A Canonical Test Problem for Computational Electromagnetics (CEM): Propagation in a Parallel-Plate Waveguide Gökhan Apaydin1 and Levent Sevgi2 Electrical-Electronics Engineering Department Zirve University, Kizilhisar Campus, 27260, Gaziantep, Turkey E-mail:
[email protected] 1
Electronics and Communications Engineering Department Doğuş University, Zeamet Sokak, No. 21, Acıbadem – Kadıköy, 34722 Istanbul, Turkey E-mail:
[email protected] 2
Abstract This paper aims to provide a tutorial on computational electromagnetics (CEM), and simple MATLAB codes for sophisticated investigation of analytical and well-known numerical models. The problem of propagation inside a parallelplate waveguide is used for this purpose. Keywords: Calibration; computational electromagnetics; CEM; FDTD; finite-difference time-domain; Gaussian beam; Green’s function; image method; MATLAB; moment methods; MoM; mode summation; parallel plate waveguide; propagation; rays; simulation; split-step parabolic equation; SSPE; validation; verification; visualization
E
1. Introduction
lectromagnetic (EM) problems (from nanoscale to kilometer-wide systems) are complex in nature, but the theory is well established with Maxwell’s equations (a few of many classical electromagnetics books may be listed as [13]). The strategies for the solution of such problems may be grouped into three: analytical modeling, numerical simulations, and measurements. Measurements in electromagnetics are time consuming, expensive, and, in most cases, extremely difficult to do. A limited number of analytical solutions are available only for a few, highly idealized problems. Numerical simulation therefore becomes the only means for almost all real-life electromagnetic engineering problems. This is why electromagnetics modeling and simulation (EM-MODSIM) has made significant progress for the last couple of decades. EM-MODSIM has been taught for sometime worldwide in many universities. A few books about EM-MODSIM may be listed as [4-9]. EM-MODSIM approaches may be divided into two: analytical-model-based and numerical-model-based approaches [10-12]. The critical issue in EM-MODSIM is the (model) validation, (data) verification, and (code) calibration
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(VV&C) [13]. The best validation, verification, and calibration method is to compare numerical models with accurately computed analytical models. In this context, high-frequency asymptotics (HFA) play an essential role in generating reference data [14-20]. High-frequency asymptotics models – such as Geometric Optics (GO), the Geometrical Theory of Diffraction (GTD), Physical Optics (PO), the Physical Theory of Diffraction (PTD), and the Uniform Theory of Diffraction (UTD) – with their advantages and disadvantages and regions of validity not only yield reference data, but also give physical insight into understanding electromagnetic wave scattering pieces. The two high-frequency asymptotics tutorials published in this column [21, 22] review fundamental diffraction models. In terms of solution approaches, electromagnetics problems may be grouped into three: (i) radiation/antenna problems, (ii) scattering and radar cross section (RCS) modeling, and (iii) wave-guiding structures. These all start with Maxwell’s equations and the definition of boundary conditions (BC). For (i), it is better to define auxiliary scalar and vector potential functions, and to solve the equations that contain these auxiliary functions. This is better because components of the vector potential fit into practical excitations.
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The scattered-field representation is used in RCS modeling, because the excitation (i.e., the incident field) is elsewhere. Finally, Maxwell’s equations are decomposed into transverse and longitudinal components in guided-wave problems. This not only yields decomposed fields, but also scalarizes and solves the problem. Eigenfunctions and eigenvalues are derived from transverse components; excitation coefficients are obtained from longitudinal components. Completeness and orthonormalization relate these two via the well-known SturmLiouville equation [23]. Most EM-MODSIM studies, commercial or educational, are based on one of the listed models: •
The Method of Moments (MoM) discretizes the object under investigation into N pieces, called segments or patches, and constructs an N by N system of equations using the Green’s function of the problem [24]. The approach is closed form and stable, but suffers from a lack of large memory and high-speed computers.
•
The Parabolic Equation (PE) Method, based on either finite-element (FE) or split-step (SSPE) discretization, is widely used in complex propagation modeling [25].
•
The Finite-Difference Time-Domain (FDTD) Method discretizes the physical environment into small cells by replacing partial derivatives with their finite-difference equivalents [26-28]. It is open form and iterative, and therefore suffers from stability.
•
The Transmission-Line Matrix (TLM) Method uses the circuit equivalents of Maxwell’s equations [29, 30]. This is also open form and iterative, with similar stability problems.
This tutorial aims to review fundamental EM-MODSIM concepts and the widely used numerical models mentioned above through a reliable validation, verification, and calibration procedure. The philosophy is to keep it as simple as possible, so that it can be used as a basic computational electromagnetics note, and to supply simple MATLAB codes so that even beginners can use them. In order to achieve these tasks, one of the most simple and widely used structures is chosen as the EM-MODSIM scenario: Propagation inside a parallelplate waveguide with perfectly electrically conducting (PEC) boundaries in two dimensions (2D) [31, 32]. The same can also be applied to a wedge waveguide [33]. Propagation inside a parallel-plate waveguide is an interesting electromagnetics problem where both analytical and numerical models can be tested, one against the other: •
First of all, the structure in two dimensions is nonphysical, and acts as a low pass filter (LPF).
•
The Green’s function solution (i.e., the electromagnetic response of a line source) is exact, but
requires an infinite number of eigenfunction (mode) summations [31]. This is a numerical challenge, especially in the near vicinity of the line source, but it is excellent for the analysis of truncation errors. •
Eigenfunctions (modes) are grouped into two types: propagating modes, with real eigenvalues; and evanescent modes, with imaginary eigenvalues. The number of propagating modes depends on the frequency and the width of the waveguide.
•
A tilted directional antenna can also be located inside and can be modeled in terms of modes, but the modal-excitation coefficients are now complex. This is another numerical challenge, especially at high frequencies, when the number of propagating modes is high.
•
The modes (eigenfunctions) are global and therefore do not suffer from local problems, but extraction of modal excitation coefficients is crucial when generating reference solutions.
•
An analytical, exact solution can also be constructed in terms of rays, which are local wave pieces: again, an infinite ray summation is required for the linesource excitation. This may be achieved via either an eigenray specification or ray shooting. The solution of eigenray equations is difficult, especially for rays with a high number of reflections from top and bottom boundaries [31]. Ray shooting is very time consuming and is highly sensitive to source/ observer locations. Their computer implementations are both numerical challenges.
•
Another ray-based analytical solution is the Image Method (IM). The Image Method overcomes the numerical difficulties of both the eigenray-extraction and ray-shooting approaches, but a very high number of images is required for accurate field computations.
•
One-way, finite-element (FE) and/or fast Fourier transformation (FFT) based parabolic-equation models have been used in propagation modeling in wave-guiding environments [13, 34, 35]. The parabolic-equation models can easily handle tilted and directed excitation antennas. However, one needs to use wide-angle parabolic-equation models, since narrow-angle models suffer paraxial propagation restriction (i.e., they are valid up to 10°-15° vertical propagation angles).
•
The two powerful time-domain methods (FDTD and the Transmission-Line Method) may also be used in modeling propagation inside a parallelplate waveguide. This is possible even for very long ranges by using sliding-window approaches [36, 37]. Both line-source and short dipoles can be used for wave excitation in these models, but special
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care is essential for modeling a tilted and directional transmitting antenna. •
•
Finally, the MoM can be used in propagation modeling inside a parallel-plate waveguide. This also requires a special treatment because the MoM has rarely been used in resonating structures (a novel approach, called Mi-MoM, was introduced in [38]). The MoM is based on the solution of an N by N system of equations that yields currents induced on the segments. The procedure is first applied, and currents induced by the source are obtained. Consecutive applications yield segment currents induced by other segment currents. This has to be continued until their contributions diminish.
Figure 1. A parallel-plate waveguide, and the source and observer locations.
TE z and TM z as
A hybrid, MoM plus Image Method (MoM-IM) increases both accuracy and computation time. In this case, an N by N system of equations is solved once for segment currents induced by the source. The rest is handled via the Image Method.
TE z equations:
In summary, a one-semester computational-electromagnetics lecture may be built on the above-mentioned analytical and numerical methods through a single structure: the parallelplate waveguide.
∂E y = M x + jωµ0 H x , ∂z
(2a)
∂H x ∂H z − =J y + jωε 0 E y , ∂z ∂x
(2b)
∂E y ∂x
2. Problem Statement and Analytical Solutions
Assuming that the medium between the plates is free space and isotropic, Maxwell’s curl equations, under exp ( jω t ) time dependence, are given as ∇ × E = − jωµ0 H − M ,
(1a)
∇ = × H jωε 0 E + J ,
(1b)
where boldface symbols denote vector quantities. The electric and magnetic field intensities are E and H, the non-phased electric and magnetic source currents are J and M, ω is the angular frequency, and ε 0 and µ0 are the free-space permittivity and permeability. Considering that there is no variation of field with respect to y ( ∂ ∂y ≡ 0 ), the vector equations in Equation (1) are respectively reduced to scalar equations for
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(2c)
TM z equations:
The two-dimensional parallel-plate waveguide is pictured in Figure 1. Here, x and z are the transverse and longitudinal coordinates, respectively. The structure is infinite along the y direction ( ∂ ∂y ≡ 0 ). The height of the waveguide is a. The PEC boundaries are assumed for Dirichlet-type boundary conditions for the TE z (transverse electric with respect to z) problem and for Neumann-type boundary conditions for the TM z (transverse magnetic with respect to z) problem (see [39] for TE/TM discussions).
292
= − M z − jωµ0 H z ,
∂H y ∂z
= − J x − jωε 0 E x ,
(3a)
∂Ex ∂Ez − = − M y − jωµ0 H y , ∂z ∂x
(3b)
∂H y = J z + jωε 0 Ez . ∂x
(3c)
These equations can be combined into wave equations, as follows: TE z wave equation: ∂ 2 ∂M z ∂M x ∂2 jωµ0 J y − + + k02 E= + , (4) 2 y 2 ∂x ∂z ∂z ∂x with boundary conditions E = 0 y = Ey 0
at= x 0,= x a,
(5a)
as z → ±∞ .
(5b)
The remaining two TE z field components can be derived from
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= Hx
∂E y − Mx , jωµ0 ∂z 1
1 ∂E y − + Mz . Hz = jωµ0 ∂x
(6a)
(6b)
g ( x, z= ; x′, z ′ ) 0= at x 0, a , ( TE z )
(11a)
∂ g ( x, z= ; x′, z ′ ) 0= at x 0, a , ( TM z ) ∂x
(11b)
g ( x, z;= x′, z ′ ) 0 as z → ±∞ .
(11c)
TM z wave equation:
Here, ( x′, z ′ ) and ( x, z ) specify source and observation points,
2 ∂J ∂J ∂2 ∂ 2 jωε 0 M y + z − x , (7) = 2 + 2 + k0 H y ∂ x ∂z ∂z ∂x
The Green’s function, g ( x, z; x′, z ′ ) , can be obtained as g ( x,z;= x′,z ′ ) g ( z; z ′ ) +
with boundary conditions ∂H y = 0 ∂x = Hy 0
x 0,= x a, at= as z → ±∞ .
respectively; and δ ( x ) and δ ( z ) are the Dirac delta functions.
(12)
(8a)
g ( z; z ′ ) = 0,
(8b)
B ( x ) = sin ( x ) ,
The remaining two TM z field components can be derived from 1 ∂H y − + Jx , Ex = jωε 0 ∂z = Ez
∂H y − Jz . jωε 0 ∂x 1
2 ∞ e − jk zm |z − z'| B ( k xm x ) B ( k xm x ′ ) ∑ a m =1 2 jk zm
g ( z; z ′ ) =
(9a)
1 e − jk0|z − z'| , a 2 jk0 ( TM z )
B ( x ) = cos ( x ) , (9b)
Here,= k0 2= π λ ω ε 0 µ0 is the free-space wavenumber, and λ is the free-space wavelength. Since the TE z and TM z sets are decoupled, each can be excited independently of the other by appropriate selection of the sources, J and M. The line sources M x , M z , J y excite the TE z set, whereas the line sources M y , J x , J z excite the TM z set. Further simplification can be obtained by setting the source components M x = 0 , M z = 0 for the TE z set, and J x = 0 , J z = 0 for the TM z set. These simplifications will be assumed in what follows.
2.1 Green’s Function in Terms of Mode Summation
( TE z )
(13a)
(13b)
2 where k xm = mπ a , k= k02 − k xm . The line-sourcezm excited fields are then given by either E y = jωµ0 g or
H y = jωε 0 g for the TE z case and TM z case, respectively. A short MATLAB code was prepared for the calculation of the field distribution inside the parallel-plate waveguide in terms of mode summation for both polarizations. Figure 2 lists this code. An example is shown in Figure 3. Here, the field as a function of range inside a 1 m wide plate at 0.7 m is pictured. The source was at a height of 0.3 m. The number of propagating modes for the sets of parameters listed in the figure was 15. The two curves belonged to 15-mode and 100-mode summations. As observed, at a distance beyond 0.5 m (i.e., after three to four wavelengths in distance), only propagating modes contributed. Figure 4 displays the field as a function of height at two different distances inside the same plate. As observed, the contribution of only propagating modes at a distance of two wavelengths was not enough to build the correct field.
The Green’s function problem associated with both the TE z set (when M = M = 0 ) and the TM z set (when x z J= J = 0 ) is defi ned by the equation x z
2.2 The Mode Summation for a Directive Antenna (Tilted Gaussian Source)
∂ 2 ∂2 2 2 + 2 + k0 g ( x, z; x′, z ′ ) = δ ( x − x′ ) δ ( z − z ′ ) , ∂z ∂x (10)
A directive antenna is used in many propagation applications. This is modeled by using a vertical, tilted Gaussian function in analytical and numerical simulations. This tilted Gaussian source inside a PEC parallel-plate waveguide at z = z ′ may be represented in terms of a modal summation as
with boundary conditions
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Figure 2. A MATLAB module for the calculation of the field as a function of the range or height at a given height or range using mode summation.
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a
cm = vm ∫ f ( x, z ) B ( k xm x ) dx .
(16)
0
The initial field profile, f ( x, z ′) , at z = z ′ is generated from a tilted Gaussian pattern: ( x − x′) 2 f ( x, 0 ) = exp − jk0 x sin θ elv − , w2
Figure 3. The magnetic field as a function of range with the Green’s function: TM z case, a = 1 m, z ′ = 0 , x′ = 0.3 m, x = 0.7 m, k0 a = 50 ; (solid) only propagating modes (15 modes); (dashed) 100 modes.
(17)
where w = 2 ln 2 k0 sin (θbw 2 ) . The tilted antenna pattern is specified by its vertical position ( x′ ), beamwidth ( θbw ), and tilt (elevation) angle ( θ elv ). Note that B again shows either a sine or cosine function behavior, starting from either m0 = 1 or m0 = 0 for horizontal ( TE z ) and vertical ( TM z ) polarizations, respectively. The number of modes would be finite for numerical computation. It is common to choose a vertically extending Gaussian function with arbitrary location, having a vertical elevation angle in the range of ±90°. Note that the modal excitation coefficient, cm , is real for a real source function without any tilt, and becomes complex if there is a tilted source.
2.3 The Eigenray Representation Modes are global wave pieces, and can be used to calculate fields at any point. They do not contain signatures of the local paths. On the other hand, rays that are local wave objects trace all the paths between the source and observer. Fields inside a parallel-plate waveguide can also be constructed in terms of rays [31, 32]. The integral representation of the eigenfunctions can be used for this purpose. The integral representation for eigenfunctions can be restructured as Figure 4. The magnetic field as a function of height with the Green’s function: TM z case, a = 1 m, z ′ = 0 , x′ = 0.3 m, k0 a = 50 ; (a) at z = 2λ , (b) at z = 20λ ; (solid) only propagating modes (15 modes), (dashed) 100 modes.
f ( x, z ′ ) =
M
∑
m = m0
cm ( z ′ ) vm B ( k xm x ) ,
(14)
where M is the highest mode that should be included for the specified excitation, ν m is the normalization constant, a vm = ∫ B 2 ( k xm x ) dx x =0
g ( x,z; x',z' ) =
(18)
in a manner that expresses the x-domain Green’s function behavior in terms of multiply reflected traveling plane waves between the boundaries at x = 0 and x = a , rather than the sinusoidal standing waves. Here, x> ( x< ) represent the max (min) of ( x′, x ) . This is achieved by expanding the resonant denominator in the integrand of Equation (18) into an infinite power series: ∞
1 Ω (1 − Ω )−= ∑ Ωn ,=
−1/2
,
(15)
and cm ( z ′ ) is the modal excitation coefficient, numerically derived from the vertical orthonormality condition as
n =0
exp ( − j 2k x a ) ,
(19)
obtained by writing sin = ( k x a ) ( 0.5 j )(1 − Ω ) exp ( jk x a ) . This series can be ordered into four infinite periodic “image arrays” in the x domain indexed by i. Interchanging the orders of summation and integration, each of the integrals represents
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+∞ sin k x sin k a − x ( x ) − jk z (z − z') e dk z ∫ 2ð −∞ k x sin ( k x a )
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an exact two-dimensional “image” line-source Green’s function, gˆ n(i ) , in free space, which can be identified asymptotically with a ray-optical field reaching the observer along a direct or multiply reflected ray path. This is frequently referred to as an “eigenray” from a specified source point to a specified observation point. Following this procedure, one obtains the Green’s function fields as 4
∞
g ( x, z; x′, z ′ ) = ∑ ∑
=i 1 = n 0
(
where gˆ n(i ) x, z; xn(i ) , z ′
)
gˆ n(i )
(
x, z; xn(i ) , z ′
),
is the “image” line-source Green’s
gˆ n(3) must be negative for the TE z case. In the high-frequency range, Equation (20) can be approximated as 2 jπ g ( x, z; x′, z ′ ) ≈ ∑ ∑ ( 4π =i 1 =n 0 k q ′′ i ) W (i ) n 0 n 4
1/2
∞
(
)
e
(
− jk0 qn( i ) Wn( i )
)
(21)
qn(1) = ( x> − x< ) cos W + z sin W + 2an cos W ,
(22a)
qn(2) = ( x> + x< ) cos W + z sin W + 2an cos W ,
(22b)
qn(3) = − ( x> + x< ) cos W + z sin W + 2a ( n + 1) cos W , (22c)
i = 2 : n reflections at upper boundary and n + 1 reflections at lower boundary (down, up),
•
i = 3 : n + 1 reflections at upper boundary and n reflections at lower boundary (up, down),
•
i = 4 : n + 1 reflections at both boundaries (up, up for x> = x′ , x< = x , and down, down for x> = x , x< = x′ ).
Here, the angular spectrum variable, W, denotes the angle between the ray and the normal to the boundaries, whereas n is the number of reflections from both boundaries. The eigenrays emanating from the source and reaching the observer are determined from the eigenray (stationary phase) condition [31, 32] d (i) qn (W ) =0 ⇒ W =Wn(i) , i =1, 2 ,3, 4 . dW
Figure 5 lists a short MATLAB script for the calculation of the field inside the parallel-plate waveguide with the eigenray summation model for the line source. Eigenrays for n = 0 (i.e., the first four eigenrays) and n = 1 (i.e., the first eight eigenrays) are plotted in Figure 6.
The most difficult task of the eigenray representation is the numerical solution of the stationary phase Equation (23). This becomes a real challenge as the number of top/bottom boundary reflections increases. Another approach in the ray model which smoothes numerical computations is to use the Image Method (IM). The Image Method can be used to satisfy an infinite number of reflections with an infinite number of images between two plates. The images are placed with respect to the position of the plates. The unit field ( E y for the TE z case; H y for the TM z case) can be created by an incident point source at ( z ′, x′ ) as
qn(4) = − ( x> − x< ) cos W + z sin W + 2a ( n + 1) cos W . (22d)
(23)
The four eigenray species and their trajectories are organized according to their departure directions (up and down) at the source, xs , and their arrival directions (up and down) at the observer, xo :
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•
2.4 A Hybrid Ray + Image Method
where a double prime in the denominator denotes the second derivative with respect to the argument. The four different ray species are defined by the following phase accumulations [31, 32]:
296
i = 1 : n reflections at both boundaries (down, down for x> = x′ , x< = x , and up, up for x> = x , x< = x′ ),
(20)
function in free space. Note that the contribution of gˆ n(2) and
j
•
Fi =
e − jk0 r k0 r
,
(24)
( x − x′ )2 + ( z − z ′ )2 . The first image sources appear at ( z ′, − x′ ) with respect to the x = 0 plate, and at ( z ′, 2a − x′ ) with respect to the x = a plate.
where r =
Figure 7 lists a short MATLAB script for the calculation of the field inside the parallel-plate waveguide in terms of the Image Method. A triple comparison among mode summation, eigenray summation, and the Image Method for the line-source excitation was done, and the results are given in Figure 8. The fig-
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Figure 5 continued on next page
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Figure 5. A MATLAB module for the calculation of the field as a function of range or height at a given height or range using an eigenray representation.
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Figure 6. Eigenrays with (a) n = 0 , (b) n = 1 inside a 1 m wide by 300 m long parallel-plate waveguide. The source height was 0.3 m, the receiver height was 0.7 m.
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Figure 7. A MATLAB module for the calculation of the field as a function of the range or height at a given height or range using the ray+image method.
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initial vertical-field profile. This initial field is then propagated longitudinally, and the transverse field profile at the next range is obtained from
{
}
u ( z + ∆z , x ) = F −1 C ( k x ) F u ( z , x ) ,
(
(26)
)
ˆ 2 . Here, F refers to where C (k= ) exp jk 2 ∆z k0 + k02 − ck the discrete sine/cosine transform (DST/DCT) while considering ( TE z / TM z ) boundary conditions, and k x is the spectral variable. The parameter cˆ is 0 (1) for the narrow (wide) angle model.
Figure 8. The electric field as a function of range with (solid) Green’s function (100 modes), (dashed) eigenray representation ( n = 50 ), (dashed-dotted) ray+image method (20 images), for the TE z case, a = 1 m, z ′ = 0 , x′ = 0.3 m, x = 0.5 m, k0 a = 50 . ure shows the field as a function of range for specified source and observer heights. Here, 100 modes were used in the mode summation; the eigenray summation was carried out until there were 50 reflections from both boundaries; and 20 images were used in the Image Method. As observed, the agreement among the results was very impressive.
3. Numerical Models
It was shown in the previous section that analytical reference data can be generated using any of the three models; mode or eigenray summations, or the Image Method. Three wellknown numerical models are discussed in this section.
3.1 Split-Step Parabolic-Equation Model The parabolic-equation (PE) wave propagator for a parallel-plate waveguide filled with air uses the parabolic equation ∂2 ∂2 ∂ 0, 2 + 2 − 2 jk0 u ( z , x ) = ∂z ∂z ∂x
(25)
under exp ( jω t ) time dependence. It represents one-way forward propagation under the paraxial approximation (i.e., nearaxial propagation). Since the direction of wave propagation is predominantly along the z axis, the reduced function, u ( z , x ) , is used by separating the rapidly varying phase term from ψ ( x= , z ) exp ( − jk0 z ) u ( z , x ) , which corresponds to either electric or magnetic fields for horizontal ( TE z ) and vertical ( TM z ) polarizations, respectively.
The new profile, u ( z + ∆z , x ) , is then used in Equation (26) as the initial profile, and the procedure is repeated; the profile u ( z + 2∆z , x ) at the next range is obtained. The procedure is repeated continuously until the propagator reaches the desired range. Figure 9 lists a short MATLAB script for the calculation of split-step parabolic-equation fields. An example produced with Figure 9 is given in Figure 10. The field as a function of range and height, simulated via narrow- and wideangle split-step parabolic-equation models, are shown in Figures 10a and 10b, respectively. A downward-tilted Gaussian beam hit the bottom plate first, and was then bounced back and hit the upper plate. As observed, there was a significant difference between these two plots. In order to see which was accurate, the same field plot as a function of range and height was produced and reference data was generated via the modesummation model. This is shown in Figure 10c. As observed, the agreement between the wide-angle split-step parabolicequation (Figure 10b) and the reference data (Figure 10c) was impressive. Two vertical cuts were taken from these plots and are shown separately in Figure 11 as the field as a function of height at a constant range. These plots showed that the agreement between the analytical reference solution and the wide-angle split-step parabolic equation was better, and that there was disagreement with the narrow-angle split-step parabolic equation.
3.2 Finite-Difference Time-Domain Model The two-dimensional parallel-plate waveguide can also be modeled with the FDTD method (see [26-28] for details of FDTD modeling). A sample virtual tool presented in [40] can be used for this purpose. The time-domain versions of the two-dimensional Maxwell’s equations in Equations (2)-(3) are given as TE z case: ∂E y 1 ∂H x ∂H z = − ∂t ε 0 ∂z ∂x
The split-step parabolic-equation solution (SSPE) uses a longitudinally marching procedure with the use of the discrete Fourier transform. First, an antenna pattern is injected as the IEEE Antennas and Propagation Magazine, Vol. 54, No. 4, August 2012
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∂H z 1 ∂E y = − , ∂t µ0 ∂x
,
(27a)
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Figure 9. A MATLAB module for the calculation of the field as a function of the range or height at a given height or range using the split-step parabolicequation method.
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n −1/2 − H ( E y )i,k = ( E y )i,k + ε ∆∆t z ( H x )in,−k1/2 +1/2 ( x )i , k −1/2 n
n −1
0
∆t 1/2 − , (29a) ( H )n−1/2 − ( H z )in−−1/2, k ε 0 ∆x z i +1/2,k 1/2 1/2 = − ( H z )in++1/2, ( H z )in+−1/2, k k
n −1/2 H = + ( H x )in,+k1/2 +1/2 ( x )i , k +1/2
∆t E µ0 ∆x y
( )i +1,k − ( E y )i,k ,
∆t Ey
µ0 ∆z
n
n
(29b)
( )i,k +1 − ( E y )i,k , n
n
(29c)
TM z case: n −1/2 − E ( H y )i,k = ( H y )i,k − µ∆∆t z ( Ex )in,−k1/2 +1/2 ( x )i , k −1/2 n
n −1
0
∆t 1/2 , + ( E )n−1/2 − ( Ez )in−−1/2, k µ0 ∆x z i +1/2,k 1/2 1/2 = + ( Ez )in++1/2, ( Ez )in+−1/2, k k
Figure 10. The electric field as a function of range and height with a tilted Gaussian beam: (a) narrow split-step parabolic equation, (b) wide split-step parabolic equation, (c) Green’s function (31 modes): TE z case, a = 1 m, z ′ = 0 , x′ = 0.4 m, k0 a = 50 , θbw= 30° , θ elv =−10° .
∂H x 1 ∂E y , = ∂t µ0 ∂z
(27c)
TM z case: ∂H y ∂t
1 ∂Ex ∂Ez , = − − µ0 ∂z ∂x
(28a)
∂Ez 1 ∂H y = , ∂t ε 0 ∂x
(28b)
∂E x 1 ∂H y . = − ∂t ε 0 ∂z
(28c)
The FDTD method is based on the discretization of these equations directly in the time domain, where the physical geometry is divided into small cells. Both time and spatial partial derivatives are handled with the finite central-difference approximation, and the solution is obtained with a marching scheme in iterative form as TE z case:
n −1/2 E = − ( Ex )in,+k1/2 +1/2 ( x )i , k +1/2
n
∆t H ε 0 ∆z y
n
( )i +1,k − ( H y )i,k , n
(30b)
( )i,k +1 − ( H y )i,k . n
(30c)
The characteristics of the medium are defined in terms of permittivity, conductivity, and permeability. The electric- and magnetic-field components are calculated at different locations of each cell (see Figure 1 in [39] for the cell structures). Beside the spatial differences in field components, there is also a halftime-step difference between electric- and magnetic-field components. In order to provide good accuracy, the time step should satisfy Courant limit, and the sampling space should be smaller than the wavelength (such as ∆x < λ 10 , ∆z < λ 10 ). Using the FDTD, although it is possible to analyze electromagnetics fields in the time domain for a broad range of frequencies with one simulation, large memory size and highspeed computer requirements should be taken into consideration. The FDTD has been extensively used in complex electromagnetic simulations for the last few decades. The field as a function of range and height at three different time instants for a tilted Gaussian beam, computed with the FDTD for the TE z polarization, is shown in Figure 12. As observed, a down-tilted Gaussian beam first hit the bottom plate, and interference patterns were formed from the interaction of direct and bottom-reflected beams. The beams then hit and were reflected from the top plate. The interaction of the direct beam with both the bottom- and top-reflected beams was clearly observed from the FDTD simulation.
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∆t H ε 0 ∆x y
(30a)
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Figure 11. The electric field as a function of height for a tilted Gaussian beam: TE z case, a = 1 m, z ′ = 0 , x′ = 0.4 m, k0 a = 50 , θbw= 30° , θ elv =−10° ; (a) at z = 2λ , (b) at z = 20λ ; (solid) Green’s function (31 modes), (dashed) narrow split-step parabolic equation, (dashed-dotted) wide split-step parabolic-equation.
Figure 13. The electric field as a function of the height for a tilted Gaussian beam: TE z case, a = 1 m, z ′ = 0 , x′ = 0.5 m, k0 a = 50 , θbw= 40° , θ elv =−20° ; (a) at z = 2λ , (b) at z = 20λ ; (solid) Green’s function (42 modes), (dashed) split-step parabolic-equation, (dashed-dotted) FDTD dt 1.14e − 11 s. dx = dz = 0.5 m,=
Figure 14. A parallel-plate waveguide and the source and observer points for the MoM ( ñm : source segment, ñn : observer segment).
Figure 12. The field as a function of the range and height at three different time instances for a tilted Gaussian beam computed with the FDTD: TE z case, a = 1 m, z ′ = 0 , x′ = 0.5 m, k0 a = 50 , θbw= 40° , θ elv =−20° = ; ta 3.42e − 9 s, = tb 6.84e − 9 s, = tc 1.03e − 9 s. 304
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Figure 13 shows electric field as a function of height for a tilted Gaussian beam for the TE z polarization computed with the mode summation, the split-step parabolic-equation, and the FDTD. Note that the mode summation and split-step parabolicequation are frequency domain methods, and directly yielded the results. On the other hand, the time data was accumulated at the specified observation points during the FDTD simulations, and the frequency-domain result was then obtained offline via the fast Fourier transform (FFT).
3.3 Parallel-Plate Waveguide and Method of Moments (MoM) The Method of Moments (MoM) [24] is one of the oldest numerical electromagnetic (EM) methods. In this method, a discrete model of the object under investigation is first created from small (compared to a wavelength) pieces, called segments or patches. An N × N system of equations is then built with N unknown segment/patch currents, N known segment voltages calculated from the Green’s function of the problem, and known (analytically, but requiring numerical computation) N × N segment/patch impedances (this requires N 3 operations). The model is closed-form and stable, but necessitates large memory and high-speed computers, especially for high-frequency applications. It also requires the derivation of the Green’s function of the problem. Propagation modeling inside waveguides with irregular and lossy boundaries has become very important because of signaling requirements through railway tunnels, communications in mines, screening in printed-circuit boards (PCS), etc. The MoM can be used to find the propagation of horizontally ( TE z case) and vertically ( TM z case) polarized waves by using the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE), respectively. Unfortunately, the MoM suffers from resonances in these wave-guiding structures [44], and therefore its direct application is a challenge. Recently, a novel Multiple MoM Model (Mi-MoM) was introduced for this purpose [38]. In this method, the integral equation is converted to the corresponding matrix equation via the discretization of the plates at x = 0 and x = a (see Figure 14). The lengths of the line segments on the plates are chosen to be less than one-tenth of the wavelength, to satisfy the surface current being constant on each line segment. An N × N system of equations is then constructed with the unknowns of N segment currents, and the system is solved numerically. The point-matching MoM technique [24, 42] – with the closed-form matrix equation [V ] = [ Z ] [ I ] , where [ I ] contains
the unknown segment currents, [V ] corresponds to the excitation-voltage matrix using the incident field evaluated at the midpoints of each line segment, and [ Z ] is the N × N impedance matrix for the PEC plate – can be used. Using segment currents, the scattered fields are obtained. Finally,
fields at a specified observation point are calculated in terms of the incident field and scattered fields caused by the sourceinduced segment currents on plates. The necessary equations for the elements of the excitation-voltage matrix and the impedance matrix and scattered fields are summarized as [38, 41, 42] follows: TE z case: e − jkd m Vm = E − E inc ñ = − , ( ) y m 0 k0 d m = dm
Z nm
2
2
x ( ñm ) − x′ + z ( ñm ) − z ′) ,
(31b)
k0η0 ∆z (2) ,m≠n − 4 H 0 ( k0 ñ n − ñ m ) ≅ − k0η0 ∆z 1 − j 2 log γ k0 ∆z , m = n 4e 4 π (32)
E yscat ( ñn ) ≅ −
k0 Z 0 ∆z N ∑ I m H 0(2) ( k0 ñn − ñm ) , (33) 4 m =1
= E tot E yscat + E inc y y ,
(34)
TM z case: E e − jkd m Vm = − H inc − 0 , y ( ñm ) = η 0 k0 d m = dm
2
(35a)
2
x ( ñm ) − x′ + z ( ñm ) − z ′) ,
k0 ∆z (2) H1 ( k0 ñn − ñm j Z nm ≅ 4 0.5
H yscat ( ñn ) ≅ j
(35b)
) ( nˆ m ñˆ nm )
,m≠n ,m=n
(36)
k0 ∆z N ∑ I m H1(2) ( k0 ñn − ñm 4 m =1
) ( nˆ m ñˆ nm )
(37)
= H tot H yscat + H inc y y ,
(38)
where ∆z is the segment length; η0 ≈ 120π is the intrinsic
impedance of free space; H 0(2) and H1(2) are the second kind Hankel functions with order zero and one, respectively; γ ≈ 1.781 is the exponential of the Euler constant; nˆ m denotes the unit normal vector of the plate at ñm ; and ñˆ nm is the unit vector in the direction from source ñm to the receiving element, ñn .
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(31a)
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Figure 15 continued on next page
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Figure 15 continued on next page
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Figure 15. A MATLAB module for the calculation of the field as a function of the range or height at a given height or range using the MoM method. The Mi-MoM procedure for propagation modeling inside resonating structures is based on taking into account multipleinduced segment currents, and may be outlined as follows [38]: •
•
•
First, discretize the top and bottom boundaries. Use N 2 segments for the lower boundary and N 2 segments for the upper boundary. Label all segments from 1 to N such that Segment 1 and Segment N + 1 have the same horizontal (i.e., z) coordinate (i.e., are parallel to each other).
respectively. Repeat this for all segments on the bottom plate, and find the voltages on the top plate caused by the segments on the bottom plate. •
Do the same for the segments on the top plate, and find the voltages on the bottom plate caused by the segments on the top plate. This will yield a secondround [V ] .
•
Use the same impedance matrix, Z mn , and calculate second-round segment currents [ I ] from
−1
Calculate segment currents [ I ] from [ I ] = [ Z ] [V ] , and calculate the scattered and total fields using either E inc Equation (31a) or H inc in y y
[ I ] = [ Z ]−1 [V ] , and scattered and total fields using either Equation (33) or Equation (37) for the TE z and TM z polarizations, respectively.
Equation (35a) for the TE z and TM z polarizations, respectively.
•
For a given source point, calculate the distances to all segments and all segment voltages, using either E inc in Equation (31a) or H inc in Equation (34a) y y
Repeat the procedure, and find third-round segment currents, and scattered and total fields caused by these currents.
•
Repeat the whole procedure until a desired accuracy is reached.
for the TE z and TM z polarizations, respectively. This will yield [V ] . •
Calculate the impedance matrix, Z mn , from either Equation (32) or Equation (36) for the TE z and TM z polarization, respectively.
•
The segment currents induced by the external source on the top plate excite fields on segments on the bottom plate, and vice versa. For the first segment on the bottom plate, calculate the distances to all segments on the top plate and the segment voltages, using either E inc in Equation (31a) or H inc in y y Equation (34a) for the TE z and TM z polarizations,
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An alternative way is to find the first-round segment currents, and then use the image method. First, all segment currents of the upper and lower plates are obtained. The boundaries are then removed, and image segments are added with respect to the upper and lower plates. Finally, the field contributions from the currents of the segments and image-segments are superposed at the receiver. A short MATLAB script for the MiMoM calculation of field as a function of range and height at a given height/range point is given in Figure 15. The classical MoM can also be used together with the Image Method (IM) in hybrid form. This is achieved by finding segment currents once, and then using images of all segments on both sides of the boundaries. A short MATLAB script for the hybrid MoM+IM calculation of the field as a function of range and height at a given height/range is listed in Figure 16. IEEE Antennas and Propagation Magazine, Vol. 54, No. 4, August 2012
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Figure 16. A MATLAB module for the calculation of the field as a function of the range or height at a given height or range using the MoM+Image method.
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Figures 19-21 belong to comparisons for directive antennas. As observed, the agreement between the Mi-MoM results and the reference data was impressive, even for these highly resonating/oscillatory variations. The final example belongs to a more realistic case. Figure 22 presents the Mi-MoM compared to split-step parabolicequation results for propagation inside a PEC parallel-plate
Figure 17. The propagation factor (PF) as a function of range ( TM z case): (solid) mode sum, (dashed) Mi-MoM; a = 100 m, z ′ = 0 , x′ = 50 m, x = 5 m, k0 a = 209.5 .
Figure 19. The field as a function of the range and height ( TE z case): (top) mode sum with 42 modes, (bottom) MiMoM with 40 iterations; a = 1 m, z ′ = 0 , x′ = 0.3 m, k0 a = 50 , dz = dx = 0.01 m, θbw = 45 , no tilt.
Figure 18. The field as a function of the height ( TE z case): (solid) mode sum, (dashed) multi-MoM; a = 1 m, z ′ = 0 , x′ = 0.4 m, k0 a = 50 .
Two examples of the Mi-MoM procedure are given in Figures 17 and 18. Figure 17 shows the propagation factor (PF) as a function of the range at a fixed height inside the parallelplate waveguide, calculated with the mode-summation and MiMoM methods. As shown, very good agreement was obtained. As expected, Mi-MoM suffered from end-point effects (as did MoM), since segments before the first segment and after the last segment were neglected [38]. Figure 18 shows the field as a function of height at two different ranges inside the parallel-plate waveguide, calculated with the mode-summation and Mi-MoM methods. Again, the agreement was very good. Note that the agreement in Figure 17 was better than the agreement in Figure 18: this is merely because of the frequency used in these examples ( k0 a = 209.5 in Figure 17, but k0 a = 50 in Figure 18). The accuracy of the Mi-MoM solution increases with frequency (i.e., with ka ). 312
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Figure 20. The field as a function of height at x = 0.2 m: (top) TE z case, (bottom) TM z case; (solid) mode sum with 282 modes, (dashed) Mi-MoM with 50 iterations; a = 1 m, z′ = 0 , x′ = 0.4 m, k0 a = 200 , dz = dx = 0.0025 m,
θbw = 80 , no tilt.
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waveguide having irregularities on the bottom plate. The top plot shows the three-dimensional field map inside the plate, with two Gaussian-shaped irregularities on the bottom plate. The bottom plot shows the field as a function of range variations at x = 0.4 m for the TE z polarization, showing the comparison between the Mi-MoM and split-step parabolic-equation models.
4. Conclusions
Figure 21. The field as a function of height at x = 0.2 m: (top) TE z case, (bottom) TM z case; (solid) mode sum with 298 modes, (dashed) multi-MoM with 50 iterations; a = 1 m, z ′ = 0 , x′ = 0.4 m, k0 a = 200 , dz = dx = 0.0025 m, bw = 45 ,
A simple canonical structure was investigated for computational electromagnetics (CEM) lectures. Wave propagation inside a parallel-plate waveguide with non-penetrable boundaries was chosen for this purpose. Analytical solutions in terms of mode and ray summations, as well as the Image Method, were discussed, and simple MATLAB scripts were developed. Various excitation models were introduced. Fundamental numerical models, such as the Finite-Difference Time-Domain (FDTD) and Parabolic Equation (PE) Methods, and the Method of Moments (MoM), were also included.
θtilt = −20 .
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