accurate numerical modeling of microstrip junctions and discontinuities

Accurate Numerical Modeling of Microstrip Junctions and Discontinuities

Doris 1. Wu,' David C. Chang,' and Brad L. Brim2

' MIMICAD Center, Department of Electrical and Computer Engineering, University of Colorado, Boulder, Colorado 80309 2Hewlett Packard Network Measurements Division, Santa Rosa, California 95401

Received April 13, 1990: revised August 22. 1990

ABSTRACT

A general numerical solver for analyzing microstrip geometries of rectangular shape is presented in this paper. The analytical foundation of this solver is based on an integral equation approach which we formulate in the spatial domain. The unknown surface current on the microstrip is solved by the method of moments using 2D rectangular pulses as the expansion functions. Transmission-line modeling is then used to parameterize a given microstrip junction or discontinuity. Aided by a graphics interface, this solver can analyze complex structures without incurring additional analytical complexity. We illustrate the accuracy and versatility of our solver by applying it to several different microstrip discontinuities ranging from a single-stub to an interdigitated capacitor.

1. INTRODUCTION

Existing computer-aided design (CAD) packages for designing microstrip circuits have several de- sirable characteristics: they are simple to use and they perform real-time simulations. However, to yield good results it is not only crucial to have accurate models for the different junctions and discontinuities. but effects such as radiations and parasitic couplings must also be accounted for in the 4mulation process. As pointed out by many in the literature [ 1,2], the need for more accurate CAD tools has become essential in keeping pace with the latest GaAs MMIC technology.

The purpose of this paper is to describe an integral-equation-based numerical solver, which we developed and implemented in a workstation environment, capable of producing highly accurate design databases for the various bends, junctions, and closely spaced circuit elements. Although the emphasis of this paper will be on the application of our solver, we will first describe briefly the an-

alytical foundation for our solver. As will be illustrated, the strength of our solver lies in its accuracy as well as its generality.

2. ANALYTICAL PROCEDURE

2.1 Mixed Potential Integral Equation

In analyzing microstrip structures, a full two-dimensional (2D) integral equation formulation combined with the method of moments approach generally yields the most accurate results. By solving for the components of the E and H fields normal to the slab surface and imposing the boundary conditions on the surface of the upper conductor [ 3 ] , we obtain a mixed potential integration equation (MPIE) for microstrip structures. The same MPIE can also be obtained by solving for the vec- tor and scalar potentials associated with the electric field of a microstrip structure [4]. Compared to the more traditional electric-field integral-equa-

International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, Vol. 1 , No. 1, 48-58 (1991) B 1991 John Wiley &r Sons. Inc. CCC 1050- 1827 191 1010048-11$04.00 49

Accurate Numerical Modeling of Microstrip 49

tion approach, the MPIE lends itself well to spatial domain evaluation. Although the choice of evaluating the integral equation in the spectral or the spatial domain is arbitrary because the two ap- proaches are physically equivalent, they are different in the numerical implementation.

Spectral domain evaluation is most useful for treating microstrips of simple shape, where the current distributions can be expanded using a set of functions having closed-form Fourier-trans- formed counterparts. Analyses involving semi-infinitely long lines are ideal for this approach since the currents on these lines can be represented by a pair of forward and backward traveling waves, which in turn can be represented by simple Dirac delta functions in the Fourier transform domain. Therefore, in analyzing microstrip discontinuities using this approach, semi-infinitely long lines are often used as standard feed strips [5,6]. However, since the current in and near the discontinuity region is not uniform, a different set of subsectional basis functions is needed to capture the junction effect. This hybrid use of basis functions implies that each set of basis functions must be carefully defined over the structure, and different algorithms are needed to evaluate the moment integrals associated with each type of basis functions. Moreover, to obtain accurate results, this approach also requires the pre-computation of the propagation constant and/or transverse distributions of the current on each distinct feed strip [5-71.

Spatial domain evaluation provides more phys- ical insight since the problem remains in the phys- ical domain. For this approach, the Green’s functions are numerically evaluated first and treated as known functions in the integral equation. This implies that the selection of the expansion functions for the current is arbitrary, which renders the approach very versatile. Moreover, subsectional basis functions of simple form can be implemented with straightforward numerical algorithms using the spatial domain approach. Therefore, this approach provides an ideal base for a general solver. The groundwork for a MPIE- based, spatial domain microstrip solver can be attributed to Mosig and Gardiol [4]. Utilizing their basic approach, we present a numerical solver with modified algorithms and expanded generality applicable for most microstrip circuit junctions and discontinuities. As will be shown in this paper, complex structures can be treated using this solver without incurring additional analytical complexity.

The use of potentials is generally preferred in

the spatial domain approach because the associated Green’s functions are better suited for numerical evaluations. Using an elwr time convention, the MPIE can be expressed as

1 - - VV’ . G,(X, X ’ ) I ( X ‘ ) ] dS’

k:

where 5 is the unknown current density on the microstrip surface (S), and 3’ and Einc denote the scattered and incident fields, respectively. The scalar Green’s functions, G, and G,, are associated with the potentials produced by a unit current source on top of the grounded substrate. They are identified as Green’s functions of the magnetic and electric type, respectively. By evoking the continuity equation

V . f + Lwq = 0, (2)

it can be shown that G, is related to the surface current density while G, is related to the surface charge density, q. Thus, eq. (1) can also be expressed in terms of the surface current and charge densities. For a single-layer grounded dielectric slab in open space, these Green’s functions can be expressed in terms of Sommerfeld integrals as ~ 3 ~ 9 1

(3) R = IX - X’l ,

where

R is the distance between the source and obser- vation points, t is the slab thickness, and k, is the free space propagation constant. Since these ker- nels are a function R , they can be precomputed for a given range of R. Once they are known, the integral eq. (1) can be solved readily in the spatial domain.

50 WLL Chang, and Brim

2.2 Evaluation of the Green’s Functions In evaluating the Sommerfeld integrals of eqs. (3) and (4). it is well known that for large R. the prudent approach is to deform the path of integration to yield at least one residue term and a branch cut integral with exponentially decaying integrand. However, for small distances. the integration along the positive real axis is often more efficient. To achieve efficiency, we subtract out the static term of the Green’s function first [4.8] and evaluate the remaining Green’s function expres- sion numerically by deforming the contour to wrap around the branch cut [ 101.

In our solver. the static portion of the Green‘s function is handled analytically in the MPIE. The Green’s functions without the static terms are numerically evaluated at discrete points for a given range of R predetermined by the maximum di- mension of the microstrip geometry. Cubic spline approximations are then used to obtain a smooth curve fit for each of the Green‘s functions.

2.3 GalerkidMoment Method To solve the integral eq. ( 1 ) for the unknown current and charges. we use the GalerkiniMoment method. For simplicity. we choose to use 2D rectangular pulses as the expansion functions for the current. The expansion functions for the charges are obtained by applying a finite difference approximation to the continuity equation [11,12]. This yields a set of charge cells that are spatially shifted from the current cells. Similar to the current. the distribution in each charge cell also consists of a 2D rectangular pulse. To ensure consistency in our gridding process, we find it is best to divide the microstrip surface into elemen- tary cells for the charges first and generate the current cells from the charge cells. The x- and y - current cells are generated by dividing the charge cells in halves. each along the direction of the current, and combining the adjacent halves sys- tematically to generate new cells for the corresponding currents.

Along the edges of the microstrip. we have an additional boundary condition requiring the normal component of the current to be zero. To en- force this condition. we follow the procedure used in ref. [4] and introduce a narrow strip of cells along the edges normal to the current and force the current to be zero in these cells. For simplicity. this narrow strip is chosen to be half the width of the charge cells along each edge. Figure 1 is an

c h a r g e g r i d

J \ I I

I

y - c u r r e n t g r i d

Figure 1. Charge and current grids.

illustration of the x- and y-current grids resulting from a given set of charge cells.

Despite the need for multi-grids to characterize the current and charges, using 2D pulses does offer the advantage that the coupling integrals associated with the static terms, which we sub- tracted out in the numerical computation of the Green’s functions, can be integrated in closed form [13,14]. This ensures a greater accuracy in our numerical computations since the dominant con- tribution of the moment integrals generally comes from the static terms.

Using 2D rectangular pulses as the weighting functions, we reduce the integral equation to a matrix equation where the unknowns are the expansion coefficients for the surface current. In evaluating the coupling matrix, the nonstatic part of the coupling integral involves a 4D surface integration which is done numerically. Using a voltage-gap source as excitation, the unknown current is solved with standard matrix manipulations such as LU decomposition and back substitutions. Al- though the use of a voltage-gap source may be physically impractical, it does provide a simple means of exciting the microstrip. As will be shown later, the important characteristics of a microstrip junction can be extracted independent of the source.

3. APPLICATIONS

A numerical solver, which we call PATCH,’ has been implemented using the analytical procedures

‘As pointed out by one reviewer, PATCH was the name of the solver originally developed by J . Mosig. Our program should not be confused with the original one. The two solvers contain different numerical algorithms for the evaluation of the Green‘s functions and the moment integrals.


described in the previous section. Because of the choice of our basis functions, PATCH is most suit- able for analyzing rectangularly shaped microstrip structures. To retain generality, we do not make any a priori assumptions regarding the transverse distribution of the longitudinal current, nor do we neglect the transverse component of the current from the onset. On the contrary, these transverse effects can be included as needed, depending on the structure, without incurring any additional analytical complexity in our solver. Thus, the types of microstrip junctions and discontinuities we can analyze range from simple structures such as open- end, gap, and L-bend, to more complex elements such as coupled T-junction and interdigitated capacitor. The remaining portion of this paper will focus on the application of our numerical solver. In particular, we will illustrate the versatility and generality of our solver by applying it to different junctions and discontinuities of complex shape.

The usage of our solver is simplified greatly by the development of a graphics interface for drawing and gridding microstrip geometries. This graphics tool, which we call UDRAW [15], provides an input mechanism that is intuitive and easy to use. Complex structures can now be analyzed with the help of this graphics tool. UDRAW sim- plifies the gridding process by requiring the users to grid only for the charge cells. UDRAW then performs the shifting of charge and current cells internally to create the corresponding sets of cells for the x- and y-currents. Thus, for a given geometry, gridding is done only once. In UDRAW, uniform as well as nonuniform size cells are al- lowed. Therefore, charge concentration near a discontinuity can be characterized accurately by using smaller-sized cells adaptively .

Since the direct output of PATCH is the current on the microstrip surface, additional processing is needed to extract or de-embed equivalent circuit parameters from the computed current. The procedure we use to de-embed is based on transmission-line modeling and will be described briefly first. As with any electromagnetic solver, PATCH is computationally intensive. To minimize unnec- essary computation, we will carry out a convergence study to determine the optimal number of cells to use in our gridding process. To further reduce the number of unknowns, we will also examine a modified one-dimensional method for treating narrow microstrip lines with 90" corners. As will be illustrated, this method is most eEective in capturing the junction effect.

Our computing environment consists of a clus-

ter of networked HP 9000/319C workstations. Most of the numerical computations for this paper were done on these HP workstations. However, due to finite capacity, the maximum number of unknowns these workstations can accommodate is approximately 500. Computations involving more than 500 unknowns, such as those encoun- tered in the convergence study, were done using the ETA10 supercomputer resources allocated to us by the John von Neumann Supercomputing Center.

3.1. Extraction of S-Parameters In characterizing a microstrip junction or discontinuity, the quantities of interest are usually the equivalent circuit parameters, such as the scattering parameters (S-parameters). Since the S-parameters of a junction describe the relationship between waves coming in and going out of a junction, we will use a network description of the discontinuity. Enclosing the reference junction or discontinuity in a black box as illustrated in Figure 2, the S-parameters for a 2-port device can be expressed as

bl = SIlal +- S~,U?,

b2 = &aI +- &a2,

( 7 )

(8)

where the as and the bs are the voltage amplitudes associated with the waves propagating into and out of the network at each port. For a TEM line, these wave variables are related to the voltage and current of the line via the transmission-line equations

(9)

(10)

V(x) = ae-@ + be'"

Z(x) = - 1 (ae-+"" - be"""), z,.

where 2,. is the characteristic impedance of the line, and p is the propagation constant. By making the assumption that dominant-mode propagation prevails away from the juntion, we can model the current on a feed strip using eq. (10). Moreover, to ensure dominant-mode propagation, we can ar-

Figure 2. A 2-port device.

52 Wu, Chang, and Brim

tificially extend the length of the feed strip at each port by several wavelengths from the junction reference point. The normalized wave variables, a' = a/Z, and b' = blZ,, as well as the propagation constant. can be extracted by "probing" the total current away from the discontinuity. If these normalized wave variables are used directly in eqs. (7) and (8) in place of a and 6 , we obtain a set of S-parameters that are normalized implic- itly to the characteristic impedance of each line. This normalization gives us the advantage that the S-parameters can be obtained without a priori knowledge of the characteristic impedance of each line.

To minimize the effect of higher-order modes, the current within a distance of one guided wavelength from both ends of the strip is excluded from the probing process. In general, p can be found very easily by examining the current standing wave pattern. Since the distance between two adjacent peaks is one-half of the guided wavelength (Ag), f3 can be found once the guided wavelength is known. To ensure consistent values of a' and b ' , curve-fitting is used to further smooth out the effect of higher-order modes. Once p is found. we curve fit the real and imaginary parts of the current separately to a sinusoidal distribution of the form A ,,,] sin(@ + &",,,). where A,,,, and b,,,, are the optimized amplitude and phase parameters obtained from the curve-fitting routine. The sub- scripts r and m denote the real and imaginary parts of the current, respectively. The values of a' and b' can be computed from these optimized parameters by expressing the total current in terms of these two curved-fitted functions and equating it to eq. (10). Therefore. they can be computed in a straightforward manner [16]. Unlike the procedure used in ref. (171, our approach for finding the wave variables does not depend on the specific choice of x.

For a 1-port device, the ratio of b' to a' is the desired S,, parameter. For a general, nonsym- metrical, N-port device. each port must be excited individually. The current over each extension strip is then fitted separately to the transmission-line equation of eq. (10) to yield the appropriate p, a ' , and 6 ' . The S-parameters can be computed once the a' and b' are known for each port and each excitation [18].

3.2. Convergence In using PATCH, the accuracy of the simulation depends in part on the size of the cells, or the

number of cells, used in the gridding process. Al- though the accuracy improves with decreasing cell size, the computation time can increase quite dras- tically as the total number of cells is increased. To achieve an optimal balance between accuracy and computation time, we examine the convergence of our numerical simulation in order to establish a guideline on the minimum number of cells to use in the gridding process.

Using a four-wavelength-long, open-ended microstrip line as an example, we grid the strip using various numbers of cells per guided wavelength. The transverse component of the current is ne- glected for now and the transverse distribution of the longitudinal current is assumed to be uniform. The frequency of operation is arbitrarily chosen to be 10 GHz, the permittivity of the slab (q) is 9.8, slab thickness is 0.24 mm, and strip width is 0.096 mm. Exciting the strip with a constant voltage-gap source located at far end of the strip, we examine the current as well as the guided wavelength on the open-ended strip for various numbers of cells ranging from 12 to 150 per guided wavelength.

Figures 3 and 4 are plots of the real and imaginary parts of the computed current, respectively, on the strip for the different numbers of cells. As can be seen, the current distributions are sinusoidal. The imaginary part of the current, which is three orders of magnitude smaller than the real part, is directly related to radiation from the open end. Using the 150/h, case as the converged case, Table I shows the percent difference on the amplitudes of the real and imaginary parts of the current for various cell sizes. For example, using

4

75

150

...____ _. 20

12

- _._ - - - ----- - - , , . . . , . . . , . . . I . . .

0 02 0 03 0 04 0 0 5 0 06

Position Along Strip ( in m)

Figure 3. bers of cells.

Real part of the current for different num-


00120-

0

00119-

0 002 75 ................. 12

20 - 150 ----- -. - . -. - . -

:r- 0 002

0 02 0 03 0 04 0 05 0 06

Position Along Strip (in m)

Figure 4. Imaginary part of the current for different numbers of cells.

a number of 20 cells per guided wavelength can yield current amplitudes that are within 5 % of the converged results.

The second quantity we use to evaluate convergence is the guided wavelength. We extract the guided wavelength by curve-fitting the current magnitude over the mid-portion of the strip to a known distribution similar to eq. (10). Figure 5 is a plot of the computed guided wavelength as a function of number of cells. While this plot also indicates a strong convergence, the guided wavelength does not seem to be as sensitive to cell size as the current amplitude. The guided wavelength computed using the most coarse gridding, i.e., 12 per wavelength, is already within 2% of the converged result. This implies that the periodicity of the current is more stable than the magnitude.

The data shown in Table I were obtained using a 1D current approximation. To validate this approximation, we compare it to the full 2D representation. The 2D representation takes into account both the transverse distribution of the longitudinal current as well as the transverse component of the current. For this comparison, we use a section of strip two wavelengths long as an example. The frequency of operation is arbitrarily

TABLE I. Percent Difference From the Converged Results for Real and Imaginary Parts of the Current

No. of Cells % Diff. for Re(1) % Diff. for Im(1)

12 11.8 5.8 20 4.5 4.6 25 2.6 4.1 75 0.2 0.1

I 0 01 21

0 5 0 100 1 5 0 2 0 0

Number of Cells per 1

Figure 5. cells.

Guided wavelength for different numbers of

increased to 60 GHz. The slab thickness is 0.12 mm, er is 9.8, and the strip width is 0.11 mm. For the 1D case, we grid the strip using 25 cells per A, in the longitudinal direction. For the 2D case, the strip is gridded using 75 cells per A, in the longitudinal direction and 5 cells in the transverse direction. This yields a total of 1345 unknowns for both x- and y-current cells.

Figures 6 and 7 show the real and imaginary parts of the longitudinal current, respectively, for the 1D and 2D cases. The displayed current for the 2D case is the current averaged over the strip width. The results show that the difference between the two cases is less than 2%. Therefore, for most typical microstrip lines, the 1D current approximation is an adequate representation. Fur- thermore, a guideline of 20-25 cells per guided wavelength is deemed adequate for our application since it provides a 5% tolerance on the current amplitudes and a 0.5% tolerance on the guided wavelength.

3.3. Modified One-Dimensional Method In treating simple discontinuities such as an open- end or a gap, a 1D current is generally adequate. For more complex structures involving corners, bends, and change-in-widths, both transverse and longitudinal currents ar.e needed to characterize the junction accurately. However, to include the transverse effects everywhere on a microstrip structure can be costly because the total number of unknowns can become exceedingly large. To alleviate this difficulty, we examine a method which utilizes the simplicity of 1D current for nar-


3

2 -

1 -

0 -

1 -

2 -

1D

- 2'2

- - - _- - -

3 , . , . . , . . . . I , . 1 1 , . , , I

row strip line and at the same time captures the junction effect by using a 2D current in the critical region. This method. which we call a modified 1D method. is intended for treating narrow microstrip lines with 90" bends.

Using an L-bend as an example. we treat the bend as two overlapping single strips in this method and assume that the current in each single strip can be approximated by a 1D current in the longitudinal direction. Each strip is gridded independently. The portion of each strip in the overlapping region is further divided into smaller cells to give us a finer characterization of the junction. Figure 8 shows the composite current representation for the L-bend. It consists of a 2D current expansion in the junction region and a 1D current in each leg of the strip. Although the currents in both legs are treated independently in the gridding process. they are coupled together in the computational process through the charge cells.

To show the adequacy of this modified 1D current. we compare it to the full 2D current rep-

- o ' I , , , , , , , . , , , , , , . , ,

0 20' 0 c 3 2 3 c c 3 3 33:

Position Along Strip (ln m i

Figure 7. Imaginary part of the current for 1D and 2D representations.

resentation for the L-bend shown in Figure 8. For the parameters shown in Figure 8, the length of each leg is approximately one guided wavelength. The gridding for the modified 1D representation consists of 6 cells in the junction region and 20 away, yielding a total of 52 cells for both legs. For the 2D approach, we grid the bend using 23 cells in the longitudinal direction and 3 cells in the transverse direction, yielding a total of 302 unknowns for both x- and y-currents. Exciting the structure with a uniform voltage-gap source located at the top end of one leg, we compute the current for both cases.

Figures 9 and 10 show the real and imaginary parts of the input admittance, respectively, for the two cases as a function of frequency. As can be seen, the modified 1D approximation is adequate in capturing the junction effect since the plots show a good agreement between the two. There- fore, this modified 1D approach is found to be sufficient for applications such as de-embedding where only the effect of the junction is needed, not the detailed behavior of the junction currents. Additional numerical experimentations have shown that this approximation can provide a 0.5% tolerance on the guided wavelength for k , w < 0.1, where w is the width of the strip.

3.4. Examples We illustrate the accuracy of our numerical simulations by computing the S-parameters for three different microstrip elements and comparing the results to measured data. The shapes and dimen- sions of the selected structures are shown in Figure 11, To eliminate redundant gridding, each reference structure is gridded only once using the 25 cells per wavelength guideline for the highest frequency of operation. At each frequency point, we artifically add on a fixed-length, pre-gridded extension strip at each port. The S-parameters are found by probing the currents on both strips using the procedure described in Section 3.1. For all three examples, we assume a perfect dielectric slab and neglect conductor losses.

Figure 8. an L-bend.

A modified 1D current representation for


" -

0 1 -

- c E 5 0 0 -

-

-0 1 -

- 0 2

10-1 - MODlD

- MODID

2D 0

:I * . . . , . . . . , . . . . , . . . .

10-2

- 10.3

t. . _

d 1 0 - 4

1 0 - 5

4.0 4 5 5 0 5 5 6 0

Freq (GHz)

Figure 9. Comparison of the modified 1D with full 2D representation for the real part of the input admittance.

The first structure consists of a single open- ended stub connected to a section of microstrip line. A modified 1D current approximation is used for this structure. Including the three-wavelength- long extension strips, the total number of cells for this structure is 194. Figures 12 and 13 are plots of the magnitudes of SI1 and SI2, respectively, for both measured and computed data over the frequency range of 10-40 GHz. As can be seen, our response curves agree quite well with the measured data.

The second structure we tested is a 2-port, coupled-Tee junction. It consists of two open-ended stubs connected in opposite to a section of microstrip line. A modified 1D current approximation is also used for this structure. Using an extension strip length of 4h,, the total number of cells for the composite structure is 315. The corn-

L2

1 - L w 2 r t

LZ

I

W1 = 0 254 mm W2 = 0 126 mm L i = 1 708mm L z = t 5 m m ~r = 9 413 Slab thickness = 0 254 mm

Line Width ~ 0 122 mrn Slab thickness = 0 127 mm

L i =O757mm L2 = 2 921 mm 13 = 0 442 mm tr = 9 9

Finger Width ~ 0 015 nim Finger Gap = 0 005 nini Slab Thickness = 0 1 rim

1 w2 T

g = 0 017 mm t r = I 2 9

(C)

Figure 11. Three simulated structures: (a) single-Tee, (b) double-Tee, and (c) interdigitated capacitor.

puted as well as measured data for /SI11 and lSIzl are shown in Figure 14 and 15, respectively. Ex- cept for a slight shift in frequency, the comuted cruves agree well with measured data. The frequency shift in the response curves can be partially attributed to the 2% variability in the dielectric permittivity. Although the phase plots are not shown, the results are similar to the magnitude plots [19].

This double-Tee structure is particularly inter- esting because the double-resonance phenomenon observed in the ISI21 plot was unexpected. Labo- ratory experiments have shown that this phenomenon was caused by mutual coupling between the two Tees [20]. Since most existing CAD circuit simulators are lumped-element based, they failed to predict this occurrence because they cannot take this type of secondary effect into account. In contrast, we are able to predict this phenomenon accurately because effects such as radiations and mutual couplings are intrinsically accounted for in our integral-equation approach.

To illustrate the generality of our solver, we model an interdigitated capacitor as our third example. This commonly used microstrip element is


-80 ‘ 0 2 c 3 0 4 0

Freq (GHz)

Figure 12. gle-Tee.

Computed and measured l.Si,1 for the sin-

CXXPUTED

- MEASURED

_ _ - _ _

. . . . , . . . . , . . . . , . . . . , . . . .

a complex device because it is composed of many different types of discontinuities. such as open- ends, gaps, T-junctions, and coupled lines. To model this device accurately. the individual discontinuities along with their mutual interactions must be taken into account. Because our solver is comprehensive in nature. i t is capable of modeling this device accurately with no more effort than the two devices we simulated previously.

To capture the large strip-width change in the feed strip to capacitor transition, we use a 2D current throughout the reference structure except in the finger region. To reduce the number of unknowns. we also assume that the transverse component of the current in both extension strips is negligible beyond a distance of three substrate thicknesses away from the strip-to-capacitor transition. Using a length of 2.5h, for each extension strip, the total number of unknowns for this structure is 475. Figure 16 shows the comparisons be-

CONPUTED - - - - - 0 0 , . . . . , . . , , , . . , ,

1 0 2 0 30 4 0

Freq (GHz)

Figure 13. gle-Tee.

Computed and measured lS,>l for the sin-

tween measured and computed data for both ISll\ and ISl2]. A slight disagreement between the measured and computed values is visible for \Slzl and more so for lSlll. The largest discrepancy occurs in lSlll at 18 GHz with an absolute difference of 0.08 between the measured and computed values. Although a possible discrepancy between the widths of the feed strips for the measured and simulated structures was discovered later, additional numerical simulations using different strip widths showed no significant changes on the comparison. While this type of accuracy may be ac- ceptable for design comparison purposes, a finer grid in the capacitor region can always be used to further improve the results.

4. CONCLUSION

In this paper, we outlined the developmental procedure of a moment method-based general solver for microstrip structures. We also illustrated the versatility of our simulation program by applying it to three different structures. Our simulation process began with the drawing and gridding of a geometry using UDRAW. The output files created by UDRAW were used to run the general solver, PATCH. The S-parameters were then extracted from the output of PATCH using a straightforward de-embedding procedure. As we have shown, we can analyze complex structures and yield accurate results. Although the three steps described above are in separate form currently, they can be combined and automated.

While our solver can analyze complex structures with good accuracy, it does require a longer computation time compared to the conventional


MEASURED - -60,. . . . , . . . . , . . . . , . . . . , . . . .

2 7 12 1 7 2 2 2 7

Freq (GHz)

Figure 15. ble-Tee.

Measured and computed /SI21 for the dou-

lumped-element-based circuit simulators. For example, a structure with 200 unknowns would take approximately 30 minutes of CPU time per frequency point on our HP 9000/319C workstations (rated at 2 MIPS). The strength of our current solver is in its ability to model microstrip components accurately. Although our solver in its present form is not suited for real-time simulations, it has not been optimized to the fullest. For example, it does not take advantage of the uniform-sized cells often used in the extension strip sections. Comutation time can be shortened some- what if we recognize this uniformity and eliminate all repetitive computations of couplings between uniform-sized cells. Moreover, with the availabil- ity and the widespread use of workstations, parallel computations can be introduced in a networked environment to further speed up the simulation process.

, ”

COMWTED . . . . _. . . . 0 0 , . . . , . . . I . . . i . , . , . . .

0 4 8 1 2 1 6 2 0

Freq (GHz)

Figure 16. Computed and measured ISl,\ and (Sl2l for the interdigitated capacitor.

ACKNOWLEDGMENT

The measured data for the various structures we simulated were provided to us by Hewlett Pack- ard, Hughes Aircraft, and Texas Instruments. We gratefully acknowledge their support. We would also like to acknowledge JvNCC supercomputing center for allocating the requested computer resources. Lastly, all the curve-fitting algorithms used in our de-embedding procedure were provided by the NAG Fortran Library.

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BIOGRAPHY

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19. D. I . Wu and D. C. Chang, A highly-accurate numerical modeling of microstrip junctions and discontinuities. Final Report, MIMICAD Center, University of Colorado, Boulder, April 1990.

20. M. Goldfarb and A. Platzker, "The effect of electromagnetic coupling on MMIC design," Int. J. Mi- crowave Millimeter- Wave Computer-Aided Eng., to appear.

Doris 1. Wu received the BS degree from Michigan State University. East Lansing. in 1980. the MS degree from Stanford University, Stanford. California. in 1981. and the PhD degree in 1987 from the Uni- versity of Colorado. Boulder. all in electrical engineering.

From 1980 to 1983. she was a Member of Technical Staff at Bell Laboratories:

AT&T. Indianapolis. Indiana. where she was engaged in the study of radio frequency interference. From 1987 to 1988. she was with the National Institute of Standards and Technology. working in the area of field characterization and measurement. In 1989. she joined the Center for MicrowaveiMillimeter-wave Computer-Aided Design at the University of Colorado. Boul- der. as a Research Associate. Her research interests include CAD. antenna radiation. scattering. and numerical modeling.

Dr. Wu is an associate member of URSI Commission B.

David C. Chang received the PhD degree in applied physics from Harvard Uni- versity. Cambridge, Massachusetts. in 1967.

He joined the faculty in the De- partment of Electrical and Computer Engineering, University of Colorado. Boulder, in September 1967 and has been Professor of Electrical and Computer En-

gineering since 1075. and Chairman of the Department from 1982 to 1989. He is also currently the Director for the NSF

Industry/ University Cooperative Research Center for Micro- waveiMillimeter-Wave Computer-Aided Design at the Uni- versity of Colorado.

Dr . Chang has been active in electromagnetic theory, antennas. and microwave circuits research. H e has served, on various occasions. as the Associate Editor (1980-82), Coor- dinator for the Distinguished Lecturers Program (1982-88), Chair of the Ad Hoc Committee for Basic Research (1985- 88). and a member of the Administrative Committee (1985- present) of the IEEE Antennas and Propagation Society and now serves as President of A P Society. H e also has served as a member of the Technical Subcommittee on Microwave Field Theory of the I E E E Microwave Theory and Techniques So- ciety (1975-85); as a member-at-large (1982-85) and Secretary of the U.S. National Committee (1987-90), as well as Chair of the Technical Program Committee, Commission B on Fields and Waves (1983-86) of the International Union of Radio Science.

Brad Brim was born in 1959. He received his BSEE in 1982 and his MSEE in 1983 from Washington State University, Pullman. From 1983 to 1988 he was a research assistant at the University of Colorado, Boulder. His research was in the area of computational and analytical methods for planar microwave structures.

In 1988 he joined Hewlett Packard in the Network Mea- surements Division as an R&D engineer. His work has been mainly in the area of modeling of passive microwave components and circuits for computer aided engineering applications. (Photo not available at this time.)

accurate numerical modeling of microstrip junctions and discontinuities

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