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US6226599B1 - Electromagnetic wave analyzer apparatus - Google Patents
Electromagnetic wave analyzer apparatus Download PDF
US6226599B1
US6226599B1 US08/923,970 US92397097A US6226599B1 US 6226599 B1 US6226599 B1 US 6226599B1 US 92397097 A US92397097 A US 92397097A US 6226599 B1 US6226599 B1 US 6226599B1
US08/923,970
1997-03-05 Priority to JP5037697 priority Critical
1997-05-03 Priority to JP9-050376 priority
1997-09-05 Assigned to FUJITSU LIMITED reassignment FUJITSU LIMITED ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: NAMIKI, TAKEFUMI
2001-05-01 Publication of US6226599B1 publication Critical patent/US6226599B1/en
The first method performs computations on the assumption that every metal part in the calculation space is a perfect conductor that has an infinite electrical conductivity. Hereafter, this method is referred to as the “first conventional method.”
The second method deals with the meshes corresponding to a metal part in the same way as those of other objects in the calculation space, while giving a finite electrical conductivity to that part. This method is referred to as the “second conventional method.”
The third method sets a surface impedance boundary condition proposed by Beggs et al. on the surface of a metal part. Hereafter, this is referred to as the “third conventional method.”
More specifically, when the x-axis is defined as an axis in the thickness-wise direction of a metal part, the frequency-domain steady state analysis of an electromagnetic wave Ey in a direction perpendicular to the x-axis is given by E y  ( ω ) = Z s  ( ω )  H z  ( ω ) , ( Z S  ( ω ) = j   ω   μ σ ) ( 2 )
where ω is angular frequency, μ is permeability of the metal part, σ is electrical conductivity, and j is the imaginary unit.
The above-described third conventional method uses Beggs's surface impedance boundary condition, assuming that the metal part has a sufficiently large thickness compared to the skin depth δ, where δ=(2/μσω)½. This assumption may be reversely interpreted as a drawback that the third conventional method cannot be applicable to such a case where the metal thickness is not sufficiently larger than the skin depth. This happens, for example, when a researcher tries to analyze the low frequency components of an electromagnetic field.
Here, the sum I of the currents flowing through a section of the metal part 10 is given by I =  I 0  ∫ 0 d  exp  ( - 1 + j δ  x )    x =  I 0  ( δ 2  j )  ( 1 - exp  ( - d δ  2  j ) ) =  I 0  ( 1 - exp  ( - d  j   ωμσ ) ωμσ ) ( 3 )
where, I0 is the current density on the surface of the metal part 10, j is the imaginary unit, δ is the skin depth, μ is the permeability of the metal part 10, σ is the electrical conductivity of the metal part 10, and ω is an angular frequency. Surface impedance Zs(ω) is then given by Z s  ( ω ) = R S0  d  jωμσ 1 - exp  { - d  jωμσ } , ( 4 )
where Rs0 represents the sheet resistance of the metal part 10 with respect to direct current and is equal to 1/dσ. This Equation (4) shows a surface impedance boundary condition applicable to a metallic film whose thickness d is comparable to or less than the skin depth δ.
To apply the FDTD method to the computation of Equation (4), the surface impedance Zs(ω) should be changed as Z s ′ =  Z s  ( jω ) jω =  R S0  ( a jω )  ( 1 1 - exp  { - a  jω } ) =  R S0  ( a 2  jω )  { coth  ( a 2  jω ) + 1 } , ( 5 )
where a=d(μσ)½.
Incidentally, the electrical current flowing through the metal part 10 can be represented by the magnetic field. Therefore, the y-axis component Ey of the electromagnetic wave can be expressed as the product of the surface impedance Zs(ω) and the z-axis component Hz(ω) of the magnetic field as E y  ( ω ) =  Z s  ( ω )  H z  ( ω ) =  Z S ′  ( ω )  { jω   H Z  ( ω ) } . ( 6 )
This Equation (6) is used in the steady state analysis (i.e., the analysis in the frequency domain) of electromagnetic waves.
On the other hand, in order to perform a transient analysis (i.e., the analysis in the time domain) of electromagnetic waves, it is necessary to apply the inverse Fourier transform to Zs′(ω) shown in Equation (5). Zs′(t) is then obtained as Z s ′  ( t ) = F - 1  [ Z s ′  ( ω ) ] = { R s0  ( Θ 3  [ 1 , 4  π   t   j / a 2 ] + a / 4  π   t ) t > 0 0 t ≤ 0 ( 7 )
where F−1 is an inverse Fourier transform operator, and Θ3 represents an elliptic theta function. The relation between surface impedance and electromagnetic field in the time domain is then expressed by E y  ( t ) = Z s ′  ( t ) ⊗ { ∂ ∂ t  H z  ( t ) } , ( 8 )
where symbol {circle around (x)} represents a convolution operator. Accordingly, that relation can be formulated in the FDTD method by using Equation (7) as E y n =  R s0  ∫ 0 n   Δ   t  ( Θ 3  [ 1 , 4  πτ   j / a 2 ] + a / 4  πτ )  ( - ∂ ∂ r  H y n   Δ   t - r )    τ =  R s0  ∑ m = 0 n - 1  [ ∫ m - 1 2 m + 1 2  { Θ 3  [ 1 , 4  πΔ   t   α   j / a 2 ] +  a / 4  πΔ   t   α }   α  ( H y n - m + 1 2 - H y n - m - 1 2 ) ] ( 9 )
where τ=Δtα. The above discussion yields the following equation. E y n =  R S0  ∑ m = 0 n - 1 [ ∫ m - 1 2 m + 1 2  { Θ 3  [ 1 , 4  πΔ   t   αj / a 2 ] +  a / 4  πΔ   t   α }   α  ( H y n - m + 1 2 - H y n - m - 1 2 ) ] ( 10 )
This Equation (10) is subjected to the analyzing unit 2 for computing the transitional behavior of electromagnetic waves.
The present form of Equation (10), however, is inconvenient for actual computational operations, because it requires all past values of magnetic field components at every time step from the beginning of computation. Here, the integral term in Equation (10) is extracted as Z 0  ( m ) = ∫ m - 1 2 m + 1 2  { Θ 3  [ 1 , 4  πΔ   t   αj / a 2 ] + a / 4  πΔ   t   α }   α , ( 11 )
and this Equation (11) will be expanded into a Prony series as Z 0  ( m ) ≅ ∑ k - 1 N  p k   q k  m . ( 12 )
Then Equation (10) can be rewritten as follows. ∑ m - 1 n - 1  Z 0  ( m )  [ H y n - m + 1 2 - H y n - m - 1 2 ] = ∑ m - 1 n - 1  { ∑ k - 1 N  P k   q k  m }  [ H y n - m + 1 2 - H y n - m - 1 2 ] ( 13 )
Consider that this Equation (13) equals ∑ k - 1 N  Ψ k n , ( 14 )
then the following equation is finally obtained. Ψ k n = P k   q k  ( H y n -  1 2 - H y n - 3 2 ) +  q k  Ψ k n - 1  ( Ψ k 0 = Ψ k 1 = 0 ) ( 15 )
This executable form allows the analyzer to solve the equation in a stepwise manner. That is, the computation of Equation (15) requires only two past records of magnetic field components at one time step before and two time steps before the present time. Unlike Equation (10), there is no need to memorize all past values of magnetic field components calculated at every time step from the beginning of the computation. Equation (15) allows the analyzer to solve the problem by sequential operations referring to just a few past results, thus relieving the memory requirement to the computer.
Next, for a given value of N, the analyzer calculates two coefficients pk and qk for each surface so that the following approximate equation will hold. ∫ m - 1 2 m + 1 2  { Θ 3  [ 1 , 4  πΔ   t   α   j / a 2 ] + a / 4  πΔ   t   α }   α ≅ ∑ k - 1 N  P k   q k  m ( 16 )
The obtained coefficients pk and qk as such are each saved as linear arrays, p(k) and q(k), where k is an integer ranging from 1 to N.
[S5] The analyzer calculates electric fields of the cells that directly depend on the surface impedance boundary condition. More specifically, the following equation is first evaluated. Ψ t  ( k ) = P  ( k )   q  ( k )  ( H y t -  Δt 2 - H y t - 3  Δ   t 2 ) +  q  ( k )  Ψ t - Δ   t  ( k )   ( Ψ k 0 = Ψ k Δ   t = 0 ) ( 17 )
where k is an integer ranging from 1 to N, and Hy represents a magnetic field that is perpendicular to, but has no intersection with, an electric field E dependent on the surface impedance boundary condition. The electric field E is then given by E t = R s0  ∑ k = 1 N  Ψ t  ( k ) . ( 18 )
[S6] The analyzer calculates magnetic fields while adding a half time step size ΔT/2 to the time T.
The thickness of the substrate 21 is 3.0 μm, the thickness of the conductor 22 is 1.0 μm, and the thickness of the conductor 23 is 1.0 μm. The distance between the upper surface of the conductor 22 and the absorbing boundary 41 is 3.0 μm, and the width of the conductor 22 is 5.0 μm. The distance between the underside of the conductor 23 and the absorbing boundary 41 is 2.0 μm. The electrical conductivity σ of Au is 4.167×107 (S/m), and the relative dielectric constant ∈r of SiON is 5.0.
Here, the skin depth δ at a frequency f of 24 GHz is about 0.5 μm since δ=1/(πfμσ)½. This simply means that, in the analysis of electromagnetic waves having the frequencies lower than 24 GHz, the conductors 22 and 23 are not sufficiently thick in comparison with the skin depth. The present embodiment will show a typical output of the electromagnetic wave analyzer of the present invention by making a calculation of the conductor loss vs. frequency characteristics of a microstrip transmission line shown in FIG. 4. The calculated curve will be then compared with the results of the same analysis by the second and third conventional methods. Note that the analysis is conducted in a frequency range from dc to 20 GHz.
First, the following explains how to calculate the conductor losses. FIG. 5 shows a side view of the microstrip transmission line of FIG. 4. The conductor loss, loss(ω,L), is derived from two potential difference values V(Z=0) and V(Z=L) measured at two separate positions having a predetermined distance L, as loss  ( ω , L ) = R e  [ - 20   log 10  ( V  ( t , z = 0 ) V  ( t , z = L ) ) ] , ( 20 )
where F is a Fourier transform operator, and Re represents the real part of a complex number.
Here, the mesh intervals and the temporal discretization interval dt should satisfy the condition given by dt ≤ 1 c  ( 1 dx min ) 2 + ( 1 dy min ) 2 + ( 1 dz min ) 2 , ( 21 )
dt≦1.481(ƒs). (22)
Nmesh=nx×ny×nz=12×13×nz=156×nz, (23)
dt≦0.294(ƒs). (24)
Nmesh=nx×ny×nz=24×36×nz=864×nz. (25)
1. An apparatus numerically analyzing transitional behavior of an electromagnetic wave near a metal part located in a calculation space divided into meshes, using parameters such as thickness of the metal part, number of meshes, mesh sizes, dielectric cell constants, permeability of the metal part, and electrical conductivity of the metal part, for calculation parameters, comprising:
an initialization device initializing the calculation parameters and surface impedance boundary conditions to be used in a calculation of the transitional behavior of the electromagnetic wave;
an analyzing device, coupled to said initialization device, calculating the transitional behavior of the electromagnetic wave by using finite difference time domain calculations, based on a calculated surface impedance boundary condition in which a surface impedance of the metal part becomes 1/ds as an angular frequency of the electromagnetic wave approaches to zero, where d is thickness of the metal part and s is electrical conductivity of the metal part; and
an outputting device, outputting the calculated transitional behavior of the electromagnetic wave to be used in a design of an electrical device containing the metal part.
2. The analyzer apparatus according to claim 1, wherein said calculated surface impedance boundary condition used by said analyzing device is defined as a surface impedance Z(w) given by Z  ( ω ) = 1 d   σ  d  j   ω   μ   σ 1 - exp  { - d  j   ω   μ   σ } ,
where j is an imaginary unit, μ is permeability of the metal part, σ is electrical conductivity of the metal part, d is thickness of the metal part, and ω is the angular frequency.
said initialization device specifies a particular area on the metal part the calculated surface impedance boundary condition should be applied, and
said analyzing device calculates the transitional behavior of the electromagnetic wave concerning the particular area on the metal part specified by said initialization device.
4. An apparatus for numerically analyzing transitional behavior of an electromagnetic wave near a metal part located in a calculation space divided into meshes, using parameters such as thickness of the metal part, number of meshes, mesh sizes, dielectric cell constants, permeability of the metal part, and electrical conductivity of the metal part, for calculation parameters, comprising:
initialization means for initializing the calculation parameters and surface impedance boundary conditions to be used in a calculation of the transitional behavior of the electromagnetic wave;
analyzing means, coupled to said initialization means for calculating the transitional behavior of the electromagnetic wave by using finite difference time domain calculations, based on a calculated surface impedance boundary condition in which a surface impedance of the metal part becomes 1/ds as an angular frequency of the electromagnetic wave approaches to zero, where d is thickness of the metal part and s is electrical conductivity of the metal part; and
outputting means outputting the calculated transitional behavior of the electromagnetic wave to be used in a design of an electrical device containing the metal part.
US08/923,970 1997-03-05 1997-09-08 Electromagnetic wave analyzer apparatus Expired - Fee Related US6226599B1 (en)
JP5037697 1997-03-05
JP9-050376 1997-05-03
US6226599B1 true US6226599B1 (en) 2001-05-01
ID=12857173
US08/923,970 Expired - Fee Related US6226599B1 (en) 1997-03-05 1997-09-08 Electromagnetic wave analyzer apparatus
US (1) US6226599B1 (en)
US6662125B2 (en) 2001-06-26 2003-12-09 Fujitsu Limited Electromagnetic wave analyzer and program for same
US20050187723A1 (en) * 2004-02-24 2005-08-25 Fujitsu Limited Electric/magnetic field analysis method using finite difference time domain, material descriptive method in electric/magnetic analysis, electric/magnetic analysis device, analysis data generation device and storage medium
US20090306916A1 (en) * 2007-02-28 2009-12-10 Fujitsu Limited Electrical characteristic analyzing apparatus for substance on which metal-containing paint is coated
GB2470577A (en) * 2009-05-27 2010-12-01 Access Business Group Int Llc Managing eddy currents in storage devices
CN102332055A (en) * 2011-09-26 2012-01-25 南京航空航天大学 Simulative calculation method for extremely-low-frequency electromagnetic waves
1997-09-08 US US08/923,970 patent/US6226599B1/en not_active Expired - Fee Related
John H. Beggs, et al.; "Finite-Difference Time-Domain Implementation of Surface Impedance Boundary Conditions"; IEEE Transactions on Antennas and Propagation, vol. 40, No. 1, Jan. 1992, pp. 49-56.
Toshiaki Kitamura, et al.; "Analysis of Thin Film Microstrip Lines with Conductor Loss Utilizing the Finite Difference Time-Domain Method"; Faculty of Engineering, Osaka University, Suita-shi, 565 Japan, vol. J76-C-I, No. 5, pp. 173-180; May 1993 (with translation-abstract only), (Only the Abstract has been Translated).
Toshiaki Kitamura, et al.; "Analysis of Thin Film Microstrip Lines with Conductor Loss Utilizing the Finite Difference Time-Domain Method"; Faculty of Engineering, Osaka University, Suita-shi, 565 Japan, vol. J76-C-I, No. 5, pp. 173-180; May 1993 (with translation—abstract only), (Only the Abstract has been Translated).
US7089130B2 (en) 2004-02-24 2006-08-08 Fujitsu Limited Electric/magnetic field analysis method using finite difference time domain, material descriptive method in electric/magnetic analysis, electric/magnetic analysis device, analysis data generation device and storage medium
CN102332055B (en) 2011-09-26 2014-01-29 南京航空航天大学 Simulative calculation method for extremely-low-frequency electromagnetic waves
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