Electromagnetic vector potentials in isotropic non-homogeneous materials: mode equivalence and scala risation

N.K. Georgieva and W.S. Weiglhofer

Abstract: Electromagnetic mode equivalence and mode coupling are considered in the context of a tinie-domain vector potential formalism valid in a non-homogeneous isotropic lossy medium that may contain sources. General expressions for equivalent mode transformations are derived. The necessary conditions for the field scalarisation (known also as TE/TM decomposition) are outlined. A scalarisation technique based on mode equivalence is proposed and the analysis of transient problems in terms of two scalar functions (a vector potential pair of given orientation) is discussed. The scalar wave potential model and the theory of mode equivalence are illustrated through numerical examples. It is shown that the numerical approaches based on the scalar wave potential analysis offer better accuracy and reduced computational load in comparison with approaches based on direct field analysis.

1 Introduction

Vector potentials have often been considered an auxiliary mathematical concept that does not necessarily reflect a physical phenomenon. Opinions vary widely from state- ments that ‘the vector potential is a useful mathematical tool, but not a physical entity’ [I], to conclusions based on the Aharonov-Bohm effect [2] that the ‘vector potential A-jield is real‘ as it is consistent with observable phenomena at the quantum level where ‘direct’ field analysis fails [3]. The belief that electromagnetic (EM) potentials can hardly be of any use in the solutions of real engineering problems extends well into computational electromagnetics (with very few exceptions in antenna and scattering computations). The most popular time-domain approaches such as the finite difference time-domain (FDTD) method and the transmission-line method (TLM) are based on the time- domain Maxwell equations.

At the same time, the concept of EM potentials is essential in the theory of scalarisation (known also as TE/ TM decomposition)’ for different classes of linear media. The subject is extensive and the interested reader is referred to a small cross-section of it [MI, where comprehensive discussions and representative references can be found. With very few exceptions [7], the focus has been on homogeneous (albeit complex) media and, to our knowl- edge, the concept has never been developed into a computational approach to the analysis of non-homoge- neous media.

0 IEE, 2003 ZEE Proceedings online no. 20030264 ’ TE stands for transverse electric mode, and TM stands for transverse magnetic mode. doi: 10.1049/ip-map:20030264 Paper first received 2nd May and in revised form 7th November 2002 N.K. Georgieva is with the Department of Electrical and Computer Engineering, McMaster University Hamilton, Ontario L8S 4K1, Canada W.S. Weiglhofer was with the Department of Mathematics, University of Glasgow, Glasgow G12 SQW, UK (Deceased)

The EM field scalarisation in a non-homogeneous medium is related to (i) the equivalence of different modal representations of the EM field; (ii) the coupling of EM modes at material non-homogeneities. These problems, which have attracted little attention, are the subject of t h s work.

As an introduction, we briefly review the well-known problem of the TE/TM field decomposition in a homo- geneous isotropic source-free medium as well as the related terminology. The concept dates back to 1904, when E.T. Whittaker proved that ‘only two functions are actually necessary (in place of four)’, i.e. the vector and the scalar potentials (A, @)2, to describe the EM field associated with any configuration of moving or static charges [SI. A number of modern texts on classical electrodynamics [9, 101 also treat the problem of the field decomposition into TM and TE modes. The TM (with respect to the distinguished direction specified by an arbitrary unit vector C) field is described by the magnetic vector potential, A = Ai?, while the TE field is described by the electric vector potential F = Fi?. Both potentials are solutions of the vector wave equation in the time domain (or the vector Helmholtz equation in the frequency domain). In the following discussion, the scalar functions A and F are referred to as wave potentials (WPs), and the pair of collinear vector potentials is denoted as (A, q e .

The scalarisation of the EM field in a homogeneous isotropic source-free medium is straightforward. The technique reduces the Maxwell equations to two decoupled scalar wave equations for the pair (A,F)i?. The major advantage of scalarisation is that one no longer deals with vectors, and that the number of unknowns is reduced from six (the EM field components) to two (the WPs A and F ) . This can be advantageous in computational electromag- netics, where the number of unknowns determines the computational requirements of the algorithm. However, the field scalarisation in any medium more complex than homogenous isotropic and source-free is a challenge. The

*Throughout the paper, vectors are in bold, and unit vectors are denoted by 1.

IEE Proc -Microw. Antrnnus Propuy., Vol. 150, No. 3, June 2003 164

main reason is that the orientation of the EM field vectors is determined not only by the orientation of the impressed (or explicit) sources but also by the medium of propagation. The interaction of the EM field with the medium results in dielectric and/or magnetic polarisation, thus introducing induced (or implicit) sources, which in turn affect the field. The scalarisation of the field is thus related to the scalarisation of its sources, which can be carried out using similar techniques.

The theory and applications of scalar wave potentials are best understood in the light of the following two observations. First, vector potentials of different orientation can represent the same field. The problem of finding the equivalence relation between vector potentials of different orientation is referred to as mode equivalence. Second, sources of different orientation can generate the same field. This leads to the problem of source equivalence. Here, we focus on the EM mode equivalence. The theory of EM source equivalence deserves special attention and is considered elsewhere [ 1 11.

The wave potential analysis described here (i) presents a novel vector potential formalism for the EM field propaga- tion in a non-homogeneous medium; (ii) shows how mode coupling occurs in a non-homogeneous medium; (iii) gives the necessary conditions for the scalarisation of an EM problem; (iv) describes the equivalence of EM modes and its application to EM field analysis in non-homogeneous media. The basic concepts of EM mode equivalence are illustrated by numerical examples analysed with the time- domain wave-potential (TDWP) algorithm [ 121. Compar- isons with analytical results or results generated by alternative fullwave simulators [13, 141 are given.

2 Potentials and fields

Consider the vector potential functions A,, and F,: due to general EM sources defined in terms of the electric current density J, and the magnetic current density J,n. In a non- homogeneous medium, they satisfy the wave equations [12]

(1) V2Ap - S p A p + (VS,:) x F, + (OS,)@ = -Jp

V2Fc - XiIl:F,: - (V2,) x A, + (V2,)Y = -J,

where the scalar potentials @ and Y relate to A,, and F, via the generalised Lorenz gauge

In (1) and (2), the differential operators in time are3 -&;@ = U. Ai, - Xi,'€' = V . FE (2)

(3) 2, = €8, + CJe 2, = pat + CJn1

2p = 2,2, = pdtt + (cum + ~ 0 e ) a t + Geum

The properties of the medium, described in terms of the dielectric permittivity r , the magnetic permeability p , the electric conductivity ue, and the magnetic conductivity gn;,

are functions of the position in space x = (x,y,z). It is assumed that the medium has instantaneous response, thus its constitutive parameters do not depend on time. Such a non-physical assumption [16] is often made in computational time-domain algorithms when the medium is nearly dispersion-free in the frequency band of interest. The time-domain operators 2, and X,,, introduced for

Partial derivatives are denoted with a a. The order of the derivative and the variables it relates to become obvious from the subscript. 4The inagnetic conductivity is a property of fictitious matter where the instantaneous magnetic power loss density is proportional to the square of the instantaneous magnetic field, p,,,(r) = u,,P(t). H(f). It is used in computational electrodynamics to construct fictitious perfectly matched absorbers [lS].

IEE Pr~c.-Mic.~.o~t: Antennus Propuq., Vol. 150, No. 3, June 2003

convenience, also allow the direct transfer of the transient field analysis presented below into the frequency domain by replacing 2, with joE and 2,, with jwp, where E is the complex dielectric permittivity and is the complex magnetic permeability.

The vector operators (V2,) and (72,) are the gradients of the operators defined in (3):

(4) (VZ) = (V&)at + (Voe) (02,) = ( 0 P ) a t + ( V c m )

so that, for example,

( 5 ) (VX,)@ = (Vs)a,@ + (Voe)@ (V2,l) x F, = (VE) x + (Vue) x Fe

These vector operators reflect the influence of the material non-homogeneities. Notice that, according to (l), the vector potentials A,, and F, are, in general, coupled in a non- homogeneous medium. In a homogeneous medium, where (V2,) = (V2,) = 0, (1) simplifies to two decoupled wave equations:

(6) (V2 - Z,,)W = -G, W A,,, F,, G J,, J,

The equations (1) can be re-expressed in the form D x (V x A, - %,FE - V Y )

+ 2,(V x F, + %,A, + V@) = J, V x (V x F, + X,Ai, + V@)

- 2,(V x A, - 2,F, - Vul) = J, (7)

(8)

Equations (7) allow the introduction of the field vectors E = -2,Ap - U@ - V x F, H = -2,Fc - V Y + V x A,

whch fulfil

(9) %,E = V x H - J, 2,,H = -V x E - J,

Notice that (9) is the system of Maxwell equations for dispersion-free media whose constitutive relations involve time-independent constitutive parameters

D(x, t ) = E(x)E(x, t )

(10) B(x, t ) = d x ) H ( x , t> Jue(x, t ) = ce(x)E(x, t )

J U " , ( X , t ) = om(x)H(x, t )

Substitution of (8) into (9) results in another potential- to-field relation

(11) 2,E = V x (V x A, - %,FE) - J, 2,H = V x (V x F, + %,A,,) - J,

equivalent to (8). From (8) and (ll), it follows that the vector potential functions Ap and F, relate to the well- known magnetic vector potential A and electric vector potential F as

We refer to A,, and F, as modified vector potentials. Adopting the established terminology in the theory of TE/ TM field decomposition, an EM mode is one of the six elements of the total field corresponding to one of the six spatial vector potential components: TM,, (A$), TEk (FFk), where k = x, y , z. Occasionally, we will refer to the field given by a pair of collinear vector potentials (A,,, F,J C as a TMc/TEc mode.

A,, = A l p F, = F/E (12)

165

The equations in (1) and (2), together with (S), represent a complete time-domain vector potential model of the EM field in a non-homogeneous isotropic lossy medium. In its general form, it involves both A,, and F,, with their six components. It is not immediately obvious how this model can be advantageous in computations. Its usefulness becomes apparent only after its scalarisation is carried out; i.e. the six components of A,, and F, are equivalently replaced by a pair of collinear vector potentials (Akmn, of .known direction.

Consider a medium whose non-homogeneity is restricted so that the gradients of the constitutive parameters are single-component vectors parallel to the fi axis. This corresponds to a locally plane material interface of unit normal fi. For a problem described by a pair (A,,, FE,$, the terms (V'z,) x F,: and (VZ,,) x A, in (1) vanish. The terms (V2,:) @ and (VZ,,) Y are single-component vectors along 6. Thus, the two vector potentials A,,,,fi and Ft$, which are collinear with the gradients of the constitutive parameters, are not coupled and do not give rise to tan- gential vector potential components; i.e. they are sufficient by themselves.

The orientation of the vector potentials is also important when setting their boundary conditions (BCs) at conducting edges [12]. The BCs for the vector potential components, which are tangential to a perfectly conducting edge5, are well posed in the sense that they do not depend on the direction from which the edge is approached. The BCs for the vector potential components, which are orthogonal to the edge, are ill-posed since the edge is the intersection of two boundary segments with two different BCs.

In summary, an EM problem can be described by a pair of collinear vector potentials (Aw, F,)ii in regions where: (i) all sources are aligned along fi; (ii) the gradient of the constitutive parameters is along 5; and (iii) the perfectly conducting edges are along fi. Such regions will be said to have a distinguished axis 5. In such regions, the TM,, and TE,, modes can be analysed separately. Condition (i) refers to the impressed (explicit) currents. Conditions (ii) and (iii) are related to the induced (implicit) currents: the bound electric/magnetic charge distributions along fi at material non-homogeneities; and the current at &directed edges. If a region does not have a distinguished axis, the TM,, and TE,, modes are coupled through the implicit currents.

3 Scalarisation and mode equivalence

Consider a region where all current sources are aligned with the i? axis6, i.e.

J, = Jenb J, = Jmnfi (13) This assumption will keep the discussion on mode equivalence simple and separate from issues related to EM source equivalence. Assume also that the field in that region is originally described by a set of modified vector potentials, which have transverse (subscript z) and long- itudinal (subscript n) components:

A; = AiT + AP,,,fi F: = F:, + F:,,fi (14)

The objective is to scalarise this problem, i.e. to replace the original vector potentials (14) with equivalent longitudinal

5The term perfect conductor refers to both perfect electric conductor and perfect magnetic conductor. 6The equivalent transformations of arbitrarily oriented currents into currents along a distinguished axis in a non-homogeneous medium are obtained by the theory of source equivalence and scalarisation [ I I].

I66

vector potentials:

A p f i = (AEn + A;,)fi FIxfi E (F:n + F&)h (15) The sources of the original problem in terms of A; and F: are the same as the sources of the equivalent problem in terms of AlInfi and &,fi . Therefore, the (AEn1 Afn)fi pair, which replaces equivalently the original transverse compo- nents, does not have sources of its own.

The equivalence is established through the longitudinal field components E, and HI, (E, = E . 6, HI, = H .;I). The validity of mode equivalence in terms of the longitudinal field components in non-homogeneous media is proven in the Appendix, Section 7. According to, (ll), when J:, = J;, = 0, the longitudinal field components in terms of (A;,, are

%&En = an(V, . A;,) - fi ' [V, x (x,:F:T)]

XpHn = a n ( V x ' FZT) + fi . [Vr x (2,iAY;,)] (16) where V, denotes the transverse component of the V operator, V = 0, +a,$. In terms of (A;lnl Yn)fi, for which J:n = J& = 0, these field components appear as

ZEtl = -V:AEl, 2 H = -V2Fe

11 n T t:n (17)

The equivalence of (16) and (1 7) leads to a transformation of the original transverse vectors A;, and F:* into the longitudinal pair (A;(n, Cn)k

OSA&,, = -an(VT . A;,) - fi . [VT X (%FtT)]

VSA:, = -an(VT . F;,) - fi . [V, x (2,Af,,)] (18)

Alternatively, one can start from the longitudinal field components extracted from (8). Then the equivalent transformations are obtained as

2 (at in - x,,E)A;ln + ( a n % E ) @ ; = Z ( - a n @ ; - h . (VT + F" ET )]

(am - z;,)~,~ + (anzc) yz = X,(-an y: + f i . (v, + AO 1lT 11 (19)

where (a,%,) and (a,'z,,) are the longitudinal components of the gradient operators in (4). In (19), the scalar potentials are related to their respective vector potentials through the generalised Lorenz gauge, e.g.

-TI:@: = V T . Ao i t 7 - %E@: 1 anAYln (20)

The transformations (18) and (19) show that the equivalent longitudinal pair (A;lnl F z ) f i relates to the divergence and the curl of the original transverse components A;T and F:T. Notice that they are valid in generally non-homogeneous media as no assumptions were made with regard to the operators (VS,) ,and (VX,,)

From (1 8) and (19), the governing equations for the WPs (Alln, are obtained

(21 1 (03 + xi:anz,lan - x p ) ~ p n + ( a n z E ) Q n = -Jen

(03 + x p a n 2 , l a n - 2p)Fn + ( a n 2 , u ) y n = -Jnin

where 2;' and Z;' are the inverse of the 2, and 2,, operators, respectively. The WP pair (AwU2, F,:,,), defined in (1 5) and governed by (2 l), equivalently replaces the original six-component vector potentials in (1) if all explicit and implicit current densities are scalarised to produce the longitudinal source pair defined in (1 3). Its equivalent part (A$, Cn) is derived from the transverse original vectors A;, and q, through (1 8) or (1 9).

IEE Proc-Microw. Antennus Propuy., Vol. 150, No. 3, June 2003

To clarify the nature of the transverse implicit current densities induced at non-homogeneities, expressions derived from the vector equations (I) are given below

VT X 2t;FEn G-%V, X F,, G-VT dJ )In + 2 ; ; Q T 2, Idn A pn =- JLr - VT x 2 , j A p n f i + 2 p V T x A,,,; - V,dnA;;,

. + ZV,z,lanF,ii = -JIr (22) The implicit current densities at conducting edges transver- sal to i3 are obtained by setting the respective electric field components in (1 I ) equal to zero.

Thus, the TE,I/TM,I and TET2/TMT2 modes are equivalently transformed into a TE,/TM, mode. Once a solution in terms of the WP pair (AL,,?, Fml) is obtained, the field vectors can be computed by the potential-to-field relations (1 I), now reduced to

%,E, = -V;A,,, - Jen

2pHn = -V:F;:n - ~ n i i i

= VT(dnAp , l ) - VT (hzE%) 2,1HT = VT(dt16>I) + VT (GzpA/U7) (23)

The transformations (1 8) and (19) are theoretically equivalent. From a computational point of view, however, they are quite different. The 2-D Poisson equations in (18) require an implicit technique, which involves a matrix inversion. The matrix inversion can be carried out as a pre- process to the transient analysis as it is independent of time. It depends on the size of the plane orthogonal to 6, where the transformation takes place, and on the boundary conditions at its contour. The result is planar distributions of the equivalent wave potentials. On the other hand, the I-D wave equations in (19) allow a simple explicit timestepping algorithm. They require boundary conditions at both ends of the axis parallel to ii, and lead to axial distributions of the equivalent wave potentials.

4 Applications and discussion

4. I Mode equivalence and vector potential pair transitions As a special case of (18) and (19), a mode of a given orientation (AlJcl , ) I ? , can be transformed equivalently into an orthogonal mode (AjlcZ, f i : c 2 ) e 2 , where t~ l&. This provides a way to change the direction of the vector potential pair in space so that it aligns with the perfectly conducting edges and the gradients of the material constants where implicit sources are induced. This elim- inates the need for equivalent source transformations in relatively simple problems involving few nonhomogeneities. A simple practical implementation of this special case of mode equivalence is used in the time-domain wave-potential (TDWP) technique [12], which we use to illustrate the equivalence of EM modes.

The TDWP technique divides the volume analysed into domains of suitable orientation of the vector potential pair. Seamless transition between neighbouring domains is achieved at the domain mutual boundaries by the solution either of the 2-D Poisson equations (18) or of the 1-D wave equations (19). Notice that in this case the right-hand sides of (18) and (19) involve only two scalar functions, which are the WPs of the orthogonal pair supported by the neighbouring domain. Once the boundary values of the WP pair are found via (18) and/or (19), the wave equations (21) are solved inside the domain with an explicit update [12].

The mode equivalence concept is illustrated with a simple waveguide example. The cross-section of the rectangular

waveguide is 3 cm by 1.5 cm. The structure is excited with the TEol mode distribution of the E, component across one of the ports. Its waveform in time is a sine wave modulated by a Bldckman-Harris window [22]. The spectrum of the excitation waveform is shown in Fig. 1. The rectangular grid is uniform with a discretisation step Ah = 1.25 mm. The computational volume has the following size: 12Ah x 24Ah x 300Ah along the x, y and z axes, respectively (see the inset in Fig. 2). The numerical constant q = cAt/Ah is set at q = lj2. Here, c = ( , U ~ E O ) - ” ~ and At is the discretisation step in time. The computational volume is divided into two domains. The input domain supports the (AL,-, FFz) pair. The output domain supports: (i) the (AJlz, Fez) pair in the first simulation, (ii) the ( A , , , Fry) pair in the second simulation, and (iii) the (AILy, &) pair in the third simulation (see Fig. 2). The transition between the two domains uses the mode equivalence formulas in (18) and (19), with d,2, = 0 and d,2, = 0. In this case, the mutual boundary is a rectangle at z = 15OAh. Its contour follows the perfectly conducting walls of the waveguide, where the WPs satisfy either Neumann or Dirichlet BCs.

1 .o

0.8

0.1 o.2 t 0

3 4 5 6 7 8 9 10 11 12 frequency, GHz

Fig. 1 rectangular waveguide

Normalised spectrum of the excitation waveform for the

1 30 2400

1600 1 bJ

3 cm

27

24

21

18 b

0 15 %

12 3

5 6 7 8 9 1 0-

Impedance and wavelength comnputations ,for a waveguide

frequency, GHz

Fig. 2 using an output domain with thfee dqjerent vector potential pairs analytical results (- ); TDWP (-, G-I+E-)

The boundary values of the pairs (ALlx, Fa)k and ( A L l j , cy)$, which are (13) tangential to the domain boundary, are computed through (19) with an explicit time-stepping update. The boundary values of the (Apz, &,z)i pair, which is normal to the boundary, are computed via the 2-D Poisson equations (18). The solution is based on the

167 IEE Proc.-Microw. Aiitrnnns Propug., Vol. 150, No. 3, June 2003

finite difference discretisation of the 2-D Laplace operator using the rectangular grid of the TDWP algorithm. Inside the domains, the WE% are computed via the wave equations (21).

A seamless transition from the input domain to the output domain is achieved and there are no spurious numerical reflections from the domain's boundary in all three simulations. This is well seen from the waveforms of the E, field component recorded during the three simula- tions at an observation point x0=(12Ah, 6Ah, 200Ah) belonging to the output domain (see Fig. 3). The waveforms show a clear incident pulse with no reflections. They are indistinguishable from each other, thus confirming the equivalence of the field representation in terms of the three mutually orthogonal WP pairs. Notice that the dominant TEol mode, which exists in this frequency range, is fully described by only one WP: either F,,, or E " , or A,lw, due to its independence from the x-coordinate.

20

10 151 I! I

-20 1 0 500 1000 1500 2000 2500 3000

time-step

Fig. 3 Time-suniples of Eh(t) recorded ut % I = (12Ah, 6Ah, 200Ah) in three dcferent simulations when the output domain supports the potentiuls F,,, A,,,, und FE],

To verify the accuracy of the transient potential analysis, the guided wavelength A, is calculated from the shft between the phase spectra of the pulse waveforms recorded at two different locations along the direction of propagation z. The wave impedance Z,,, is also calculated as the ratio of the amplitude spectra of the E, and the HJ, waveforms recorded at the observation point x,. The results from the three TDWP simulations are identical. They are plotted in Fig. 2 together with the analytically computed and Z,,.. The agreement is excellent.

4.2 The wave potentials at wedges and the accuracy of the solution The computational advantages of the potential analysis implemented in the TDWP algorithm are rooted in the reduction of the number of unknowns computed at each point of space-time in comparison with the FDTD or the TLM algorithms. The CPU time and memory requirements are approximately 66% of those associated with the above mentioned methods [12].

Here, we focus on another problem, which has not been addressed so far; namely, the accuracy of a computational approach based on the wave potential formalism. It is well known that the singularity of the EM field components transversal to perfectly conducting edges [ 171 is not correctly represented on finite grids. This causes deterioration of the accuracy of the computational approaches based on finite- difference or finite-element discretization schemes (see, for example, [IS] and [19]). Numerous approaches have been

proposed to deal with this problem ranging from quasi- static modifications in the edge vicinity [20] to hybrid FDTD/wavelet techniques [21].

In the vicinity of a wedge (see Fig. 4), the singular transversal field components depend on the distance r from the edge as - r"- ' , where 0.5<v<1 [17]. The parameter v depends on the wedge angle a so that v = O S when a=O (half-infinite conducting plane), and v = 1 when a = 180" (electric wall). On the other hand, neither the longitudinal nor the transversal (to the wedge) vector potentials are singular. Consider the following cases.

Fig. 4 wedge and the associated cylindrical coordinate system

The geometry of the z-directed perfectly conducting metallic

4.2. I The pair (Apn/ Fen) is along the edge i.e. h = z: The transversal E, component relates to the potentials as shown in the thn-d equation of (23). In cylindrical coordinates

%,E, =f[d,(d,A,,) - r-I%,d,F,] (24) + @[.-'a,(dzA,) + za,&]

Assuming that A,, and KZ depend on r as

A,, - rq F,, - rq + C,. (25) and substituting (25) into (24) leads to the conclusion that y = v. Here, C,. is any constant independent of Y. This confirms that A,, and &, depend on the distance from the wedge as - r" and - rv + C,., respectively, where 0.5 5 v 5 1. The A , potential goes smoothly to zero as the observation point approaches the wedge. This ensures a vanishing E=, component (E , N r") . The F, potential approaches a finite value, C,, see (25). This ensures proper behaviour of, H,, Hz - r" + C, [17].

4.2.2 The pair (Apn/ FE,,) is transverse to the edge, e.g. h = ?: Expressing the E, field component according to (23) as

%,.E, = dz,.A,, + r-'a,Z,F,, (26) and keeping in mind that E, - r", it is concluded that A,,. and F,, are not singular and that their dependence on r in the vicinity of the edge is - r"+' .

Due to the smooth spatial distribution of the WPs at edges, the scalar wave potential analysis on finite discrete grids has better accuracy than the direct field analysis based on the Maxwell equations when the grid parameters in space-time are identical. This is demonstrated through the analysis of an H-plane waveguide filter based on the design described in [23]. The geometry of the filter is given in Fig. 5. A bandlimited excitation in the frequency band from

168 IEE Proc.-Microw. Antennus Propug.. Vol. 150, No. 3, June 2003

, U 17 4244 , B 157988 S 062230 Wi 130683 Wz I I 8237 W3 I I 2014 W, 11 2014 L I 161798 L z I6 I798 L ? 168021 all dimensions in mm

Fig. 5 Geometry und ditnensions of the H-plune waveguide$lter

5 GHz to 10 GHz is used in order to cover the passband of the filter, which is from 5.2GHz to 9SGHz. The S- parameters of the dominant TEol mode are extracted from the transient waveforms of the E, component distribution recorded in the input and output waveguide sections using Fourier transform in time and along the y coordinate.

The results for the reflection coefficient and the insertion loss obtained through the scalar wave potential analysis are plotted in Fig. 6, together with a reference solution generated by the FEM-based frequency-domain commer- cial EM simulator Agilent HFSS [13]. The reference solution was obtained after the mesh refinement was set to achieve convergence error below 0.5%. For comparison, the reflection coefficient and the insertion loss obtained with the FDTD algorithm are also plotted. An in-house FDTD simulator is used based on the software provided with [14]. The TDWP and the FDTD solutions were obtained on identical grids. The grid is uniform with a discretisation step Ah = 0.6223 mm, and q = cAt/Ah = lj2. The computational domain has the following size: 26Ah x S6Ah x 700Ah. From Fig. 6, it is obvious that the accuracy of the potential analysis is significantly better than that of the FDTD analysis.

1 .o 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

frequency, GHz

Fig. 6 Reflection coefficient and insertion loss of the H-plane wuveguide $Iter: comparison with FEM-based frequency domain simulation (13 J and with the FDTD siniulution

4.3 The application of implicit source transformations in scalar wave analysis The scalar wave potential technique is not limited to problems with a single distinguished axis ii. The implicit sources induced at conducting edges transversal to i? and at material non-homogeneities whose gradient is transversal to fi, can be equivalently transformed into &oriented sources using the source scalarisation technique proposed in [ 1 11. This technique allows the use of a vector potential pair of

fixed direction throughout the structure analysed regardless of the present non-homogeneities.

Here, we show the results from the analysis of a waveguide iris. The geometry and dimensions of the structure are given in the inset of Fig. 7. The iris is infinitesimally thin. The F, wave potential is excited using a bandlimited waveform so that a single TEOl mode is launched at one of the ports. The rectangular iris has edges along the x- and y-axes, which are both orthogonal to the direction of the excited potential. The implicit sources induced at the edges are transformed to produce single- component electric and magnetic equivalent current densities oriented along 2 . Thus, the problem requires the simultaneous solution of the wave equations of the (‘4p2, &)i pair.

I I Srll (Agilent HFSS) I - - l S ~ l l (Agilent HFSS)I __.

0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5

frequency (GHz)

Fig. 7 iris: compurison with the results of Ayilent HFSS [13/

ReJtection coejjcient and insertion loss of the rectangular

The reflection coefficient and the insertion loss of the iris are plotted in Fig. 7 together with the results obtained with Agilent HFSS [13]. Very good agreement is observed.

5 Conclusions

Equivalent EM mode transformations are derived from the vector potential formalism in a non-homogeneous isotropic lossy medium. The scalarisation with respect to a distin- guished axis is carried out. Practical applications of the theory of mode equivalence and field scalarisation in a computational algorithm are outlined. It is shown that (i) fullwave analysis of a non-homogeneous medium is possible in terms of two scalar wave potentials, and (ii) the scalar wave potential technique offers better accuracy and reduced computational requirements in comparison with techniques based on direct field solutions. The advantages of the potential analysis over the direct field analysis are illustrated with time-domain computational methods. Similar results can be expected in frequency-domain analysis based on scalar wave potentials since the improved accuracy is due to their smoother distribution at sharp discontinuities.

6 References

1

2

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7 Appendix

Assume that two sets of solutions exist in a given region, (E,, HI) and (E2, H2), such that they have the same sources, the same BCs and the same initial conditions (ICs). Both sets satisfy the Maxwell equations. An additional require- ment is imposed: both fields have their components along the distinguished axis ii identical at every point of the analysed volume:

Eln 1 E 2 n , H 1 n 1 H2n (27)

Ed 1 E1 ~ E>; Hd H1 - HZ (28)

Then, the difference field

is a source-free field with homogeneous BCs/ICs. It is described by

Edil = 0; Hdf, = 0 (29)

where

(33) (a f l z f i ) = [ ( a f l E ) a / + a,,oelfi (af7qlq = [ ( a , P ) a + anOnz]f i

The system of 1-D wave equations (32) provides the proof for the equivalence of the (E,, HI) field and the (E2, H2) field. It has only a trivial solution when complemented by homogeneous BCs/ICs. Thus, if the two solutions (El, HI) and (Ez, H2) have identical longitudinal components, then all their components are identical provided that: (i) they satisfy the Maxwell equations; (ii) they have the same BCs, ICs and sources.

170 IEE Proc.-Microu: Antenrici.s Propug., Vol. 150, N o 3, Jnne 2003