On the surface impedance used to model the conductor losses of microstrip structures

I.P. Theron J.H. Cloete

Indexing terms: Numerical analysis, Microstrip structures, Equivalent surface impedance, Tangentialfield, Integral equation

Abstract: In the numerical analysis of microstrip structures their finite conductivity is often mod- elled by using an equivalent surface impedance to relate the nonzero tangential electric field on the conductor surface to an equivalent electric surface current. The fields on top and below the patch are not necessarily the same and to account for this discontinuous tangential electric field a strict equivalence formulation should include magnetic surface currents. However, it is common practice to neglect them, presumably because they con- siderably complicate the surface integral equation and its numerical solution. Here it is shown that this practice leads to the use of an incorrect, although conservative, surface impedance. A basic assumption is that 6 4 A << I, with 6 the skin depth, A the conductor thickness, and I the free space wavelength.

List of symbols

E, = excitation field Et, = tangential component of the excitation field J, = equivalent electric surface current M, = equivalent magnetic surface current Erj = scattered tangential field due to J, E,, = scattered tangential field due to M , E, = total tangential field on the patch (E, = E,, + EIj

E: = total tangential field on the patch neglecting the

E , = y-component of the field due to J, E,, = y-component of the field due to M , E , = x-component of the field due to J, E,, = x-component of the field due to M , Z , = surface impedance of each side of the finite thick-

ness physical patch Z: =effective surface impedance of the zero thickness

equivalent patch, calculated using the super- position of the electric equivalent currents flowing on the two sides

I = free space wavelength 6 = skin depth [a = J(l/nfpa)] A = thickness of the patch

+ Em)

scattered field due to M, (E; = E,, + Etj)

~

0 IEE, 1995 Paper 16528 (Ell) , received 13th January 1994 The authors are with the Department of Electrical and Electronic Engineering, University of Stellenbosch, Stellenbosch 7600, South Africa

IEE Proc.-Microw. Antennas Propag., Vol. 142, No. I , February 1995

As can be seen above, vectors are written in bold font with tangential vectors denoted by the subscript 't'. A superscript '+' will be used to indicate quantities on top of the patch, while a '-' superscript will ,denote the bottom of the patch. The time convention is e'"''.

1 Introduction

Although the work described in this paper was stimu- lated by the literature on microstrip antennas the term 'patch' will be used generically to describe arbitrary planar microstrip conducting surfaces.

Various rigorous integral equation formulations for the analysis of microstrip patches are available [l, 21. To make such analyses tractable the following assumptions are common: (i) perfectly flat, smooth and parallel boundary surfaces; (ii) ground plane and substrates of infinite transverse extent; (iii) perfectly conducting ground plane; (iv) electrically thin substrates such that only the dominant surface wave mode is not cut off; (v) homogeneous, isotropic dielectrics with low loss; (vi) zero thickness patches; and (vii) electrically perfect conducting patches, thus E, 0.

In a departure from the last assumption, Mosig et al. [l], van Deventer et al. [3], Splitt [4] and others account for the finite conductivity of the patches by modelling them as an impedance surface which is assumed to satisfy the boundary condition

E, = 9, J, (1)

with the surface impedance, T,, as yet undefined. This relationship, known as the Leontovich boundary condi- tion [SI, contains certain implicit assumptions which are the main subject of the paper.

A rigorous equivalence formulation requires electric and magnetic surface currents, J, and M , , to be impressed on the surface area of the volume occupied by the patches. These magnetic currents are commonly neglected in microstrip antenna analysis [4, 61, presum- ably because they complicate the integral equation for- mulation without contributing significantly to accuracy. While intuitively plausible, quantitative confirmation that the equivalent magnetic currents are indeed negligible has not been found in the literature. It is easy to show that lM,I/l J,I x 1/08 6 1 for good ohmic conductors

The authors would like to thank one of the refer- ees for stimulating us to interpret our results in terms of Poynting's theorem.

35

..,

like copper, at microwave frequencies, but it is not neces- sarily obvious that the $fields radiated by M,, even if much smaller than the fields of J,, are negligible when imposing the boundary condition on the tangential elec- tric field at the conductor surface. Also, since the surface impedance increases with frequency due to the skin effect, the importance of M, will be enhanced at millimetre wavelengths.

In a moment method formulation, employing either the electric field or mixed potential integral equation, only the tangential fields on the surface of the patches are needed for the computation of the current distribution. The total fields on the patch are the sum of the excitation fields, which are the fields excited by the source in the absence of the patches, and the scattered fields, which are radiated by the equivalent sources representing the patches. The purpose of this paper is to study the contri- bution of the magnetic current to the tangential electric component of this field, and to determine the implica- tions on the surface impedance in eqn. 1. The comparison has two facets.

First, the scattered tangential electric field, E,, , due to the magnetic surface current is compared to E,!, which is scattered by the electric surface current. This is done by comparing the fields scattered by a single moment method basis function and showing that the contribution of the magnetic current is very small for copper at 8 GHz.

Secondly, E,, is compared to the total tangential elec- tric field on the patch, E,. This comparison will reveal that, despite its typically small magnitude at microwave frequencies compared to Etj , the electric field scattered by tlpe-magnetic surface current plays a crucial role in ensuring that the total tangential field, which is also very small, satisfies the boundary condition at the surface of the patch. This analysis is then extended to show that the correct value of the effective surface impedance of the equivalent patch is dependent on the values of the magnetic surface currents flowing on either side of the patch. This fact has apparently not been noted in the microstrip literature dealing with the modelling of finite conductivity.

2 The scattered fields

2.1 Equivalence formulation Fig. 1 gives the topology of a patch on a microstrip sub- strate. The skin depth, 6, is assumed to be much smaller

region 0 Po. Eo

J; t '///////// t - d

0 - w Fig. 1

than the thickness of the conductor, A, which in turn is assumed to be much smaller than the free space wave- length, A. The surface currents J,' are defined equal to the total volume currents per unit width flowing below each surface. These are found, as usual, by integration of the

Side view of the structure under consideration

36

volume conduction current density [7, p. 1521. Since 6 4 A the volume currents below each surface are assumed to decay to zero and not mingle.

From Ampkre's law, and assuming negligible displace- ment current density in the conductor - for good ohmic conductors this assumption is valid well beyond the milli- metre wave regime [7, p. 2831 - it follows that

J,' = r i * x Hi (2)

where H' is the exact magnetic field at each surface [7, p. 1551.

Note that eqn. 2 conveys two independent vector equations, with the superscript ' + ' denoting the current and magnetic field distribution on the top surface of the physical patch, i.e. the air side of the interface, and the '-' superscript the same for the bottom of the patch. This convention is used throughout the paper.

The surface impedance, Z,, is then defined by the linear relationship [7, p. 1521,

E: = Z , J,' (3) between the tangential electric field at the conductor surface and the equivalent surface current. This is typic- ally assumed, as in this paper, to be the internal imped- ance of a semi-infinite conducting slab Z , = (1 + j )R, = (1 +j)(l/a6) = (1 +j),/(n&,,/a) [7, p. 1521. However, for conductors which are not many skin depths thick the approach of Van Deventer et al. [3] is preferred. Implicit in eqn. 3 is the assumption that the surface impedance is the same for both sides of the conductor. In practice this may not be accurate due to a possible difference in surface roughness and the unprotected side being in contact with the atmosphere.

Note that the symbols representing the surface imped- ance in eqn. 1 and eqn. 3 are deliberately different, since 2, is undefined while Z , has a well-defined physical meaning.

From the electromagnetic equivalence theorem [S, pp. 106-1101, [9, pp. 34-44, 134-1391, the physical strip in Fig. 1 can be replaced with the medium of region 0, by impressing the correct electric and magnetic surface cur- rents on the top and bottom surfaces of the volume. (It will be assumed that the edge currents are negligible.) The currents can be chosen such that the fields external to the strip will remain unchanged while a null field is maintained within the volume. This requires impressing electric and magnetic surface currents J$ = ri* x H* and M,' = - r i * x E*. Clearly J,' is identical to the equivalent electric surface current for the physical geometry, eqn. 2, and satisfies eqn. 3.

Thus, using eqn. 3, the magnetic surface current can be expressed as

where r i+ = i and r i - = -i. Hence it clear that when the Leontovich boundary condition is invoked to model finite conductivity the impressed electric and magnetic currents on each surface are inextricably related.

The necessity for the magnetic surface currents can also be argued physically as follows. Penetration of the fields into the conductor is a diffusion process with both attenuation constant and phase constant equal to 1/6 [7, p. 1511. Thus although the physical electric volume cur- rents decay exponentially to negligible amplitudes only a few skin depths below the surface, there is a very signifi- cant phase shift of 1 radian, or approximately 57 degrees, per distance of one skin depth. This means that the 'skin' is electromagnetically thick, despite the fact that

M,' = -i* x Z,J,' (4)

I E E Proc.-Microw. Antennas Propag., Vol. 142, No. 1, February I995

6 Q A 4 1, with 6 the skin depth and A the conductor thickness. Hence the electrical volume current flowing below the surface of the conductor cannot be collapsed to an electrical current sheet without introducing a mag- netic current sheet.

The next step in the equivalence formulation is to model the patch as a thin, flat volume of free space with the impressed currents flowing on the upper and lower boundaries. Because the conductor thickness is a small fraction of the free-space wavelength, A 4 1, it is electro- magnetically very thin and the phase difference due to the spatial separation between the two surface current sheets is negligible. Hence they can be superimposed with negli- gible error to yield the final equivalence in which a single electric surface current sheet

J, J: + J,- (5)

M, E M: + M,- = x Z,(J,- - J:) (6)

and a single magnetic surface current sheet

are impressed on the interface between the two dielectrics in the place of the conducting patch. Now the finite thickness patch is very accurately modelled by two co- located surface current sheets.

Eqn. 6 reveals that the equivalent magnetic surface current is zero only if the electric surface currents on the top and bottom surfaces are everywhere identical. From physical considerations this is clearly not true in general; consider for example a practical patch in the transmit mode where the fields on the bottom surface are usually much greater than on top [lo].

The rest of the paper considers the consequences of a non-zero magnetic surface current in the equivalence for- mulation.

2.2 Comparison of the scattered fields due to J, and

To compare the electric fields, Etj and E,,, scattered by J, and M, the contributions of a single basis function centred at the origin of the coordinate system are calcu- lated, since any current distribution can be found as a superposition of such currents. To compare the fields at the origin, the basis function must be smooth. If it is not, its divergence, and thus also the surface charge, will have a discontinuity causing a locally infinite field, which cannot physically be the case. Also, for ease of analysis, the basis function must have a relatively simple Fourier transform. To cover arbitrary patch shapes, a subdomain basis function is preferred. The form of the basis function is chosen as discussed in the Appendix.

The first step towards finding the scattered fields is the calculation of the Green’s functions. For a single layer substrate above a perfectly conducting ground plane, assuming a y-directed electric dipole and an x-directed magnetic dipole, imposing the boundary conditions in the spectral domain [l, 2, 113 results in

Ms

jk ,k ,Z,{k, , cos k,,d + jk , , sin k,,d} I~ m m

i: . = XJ

KO lee 1,

jZ , {k: T, - k;(k,, cos kZld + j k , , sin k, ,d)}

j(&, - l )kx k, cos k,,d

G y j = -

G,, = -

G y m =

ko Z e Tm

Ze Tm j { (&, - l)kZ cos k,,d - k,, Tee sin k, ,d)

Ze T m I E E Proc.-Microw. Antennas Propag., Vol. 142, No. I , February 1995

At the source point the y-component of the_electric field of the magnetic dipole is discontinuous and G,, gives the field at z = 0-, i.e. immediately below the air-dielectric interface. The subscripts j or rn designate the electric or magnetic dipole as source, and k , = q / ( p , ~ , ) ; k , =

J(k: - k: - k;); Tee = Te/sin k, ,d; T, = kZl cos k,,d + jk , , sin k,,d; T, = E, k,, cos k,,d + j k , , sin k,,d.

The spatial domain Green’s functions can be found by applying the inverse Fourier transform to each of the above, but since this results in singular fields, the scat- tered fields are calculated for the entire basis function by multiplying with the Fourier transform of the basis func- tion before doing the inverse Fourier transform [2].

J(&r)kO; z o = J(pO/&O) ; kz0 = J ( k i - kZ - k;); kz1 =

2.3 Numerical example The fields scattered by a single basis function are calcu- lated as discussed above. The segment length, w, is chosen as 0.051, which is somewhat smaller than typically used in moment method analysis. Since microstrip patch antennas are especially practical at frequencies ranging from approximately 1 to 20 GHz the frequency selected for detailed analysis was 8 GHz, more or less in the middle of this band. The relative permittivity was E, = 2.5, and the thickness of the dielectric was chosen to be small enough that only the first surface wave mode pro- pagates at 8 GHz, thus d = 1.5 mm or 0.041. The surface wave manifests itself as a zero of T, , which causes a pole singularity in the Green’s function. This pole is found in terms of the variable k, where k i = k: + k;. The doubly infinite integral over -CO < k, < CO, - CO < k, < CO is then transformed to an integral over 0 < k , < CO, 0 < 8 < 2n and the contribution of the pole is included using Cauchy’s theorem. The fields on the surface of the patch are calculated as a function of p in a polar coordin- ate system at a fixed angle, 4 = n/6. This is done since the fields contain some terms which are zero on either the x or the y-axis, and by calculating the values at this angle, all the terms contribute to the fields.

In practice the surface impedance of copper microstrip is significantly higher than the value for a semi-infinite medium of pure copper [12]. Here the effective con- ductivity of the microstrip is conservatively assumed to be 0.2 times that of pure copper; i.e. CJ = 0.2 x 5.8 x lo7 S/m was used in calculating 2, = (1 + j ) ( l /ud) . Because I E,, I/[ Etj I a I M, 1 / 1 J, I and, from eqn. 4, I M, I / [ J, I a J(l/a) it follows that I E,, [/I Etj I oc J(l/a). Hence the numerical results can easily be scaled for dif- ferent conductivities.

Fig. 2 gives the magnitudes of the field components due to a y-directed electric current cell and an x-directed magnetic current cell, at 8 GHz. It was assumed that no currents flow on top of the patch, which is physically rea- listic for large patches; hence from eqn. 5 and eqn. 6 J, = J,- and M, = i x 2, J,- = i x 2, J , . The field due to the magnetic current is calculated for the bottom of the patch. It is then scaled by a factor of 100 for easier com- parison, showing that the magnetic current radiates a contribution which is almost everywhere much less than 1% of the corresponding component radiated by the elec- tric current. (Due to the scale factor, the ratio would be exactly 1 % if the lines touch.)

For the x-components, the field of the magnetic current relative to that of the electric current is largest in the region 0.2-0.31, attaining a relative magnitude of slightly less than 1 Yo. Here the absolute value of the mag- netic current field is, however, much lower than the maximum value of the electric current field. Away from

3 1

the edges of a practically sized patch the currents pre- sumably change gradually with distance. Thus the rela- tively low field of the electric current in the region

\+ '

10-31 a 3 ~~~~~~1 8 I ~ ~ ~ ~ ~ ~ 1 a -#ad 10-2 10-1 100 IO'

P I A

Fig. 2 Comparison of the electricfields radiated by the basis function with J, y-directed and M, x-directed. The magnetically radiatedfields are scaled up by a factor of 100

~ I & I Erj I

WE;i,I 1wIELI . . . . . . . ~

d = 1.5 mm, /= 8 GHz, E, = 2.5, M,, = - Z , J , y . Z, = J ( S ) times the value for a semi-infinite medium of pure Cu, such that Z, = (1 + Jl(l/ah) = (1 + j )R, where R, = 0.0522 fl

0.2-0.31 from the centre of the cell will tend to be domi- nated by the peak fields of nearby basis functions. The magnetic current thus make their biggest contribution at their peak in Fig. 2 and not where the ratio between the two fields is a maximum. Note that E, goes to zero as p tends to zero, as it should for a y-directed horizontal elec- tric dipole (HED).

Theament tangential to the edge of the patch can be quite large at the edges [13, pp. 242-2511. The current then decays rapidly away from the edge, seemingly ren- dering the argument of the previous paragraph invalid at the edges. This decay is, however, expected to be very rapid over the width of a normal basis function and thus the edge should not contribute much to the vahe of the current on the basis function nearest to the edge. This phenomenon is evident in the graphical results presented by Mosig et al. [l, p. 4421, where the computed current associated with each subdomain basis function changes very slowly in the direction orthogonal to the edge of the patch. Thus the effect of the edge singularities on the rela- tive strength of the electric and magnetic current distribu- tions is expected to be small in a moment-method formulation.

For the case of the y-components, the field radiated by the magnetic current is always less than 1% of the field radiated by the electric current. In this case the fields have finite values as p + 0 which is also the maximum of the fields. The ratio of I E,, I to I E , I at this point is larger, by a factor of approximately 4 (or 12 dB), than that of I E,, 1 to 1 E, I at its maximum. Thus I E,,JEYj 1 provides a good worst-case estimate of the order of the relative contribution of the magnetic current in a moment method formulation. In the case studied here this ratio is about 0.015%.

From this analysis it is tempting to conclude that the contribution of the magnetic current to the tangential scattered field may be neglected for good conductors like copper at typical microwave frequencies. However, as discussed in the following section, it is their contribution to the total tangential field, comprising the sum of the scattered and the excitation field, that must be considered in the context of the boundary condition. This is done in the following section where it is shown that the common

use of 2, = 2, in the boundary condition eqn. 1 is incor- rect.

3 The effective surface impedance

3.1 The contribution of the magnetically scattered field to the total tangential field

Suppose the scattered tangential field due to the magnetic current can indeed be ignored. This means that only E; need be considered, where E; is the sum of the tangential components of the excitation field and the scattered field due to the superimposed electric surface currents

Since both E,, and Etj are continuous across the interface, E; will also be continuous. Thus, setting E: = E; in eqn. 3 requires the electric surface currents flowing on the top and bottom of the patch to be identical. This is physically unrealistic for practical patches and the magnetic current is essential to enforce the required discontinuity, which contradicts the above supposition.

The correct electric tangential field, E:, which is dis- continuous, can be written as

with E;, defined by eqn. 7, continuous in the normal direction at surface of the air-dielectric interface and the discontinuity provided by E&. The importance of this discontinuity, albeit very small, in determining the correct value for the effective surface impedance of the patch is shown below.

E; = E,, + Etj (7)

E: E,, + Etj + E& = E; + E& (8)

3.2 The Leontovich boundary condition To model the effects of finite conductivity of practical microstrip the Leontovich boundary condition is com- monly invoked, for example [l, 4,6], as follows

There are three implicit assumptions in this expression. First, that J, is the superposition of the equivalent electri- cal surface currents flowing on the top and bottom sur- faces of the conductor. This assumption is valid and consistent with eqn. 5. The second assumption is that the total tangential electric field above the equivalent patch is identical to the total tangential electric field below the patch; this is implicit in the use of the single symbol E, without f subscripts. In the context of eqn. 8 the assumption of continuity implies that the fields E& scat- tered by the magnetic surface currents are negligible. This approximation can be stated explicitly by restating eqn. 9 as follows in the notation of eqn. 8,

The last assumption, now implicit in eqn. 10 is that the impedance relating E; to J, in eqn. 10 is Z,, the surface impedance of the physical conductor which is defined in eqn. 3. As shown below this assumption is incorrect in general.

The proof starts by replacing Z,, which has the well- defined physical meaning discussed in Section 2.1, in eqn. 10 by defining an efective surface impedance, Z[, . Thus

E, = Z, J, (9)

E: x E; = Z,J, (10)

E; %' Z[, J, (1 1) Next, by studying some special cases, it is shown that in general Zi # Z,, and that the correct effective surface impedance is largely determined by the magnetically scat- tered fields, E L , which were neglected when writing eqn. 10.

I E E Proc.-Microw. Antennas Propag., Vol. 142, No . I , February 1995 38

3.3 The effective surface impedance for some special cases

Consider first the situation where the topology is such that the total tangential electric fields on the top and bottom of the patch are identical. By eqn. 8 this implies that E& = 0, requiring M, = 0. By eqn. 6 equal electric currents must then flow on the two sides, or J,’ = J,-. Thus from eqn. 5 J,’ = J; = J,/2 and using E: = E; (since by assumption E& = 0) with eqn. 3 yields E; = (ZJ2)JS. So 2; = 2 6 2 when the magnetic current is pre- cisely zero and the popular boundary condition of eqn. 9 results in a factor 2 error in the tangential electric field.

In the opposite extreme, which can be argued to be a good approximation for a practical radiating patch, the total tangential field on the topside is negligible com- pared to the field on the bottom. Then J,‘ = 0 and J,- = J, which means that all the current flows on the bottom of the patch. From eqns. 3 and 8, Et- = E; + Et; = 2, J, , and E: = E; + E& = 0. This, with eqn. 11, yields

The validity of using 2, in the boundary condition now depends on the values of Eh, and E& produced by M,. (Note that in this case Eh, - E& = 2, J, .)

Consider the unlikely case where the magnetic current radiates only towards the bottom, i.e. only into the sub- strate; then E& = 0 and eqn. 12 results in 2: = 0. In the other extreme it radiates only into the free space region above the substrate, so E , = 0 and eqn. 12 yields 2: = 2,. If the magnetic current radiates with equal amplitude to both sides, Et; = -E& = Z , JJ2, resulting in Z: = 262 as for the case M , = 0 discussed earlier.

The ratio for the magnetically scattered fields provided by the numerical example for copper at 8 GHz is 1 E& I : I E , 1 z 3 : 2. This result suggests a boundary condition for 2: somewhere between 2, and 2 j 2 for practical patches.

The following development using Poynting’s theorem is intended to clarify the physical meaning of results such as 2: = 2 , / 2 or 2: = 0.

3.4 Interpretation using Po ynting ’s theorem After defining the complex Poynting vector at an arbit- rary point on the patch surface as

S - E , x H: (13)

the scalar complex power density, S, flowing into the path from both sides at an arbitrary point can be found as follows

s = s+ + s- = -i. (E: x H:*) + i. (E; x H;*) (14)

In the following development S is derived in two ways. First, in terms of the fields and currents of the physical patch, and then using the fields and currents which describe the equivalent patch. The two expressions must be equal, and since the physical surface impedance, Z , , appears in one and the effective surface impedance, 2: , in the other, a relationship between 2, and 2: is found.

For the physical patch the tangential electric and mag- netic fields in eqn. 14 are given in terms of the surface currents and the surface impedance by eqn. 2 and eqn. 3. The resulting complex power density is

S = Z,( I J,’ 1’ + I J; 1’) (15)

I E E Proc.-Microw. Antennas Propag., Vol. 142, No . I , February 1995

From this the average power density is S,, = 3 Re [SI = +R,( I J,’ 1’ + I J; l’), where R, = Re [Z,]. This also follows directly from the well-known result for planar conductors [7, p. 1551.

To find S in terms of the equivalence formulation use eqn. 8 and eqn. 11 to write the tangential electric fields as

E: = ZLJ, + E& (16)

and remembering that the equivalent surface current Js is defined as the superposition of the currents J,’ and J,- flowing on the top and bottom surfaces. Then, using eqn. 2, to relate the magnetic field to the surface current, eqn. 14 yields

S = ZI, I J, 1’ + E& . J,‘* + Eh, . J,-* (17) Because 2, occurs in eqn. 15 and 2: in eqn. 17, it is pos- sible to interpret the results of the previous section. In doing so it is important to note that I J, 1’ = (J,’ + Js-) . (JJ + J l ) * = I J,‘ 1’ + I J; 1’ + 2 Re [J,’ . J; *] # I J,’ 1’ + I J,- 1’. Thus the magnitude squared of the equivalent

current defined by eqn. 5 is not equal to the sum of the magnitudes squared of the top and bottom surface cur- rents.

For the case considered in Section 3.3 where it was assumed that the currents flowing on top and beneath the patch are everywhere identical, such that J: = J,. and E& = 0, the following result is obtained from eqns. 15 and 17,

S = 22,l J,’I’ = 42:1J,’ 1’ (18)

Hence 2: = 2 , /2 as before. It is now clear that for this case the use of 2, in the Leontovich boundary condition overestimates the complex power flowing into the con- ductor by a factor 2, or 3 dB, because the magnitude squared of J, is twice the sum of the magnitudessquared of its components, J,‘ and J,-.

Next consider the case where it was assumed that J,’ = 0 and E& = 0; i.e. current only flows on the bottom surface and the magnetic current only radiates into the substrate. It was found in Section 3.3 that 2: = 0. However this does not mean that no power flows into the patch, because from eqn. 17 the complex power density is S = Et; . J;* # 0. This result can be further interpreted by equating it to S = 2,l J,- 1’ from eqn. 15 and deducing that the magnetically scattered electric field tangential to the bottom of the patch is E; = Z , J,-. So here the use of 2, in the Leontovich boundary condition is indeed the correct choice, but serendipitously so because it is used in an analysis which never decomposes J, into its components J,’ and J,-. Note also that the meaning of ZI, = 0, or equivalently E; = 0, is that the excitation field is everywhere precisely cancelled by the electrically scattered field, and that the magnetically scat- tered field accounts for all the power dissipated in the patch.

4 Conclusions

It has been shown that the popular use of Z, in the Leon- tovich boundary condition, eqn. 9 to model finite con- ductivity is inaccurate because the magnetic surface current and its associated field is neglected in the equival- ence model of the patch.

This was done by introducing an effective surface impedance 2: which relates the continuous component of the tangential electric field, E; to the equivalent surface

39

current J, = J,’ + J,- and considering some hypothetical current distributions. Also, a numerical analysis was done for a single moment method basis function representing copper at 8 GHz; this quantified the relative contribution of the magnetically scattered electric field, which accounts for the physically necessary discontinuity in the tangential electric field on either side of the cell.

To conclude we consider the engineering implications of the implicit assumption ZL = Z, .

From eqns. 15 and 17 the effective surface impedance, Z: can be written in terms of the surface impedance of one side of the patch, 2, as

After studying this result, which can be interpreted as giving Z[, for an individual moment method cell, we con- jecture that Re [Zl] < Re [ZJ for every cell in the model of an arbitrary patch. Then the implication is that using eqn. 17 with Z , instead of Z: results in the conductor losses being overestimated. In other words, the assump- tion E: N Z, J,, or equivalently Z: = Z , , in the moment method analysis of microstrip patches is a conservative engineering practice, and thus justifiable.

5

1

2

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12 RICHARDS, W.F., LO, Y.T., and BREWER, J.: ‘A simple experi- mental method for separating loss parameters of a microstrip antenna’, IEEE Trans., 1981, AP-29, pp. 150-151

13 JAMES, J.R., HALL, P.S., and WOOD, C.: ‘Microstrip antenna theory and design’ (Peter Peregrinus, 1981)

6 Appendix: Basis functions

To comply with the requirements discussed in Section 2.2 polynomial basis functions, B, where chosen to be ident- ical for the x- and y-coordinates, yielding separable dependence. (This greatly simplifies calculating the Fourier transform.) It was required that B must be sym- metric and smooth around the origin and that its deriv- ative be zero normal to the edges - yielding a smooth distribution of charge. Doing this over only two segments as with the traditional rooftop basis functions will result in the derivative of the current being zero at all sample points. Thus it was expanded to cover four sections for each coordinate direction - in other word a square sub- domain of 16 segments. The form of the resulting poly- nomial is shown in Fig. 3.

Fig. 3 Theform ofthe basisfunction:f(t) = 4 I t /w l 3 - I t /w 1’ + 1

The basis function was thus chosen as

-2w Q x d 2 w -2w < y Q 2w {

with Fourier transform

(2 sin’k, w - k, w sin 2k, w) 9

B(k,, k,,) = - w6k: k;

x (2 sin2k, w - k, w sin 2k, w)

40 I E E Proc.-Microw. Antennas Propag., Vol. 142, N o . I , February 199.5