A More Realistic Power-Ground Plane

4.3.1 Geometry

Fig. 4.9 shows a more realistic power-ground plane compare to the one in the previous section. The geometric parameters are shown in Fig. 4.9a where the top-right corner of a square metal plate with w= l=50mmis cut from

(

35,50

)

to

x y

z

w

l h Port

ε

r

0 10 20 30 40 50

0 10 20 30 40 50

(b)

(a) Unit: mm

Fig. 4.9. A more realistic power-ground plane with a clipped corner and an aperture on

the top metal. (a) The geometric parameters of the power-ground plane. (b) The

mesh setting for simulation.

(

50,20

)

. An aperture with a width 15mm and a height 10mm is also located on the

same plate with its center at

(

17.5,20

)

. The power-ground planes are separated by a dielectric slab with a relative dielectric constant

ε

_r =4.2 and a height h=1.6mm. Two different sources with their responses shown in Fig. 4.10 are injected to the port with coordinate

(

20,35

)

. Fig. 4.10a and Fig. 4.10b shows the time and frequency

0 0.1 0.2 0.3 0.4 0.2

0.4 0.6 0.8 1

Time (ns)

Isrc(mA)

(a) (b)

0 2 4 6 8 10

0.1 0.2

Frequency (GHz)

|Is| (mA)

0 0.1 0.2 0.3 0.4

0.1

Time (ns)

Isrc(mA)

(c) (d)

-0.1 0 2 4 6 8 10

0.1 0.2

Frequency (GHz)

|Is| (mA)

Fig. 4.10. (a) The time response of a standard Gaussian pulse and (b) its frequency

response; and (c) the time response of a modulated Gaussian pulse with zero DC

level and (d) its frequency response.

responses of the first current source with a standard Gaussian pulse response

( )

(

0 ^{2 2}

)

0 exp _d

s I t t t

I = − − , where

I

₀ =1mA, ps

t

₀ =167 , and

t

_d =33ps . This source is modulated by a 1GHz sine wave for the second source with a zero DC level, which has the time and frequency responses as shown in Fig. 4.10c and Fig. 4.10d, respectively. The input impedance at the same port is then calculated by frequency transforming the time responses of the voltage and current with FFT.

4.3.2 Simulation Results

Fig. 4.11 shows the magnitude of the frequency responses of the input impedance at the port shown in Fig. 4.9. The solid line is obtained by directly inversing the Y-matrix in frequency domain. The other two lines are the results by time-domain iteration and FFT with different sources of excitation. The result plotted in dashed line is excited by the first source, in which a DC level at stead-state in time-domain is transformed into a delta response at zero frequency. Since most energy is confined at DC, the precision of the results obtained at other frequencies is limited. The result of the second excitation, a modulated Gaussian current pulse with zero DC level, is plotted in a dash-dotted line. The sharp transition at zero frequency no longer exists and responses at other frequencies become closer to the solid line.

0 0.5 1 1.5 2 2.5 3 0

50 100 150 200 250 300 350

Frequency (GHz)

|Zin| (Ohm)

Frequency Time+FFT Time+FFT with 0-DC source

Fig. 4.11. The magnitude of the frequency responses of the input impedance at the port

shown in Fig. 4.9. The solid line is obtained by direct inversing the Y-matrix in

frequency domain. The dashed and dash-dotted lines are the results by

time-domain iteration and FFT with two different sources of excitation.

The result of the hybrid method is also obtained and compared. Fig. 4.12 shows the magnitude of the frequency responses of the input impedance at the port shown in Fig. 4.9 obtained by direct time-domain iteration (solid line) and the hybrid method (dashed line) with FFT. As shown in the figure, the hybrid method is able to

0 0.5 1 1.5 2 2.5 3 0

50 100 150 200 250 300 350

Frequency (GHz)

|Zin| (Ohm)

Direct Hybrid

Fig. 4.12. The magnitude of the frequency responses of the input impedance at the port

shown in Fig. 4.9 obtained by direct time-domain iteration (solid line) and the

hybrid method (dashed line) with FFT.

extract the major resonant frequencies of the structure. However the two responses are quite different at low frequencies. This is due to the nature of the model-order reduction process in the hybrid method in which the resonant modes were extracted for the reconstruction of late-time responses. Since the pole at DC cannot be treated as a “resonant mode”, it cannot be extracted by the hybrid method.

4.3.3 Modal Patterns

The modal patterns of the first two modes extracted by the hybrid method are shown in Fig. 4.13a and Fig. 4.13b. The first mode, as shown in Fig. 4.13a, is basically a variation of the first mode extracted from the structure with no defected metal plate shown in Fig. 4.3 by the same method. However the combination of the two degenerated modes no longer exists since symmetry is destroyed by the cut corner. The second mode extracted in this case shows a quite different pattern as the one extracted from the structure shown in Fig. 4.3, because the symmetry for this mode is heavily destroyed by both the cut corner and the aperture.

The modal patterns for the same structures are also extracted by Ansoft^® HFSS^™, a popular commercial full-wave simulation software package, for verification. The resulting modal patterns of same modes as Fig. 4.13a and Fig. 4.13b are shown in Fig. 4.13c and Fig. 4.13d, respectively, in which the magnitude and direction of electric field in the structure at different points are represented by the length and direction of arrows, respectively. It is obvious that the modal patterns for the first two modes extracted by the hybrid method are in good coherence to those extracted by HFSS. For example, for the first mode, both Fig. 4.13a and Fig. 4.13c

0 10 20 30 40 50 0

20 40 -0.460 -0.2 0 0.2 0.4

0 1020304050 10 0

3020 50 40

-0.4 -0.2 0 0.2 0.4

(b) (a)

Mode 1 (Hybrid Method) Mode 2 (Hybrid Method)

(d) (c)

Mode 1 (HFSS) Mode 2 (HFSS)

Fig. 4.13. (a) The first mode and (b) the second mode of the structure shown in Fig. 4.9,

which are extracted by the hybrid method; and (c) the first mode and (b) the

second mode of the same structure extracted by Ansoft^® HFSS^™.

show that the strongest field occurs at the two corners next to the one cut, with opposite directions. The field decrease gradually along the diagonal connecting these two corner and zeroes occur along a line connecting the cut corner and the aperture.

4.4 Convergence and Complexity

The hybrid method proposed in this chapter is composed of three major parts, the Delaunay-Vononoi Modeling of Power-Ground Planes, the time-domain iteration, and model order reduction by Krylov subspace method. The convergence with respect to the three parts will be discussed separately. A self convergence test will also be given. The complexity analysis is based on the step-by-step algorithm provided in 4.1. Both the complexity of computation and memory overhead will be considered.

4.4.1 Convergence

The convergence analysis begins with the Delaunay-Vononoi modeling of power-ground planes, where the mesh arrangement reveals that the accuracy is associated with the longest distance between any two connected virtual or source ports. As for the probing frequencies, its accuracy is ranged from 0.20 to 0.25 of

wavelength interested [22].

Taking the simulation in 4.2 for example, the longest distance between the ports

is 20 2mm, which means the highest frequency of accuracy results is ranged from 2GHz to 2.5GHz, according to the simulation parameters for this case. Inaccurate results such as spurious modes may arise above this frequency range, which can be observed in the simulation results.

As mentioned in [22], the domain of Delaunay-Vononoi modeling of power- ground planes reduced identically to the rectangular FDTD grid when the Voronoi tesserlation becomes rectangular shape. Since the time-domain iteration based on this model, which is discussed in 4.1, is also updated with explicit equations in the same manner as FDTD, therefore the stability constraint in the Courant number

≤1 Δ

=

c

Δ

t

S

, (4.10)

must be satisfied, where Δ is the smallest distance crossing a cell from one grid point to another [2].

As described in step 5) in the step-by-step implementation that summarized 4.1, in order to determine the number of modes that are needed to be extracted precisely,

the expansion coefficients of the eigenmodes are compared. After the Krylov subspace of order m is constructed, an approximate eigensolution set of the original system can be found by (2.20). The expansion coefficient

a associated with

_i

θ is

_i calculated by taking the inner product with the voltage vectors

V at the

ⁿ⁰

n -th

₀ time step of FDTD iteration that the Lanczos algorithm starts. Terms with small expansion coefficient can then be dropped safely.

In order to verify the self convergence of the hybrid method proposed in this chapter, a set of three mesh settings from coarser to finer for the structure shown in

Fig. 4.3 is arranged as shown in Fig. 4.14. The space division in both x and y directions are 10mm, 5mm, and 10/3mm for Mesh1 in Fig. 4.14a, Mesh2 in Fig.

4.14b, and Mesh3 in Fig. 4.14c, respectively. A Gaussian current pulse

( )

(

0 ^{2 2}

)

0 exp _d

s I t t t

I = − − , with

I

₀ =1mA, ns

t

₀ =10 , and

t

_d =2ns is injected to excite the structure.

The frequency responses of the input impedances at the incident port denoted by black dots in Fig 4.13 for all three mesh settings are calculated by frequency transforming the late-time response obtained by the hybrid methods with FFT.

Results are plotted in Fig. 4.15, where the dotted, dashed, and solid lines are obtained

(a) (b)

0 10 20 30 40 50

10 20 30 40 50

0 10 20 30 40 50

10 20 30 40 50

0 10 20 30 40 50

10 20 30 40 50

(c)

Fig. 4.14. (a) Mesh1, a coarse mesh setting, (b) Mesh2, a marginal mesh setting, and (c)

Mesh3, a fine mesh setting arranged for the verification of self convergence of

the hybrid method proposed in this chapter.

0 0.5 1 1.5 2 2.5 3 0

100 200 300 400 500 600

Mesh1 Mesh2 Mesh3

Frequency (GHz)

|Zin| (Ohm)

Fig. 4.15. Frequency responses of the input impedances at the incident port denoted by

black dots in Fig 4.13 for mesh settings Mesh1 (dotted line), Mesh 2 (dashed

line), and Mesh3 (solid line).

with mesh settings Mesh1, Mesh2, and Mesh3, respectively. It is obvious that as the mesh settings become finer, both resonant frequencies obtained converge to fixed values.

Modal patterns are also extracted. Fig. 4.16 shows the modal patterns of the first

0 1020 3040 50

Fig. 4.16. Modal patterns extracted by the hybrid method. (a) The first mode and (b) the

second mode extracted from Mesh1; and (c) The first mode and (d) the second

mode extracted from Mesh2.

two modes extracted by the hybrid method for mesh settings Mesh1 and Mesh2. Fig.

4.16a and Fig. 4.16c shows the modal patterns of the first mode extracted from Mesh1 and Mesh2, respectively. It is obvious that these two patterns are identical.

The second modal patterns extracted from Mesh1 (Fig. 4.16b) and Mesh2 (Fig. 4.16d) can also be recognized as the same pattern except for an inverted phase.

4.4.2 Complexity

As discussed in 3.4.2, a frequency response with a higher resolution in frequency requires a longer time period of steady-state response in time-domain. The previous subsection also mentioned that for the hybrid method proposed in this chapter to converge, a finer mesh setting in space leads to a smaller division in time as traditional FDTD simulation does. As a result, more simulation time is needed for better frequency resolution.

Assume that a mesh setting for some structure is determined for simulation by the hybrid method. According to the step-by-step algorithm that concludes 4.1, an equivalent circuit network for a structure is firstly constructed using Delaunay- Vononoi modeling. If a network of total number P nodes and branches is constructed,

( )

P

O basic arithmetic operations, e.g., multiplications and accumulations, is required for computing the values of capacitors and inductors.

After constructing the equivalent circuit network, a total number P of voltage and current values is required to be updated in every time step for the time-domain iteration. If L time steps are took for the sources fade to zero and the modal patterns begin to appear and after that N time steps are performed for a satisfactory frequency resolution. The overall computation time for the time-domain iteration is linearly

proportional to the number of field points and total time step computed, or

( )

(

P L N

)

O ⋅ + .

For the proposed hybrid method, normal time-domain iteration is firstly applied for the same L time steps. Krylov subspace method is then constructed for model-order reduction. Assume that a necessary number of voltage vectors are stored for the model order reduction process to converge without penalty and Q modes need to be extracted precisely and the Lanczos algorithm converges at the M-th time step.

For constructing a one-order-larger Krylov subspace, at most Q modal expansion coefficients are obtained and the eigensolution of the m×m tridiagonal matrix is also solved for the convergence criteria.

After that, the frequency response at a single field point with the same resolution can be directly obtained in O

( )

N basic arithmetic operations. The overall complexity of the hybrid method is therefore

^O ( ^P

^⋅

^L

⁺

^Q

^⋅

^N

⁺

^M

²

)

. In the usual

cases,

Q

≤

M

<<

L

<<

N

<<

P

, thus the normal time-domain iteration has an

(

P N

)

O ⋅ complexity, where as the hybrid method reduces the complexity to

(

P L

)

O ⋅ .

When penalty occurs, however, the overall model order reduction process is repeated with a larger number of voltage vectors. This means that another

^O ( ) ^M

²

operation is required for the new model order reduction process to complete.

Although an

^O ( ) ^M

² operation is a small part of the overall hybrid method, doing the same operation repeatedly with only another starting value is still a time-wasting job if penalty occurs too often.

Penalty can be avoided by pre-storing a larger number of voltage vectors for the construction of Krylov subspace. If plenty of voltage vectors are pre-stored, penalty never occurs. It is obvious that every vector pre-stored requires an O

( )

P of memory spaces and a minimum memory space of O

(

M⋅P

)

is required for the model order reduction process to converge without penalty. Since this minimum size of memory

space is unknown before completing the model order reduction process, overestimating the number of voltage vectors is necessary. If this number is chosen too large such that secondary storage is used, the overall performance will be seriously reduced by the model order reduction process.

4.5 Summary of the Chapter

A hybrid method combining the Delaunay-Vononoi modeling of power-ground planes in time-domain and Krylov subspace method in proposed in this chapter.

Taking advantage of the space information, only few time-domain iterations before the sources fade to zero is required for extracting the excited modes and reconstructing the late time response by analytic expression. With a simple example and a more realistic case, the correctness, efficiency, convergence, and complexity of this method have been verified.

95

5 5

Conclusions

YBRID methods that able to construct the late-time response of a system have been proposed in this thesis. In general, Krylov subspace method based model order reduction technique is applied to time domain full-wave electromagnetic simulations after the sources fade to zero for extracting the active modes in the system. Late-time responses are then constructed by the linear combination of the extracted modes.

This chapter concludes the thesis by firstly provide a summary of the work, where the basic concept and typical application of the hybrid methods proposed in this thesis will be briefly reviewed. Several suggestions for the future work, according to the limitations and disadvantages in the present development of the proposed hybrid methods, will then be given for those interested in the further study of these methods.

H

在文檔中 時域全波電磁模擬的模型降階與遲時響應 (頁 94-114)