KR101742116B1 - Device and Method for Analyzing Electromagnetic Wave of Dispesive Dielectric Material Using Higher-order Complex Rational Function - Google Patents

Device and Method for Analyzing Electromagnetic Wave of Dispesive Dielectric Material Using Higher-order Complex Rational Function Download PDF

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KR101742116B1
KR101742116B1 KR1020150117568A KR20150117568A KR101742116B1 KR 101742116 B1 KR101742116 B1 KR 101742116B1 KR 1020150117568 A KR1020150117568 A KR 1020150117568A KR 20150117568 A KR20150117568 A KR 20150117568A KR 101742116 B1 KR101742116 B1 KR 101742116B1
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equation
dielectric
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dispersion characteristic
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정경영
하상규
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한양대학교 산학협력단
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R29/00—Arrangements for measuring or indicating electric quantities not covered by groups G01R19/00 - G01R27/00
    • G01R29/08—Measuring electromagnetic field characteristics
    • G01R29/0864—Measuring electromagnetic field characteristics characterised by constructional or functional features
    • G01R29/0892—Details related to signal analysis or treatment; presenting results, e.g. displays; measuring specific signal features other than field strength, e.g. polarisation, field modes, phase, envelope, maximum value
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R23/00—Arrangements for measuring frequencies; Arrangements for analysing frequency spectra
    • G01R23/16—Spectrum analysis; Fourier analysis
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R29/00—Arrangements for measuring or indicating electric quantities not covered by groups G01R19/00 - G01R27/00
    • G01R29/08—Measuring electromagnetic field characteristics

Abstract

A dielectric electromagnetic wave analyzing apparatus having a dispersion characteristic using a higher order complex fractional functional equation according to an embodiment of the present invention includes a pre-processing unit for measuring frequency of a dielectric or acquiring dispersion characteristics of the dielectric through literature; An error function representing the fitting error of the dispersion characteristic is generated using a complex fraction function having a frequency function expressed by a real part and an imaginary part of the dispersion characteristic of the dielectric and a plurality of coefficients representing dispersion characteristics of the dielectric material for each frequency ; And calculating a coefficient of the complex fraction function equation using a condition that the rate of change of the error function is minimized and applying the calculated coefficient to the complex fraction function equation to obtain a dispersion characteristic for numerical analysis of time- And a dispersion characteristic modeling unit for performing modeling.

Description

BACKGROUND OF THE INVENTION Field of the Invention [0001] The present invention relates to a dielectric electromagnetic wave analyzing apparatus and a dielectric electromagnetic wave analyzing apparatus having a dispersion characteristic using a high-

Embodiments of the present invention relate to electromagnetic numerical analysis techniques for dielectric materials having a dispersion characteristic using a high-order complex fractional function (CRF).

Recently, research into the convergence of information technology and medical technology based on electromagnetic waves has been progressing actively due to the entry of aging society and increasing interest in health. Finite-Difference Time-Domain (FDTD) has been applied to bio-related technologies such as human body's absorbed power and temperature rise of mobile phones and human body's influence on exposure to broadband electromagnetic waves. FDTD does not require matrix computation, and has the advantage of knowing characteristics of a very wide frequency band with a single simulation. FDTD is also widely used in nanotechnology and energy technology, as well as biotechnology, because it easily disassembles the Maxwell Curl equation, making it very easy to model complex and diverse structures.

The human body has a dispersion characteristic in which the permittivity changes according to the frequency. In 1996, electrical characteristics of human tissue were recorded by Gabriel, and a fourth-order Cole-Cole model was used to express the dispersion characteristics at 10 Hz to 100 GHz. However, in order to apply the fourth-order Cole-Cole model to the time-domain electromagnetic wave numerical analysis technique FDTD, the field components to be updated are greatly increased, and the calculation efficiency (memory and calculation time) remarkably drops.

On the other hand, studies to apply the first-order Cole-Cole model to the FDTD to improve the calculation efficiency have been published, but still require a considerable amount of computation time and memory. In addition, the first Debye model expresses the dispersion characteristics of the human tissue well in the UWB band (3.1 ~ 10.6 GHz), but it can not be used because the variation of human body dispersion is very high at frequencies below 2.5 GHz. Secondary Debye models that do not contain conductor losses can be modeled up to the DC to 1.5 GHz band but are less accurate and the second Debye model with conductor losses must solve the nonlinear optimization problem to find the coefficients, There is a disadvantage that accurate modeling is impossible.

Related arts are disclosed in Japanese Patent Application Laid-Open No. 10-2002-0082665 entitled METHOD AND APPARATUS FOR METHOD AND APPARATUS FOR MEASURING FUNDAMENTAL BANDWIDTH FLEXIBILITY RANGE OF FLANGE, published on October 31, 2002.

One embodiment of the present invention is a dielectric electromagnetic wave analysis method having a dispersion characteristic using a higher order complex fraction function formula that can improve the accuracy and efficiency of time domain electromagnetic wave numerical analysis by modeling dispersion characteristics of dielectrics using a complex fraction function equation Device.

The problems to be solved by the present invention are not limited to the above-mentioned problem (s), and another problem (s) not mentioned can be clearly understood by those skilled in the art from the following description.

A dielectric electromagnetic wave analyzing apparatus having a dispersion characteristic using a high-order complex fractional function formula according to an embodiment of the present invention includes a dispersion function of a dielectric, a frequency function expressed by a real part and an imaginary part, and a plurality of coefficients A function generating unit for generating an error function representing a fitting error of the dispersion characteristic using a complex fractional function equation having a dispersion function; And calculating a coefficient of the complex fraction function equation using a condition that the rate of change of the error function is minimized and applying the calculated coefficient to the complex fraction function equation to obtain a dispersion characteristic for numerical analysis of time- And a dispersion characteristic modeling unit for performing modeling of the dispersion function modeling unit, wherein the dispersion characteristic modeling unit comprises first to seventh refinement units for redefining an updating equation for an electric field multiplied by the complex fractional function equation, Wherein the first processing unit defines the constitutive relation expressing the dispersion characteristics of the dielectric by an inverse Fourier transform and a central difference method and defines the equation as an update equation relating to an electric field, And the magnetic field and the electric field Wherein the update equation for the magnetic field is defined as a relational expression of the dispersion characteristic of the dielectric and the magnetic field in consideration of the dispersion characteristic of the dielectric after the update equation relating to the magnetic field is obtained by a relational expression, Calculating an update equation relating to the dispersion characteristics of the dielectric by using a center difference method and redefining an update equation for the electric field by using a state space signal process for improving the memory requirement, do.

Wherein the complex fractional function expression includes coefficients existing in a denominator of the complex fractional function equation that are not multiplied by the frequency and exist singularly in order to prevent divergence when the frequency is 0, It can have a nonzero constant value.

It is preferable that the coefficient existing alone has a constant value of 1.

The complex fractional function equation is a fourth-order complex fraction function (? R, 4CRF (?)) Of Equation (3)

Figure 112015080968809-pat00001

Here, A 0 , A 1 , A 2 , A 3 , A 4 , B 0 , B 1 , B 2 , B 3 and B 4 are coefficients, ω is a frequency and j is an imaginary number. In order to prevent divergence, B 0 is preferably a constant value other than 0 or a constant value 1.

The function generation unit multiplies the fitting error according to the numerical difference between the real and imaginary parts and the numerical difference of the frequency functions by the dispersion characteristic of the dielectric material according to the sampling frequency and expresses the fitting error as the square root of the absolute value Thereby generating the error function.

The dispersion characteristic modeling unit may calculate the rate of change of the error function based on the magnitude of the error function for each sampling frequency and calculate each coefficient of the complex fraction function formula using a condition that the rate of change of the error function is zero .

The dispersion characteristic modeling unit may calculate the rate of change of the error function by differentiating the magnitude of the error function with each coefficient of the complex fraction function formula.

The dispersion characteristic modeling unit may perform a matrix transformation of each coefficient-based equation based on a condition that the rate of change of the error function becomes 0 to calculate each coefficient of the complex fraction function equation.

A dielectric electromagnetic wave analyzing apparatus having a dispersion characteristic using a high-order complex fractional function expression is characterized in that, in order to apply each coefficient of the complex fractional function equation to numerical analysis of time-domain electromagnetic waves of the dielectric, satisfying the stability condition using the stability polynomial, And a stability evaluation unit for evaluating the stability of the image.

Wherein the stability evaluation unit calculates the stability polynomial by applying a central difference method to a constitutive relation and a Maxwell's equation that express the complex fractional function expression in the frequency domain in a time domain, and determines that the absolute value of all roots of the stability polynomial is 1 The stability of each coefficient can be evaluated as a stable state in which divergence does not occur.

A dielectric electromagnetic wave analysis method having a dispersion characteristic using a high-order complex fraction function formula according to an embodiment of the present invention is a dielectric electromagnetic wave analyzing apparatus, in which a dispersion function of the dielectric is expressed by a frequency function expressed by a real part and an imaginary part, Generating an error function representing a fitting error of the dispersion characteristic by using a complex fractional function expression having a plurality of coefficients indicating dispersion characteristics, using a condition that the rate of change of the error function is minimum in the dielectric electromagnetic wave analysis apparatus Calculating each coefficient of the complex fraction function equation; And performing dispersion characteristic modeling for time-domain electromagnetic wave numerical analysis of the dielectric by applying the calculated respective coefficients to the complex fraction function equation in the dielectric electromagnetic wave analysis apparatus, wherein the dispersion characteristic modeling Redefining an update equation for an electric field multiplied by the complex fractional function equation to improve a memory requirement when performing the distributed feature modeling, the step of redefining an update equation for the electric field comprises: Defining constitutive relations expressing dispersion characteristics as update equations related to electric fields by rearranging in time domain using Fourier inverse transform and center difference method; An updating equation for the magnetic field is obtained by using a center differential method in the time and space domain, and the updating equation related to the magnetic field is obtained by considering the dispersion characteristic of the dielectric, Defining as a relational expression; And an update equation for the dispersion characteristics of the dielectric is obtained by using a center difference method in a time and space domain. Then, an update equation for the electric field is obtained by using state space signal processing (state-space) And redefining the value.

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The performing the dispersion characteristic modeling may include calculating a rate of change of the error function based on the magnitude of the error function for each sampling frequency; And calculating each coefficient of the complex fraction function formula using a condition that the rate of change of the error function becomes zero.

A dielectric electromagnetic wave analyzing method having a dispersion characteristic using a high order complex fraction function formula according to an embodiment of the present invention is a dielectric electromagnetic wave analyzing apparatus in which, in order to apply each coefficient of the complex fraction function equation to time domain electromagnetic wave numerical analysis of the dielectric, And evaluating the stability of each of the coefficients through satisfaction of the stability condition using the stability polynomial, wherein the step of evaluating the stability includes constructing the complex fraction function expression in the frequency domain as a constitutive relation in time domain, Calculating a stability polynomial by applying a central difference method to a Maxwell equation; And evaluating the stability of each coefficient as a stable state in which divergence does not occur when the absolute values of all the roots of the stability polynomial satisfy conditions equal to or less than 1.

The details of other embodiments are included in the detailed description and the accompanying drawings.

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According to one embodiment of the present invention, the accuracy and efficiency of time-domain electromagnetic wave numerical analysis can be improved by modeling the dispersion characteristics of a dielectric using a complex fractional function.

According to one embodiment of the present invention, memory requirements can be improved by redefining update equations for an electric field using state space signaling (state-space). For example, memory efficiency can be improved by 33% compared to existing algorithms.

According to one embodiment of the present invention, when the stability of each coefficient of a complex fraction function is evaluated through the satisfaction of the stability condition using the stability polynomial, each coefficient of the complex fraction function is applied to the time domain electromagnetic wave numerical analysis of the dielectric So that divergence does not occur.

FIG. 1 is a block diagram illustrating a dielectric electromagnetic wave analyzing apparatus having a dispersion characteristic using a high-order complex fractional function equation according to an embodiment of the present invention. Referring to FIG.
FIG. 2 is a block diagram showing a detailed configuration of a dispersion characteristic modeling unit for performing a process of redefining an update equation relating to an electric field in an embodiment of the present invention. FIG.
FIG. 3 and FIG. 4 are flowcharts for explaining a dielectric electromagnetic wave analysis method having a dispersion characteristic using a high-order complex fraction function equation according to an embodiment of the present invention.
FIG. 5 is a graph showing the results of dispersion modeling of human tissues (dry skin) (showing coefficients and error rates).
FIG. 6 is a graph showing the results of numerical analysis of eye information according to the result of dispersion modeling of human tissue (dry skin).
7 is a graph showing simulation results of dispersion modeling of human tissue (dry skin).

BRIEF DESCRIPTION OF THE DRAWINGS The advantages and / or features of the present invention, and how to accomplish them, will become apparent with reference to the embodiments described in detail below with reference to the accompanying drawings. It should be understood, however, that the invention is not limited to the disclosed embodiments, but is capable of many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, To fully disclose the scope of the invention to those skilled in the art, and the invention is only defined by the scope of the claims. Like reference numerals refer to like elements throughout the specification.

Hereinafter, embodiments of the present invention will be described in detail with reference to the accompanying drawings.

FIG. 1 is a block diagram illustrating a dielectric electromagnetic wave analyzing apparatus having a dispersion characteristic using a high-order complex fractional function equation according to an embodiment of the present invention. Referring to FIG.

1, the dielectric bus characteristics modeling apparatus 100 includes a preprocessing unit 110, a function generating unit 120, a dispersion characteristic modeling unit 130, a stability evaluation unit 140, and a control unit 150 .

The preprocessing unit 110 measures the frequency of the dielectric or obtains the dispersion characteristics of the dielectric through literature. That is, the preprocessing unit 110 can express the dispersion characteristics (ε r, DATA ) of various dielectrics obtained through measurements or documents as a real part (R) and an imaginary part (I) of a frequency function as shown in the following equation have.

Figure 112015080968809-pat00002

Where ε r and DATA represent the dispersion characteristics of the dielectric, ω represents the frequency, R represents the real part, and I represents the imaginary part.

The function generating unit 120 generates an error function indicating a fitting error of the dispersion characteristic using a frequency function (Equation 1) and a complex fraction function expression expressed by a real part and an imaginary part, X).

Here, the complex fractional function expression has a plurality of coefficients indicating the dispersion characteristics of the dielectric material by the frequency, and its general expression is expressed by the following equation (2).

Figure 112015080968809-pat00003

Here, ε r and NCRF represent N-th order complex fractional functional expressions, N represents the order of the function expression, A and B represent the coefficients of the N-th order complex fractional function expression, j represents an imaginary number, and ω represents a frequency.

In one embodiment of the present invention, an N-th order complex fractional function expression ( r, NCRF ) is used to express dispersion characteristics of various dielectric materials in an ultra-wideband, and for example, a quaternary complex fraction function expression (? r, 4CRF ) can be used.

Figure 112015080968809-pat00004

Here, ε r, 4 CRF is quaternary denotes the complex fraction function formula, A 0, A 1, A 2, A 3, A 4, B 0, B 1, B 2, B 3, B 4 is a coefficient, ω is The frequency, j, represents the imaginary number.

In this case, the complex fractional function equation includes coefficients existing in a denominator of the complex fraction function equation, which do not multiply with the frequency, and exist singly, in order to prevent divergence when the frequency is 0, The coefficient may have a non-zero constant value, preferably a constant value of one.

In other words, in Equation (3), B 0 may have a non-zero constant value to prevent divergence when the frequency is 0, preferably a constant value of 1.

Accordingly, when B 0 is set to 1 in Equation (3) and the fourth-order complex fractional function expression is represented by a real part and an imaginary part, the following equation (4) is obtained.

Figure 112015080968809-pat00005

That is, the function generator 120 calculates the fitting error e according to the numerical difference of the frequency function (Equation 1) and the mathematical expression (4) representing the complex fraction function expression as the real part and the imaginary part, Can be expressed by the following equation (5). The function generator 120 may generate the error function by multiplying the fitting error e by the dispersion characteristic of the dielectric by the sampling frequency?, And expressing the result by the square root of the absolute value.

In other words, the function generator 120 rearranges Equation (5) by multiplication and square root calculation of an absolute value to obtain an error function (X) representing the fitting error (e) as shown in Equation (6) .

Figure 112015080968809-pat00006

Figure 112015080968809-pat00007

Here, X represents an error function, D represents a total velocity, e represents a fitting error,? K represents a sampling frequency, and M represents a maximum sampling index.

The dispersion characteristic modeling unit 130 calculates each coefficient of the complex fraction function equation using a condition that the rate of change of the error function is minimized, applies the calculated coefficient to the complex fraction function equation, We perform dispersion characteristic modeling for numerical analysis of electromagnetic field.

At this time, the dispersion characteristic modeling unit 130 may calculate the rate of change of the error function based on the magnitude of the error function for each sampling frequency. That is, the dispersion characteristic modeling unit 130 calculates the rate of change of the error function based on the magnitude of the error function, and calculates each coefficient of the complex fraction function equation using the condition that the rate of change of the error function becomes zero can do.

In other words, the dispersion characteristic modeling unit 130 calculates the rate of change of the error function by differentiating the magnitude of the error function with each coefficient of the complex fraction function formula, and using the condition that the rate of change of the error function becomes 0 The respective coefficients of the complex fractional function equation can be calculated.

At this time, the dispersion characteristic modeling unit 130 may calculate each coefficient of the complex fraction function equation by matrix-transforming each coefficient-based equation based on a condition that the rate of change of the error function becomes zero.

For example, the dispersion characteristic modeling unit 130 can be expressed as Equation (7) by applying a condition that the rate of change of the error function becomes zero for each coefficient of the complex fraction function formula, 7 can be transformed into a 9 * 9 matrix as shown in Equation (8) to calculate each coefficient of the complex fractional function equation.

Figure 112015080968809-pat00008

Here, m represents the maximum sampling index as in the case of M in Equation (6). Also,

Figure 112015080968809-pat00009
,
Figure 112015080968809-pat00010
,
Figure 112015080968809-pat00011
,
Figure 112015080968809-pat00012
.

Figure 112015080968809-pat00013

here,

Figure 112015080968809-pat00014
,
Figure 112015080968809-pat00015
,
Figure 112015080968809-pat00016
,
Figure 112015080968809-pat00017
,
Figure 112015080968809-pat00018
,
Figure 112015080968809-pat00019
.

The stability evaluation unit 140 may evaluate the stability of each coefficient through the satisfaction of the stability condition using the stability polynomial so as to apply each coefficient of the complex fraction function equation to the time domain electromagnetic wave numerical analysis of the dielectric. The stability evaluation unit 140 may calculate the stability polynomial by applying a central difference method to a constitutive relation and a Maxwell equation that express the complex fractional function in the frequency domain in a time domain. The stability polynomial can be expressed as Equation (9).

Figure 112015080968809-pat00020

Here, α 0 = A 0 t 4 , α 1 = A 1 t 3, α 2 = A 2 t 2, α 3 = A 3 t, α 4 = A 4, β 0 = B 0 t 4, β 1 = B 1 t 3 , β 2 = B 2 t 2 , β 3 = B 3 t, and β 4 = B 4 . Also,

Figure 112015080968809-pat00021
Lt;
Figure 112015080968809-pat00022
Represents the numerical wave number. The absolute value of all the roots of the stability polynomial S (Z) is preferably less than or equal to 1 as a condition for securing the numerical stability.

That is, when the absolute values of all the roots of the stability polynomial S (Z) satisfy a condition less than or equal to 1, the stability evaluation unit 140 determines the stability of each coefficient of the complex fraction function equation It can be evaluated as being in a stable state.

Hereinafter, the derivation of the stability polynomial S (Z) will be described in detail with reference to equations (10) to (22).

Deriving the wave equation using Maxwell's curl equation is shown in Equation 10 below.

Figure 112015080968809-pat00023

Where D is the total flux density, E is the electric field, t is the time, and ε 0 is the permittivity in free space.

The Von Neumann technique based on Fourier series expansion is defined as Equation (11) below.

Figure 112015080968809-pat00024

By applying the center difference method to Equation (10), an approximate expression expressed by the center averaging operator can be obtained. Then, in order to simplify the calculation while substituting the Fourier series expansion-based von Neumann technique (Equation 11) into the approximation equation, the complex density of electric flux density and the electric field

Figure 112017000153427-pat00025
And
Figure 112017000153427-pat00026
) And the eigenvalues of the center averaging operator are rearranged and rearranged to obtain the following expression (12). here
Figure 112017000153427-pat00027
Is a complex amplitude,
Figure 112017000153427-pat00028
Is an amplification factor, i, j, k are spatial gratings,
Figure 112017000153427-pat00029
,
Figure 112017000153427-pat00030
,
Figure 112017000153427-pat00031
Is divided into space division,
Figure 112017000153427-pat00032
,
Figure 112017000153427-pat00033
,
Figure 112017000153427-pat00034
Is a numerical wave number.

Figure 112015080968809-pat00035

The constitutive relation of the fourth-order complex fractional function (CRF) in the time domain is expressed by Equation (13) below.

Figure 112015080968809-pat00036

Substituting Equation (14) representing a partial differentiation operator with respect to time into Equation (13), Equation (15) is obtained and Equation (17) is summarized using the center difference method and the central averaging operator of Equation (16).

Figure 112015080968809-pat00037

Here, δ t represents the center-mean operator for time, and μ t represents the center-mean operator for space.

Figure 112015080968809-pat00038

Figure 112015080968809-pat00039

Figure 112015080968809-pat00040

If Equation (17) is rewritten, Equation (18) is obtained.

Figure 112015080968809-pat00041

Expression (12) and Expression (18) can be expressed by the following mathematical expression (19).

Figure 112015080968809-pat00042

Here, P is a coefficient added to the total speed density D 0 in Equation (18), and Q is a coefficient added to the electric field E 0 in Equation (18).

Since the determinant of Equation (19) must be 0 in order for Equation (19) to have a solution, Equation (20) is expressed as Equation (20) below. Substituting the matrix of P and Q into Equation .

Figure 112015080968809-pat00043

Figure 112015080968809-pat00044

Equation (21) is summarized for each degree of Z, and a stability polynomial expressed by Equation (22) below can be obtained.

Figure 112015080968809-pat00045

In the stability polynomial S (Z) of Equation (22), the stability condition is a condition where all of | Z | values are equal to or less than 1, and if it is satisfied, it can be determined that the fourth- On the other hand, if any one of the six | Z | values does not satisfy the stability condition, it can be determined that the fourth-order complex fractional function diverges.

The control unit 150 controls the dielectric bus characteristics modeling apparatus 100 such as the preprocessing unit 110, the function generating unit 120, the dispersion characteristic modeling unit 130, the stability evaluation unit 140, The operation can be controlled as a whole.

For example, if the stability evaluation unit 140 determines that each coefficient of the complex fraction function formula to be applied to the dielectric is unstable (divergent), the control unit 150 may control the dispersion characteristic modeling unit 130 not to perform dispersion characteristic modeling have.

Hereinafter, the process of redefining the update equation for the electric field will be described in detail with reference to FIG. 2 and Equations 23 to 30. 2 is a block diagram illustrating a detailed configuration of a dispersion characteristic modeling unit that performs a process of redefining an update equation relating to an electric field in an embodiment of the present invention.

As shown in FIG. 2, the dispersion characteristic modeling unit 130 may include first to eighth embodiments for redefining an updating equation relating to an electric field multiplied by the complex fraction function equation, for improving the memory requirement, And may include third processing units 210, 220, and 230.

The first processing unit 210 may define a constitutive relation expressing the dispersion characteristics of the dielectric in the time domain using Fourier inverse transform and center difference method and define it as an update equation related to the electric field.

Specifically, for the realization of the ultra-wideband electromagnetic wave analysis algorithm based on the fourth-order complex fractional function dispersion modeling, the flux density (D) of the dielectric material is expressed by the following Equation 23 using the constitutive relation.

Figure 112015080968809-pat00046

The above equation (24) is obtained by multiplying both sides of Equation (23) by the demarcation term and rearranging in the time domain using the Fourier inverse transform and the center difference method.

Figure 112015080968809-pat00047

The equation (24) can be expressed by the following equation (25) as the final update equation for the electric field. In the case of Equation (25)

Figure 112015080968809-pat00048
To update the value
Figure 112015080968809-pat00049
value,
Figure 112015080968809-pat00050
value,
Figure 112015080968809-pat00051
value,
Figure 112015080968809-pat00052
value,
Figure 112015080968809-pat00053
value,
Figure 112015080968809-pat00054
value,
Figure 112015080968809-pat00055
value,
Figure 112015080968809-pat00056
value,
Figure 112015080968809-pat00057
Nine memories are needed to store each value.

Figure 112015080968809-pat00058

The second processing unit 220 obtains an update equation related to the magnetic field by a relational expression between a magnetic field and an electric field using a center difference method in time and space regions and then calculates an update equation related to the magnetic field Is defined as a relational expression of the dispersion characteristic of the dielectric and the magnetic field.

Specifically, the updating equation for the magnetic field is a relational expression of the magnetic field H and the electric field E, and is expressed by Equation 26 below.

Figure 112015080968809-pat00059

Where mu is the permeability, t is the time,

Figure 112015080968809-pat00060
Is a del operator
Figure 112015080968809-pat00061
.

The update equation related to the magnetic field H is obtained by the following equation (27) using the center difference method in the time and space domain.

Figure 112015080968809-pat00062

In order to consider the dispersion characteristics of complex and various dielectric materials, And the magnetic field (H).

Figure 112015080968809-pat00063

The third processing unit 230 obtains an update equation related to the total flux density of the dielectric by using a center difference method in a time and space domain, and then uses the state space signal processing (state-space) To redefine the update equation for the electric field.

More specifically, the total density D Is calculated by the following equation (29) using the center difference method in the time and space domain.

Figure 112015080968809-pat00064

To improve the memory requirement, the update equation for the electric field using the state-space signal processing algorithm is expressed as Equation 30 below.

According to Equation 30,

Figure 112015080968809-pat00066
Through a formula for the value
Figure 112015080968809-pat00067
Update the value,
Figure 112015080968809-pat00068
Value above it
Figure 112015080968809-pat00069
Lt; RTI ID = 0.0 >
Figure 112015080968809-pat00070
Into the value
Figure 112015080968809-pat00071
Update the value,
Figure 112015080968809-pat00072
The value is located in the middle
Figure 112015080968809-pat00073
Lt; RTI ID = 0.0 > Into the value
Figure 112015080968809-pat00075
Value, and the updated
Figure 112015080968809-pat00076
The value is located above the middle
Figure 112015080968809-pat00077
Lt; RTI ID = 0.0 >
Figure 112015080968809-pat00078
Into the value
Figure 112015080968809-pat00079
Value, and the updated
Figure 112015080968809-pat00080
Value at the top
Figure 112015080968809-pat00081
Lt; RTI ID = 0.0 >
Figure 112015080968809-pat00082
Into the value
Figure 112015080968809-pat00083
Value.

At this time,

Figure 112015080968809-pat00084
For update operations on values
Figure 112015080968809-pat00085
A first memory for storing a value,
Figure 112015080968809-pat00086
A second memory for storing a value and
Figure 112015080968809-pat00087
A third memory for storing the value is needed.

Also,

Figure 112015080968809-pat00088
For update operations on values
Figure 112015080968809-pat00089
A first memory for storing a value,
Figure 112015080968809-pat00090
A second memory for storing a value,
Figure 112015080968809-pat00091
A third memory for storing a value,
Figure 112015080968809-pat00092
A fourth memory for storing the value is needed. At this time, the second memory
Figure 112015080968809-pat00093
After the update of the value is completed,
Figure 112015080968809-pat00094
Value, and the third memory stores
Figure 112015080968809-pat00095
After the update of the value is completed,
Figure 112015080968809-pat00096
Value.

Also,

Figure 112015080968809-pat00097
For update operations on values
Figure 112015080968809-pat00098
A fourth memory for storing a value,
Figure 112015080968809-pat00099
A second memory for storing a value,
Figure 112015080968809-pat00100
A third memory for storing a value and
Figure 112015080968809-pat00101
A fifth memory for storing the value is needed. At this time, the second memory
Figure 112015080968809-pat00102
After the update of the value is completed,
Figure 112015080968809-pat00103
Value, and the third memory stores
Figure 112015080968809-pat00104
After the update of the value is completed,
Figure 112015080968809-pat00105
Value.

Also,

Figure 112015080968809-pat00106
For update operations on values
Figure 112015080968809-pat00107
A fifth memory for storing a value,
Figure 112015080968809-pat00108
A second memory for storing a value,
Figure 112015080968809-pat00109
A third memory for storing a value and
Figure 112015080968809-pat00110
A sixth memory for storing a value is required. At this time, the second memory
Figure 112015080968809-pat00111
After the update of the value is completed,
Figure 112015080968809-pat00112
Value, and the third memory stores
Figure 112015080968809-pat00113
After the update of the value is completed,
Figure 112015080968809-pat00114
Value.

Finally,

Figure 112015080968809-pat00115
For update operations on values
Figure 112015080968809-pat00116
A sixth memory for storing a value,
Figure 112015080968809-pat00117
A second memory for storing a value,
Figure 112015080968809-pat00118
A third memory for storing the value is needed. At this time, the second memory
Figure 112015080968809-pat00119
After the update of the value is completed,
Figure 112015080968809-pat00120
Value, and the third memory stores
Figure 112015080968809-pat00121
After the update of the value is completed,
Figure 112015080968809-pat00122
Value.

That is, according to an embodiment of the present invention,

Figure 112015080968809-pat00123
value,
Figure 112015080968809-pat00124
value,
Figure 112015080968809-pat00125
value,
Figure 112015080968809-pat00126
Value and
Figure 112015080968809-pat00127
By updating the values in a sequential manner,
Figure 112015080968809-pat00128
By updating the value, only six memories are used
Figure 112015080968809-pat00129
Value, and the memory efficiency is improved as compared with the conventional method of Equation 25, which requires nine memories.

FIG. 3 and FIG. 4 are flowcharts for explaining a dielectric electromagnetic wave analysis method having a dispersion characteristic using a high-order complex fraction function equation according to an embodiment of the present invention. Here, the dielectric electromagnetic wave analysis method may be performed by the dielectric electromagnetic wave analysis apparatus 100 of FIG.

Referring to FIG. 3, in step 310, the dielectric electromagnetic wave analysis apparatus obtains dispersion characteristics of the dielectric by measuring frequency of the dielectric or through literature.

Next, in step 320, the dielectric electromagnetic wave analysis apparatus calculates a dispersion function of the dielectric by using a complex fraction function formula having a frequency function expressed by a real part and an imaginary part, and a plurality of coefficients representing a dispersion characteristic of the dielectric by frequency , And generates an error function indicating the fitting error of the dispersion characteristic.

Next, in step 330, the dielectric electromagnetic wave analyzing apparatus calculates the respective coefficients of the complex fractional function equation using a condition that the rate of change of the error function is minimized.

Next, in step 340, the dielectric electromagnetic wave analysis apparatus determines whether or not each coefficient of the complex fraction function equation is applied to the numerical analysis of time-domain electromagnetic waves of the dielectric by using the stability condition using the stability polynomial, Stability is evaluated.

4, in step 410, the dielectric electromagnetic wave analysis apparatus applies a central difference method to a constitutive relation and a Maxwell's equation, which express the complex fraction function equation in the frequency domain in a time domain, Polynomial can be calculated. If the absolute value of all the roots of the stability polynomial satisfies the condition that the absolute value of the root of the stability polynomial is less than or equal to 1 (the "Yes" direction of 420), the dielectric electromagnetic wave analysis apparatus, at step 430, It can be evaluated as a stable state that does not occur. On the other hand, if the condition of 420 is not satisfied ("No" direction of 420), the present embodiment can be ended.

Next, referring again to FIG. 3, in step 350, the dielectric electromagnetic wave analysis apparatus applies the calculated respective coefficients to the complex fraction function equation to perform dispersion characteristic modeling for time domain electromagnetic wave numerical analysis of the dielectric .

FIG. 5 is a graph showing the results of dispersion modeling of human tissues (dry skin) (showing coefficients and error rates). According to one embodiment of the invention, in Figure 5, each coefficient A 0 = 1.062593e + 03, A 1 = 4.179049e-05, A 2 = 1.393358e-13, A 3 = 4.318064e-23, A 4 = 1.048884e-34, B 1 = 7.411793e-08, B 2 = 9.702195e-16, B 3 = 1.077459e-24, B 4 = 1.038490e-35 and err = 1.193%.

FIG. 6 is a graph showing the results of numerical analysis of eye information according to the result of dispersion modeling of human tissue (dry skin). As shown in FIG. 6, according to an embodiment of the present invention, the numerical stability of the human tissue (dry skin) is analyzed. As a result, the dispersion characteristic modeling algorithm of the present invention satisfies the stability polynomial | Z | It can be confirmed that it has numerical stability.

7 is a graph showing simulation results of dispersion modeling of human tissue (dry skin). In FIG. 7, a solid line indicates a simulation result according to an ultrawideband finite difference time-domain electromagnetic wave technique, and a symbol indicates a simulation result according to a frequency domain analysis. This confirms that both simulation results are the same.

Embodiments of the present invention include computer readable media including program instructions for performing various computer implemented operations. The computer-readable medium may include program instructions, local data files, local data structures, etc., alone or in combination. The media may be those specially designed and constructed for the present invention or may be those known to those skilled in the computer software. Examples of computer-readable media include magnetic media such as hard disks, floppy disks and magnetic tape, optical recording media such as CD-ROMs and DVDs, magneto-optical media such as floppy disks, and ROMs, And hardware devices specifically configured to store and execute the same program instructions. Examples of program instructions include machine language code such as those produced by a compiler, as well as high-level language code that can be executed by a computer using an interpreter or the like.

While the present invention has been described in connection with what is presently considered to be practical exemplary embodiments, it is to be understood that the invention is not limited to the disclosed embodiments. Therefore, the scope of the present invention should not be limited to the described embodiments, but should be determined by the scope of the appended claims and equivalents thereof.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the invention is not limited to the disclosed exemplary embodiments, but, on the contrary, Modification is possible. Accordingly, the spirit of the present invention should be understood only by the appended claims, and all equivalent or equivalent variations thereof are included in the scope of the present invention.

110:
120: function generation unit
130: dispersion characteristic modeling unit
140: Stability Evaluation Unit
150:
210: first processing section
220: second processing section
230: third processing section

Claims (15)

A function for generating an error function representing a fitting error of the dispersion characteristic by using a complex function function having a frequency function represented by a real part and an imaginary part of a dispersion characteristic of the dielectric and a plurality of coefficients representing dispersion characteristics of the dielectric material for each frequency, Generating unit; And
Calculating a plurality of coefficients of the complex fraction function equation using a condition that the rate of change of the error function is minimized and applying the calculated coefficients to the complex fraction function equation to perform a dispersion characteristic modeling And a dispersion characteristic modeling unit for performing the dispersion characteristic modeling,
The dispersion characteristic modeling unit
And a first to a third processing unit for redefining an updating equation related to an electric field multiplied by the complex fractional function expression to improve the memory requirement when performing the distributed characteristic modeling,
The first processing unit
The constitutive relation expressing the dispersion characteristics of the dielectric is defined as an update equation relating to the electric field by summarizing in time domain using Fourier inverse transform and center difference method,
The second processing unit
An updating equation for the magnetic field is obtained by using a center differential method in the time and space domain, and the updating equation related to the magnetic field is obtained by considering the dispersion characteristic of the dielectric, And is defined as a relational expression,
The third processing unit
An update equation related to the dispersion characteristics of the dielectric is obtained using a center difference method in a time and space domain, and an update equation related to the electric field is obtained by using state space signal processing (state space) Wherein the first and second complex fractional function equations are redefined.
The method according to claim 1,
The complex fractional function equation
A coefficient that is not multiplied by the frequency of the denominator of the complex fraction function equation and exists alone in order to prevent divergence when the frequency is 0,
Wherein the coefficient existing alone has a non-zero constant value. ≪ RTI ID = 0.0 > 11. < / RTI >
3. The method of claim 2,
The only coefficient present is
And a constant value of 1. The dielectric electromagnetic wave analysis apparatus according to claim 1,
The method according to claim 1,
The complex fractional function equation
Order quadratic complex fraction function (? R, 4CRF (w)) of Equation (3)
&Quot; (3) "
Figure 112016074001250-pat00130

Here, A 0, A 1, A 2, A 3, A 4, B 0, B 1, B 2, B 3, B 4 is a coefficient,
Figure 112016074001250-pat00131
Is a frequency, j is an imaginary number,
Wherein B 0 is a constant value other than 0 or a constant value of 1 to prevent divergence when the frequency is 0. 2. The dielectric electromagnetic wave analysis apparatus according to claim 1,
The method according to claim 1,
The function generating unit
The fitting error according to the numerical difference between the real part and the imaginary part and the fitting error according to the numerical difference of the frequency function are multiplied by the dispersion characteristic of the dielectric by sampling frequency and then expressed as the square root of the absolute value, Wherein the dielectric waveguide has a dispersion characteristic using a high-order complex fraction function.
The method according to claim 1,
The dispersion characteristic modeling unit
Calculating a rate of change of the error function based on the magnitude of the error function for each sampling frequency, and calculating each coefficient of the complex fraction function formula using a condition that the rate of change of the error function is zero. Dielectric Electromagnetic Wave Analysis System with Dispersion Characteristics Using Functional Equation.
The method according to claim 6,
The dispersion characteristic modeling unit
And a rate of change of the error function is calculated by differentiating the magnitude of the error function with each coefficient of the complex fraction function expression.
The method according to claim 6,
The dispersion characteristic modeling unit
Wherein the coefficient of each of the coefficients based on a condition that the rate of change of the error function is zero is subjected to matrix transformation to calculate each coefficient of the complex fractional function expression. .
The method according to claim 1,
A stability evaluation unit for evaluating the stability of each coefficient through satisfaction of the stability condition using the stability polynomial to apply each coefficient of the complex fraction function equation to numerical analysis of the time domain electromagnetic wave of the dielectric,
Further comprising: a dielectric waveguide having a dispersion function using a higher-order complex fraction function.
10. The method of claim 9,
The stability evaluation unit
Calculating a stability polynomial by applying a central difference method to a constitutive relation and a Maxwell equation expressing the complex fractional function expression in a frequency domain in a time domain and calculating an absolute value of all the roots of the stability polynomial less than or equal to 1 Wherein when the condition is satisfied, the stability of each coefficient is evaluated to be a stable state in which divergence does not occur, and a dielectric electromagnetic wave analyzing apparatus having a dispersion characteristic using a higher-order complex fraction function.
delete In a dielectric electromagnetic wave analysis apparatus, a complex fractional function expression having a dispersion function of a dielectric body as a real part and an imaginary part and a plurality of coefficients representing dispersion characteristics of the dielectric body for each frequency, Generating an error function;
Calculating each coefficient of the complex fractional function equation using the condition that the rate of change of the error function is minimized in the dielectric electromagnetic wave analyzing apparatus; And
And performing dispersion characteristic modeling for numerical analysis of time-domain electromagnetic waves of the dielectric by applying the calculated respective coefficients to the complex fraction function equation in the dielectric electromagnetic wave analyzing apparatus,
The step of performing the dispersion characteristic modeling
Redefining an update equation for an electric field multiplied by the complex fractional function equation to improve memory requirements when performing the distributed feature modeling,
The step of redefining the update equation for the electric field
Defining a constitutive relation expressing the dispersion characteristics of the dielectric by an inverse Fourier transform and a central difference method in a time domain and defining it as an update equation relating to an electric field;
An updating equation for the magnetic field is obtained by using a center differential method in the time and space domain, and the updating equation related to the magnetic field is obtained by considering the dispersion characteristic of the dielectric, Defining as a relational expression; And
An update equation related to the dispersion characteristics of the dielectric is obtained using a center difference method in a time and space domain, and an update equation related to the electric field is obtained by using state space signal processing (state space) Wherein the method further comprises the step of redefining the dielectric property of the dielectric material.
13. The method of claim 12,
The step of performing the dispersion characteristic modeling
Calculating a rate of change of the error function based on the magnitude of the error function for each sampling frequency; And
Calculating each coefficient of the complex fraction function formula using a condition that the rate of change of the error function is 0
Wherein the dielectric waveguide has a dispersion characteristic using a high-order complex fraction function.
13. The method of claim 12,
Evaluating the stability of each coefficient through the satisfaction of the stability condition using the stability polynomial to apply the coefficients of the complex fraction function equation to the time domain electromagnetic wave numerical analysis of the dielectric in the dielectric electromagnetic wave analysis apparatus
Further comprising:
The step of evaluating the stability
Calculating the stability polynomial by applying a central difference method to a constitutive relation and a Maxwell equation that express the complex fractional function equation in a frequency domain in a time domain; And
Evaluating the stability of each coefficient as a stable state in which divergence does not occur when the absolute value of all the roots of the stability polynomial satisfies a condition less than or equal to 1
Wherein the dielectric waveguide has a dispersion characteristic using a high-order complex fraction function.
delete
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