RT. This was used to make predictions based on theoretical formulas. The theoretical prediction and the calculation result are in good agreement, and it can be seen that the calculation is correct.

FIG. 10B shows an example of numerical calculation when CPE1005 is T = 0.318 [F ^0.5 / Ω ^0.5 ] and α = 0.5. The horizontal and vertical axes in FIG. 10B are the same as those in FIG. 10A. When α is 0.5, the current is expected to have a tail as compared with the case where α = 1 by fractional differentiation. In FIG. 10B, as expected, a calculation result in which the current has a tail is obtained, and it can be confirmed that the calculation is correct.
By using this method in this way, even when the input waveform behaves like a heavisite, that is, when solving in the frequency domain, a very large number of frequency components are required, a solution can be easily obtained. It was shown that a circuit including an element other than CPE such as a resistor can be solved.

[Example 5]
In this embodiment, the difference between Expression (6) and Expression (7) will be described. In this embodiment, a circuit in which a resistor and a CPE are arranged in parallel as shown in FIG. 3 and a circuit in which a resistor and a capacitor are arranged in parallel as shown in FIG. 11 are used. Assume that a potential of V ₃₀₁ = 10 t [V] is applied to the terminals 301 and 1301 as shown in FIG. The potentials of the terminal 302 and the terminal 1302 are fixed at 0 [V]. Reference numerals 303 and 1303 denote resistances, and the resistance value R = 1 [Ω]. 304 indicates CPE, T is a function of the potential difference applied to both ends, and T = exp (0.01ΔV ₃₀₄ ) [F], α = 1.0. However, ΔV ₃₀₄ is an absolute value of a potential difference applied to the CPE ₃₀₄ .
Reference numeral ₁₃₀₄ denotes a capacitor, and the capacitance C is assumed to be a function of a potential difference applied to both ends, and C = exp (0.01ΔV ₁₃₀₄ ) [F]. However, ΔV ₁₃₀₄ is the absolute value of the potential difference applied to the capacitor 1304.
The direction in which the current flows from the terminal 301 to the terminal 302 and the direction in which the current flows from the terminal 1301 to the terminal 1302 is positive.
In the case of the present embodiment, all potentials are given as boundary conditions, so that only the branch current needs to be obtained in step C.
The start time is t = 0 [sec], and the end time is t = 10 [sec]. The time increment is Δt = 1.0 × 10 ⁻² [sec].

FIG. 13 shows an example of numerical calculation of the circuit shown in FIG. 3 under the above conditions for each of the cases where the expression (6) and the expression (7) are used. An example of numerical calculation of the circuit shown in FIG. 11 under the above conditions is also shown in FIG. The horizontal axis of FIG. 13 is time [sec], and the vertical axis is current [A] (sum of displacement current and conduction current).
As already discussed, CPE needs to match the capacitor when α = 1. Referring to FIG. 13, when the equation (7) is used, it matches the capacitor, but when the equation (6) is used, it does not match the capacitor. Therefore, when T is a function of the potential difference applied to both ends, that is, when it is a function depending on time, it can be understood that the current can be calculated more accurately by using the equation (7).

[Example 6]
In this embodiment, the difference between Expression (6) and Expression (7) when α is not 1 will be described. Basically, the circuit configuration and calculation conditions described in the fifth embodiment are used, but the calculation is performed by changing the physical property values of the CPE.

First, the physical property values of CPE304 are T = 1.0 [F / sec ^0.2 ] and α = 0.8. FIG. 14A shows an example of numerical calculation of the circuit shown in FIG. 3 under the above conditions for each of the cases using Equation (6) and Equation (7). The horizontal axis of FIG. 14A is time [sec], and the vertical axis is current [A] (sum of displacement current and conductive current).
As shown in FIG. 14A, when T does not depend on time, the calculation results of Expression (6) and Expression (7) coincide with each other, and it can be understood that either may be used for calculation.

Next, the physical property values of CPE304 are set to T = exp (0.01ΔV ₃₀₄ ) [F / sec ^0.2 ] and α = 0.8. FIG. 14B shows an example of numerical calculation of the circuit shown in FIG. 3 under the above conditions for each of the cases where Expression (6) and Expression (7) are used. The horizontal axis of FIG. 14B is time [sec], and the vertical axis is current [A] (sum of displacement current and conduction current).
As shown in FIG. 14B, when T depends on time, the calculation results of Expression (6) and Expression (7) are different. If reference is made to the case where α = 1, Expression (7) is used. It seems that the current can be calculated more accurately.

303, 504, 506, 1004, 1303: Resistor 304, 505, 507, 703, 1005: CPE
1304: Capacitor

Claims (8)

A numerical calculation method for solving an electric circuit including CPE (Constant Phase Element) as a circuit element in a time domain,
An acquisition step in which a computer acquires at least an input voltage applied to the electric circuit as a boundary condition;
A first calculation step in which a computer calculates a branch current flowing in the CPE based on the boundary condition;
A second calculation step in which a computer calculates a branch current flowing in a circuit element other than the CPE based on the boundary condition;
The computer has a step of obtaining an unknown variable in the time domain using the Kirchhoff's current law and Kirchhoff's voltage law from the boundary condition and the calculation results in the first and second calculation steps. Numerical calculation method. The numerical calculation method according to claim 1,
When CPE exists between a terminal 0 having a potential of V ₀ and a terminal 1 having a potential of V ₁ and having physical property values α and T,
In the first calculation step, a is the calculation start time, the direction flowing from the terminal 0 to the terminal 1 is positive, and the branch current I flowing through the CPE at the time t is _expressed using the fractional differential _a D _t ^α .
I = _a D _t ^α {T (V ₀ −V ₁ )}
A numerical calculation method characterized by calculating. The numerical calculation method according to claim 1,
When CPE exists between a terminal 0 having a potential of V ₀ and a terminal 1 having a potential of V ₁ and having physical property values α and T,
In the first calculation step, a is the calculation start time, the direction flowing from the terminal 0 to the terminal 1 is positive, and the branch current I flowing through the CPE at the time t is _expressed using the fractional differential _a D _t ^α .
I = T _a D _t ^α (V ₀ −V ₁ )
A numerical calculation method characterized by calculating.   A program for causing a computer to execute each step of the numerical calculation method according to claim 1.   A computer-readable recording medium on which the program according to claim 4 is recorded. A numerical calculation apparatus for solving an electric circuit including CPE (Constant Phase Element) as a circuit element in a time domain,
Obtaining means for obtaining at least an input voltage applied to the electric circuit as a boundary condition;
First calculating means for calculating a branch current flowing in the CPE based on the boundary condition;
Second calculation means for calculating a branch current flowing in a circuit element other than the CPE based on the boundary condition;
Means for determining an unknown variable in the time domain using Kirchhoff's current law and Kirchhoff's voltage law from the boundary conditions and the calculation results of the first and second calculation means;
A numerical calculation device comprising: The numerical calculation device according to claim 6,
When CPE exists between a terminal 0 having a potential of V ₀ and a terminal 1 having a potential of V ₁ and having physical property values α and T,
The first calculation means uses a fractional differential _a D _t ^α to calculate the branch current I flowing in the CPE at time t, where a is the calculation start time, the direction from terminal 0 to terminal 1 is positive,
I = _a D _t ^α {T (V ₀ −V ₁ )}
A numerical calculation device characterized by calculating. The numerical calculation device according to claim 6,
When CPE exists between a terminal 0 having a potential of V ₀ and a terminal 1 having a potential of V ₁ and having physical property values α and T,
The first calculation means uses a fractional differential _a D _t ^α to calculate the branch current I flowing in the CPE at time t, where a is the calculation start time, the direction from terminal 0 to terminal 1 is positive,
I = T _a D _t ^α (V ₀ −V ₁ )
A numerical calculation device characterized by calculating.

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