Modal solution of generalised planar transmission lines

E.E. Hassan

Abstract: The problem of a generalised planar transmission line is considered, and a detailed theory of a complete modal solution of the problem is presented. The technique is based on employing the reciprocity theorem in conjunction with the Raleigh-Ritz optimisation procedure. A full solution of both the propagating modes and the field distribution everywhere along the structure is obtained. The technique is then applied to a three-layer structure with conducting strips along each interface. Several configurations are treated and the results presented compare favorably with other results available in the literature.

1 Introduction

Generalised planar microstrip shielded lines, which have various shapes and designs such as single-strip dual- dielectric planes, dual-strip dual-dielectric planes or different broadside coupled structures, have been the focus of numerous work over the last few decades due to their various applications in the design of MIC and MMIC circuit devices. The work in tlus area is primarily concentrated around the TEM-like modes, employing mainly transmission line (TL)-like analysis [ 1,2], or different numerical techniques such as the finite element method FEM [2], finite difference time domain FDTD [3], spectral domain method SDM [4], or quasi-TEM analysis [5]. Various other enhanced techniques have been reported and may be found in the references of the above publications. To the best of the author's knowledge, no rigorous complete modal analysis of this problem has been attempted. Such a solution is necessary for a thorough understanding of the field distribution, different lugher-order modes and impe- dance distribution across any specific structure under the generalised category of a planar transmission line. In ths paper, a generalised technique developed partially more than a decade ago [6] to solve the problem of a shielded microstrip transmission line, is utilised to obtain a complete field solution and propagation modes of a generalised shielded planar transmission line, with multidielectric layers and multiconducting strips placed at different interfaces at arbitrary spacing and levels throughout the structure. The technique is based on utilising a variational expression for the travelling wave fields developed by Ramsey [7], in his treatment of travelling wave antennas. This expression can be applied to the present problem and has been used previously by the author [6].

In the following, the general theory is developed first, followed by application to specific cases, with some of the results presented and compared to available results in the literature.

0 IEE. 2003 IEE Priiceedings online no. 20030470 doi: IO. 1049/ip-niap:20030470 Paper first received 16th December 2001 and in revised form 14th October 2002 The author is with the Electrical Engineering Department, King- Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

2 Theory

Consider the configuration shown in Fig. 1, which repre- sents a generalised transmission line composed of a shielded rectangular guide containing A4 non-conducting layers of different arbitrary thickness and dielectric constants. Let the dielectric constant of the ith layer be E,. Furthermore, assume that a number of vanishingly thin conducting strips, each of arbitrary width, are placed at each dielectric interface at arbitrary separation distances. Let L be the guide width, d be its height and d, be the distance from the base of the structure to the layer i - (i + 1) interface. We start by assuming an electric and magnetic scalar potential q ~ ( I ) ( x , y ) and 4 ( ' ) ( x , y ) in each layer i of the form:

for all i = 2 to ( M - 1). The term exp(-jpz) is understood and is omitted, with p being the unknown propagation

'I I I I

L

Fig. 1 Structure of generalised transmission line

d

- X

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constant. In addition, yn = nn/L with n running from 0 to

CO,

For i = 1, and because of the boundary conditions at the metal base of the structure, Bill and D!)both vanish. Similarly for i = M , both BiM)and D(E"')also vanish. In addition, for the layer i = M , the variable y must be replaced by (d-y) in (1).

Using the relations [SI:

= d m , a n d k; = w2pOe;.

all tangential field components may be obtained every- where, and these will take the form:

+ B t ) cosh cc$)y) sin ynx

M

+B!)y, cosh @ y ) cos ynx

+D$)at) cosh a;)y) cos ynx (26)

For the layer i = M , (d - y ) replaces y and the proper sign for the derivative d / d y is incorporated. Equation (2) contains four sets of unknowns, A!) , B!), C!' and D!), for each layer except layer 1 and layer M where, as stated earlier, B?), Di') and BiM', DiM) are identically zero.

Now, define

f$)-(z+l)(n) = L L q - ( l + 1 ) cos ynx& (3a)

zjl)-(l+l) (U) = E:)-('+') sin ynx & (36)

where the fields under the integral signs are those at the interface i - ( i + 1). Multiplying both sides of (2u) by sin y,x and (26) by cos ymx and integrating along the

132

i - (i + 1) interface between the limits 0 and L, the summation signs drop because of the orthogonality and one gets

Zii)-('+')(n) = (A!)?, sinh ~ ! ) d ; + B!)y, cosh a t )d ; )L /2

+ ?!!! (uf)C:) sinh .;Idi + M!:)D!;) cosh $)d;)L/2 (4a) B

+ B!:) cosh a;)di)L/2 (4b)

Notice that we used the field representation at layer i evaluated at y = d; to represent the tangential electric field at the interface i - (i + 1) . Notice also that the term Li2 should be replaced by L for the case n = 0. Now, repeat the above using the same field expression of region i evaluated at interface (i - 1 ) - i to obtain another two equations similar to (4u) and (4b), where d; is replaced by d;-l in the right-hand side of the equations, while the left-hand side

respectively. We then have four equations in four unknowns A t ) , B t ) , Ct) and D!:), each of which may be obtained as a linear function ofz:+('+') ( n ) , zj+(;+l) ( n ) , I:'-')-(') ( n ) and fii-')-(') ( n ) , using simple matrix algebra. Accordingly, one gets an equation of the form

should now take the form Z:;-')-(')(n) and fii-')-(;) ( n ) ,

A!) =fA(:) ( P , d,, d( i - l ) , n)z;;)-(;+')(n)

+ f t i ( p , d;, d(;-,), n)f;i)-(;+l)(n)

+ f ~ ~ ( p , d ; , d ( ; - l ) , H ) f ~ i - l ) - ( ; ) ( n )

+ f~~(P,d; ,d( ; - , ) ,n)zJ;)-( ;+ ' ) ( n ) + ~ ~ ( p , d ; , d ( ; - l ) , n ) f , $ l - l ) - ( i ) ( n )

+ f$. i ; ' (P,di4-]) , n)I;i-')-(i) ( n )

+ & ( f i , di, d( ; - ' ) , n)z;+(i+l) ( n )

+ f$i (8, d;, d( '&]) , n)f:l-I)-(i) ( n )

+ f$j(f i,d;, + I ) , n)f;i-')-(;) ( n )

+ fj;(p,d;,d(/-l), n)zp-(;+')(n)

+ & ( P , d;, d(i& 1) , n)z$i- I )-(;) ( n )

+ j ~ ~ ( ~ , d j , d ( i - l ) , n ) ~ ~ l - l ) - ( i ) ( n ) ( 5 )

+ f , ' i )(~, d;,d(;-I) , n)zJi-')-(;)(n)

B$) =& (p , d;, d(i- I ) , n)Z$+(i+l) ( n )

c:) =f!!(f i, d;, d+,), n)IJ+(i+l)(n)

D(; i ) - - fDl ( ~ , d ; , d ( ; - ~ ) , n)I;+('+')(n)

This completes the first part of the solution.

2. I Key solution: application of reciprocity theorem Ramsey has used the reciprocity theorem [7] to show that for a wave travelling in the z-direction, the integration x over a closed boundary s in the x-y plane of the form

x = /ooey-thc { E , ( H ~ -H:) + EZ(H; - H . ) > & ( 6 ) boundaries

IEE Pruc.-Microw. Antennas Propopoy., Vol. 150, No. 3, June 2003

is identically zero. Here, H;, H; and H,b, H: are the magnetic fields at both sides of the boundary, which fit the assumed E, and E,. Ths expression is stationary for the propagation constant p [7]. We will now apply (6) M-l times over the M-1 interfaces between successive layers within the structure. Each path s is made of the i - (i + 1) layer interface and is then completed over the guide conducting surface in the x-y plane. Using (2c) and ( 2 4 for the magnetic fields on both sides of the i - (i + 1) interface (i.e. the magnetic fields at layer i and layer ( i + l)), one finally obtains for the i - (i + I ) interface:

(7)

Employing ( 3 4 , (3b) and (5) in the above equation, one gets for each interface an equation of the form

where the coefficients ylJ are functions of P, n and the positions 4.

Notice that the function x(')-('+') for each layer interface contains field integrals pertaining to layers i- 1, i and i+ 1 in addition to the propagation constant p. Our next task is to

seek a reasonable representation of the fields at each layer, and then utilise (8) to obtain both the field solution and the propagation constant along the structure. This is detailed in the next Section.

2.2 Solution of problem The solution is based on a choice of an appropriate basis function to represent the unknowns Et)-('+') and E;)-('+')at each interface as a function of the coordinate x. A thorough look at the field expansion of (2) leads to assumption of an expansion of the form

I P

q= 1

where P and Q are some suitable integers. The constants V(i)-(i+l) represent the relative magnitudes of the fields at different layers. These are embedded in the coefficients up+ 1) and b!)-('+'j and need not be explicitly inserted. Furthermore, the function g(x) is designed such that the field perpendicular to each conducting edge must approach infinity at the edge. Ths helps in achieving a faster stable solution [4, 91. Without loss of generality, the constant V(1)-(2) is taken as unity and the complete solution must solve the rest of V(i)-(i+') for all i, in addition to the elements of the two sets [a,] and [bq].

Next, (9) and (lo), with the help of (3, are inserted in (8). Ths will generate a set of M- 1 equations over all interfaces between different dielectric layers. Our h a 1 aim now is to generate a suitable set of equations for the unknowns [a,], [bq] and Vi-('+'). To this end, the variational nature of (6) will prove very useful.

2.3 Determination of fields The equations obtained after using (9) and (10) in (8) are of a variational nature. This suggests that the Rayleigh-Ritz optimisation technique may be successfully applied at each layer to generate a set of P+ Q equations for each layer in the unknowns u$)-('+')and b!'-('+'). Specifically, we will set

for each layer interface i - (i + 1). The set of (8), in addition to the sets (lla) and (llb), are sufficient to completely solve the problem and determine both the propagation constant and the field components everywhere for all propagating modes. T h s completes the solution.

2.4 Strategy of solution (i) The set (1 1) is a linear set in [a,], [bq] and [VI, where p is implicit. The solution starts by assuming a reasonable value for p and solving the set (1 1) to obtain the sets [U,] and [b4] for all layers. Such a solution will contain the elements of the set [VI as parameters in the form of a linear

interface i - ( i + 1). combination of V(i-l)-i, Vl-(z+l) and V(i+l)-(;+2) for each

IEE Proc.-Microw. Antennas Propug., Vol. 150, No. 3, June 2003 133

(ii) Resubstitute the obtained values of the set [a,] and [bq] in each This will generate M- 1 equations in the unknown elements of the set [VI. The resulting equations will take the following forms: Interface 1-2 Interface 2-3 Interface 3 4

gl (p , v(2-3)) = 0 g2(& V(2-31, v(3-4)) = 0 @(p, v(2-31, v(3-41, v(4-5)) = 0

&(p, y (k - l ) - (n - ) J / ( k ) - ( k + l )

V ( k + l ) ( k + ' ) ) = 0

V ( M - 2 ) ( M - 1 ) , V ( w - ( w ) = 0

p+-')-(M) ) = 0

Interface k<k+ 1)

Interface ( M - ~ ) - ( M - 1) yM-z(p, v ( ~ - ' ) - ( ~ - ~ ) ,

Interface (M-I ) -M gMP1 (p, v @ - ~ ) - ( ~ - ' ) ,

1.4

1.3

1.2

This form is very convenient for the solution of [VI. Start from the first equation and obtain V(2)-(3) . Using this value, move to the second equation to obtain V(3)-(4). Keep repeating the process until the equation before the last where one may obtain V ( M - l ) - ( M ) . This completes the determination of all the unknowns. Now, if p was the right choice, the last equation in the above set must be satisfied, else increment /3 by an infinitesimal value and repeat the process until the last equation is satisfied. This completes the full solution of the proposed problem.

In order to demonstrate the validity of our work, a full analysis of the problem is carried out for a three-layer transmission line with conducting strips of different width placed at the interface of each two layers.

3 Application

The above technique is applied to the configuration shown in Fig. 2, where the waveguide contains three layers, occupying regions 1, 2 and 3 as shown. The layers, are of different thickness and have dielectric constants 81, ~2 and Q, respectively. In addition, let a conducting strip of width W, be placed at the interface 1-2, with its axis displaced a distance dol from the centre of the structure. In a similar fashion, let another conducting strip of width W2 be placed at the interface 2-3, with its axis displaced a distance do2 from the central line. Following the above analysis, we may arrive at two equations ~( l ) - ( * ) and x ( ~ ) - ( ~ ) , similar to (8) for the four field integrals 1i2)-(3), Z:l)-(2) and

The various coefficients vi, are too cumbersome and quite involved and will not be reported here. In order to

;;/ - /

';,' , :,:/

I ;' ~/

-

- +-

L L12 - L l 2 I

d

- X

Fig. 2 interface

Three-luyer trunsmission line with conducting strip at each

gain confidence in our work, two forms of field repre- sentations of E, were attempted. The first field is that given by (9). The second field is assumed to take the form

with the representation for E$)-('+') as given by (10). In the above equation, D = t / P , where 1' is the subdistance across which the integration (3) is divided, x, = ( p D - D / 2 ) and rect is the standard rectangular function defined as:

= 0, elsewhere

,,' I ',," .' / , / 1.5

1.1 I I I I I I I I 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

d, l i

Fig. 3 Paruineters ure L= IO.Omm, d=8.0mm, dl=l.Omm, d2 = 2.0 mm und Wl= W, = 1.0 mm, with EI = E, = 4 ~ ~ ) und E~ = co

The terms Y(')-('+l) have been adjusted through g(x) to accommodate for the static field behaviour at every conducting edge as outlined earlier. The procedure followed after that is essentially that described in the first part of the paper. Some of the results of the fundamental modes for the configuration of Fig. 2 are presented in Fig. 3, with the parameters given therein dol = do2 = 0. These results compare well with the reproduced curves of [2] for the same parameters. Higher-order modes are investigated, and since there are no available results in the literature, comparison among the results of the two field expressions assumed according to (9) and (12) is carried out for every point. Some of the results obtained using the field expansion of (9) are presented in Table 1 for the same parameters of Fig. 3. In Table 2, a comparison between the results

Table 1: Relative propagation constant jlk, of structure in Fig. 2 with parameters of Fig. 3

dq/i Odd modes Even modes

0.0749 0.114. 0.854.1.2. 1.937 0.327, 1.778

0.1053 0.338, 0.626, 0.691, 0.7718, 0.12, 0.292, 0.4485, 1.854 0.835, 0.912, 1.06, 1.636, 1.963

0.1356 0.151, 0.590, 0.651, 0.712, 0.053, 0.097, 1.106, 1.271, 0.833, 0.875, 0.898, 1.095, 1.112, 1.14, 1.83, 1.976

1.613, 1.903

134

Table 2: Comparison of results of relative propagation constant p/b using field expansion of (9) and (12). Odd modes, d,/ A = 0.1053. Parameters are those of Fig. 3

I r3- ,

____ ~~

Equation (9) 0.338 0.626 0.691 0.772 0.835 0.912 1.06 1.636 1.963

Equation (12) 0.313 0.630 0.690 0.772 0.831 0.910 1.07 1.638 1.963

180

175

obtained via (9) and (12) is presented for the odd modes at d l / I , = 0.1053. The difference is about 2% at the highest- order modes and decreases to almost identical results as we approach the fundamental mode.

As a final example, a skew-symmetric structure in which do, = -do2 (Fig. 2) has been analysed for different strip displacements from the guide axis. The results for the two lowest-order modes are shown in Fig. 4. The results show that as the two conducting strips are separated, the difference in the propagation constants between the two lowest order modes increases substantially, which may find some application in filter design or data transmission. Figs. 5 and 6 give further results for various strip separations at different wavelengths.

r; /

- c

i-,

- O’

1.70 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

d,li

Fig. 4 Lowest order odd (top set of curves) and even (bottom set of curies) modes. In euclz set, upper curve is for centred conductors und lower curve is ,for dol = -do2 = 3 W,. Purumeters nre those of Fig. 3

Results also show that the field polarisation ~ E z / / ~ E y ~ is very weak, and Ez can be ignored in the solution for the fundamental modes (the TEM-like modes). However, its presence is necessary for a stable solution for the higher- order modes. In all cases, convergence occurred when the number of terms of the axial field is set to two, and the number of te rm of E , ranges from as low as four for the fundamental modes up to about sixteen for the higher-order modes. Furthermore, the number of spectral tenns N to result in a stable solution varies according to the modes, ranging from as low as fifteen for the lower-order modes up to forty for the higher-order. This technique, therefore, presents a much lower numerical effort compared to other techniques available in the literature. Consider, for example, the finite element method (FEM), where it is reported that the matrix size is of the order of 7500, which requires a core storage of 18 Mb [2]. The spectral domain method (SDM) yields a characteristic matrix which is relatively large for multiple slots/strips lines. If N is the number of basis functions and A4 is the number of slots/strips, the size of the matrix will be ( N x M ) x ( N x M) , which could become very large for multiple slots or strips [9].

1.94 1

. I I I I I i-$

0 0.5 1.0 1.5 2.0 2.5 3.0

--. 1.90

d1lWl

Fig. 5 Vuriution of /Ilk, j o r lowest order odd modes with displucement dol = -do* for vurious vulues of dJA. Purumeters ure those of Fig. 3

dl l i = 0 1963 . - -V- - - dl/d = 0.1356

i dl/n = 0 1660 dl/n = 0 1053

d,l> = 0 0749 I- *_ --

1.70 I I I I I I I

0 0.5 1.0 1.5 2.0 2.5 3.0

dl 1 Wl

Fig. 6 Vuriution of /Ilko for lowest order even modes with displacement dol = -do2 for vurious vulues of dill.. Purumeters are those of Fig. 3

4 Conclusions

The paper has presented a novel and numerically simple technique to determine the complete modal solution of a generalised planar transmission line with multiconductor strips imbedded in multidielectric layers. The problem has been solved using the reciprocity theorem in conjunction

IEE Ptw-Microii~. Aiitwimis Prop(/ . , Vol. 150, No. 3, J i m 2003 135

with the Raleigh-Ritz optimisation technique. The results show very good agreement with the available published results for the fundamental modes, and are extended to cover the higher-order modes and to investigate the effect of conductor displacement off the centre axis, where it is found that the difference in propagation constants between the lowest-order modes increases as the two conductors are pulled apart, a phenomenon that may find some use in microwave circuit design.

5 Acknowledgments

The author wishes to acknowledge the help offered to him by the Electrical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia during all phases of ths work.

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(11, PP. 3 4 4 0 3

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shielded microstrip transmission lines?, IEEE Truns. Mic&vuue Theory Tech., 1986, 34, (8), pp. 845-852 RUMSEY, V.: ‘Travelling wave slot antennas’, J . Appl. Phys., 1953, 24, nn 1’458-1’4h5

7 rr- - - - - - - - -

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