HYBRID METHOD FOR ANALYSE DISCONTINUITIES IN SHIELDED MICROSTRIP

HYBRID METHOD FOR ANALYSE

DISCONTINUITIES IN SHIELDED

MICROSTRIP

MOHAMMED EL AMINE EL GOUZI

Electronic and instrumentation Laboratory, Department of Physics, Faculty of science – B.P: 2121 TETOUAN / Morocco

[email protected] M. BOUSSOUIS

Electronic and instrumentation Laboratory, Department of Physics, Faculty of science – B.P: 2121 TETOUAN / Morocco

[email protected] Abstract:

A rigorous full-wave analysis is employed to analyze discontinuity in shielded Microstrip (open end, uniform bend). An accurate and efficient method of moments solution combined with the source method(SM) formulation is proposed in order to achieve a full-wave characterization of the analyzed structures. A wavelet matrix transform (WMT), operated by wavelet-like transform (WLT) allows a significant reduction of the central processing unit time and the memory storage.

1. INTRODUCTION

Analysis of discontinuities in transmission lines plays essential role in modeling complex microwave and millimeter wavelength circuits. Various microstrip discontinuities have been analyzed in recent years using a wide range of techniques such as mode-matching assuming a magnetic wall approximations in computing microstrip line modes [4], Frequency Domain Techniques using Electric Field Integral Equations based on spatial-domain analysis methods [12-5] or spectral-domain approaches [10,8]. More recently other numerical techniques such as Time-Domain Transmission Line Matrix (TLM) technique [9,20], Finite Difference Time Time-Domain (FDTD) approach [10-5] have also been applied to treat discontinuity problems. A mode matching for step discontinuities in microstrip lines have been presented in [23].

The source method developed by some authors [15,3], is a rigorous approach which permits to reduce the CPU-time requirements. The boundary conditions are verified by solving a system of linear equations. So, the iterations needed by the Eigen value methods are avoided. The source represents the feeding of the access line by means of a coaxial cable. The source is either punctual [16, 17] or defined on a small sub region of the circuit [15, 11, 3]. This last case is used in this paper: an arbitrary function describes the source current density. The solution of the system will give a model of the current density on all the circuit with the help of trial functions appropriately chosen. The Galerkin's method is used. The numerical value of the input impedance, or the impedance matrix seen by the sources, is easy to compute with a variation formula [16, 3, 7]. However, the impedance matrix generated by MOM is always a dense matrix, which often becomes computationally expensive and memory consuming, especially when the electrical size of the object becomes large [6]. To overcome this difficulty, various algorithms have been developed during the past decades, such as the fast multiple method [18–14], the adaptive integral equation method [1], the conjugate gradient-fast Fourier transform algorithm [19, 27], and the wavelet transform method [22–25].

and then the wavelet transform (WT) is applied to the impedance matrix to obtain a sparse matrix equation in wavelet domain [13–21].

The computational cost is substantially reduced by implementing the wavelet matrix transform (WMT) and a great amount of research papers about WT have been published by EM researchers.

In the mentioned work, a combination of Wavelet-Like Transforms and Source Method is used to reduce the CPU-time requirements to analysis of microstrip discontinuities.

2. Theory

2.1. Source method

To explain clearly the development of the theoretical formulation, we use to a large extent the formalism presented in [1]. The structure is a shielded metallic box (Fig. 1). The planar circuit is supposed lossless. A voltage source placed on the circuit plane is used and the characterization is made by modeling the current density with the help of appropriate trial functions.

Fig. 1 : Shielded microstrip line

A relation between the electric tangential field ET on the plane x = h and the total current density J is defined.

 

E



_T



Z

ˆ

.

J



  (1)

Z is a projection operator on the TE mn

f



and TM mn

f



modes of the empty waveguide. An inner product is defined by:

 

a



b







a



*

.

b



.

dS

(2)

Where

a



*is the conjugate vector of

a



Z

ˆ

is expressed as follows :

 

TM mn n

m

TM mn TM mn TE

mn n

m

TE mn TE

mn f g f Z f g f

Z g

Z  



  



 

, ,

. ˆ

   

(3)             

The mode impedances TE mn

Z and ZmnTM are defined by using a transverse resonance method [14] :

       Z jk0[ mn1 cot anh ( mn1 (a d1)) mn2 cot anh ( mn2 d1)

TE

mn          (4)          

 

) ( cot ))

( ( cot 1 [ 1

1 2 2

1 1

1 0

d anh d

a anh

k j

Z _mn

mn r mn

mn TM

mn _ 

 

  



Where ; 0 0 2 0 2 2 2 2 2 0 2 2 2 1 2 ) ( ) ( ) ( ) ( ) ( ) ( c f k k c n b m k c n b m r mn mn

















      

(Co is the light speed in vacuum).

The solution of the problem consists in satisfying the following boundary conditions:

              0 0     J E T

        (6)        

The planar dipole can be modeled with equivalent diagram as shown in Fig. 2:

Fig. 2 : Equivalent one port circuit.

The equation (6b) will be automatically satisfied by choosing the total current density on the circuit. The equations (1) and (6a) gives us the following relation:

 

ˆ

.

0







J

Z

       On (M)         (7) 

By using a source type formulation, the metallic strip is subdivided into two sub regions named (M) and (S) (Fig. 1). They represent the definition domains of the current density functions. An arbitrary excitation J0 is defined on the

(S) sub region, on the plane of the strip. Thecurrent density on the (M) sub region is expressed in term of a trial functions basis { gi, i = 1 to N}. The equation (7) becomes:

  0 1

.

ˆ

.

ˆ

_x

_g

_Z

_J

Z

N i i i





_

_





      

On (M)

              (8)

The unknowns xi (i = 1 to N) have to be computed by solving the equation (8). The (S) and (M) sub regions must be

inevitably separated. So this theory will be used with the following condition:  The trial functions must not be defined on the (S) sub region.

 The excitation function cannot be defined on the (M) sub region.

A source-type method permits a really fast computation of the problem by solving a deterministic system. This system is obtained by applying Galerkin's method to the equation (8). The kth line is:

  1 0

. ˆ .

ˆ _{x g g Z J}

Z g _k N i i i k      





               (9)

Then the system of matrix equation (3) can be expressed as:

A



X



B

(10)

The expression of Z given by (3) allows us to write the equation (9) under the following form (Equation 11): (a) On (M)

 













      n m TM mn TM mn TM mn k n m TE mn TE mn TE mn k n m N i i i TM mn TM mn TM mn k n m N i i i TE mn TE mn TE mn k J f Z f g J f Z f g g x f Z f g g x f Z f g , 0 , 0 , 1 , 1           (11)       

The inner products are made on the (S) and (M) sub regions which are the definition domains of the trial and source functions. The excitation term must nevertheless be chosen with the greatest precaution since it must disrupt the least possible the continuity of the current at the interface between (S) and (M). Then, the form of the source function and the size of the (S) subregion have an influence on the final computed result. When the current density is known on any point of the circuit, a one-port network can be characterized by its input impedance Zin, seen by the

source. A variational expression is associated to the equation (8) [7]:

  0 0*

0

0 ˆ( )

I I g x J Z J Z i i i in



    

              (12)

Where

I

₀ is the source current.

2.2. Choice of sources and trial functions

The modeling of a planar circuit consists of determining its behavior by calculation and programming. Efficiency and accuracy of the computation can be improved by the choice of appropriate sources and trial functions.

In this paper, a constant source has been chosen, composed of only one component in the z direction. This source function permits a good numerical convergence. Trial functions {gi} are roof top functions.

2.3. Wavelet-like transform

The principal advantage of using wavelets in the MoM is the sparsely of the calculated impedance matrices. The system of matrix equation in moment method can be expressed as:

         

A



X



b

      

         (13) 

By a WMT method, Equation (13) is transformed as

 

A

w



X

w



b

w                  ₍₁₄₎ 

Where

 

A

WAW

X

W

X

b

w

Wb

t w

t

w









_,

)

(

,

1

                (15) 

Transformation t

WAW

is a 2D wavelet transform which act on rows and columns of A respectively and is called a wavelet-like transform of A [9, 10].

The unknown X is found by solving (14).

 

X

W

WAW

Wb

t t

₍

₎

1



               ₍₁₆₎



























































































 n n

G

H

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

P

8 8 1 0 5 4 3 2 3 2 1 0 5 4 5 4 3 2 1 0 5 4 3 2 1 0 4 5 0 1 2 3 2 3 4 5 0 1 0 1 2 3 4 5 0 1 2 3 4 5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

        (17)  

At each level of decomposition, matrix Pn must constructed such that to perform decomposition only on lowpass

coefficients ( Hn ) of the previous stage. Before applying the wavelet transform a rescaling and normalizing

according to the following equation could be useful.

   

)

,

(

)

,

(

)

,

(

)

,

(

j

j

A

i

i

A

j

i

A

j

i

A



, i,j =1 :N (18)

Matrix Aw is now converted to a sparse matrix Aδ by thresholding with parameter δ. To investigate errors produced

after applying a threshold, we define two kinds of error:

 Reconstruction error: The reconstruction error is the difference between the actual elements of the matrix and the one reconstructed from the thresholded wavelet coefficients [10]. Reconstruction means to apply inverse wavelet transform to the thresholded matrix.

 Relative error of any desired solutions before and after thresholding moment matrix.

3. Numerical results

In this section, we present two examples to demonstrate the efficiency and accuracy of the present technique.

3.1. Application to uniform microstrip

As the first example, we have applied both techniques to a uniform Microstrip whose geometry is shown in Fig. 3 in equal conditions (i.e. dimensions, frequency) to compare our method with conventional MoM . We have developed a computer program in MATLAB to implement the procedures described in previous sections.

The computational resource requirements of the conventional MoM and our method are listed in Tables I , It is observed that MoM takes about 164,5 min to compute the solution on a Pentium IV PC while our method takes only 56,65 min.

Table I: RESOURCE requirement of both techniques

a = 2 cm , b = 4 cm, c = 5 cm Line length : 2.81cm , width :1cm

(50 ohms)

ε r = 10 , W/H = 1 Conventional MoM MoM based _WLT Number of

modes 3000 3000

Total solution

time 164,5 min 56,65 min

Memory (MB) 38 5.5

Fig 4: Stub input impedance (a = 2 cm, b = 4 cm, c = 5 cm, H = 1 cm, ε r , = 10).

3.2. Application to the right angle bend

The next example is Microstrip line containing a 90’ bend whose geometry is shown in Fig. 5 which is fabricated on a substrate with relative permittivity 9.8 and thickness H = 0.635 mm.

To validate our results, Magnitude of S11 of this structure are depicted and compared with that of conventional MoM

Fig. 5: Shows the geometry of a microstrip line containing a 90' bend.

After applying wavelet-like transform the ratio of biggest element to smallest element is increased about 106 times, this indicates that the matrix is sparse.

Table 2 : Sparsity and Errors Related To Different Values Of threshold

Threshold

δ Sparsity Percent Error of R(%) Solve Reconstruction Error (%) 1.00E-06 8.9367 4.56 E-06 0.00010543 1.00E-05 25.492 0.000134657 0.003418 1.00E-04 45.5 0.0056432 0.02631

1.00E-03 52.45 0.452 0.2614

1.00E-02 67.362 0.74321 2.6738

Choosing the threshold parameter δ = 0.0001, leaves reconstruction and solve errors quite small about a few tenth of percent. We must pay more attention to choose δ at frequencies around the resonance.

Table III: Resource requirement of both techniques for

a = 3.81cm , b = 12.7 cm, c = 12.7 cm

width : 0.635 mm (50 ohms)

ε r = 9.8 , H = 0.635 mm Conventional MoM MoM based _WLT Number of

modes 3000 3000

Total solution

time 66,7 hours 12,4 hours

Memory (MB) 635.45 24.4

Fig. 6: MAG (S11) of right-angle bend at box resonances a = b = 1.9 mm, c = 0.2 mm, H = 0.1 mm, W / H , = 1).

4. Conclusion

In this paper, a full wave analysis of microstrip discontinuities by moment method based wavelet-like transform was presented. We used wavelet-like transform to sparsify the moment matrix considering the errors generated in the process. We illustrated that our method has much potential in the area of the analysis and CAD of complex MMIC structures and is capable of being enhanced in a number of ways to improve computational efficiency and accuracy. Numerical results were presented for our proposed method, based on developed MATLAB codes, and good agreement with conventional MoM.

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Biographical notes

Mohammed el amine EL GOUZI , He received his M.E degree in Electronics & Telecommunication