矩量法讲义 Method of Moments and Fast Algorithms
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Wire Antennas (cont’d)
Pure Conductor Problem a
on S except on Sa Problem b
Choosing the basis function and letting: then testing with wn(r) yields a matrix equation:
• New source model
Fra Baidu bibliotek
• Less Sommerfeld integrals
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
General Formulation
• Two Conductors • Cylindrical Gap • Driving Voltage
Nanjing 210096, P. R. China
东南大学 计算电磁学研究中心
Contents
The Method of Moments: New Modeling
Wire-structure circuit problems PCB Simulation Problems
Fast Algorithms for Integral Equation Solver
Traditional definition:
n driving point
• Two source models are usually used:
Delta gap source
Most popularly used
Magnetic frill source Need an additional parameter b b
For the self term and near mutual terms in the impedance matrix, closedform expressions are obtained using the Taylor expansion. Hence, the CPU time to evaluate the impedance matrix does not increase although accurate model has been used.
we have
driving point Traditional definition of input admittance
The above conclusion is valid for arbitrarily-shaped conductors.
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
zoom
driving source lˆ
I (l)
Region b Region a
z
y x
Thus, the unknown in the EFIE is only I (l ).
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
If point-matching method is used (like the NEC code), the nabla operations are directly performed on the scalar Green’s functions, yielding six Sommerfeld integrals;
Reduction to the wire problems:
2a
We make the following assumptions:
The current flows along the axis of the wire; The testing points are on the surface of the wire; The current is only a function of l, and
If Galerkin’s method is used, the nabla operations can be performed on basis and testing functions. Hence, less Sommerfeld integrals are needed. (only three)
Method of Moments and Fast Algorithms
Tie Jun Cui
Center for Computational Electromagnetics State Key Laboratory of Millimeter Waves
Department of Radio Engineering Southeast University
Half-Wavelength Dipole
Source Model
N =21, 22.7mm
Delta-Gap Source
n
a=0.2 mm
a=5 mm
a=15 mm
1 (.0000, .0000) (.0001, .0000) (.0007, -.0003) 0 2 (.0000, .0000) (.0001, .0000) (.0009, -.0004) 0 3 (.0000, .0000) (.0001, .0000) (.0012, -.0004) 0 4 (.0000, .0000) (.0002, .0000) (.0018, -.0004) 0 5 (.0000, .0000) (.0003, .0000) (.0027, -.0004) 0 6 (.0000, .0000) (.0005, .0000) (.0043, -.0004) 0 7 (.0000, .0000) (.0010, .0000) (.0080, -.0004) 0 8 (.0000, .0000) (.0024, .0000) (.0181, -.0004) 0 9 (.0000, .0000) (.0091, .0000) (.0471, -.0004) 0 10 (.1253, .0000) (.1636, .0000) (.1978, -.0004) 0 11 (.7494, .0000) (.6488, .0000) (.4599, -.0004) 1
independent of ; The current is flowing in the longitudinal
direction of the wire ; The gap at driving point and the wire radius are
electrically small.
where
Impedance matrix Source vector
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
After Jp(r) is solved for, a variational expression of the general input admittance is defined as:
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
Why ? In the Existing Models (NEC Code):
• Point Matching : Need More Segments • More Sommerfeld Integrals (For example, 6 SI are used in the NEC Code) • Current is assumed to flow along the axis but test on the surface, or vice versa • The input impedance is not variationally defined
东南大学 计算电磁学研究中心
Method of Moments
Conventional method of moments: sometimes • Inaccurate • Expensive
New modeling: Combined with physics • New wire model • New PCB model
From the Huygens’ principle, the electric field in Problem a can be written as
Magnetic dyadic Green’s function
Electric dyadic Green’s function
东南大学 计算电磁学研究中心
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
What do we do in the new modeling?
• Galerkin’s method (roof-top)
Less segments
• Variational formula for the input impedance
Total electric and magnetic fields produced by both We can easily proof that: Hence, Yin is a variational formula about the exact solution of Jp. When Galerkin’s method is used, it is easily shown that the general input admittance can be written as
东南大学 计算电磁学研究中心
1. Wire Antennas
T. J. Cui and W. C. Chew, "Accurate Model of arbitrary wire antennas in free space, above or inside ground," IEEE Trans. on Antennas and Propagation, vol. AP-48, no. 4, pp. 482-493, Apr. 2000.
FFT-Based Fast Algorithms FMM-Based Fast Algorithms
东南大学 计算电磁学研究中心
Method of Moments
东南大学 计算电磁学研究中心
Method of Moments
MOM is very popular Integral equation solver: fewer unknowns on the source EFIE, MFIE, CFIE PEC objects: surface integral equation, triangular patch model Dielectric objects: volume integral equation Surface-wire objects: surface-wire junction Zeroth-order basis function (pulse) First-order basis function (RWG, roof-top) High-order basis function
Closed-form expression is obtained for the voltage vector in the new source model:
electric field produced by magnetic current
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
Source Vector Vn
Input admittance
Gap capacitance
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
Input Admittance
Galerkin’s Method: Jn(r)= wn(r), then
General definition of input admittance When the source vector is a Delta function:
Pure Conductor Problem a
on S except on Sa Problem b
Choosing the basis function and letting: then testing with wn(r) yields a matrix equation:
• New source model
Fra Baidu bibliotek
• Less Sommerfeld integrals
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
General Formulation
• Two Conductors • Cylindrical Gap • Driving Voltage
Nanjing 210096, P. R. China
东南大学 计算电磁学研究中心
Contents
The Method of Moments: New Modeling
Wire-structure circuit problems PCB Simulation Problems
Fast Algorithms for Integral Equation Solver
Traditional definition:
n driving point
• Two source models are usually used:
Delta gap source
Most popularly used
Magnetic frill source Need an additional parameter b b
For the self term and near mutual terms in the impedance matrix, closedform expressions are obtained using the Taylor expansion. Hence, the CPU time to evaluate the impedance matrix does not increase although accurate model has been used.
we have
driving point Traditional definition of input admittance
The above conclusion is valid for arbitrarily-shaped conductors.
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
zoom
driving source lˆ
I (l)
Region b Region a
z
y x
Thus, the unknown in the EFIE is only I (l ).
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
If point-matching method is used (like the NEC code), the nabla operations are directly performed on the scalar Green’s functions, yielding six Sommerfeld integrals;
Reduction to the wire problems:
2a
We make the following assumptions:
The current flows along the axis of the wire; The testing points are on the surface of the wire; The current is only a function of l, and
If Galerkin’s method is used, the nabla operations can be performed on basis and testing functions. Hence, less Sommerfeld integrals are needed. (only three)
Method of Moments and Fast Algorithms
Tie Jun Cui
Center for Computational Electromagnetics State Key Laboratory of Millimeter Waves
Department of Radio Engineering Southeast University
Half-Wavelength Dipole
Source Model
N =21, 22.7mm
Delta-Gap Source
n
a=0.2 mm
a=5 mm
a=15 mm
1 (.0000, .0000) (.0001, .0000) (.0007, -.0003) 0 2 (.0000, .0000) (.0001, .0000) (.0009, -.0004) 0 3 (.0000, .0000) (.0001, .0000) (.0012, -.0004) 0 4 (.0000, .0000) (.0002, .0000) (.0018, -.0004) 0 5 (.0000, .0000) (.0003, .0000) (.0027, -.0004) 0 6 (.0000, .0000) (.0005, .0000) (.0043, -.0004) 0 7 (.0000, .0000) (.0010, .0000) (.0080, -.0004) 0 8 (.0000, .0000) (.0024, .0000) (.0181, -.0004) 0 9 (.0000, .0000) (.0091, .0000) (.0471, -.0004) 0 10 (.1253, .0000) (.1636, .0000) (.1978, -.0004) 0 11 (.7494, .0000) (.6488, .0000) (.4599, -.0004) 1
independent of ; The current is flowing in the longitudinal
direction of the wire ; The gap at driving point and the wire radius are
electrically small.
where
Impedance matrix Source vector
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
After Jp(r) is solved for, a variational expression of the general input admittance is defined as:
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
Why ? In the Existing Models (NEC Code):
• Point Matching : Need More Segments • More Sommerfeld Integrals (For example, 6 SI are used in the NEC Code) • Current is assumed to flow along the axis but test on the surface, or vice versa • The input impedance is not variationally defined
东南大学 计算电磁学研究中心
Method of Moments
Conventional method of moments: sometimes • Inaccurate • Expensive
New modeling: Combined with physics • New wire model • New PCB model
From the Huygens’ principle, the electric field in Problem a can be written as
Magnetic dyadic Green’s function
Electric dyadic Green’s function
东南大学 计算电磁学研究中心
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
What do we do in the new modeling?
• Galerkin’s method (roof-top)
Less segments
• Variational formula for the input impedance
Total electric and magnetic fields produced by both We can easily proof that: Hence, Yin is a variational formula about the exact solution of Jp. When Galerkin’s method is used, it is easily shown that the general input admittance can be written as
东南大学 计算电磁学研究中心
1. Wire Antennas
T. J. Cui and W. C. Chew, "Accurate Model of arbitrary wire antennas in free space, above or inside ground," IEEE Trans. on Antennas and Propagation, vol. AP-48, no. 4, pp. 482-493, Apr. 2000.
FFT-Based Fast Algorithms FMM-Based Fast Algorithms
东南大学 计算电磁学研究中心
Method of Moments
东南大学 计算电磁学研究中心
Method of Moments
MOM is very popular Integral equation solver: fewer unknowns on the source EFIE, MFIE, CFIE PEC objects: surface integral equation, triangular patch model Dielectric objects: volume integral equation Surface-wire objects: surface-wire junction Zeroth-order basis function (pulse) First-order basis function (RWG, roof-top) High-order basis function
Closed-form expression is obtained for the voltage vector in the new source model:
electric field produced by magnetic current
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
Source Vector Vn
Input admittance
Gap capacitance
东南大学 计算电磁学研究中心
Wire Antennas (cont’d)
Input Admittance
Galerkin’s Method: Jn(r)= wn(r), then
General definition of input admittance When the source vector is a Delta function: