3-d envelope finite element (evfe) solver with pml boundary conditions for microwave integrated...

3-D envelope finite element (EVFE) solver with PMLboundary conditions for microwave integratedcircuits

W. Yao and Y.E. Wang

Abstract: A perfectly matched layer (PML) formulation is developed for 3-D envelope finiteelement (EVFE) solvers. The PML performance is tested against different numbers of layers anddifferent smax values. The results show that the PML can provide sufficient absorption of incidentwaves when proper parameters are chosen. Numerical tests also show that this 3-D EVFEalgorithm with PML boundary conditions is unconditionally stable as the time intervals increase.Finally, the 3-D EVFE solver equipped with PML boundary conditions has been applied to themodelling of microwave integrated circuits such as MMIC interconnects and on-chip parasiticstructures. Good agreements with existing simulation results are obtained with improvedcomputational efficiency.

1 Introduction

Radio frequency (RF) or monolithic microwave integratedcircuits (MMICs) have been widely used in moderncommunication systems. These integrated circuits oftenprocess digitally modulated signals where the basebandinformation modulates on the RF carrier. Traditionally,electromagnetic transient analysis tools such as the finite-difference time-domain (FDTD) or finite-element time-domain (FETD) methods have been very useful inanalysing interconnects in microwave circuits, because oftheir capability to simulate a broadband response of thestructure. However, when the signal bandwidth to carrierfrequency ratio is very small in certain components anddevices, such as a high-Q filter, traditional transient analysesare no longer computationally efficient, as the time-domainsolvers require a great number of time steps to finishsimulating the waves. To achieve the maximum efficiency ofthe time-domain solvers, it is proposed to solve the time-domain envelope of the wave through a technique called theenvelope finite element (EVFE) method [1–5]. In such amethod, the carrier information is de-embedded from thenarrowband modulated signal. Thus only the complexsignal envelope is sampled and analysed during every timestep. The simulated bandwidth of the EVFE technique canbe much smaller compared to that of its time-domainorigin, the FETD method, which results in a much lowertime-dispersion error, as shown in the numerical test [2].This also implies that much larger time intervals can be usedin the EVFE method than the FETD method. In fact, thetime intervals used in the EVFE technique are no longergoverned by the Nyquist sampling rate of the original RF

E-mail: [email protected]

The authors are with the Department of Electrical Engineering, University ofCalifornia, Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90095, USA

r The Institution of Engineering and Technology 2006

IEE Proceedings online no. 20050155

doi:10.1049/ip-map:20050155

Paper first received 1st July 2005 and in revised form 7th April 2006

IEE Proc.-Microw. Antennas Propag., Vol. 153, No. 6, December 2006

signal but that of the signal envelope. This method has beenproven to be stable through numerous numerical tests andthrough theoretical proof [1–5]. In contrast, the recentlyproposed envelope domain FDTD method [6] still requirestime intervals less than half of the carrier period in order toremain stable. It can be asserted that the EVFE is apowerful tool to simulate the transient response ofcomponents and devices in the narrowband system.The concept of the EVFE technique also makes itpossible to perform the electromagnetic (EM) andcircuit co-simulations combined with a circuit envelopesimulator [7].

EVFE techniques have been applied to various guidedwave problems [3, 4] with the first-order absorbingboundary conditions (ABC). However, it is well knownthat the better choice to terminate the computationaldomain is the perfectly matched layer (PML) boundarycondition, which has wider bandwidth and can providemore absorption of the incident waves. A perfectly matchedlayer was first introduced into the finite-difference time-domain (FDTD) method by Berenger [8]. Sacks etc. [9]suggested a new PML based on a lossy uniaxial mediumand successfully implemented it into the frequency-domainfinite-element method. Gedney [10] further developed theformulation for the FDTD method with an anisotropicperfectly matched layer and applied it to the analysis ofmicrowave circuits and antennas. Recently, a PML hasbeen successfully implemented with the FETD simulator inthe analysis of scattering problems [11] and active nonlinearmicrowave circuit modelling [12]. Here, based on theanisotropic PML concept, the 3-D PML formulation forthe EVFE technique is derived. Several numerical tests andexamples validate the good performance of the proposedPML scheme. With this high performance boundarycondition, a full 3-D EVFE solver is developed for theanalysis of microwave integrated circuits such as MMICinterconnects. Numerical test results demonstrate that theproposed technique promises both high efficiency andaccuracy for various interconnect structures, especially forthose with high-Q resonances in their transient responses.

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2 EVFE-PML formulation

In this Section, the EVFE-PML formulation is derived.Without losing generality, the 3-D PML formulationis presented, which can be easily reduced to a 2-Dformulation by setting the operator of the third axis tozero. The time-harmonic Maxwell’s equations in the PMLregions are:

r�H ¼ joe��eE þ J i

r� E ¼ �jom��mHð1Þ

Note that the permittivity and permeability of the PML areanisotropic tensors given by,

��e ¼ ��l ¼

sysz

sx0 0

0sxsz

sy0

0 0sxsy

sz

266664

377775 ð2Þ

where

si ¼ 1þ si

joe0i ¼ x; y; z ð3Þ

Assuming that there is no source excitation in the PMLregion, the vector wave equation can be derived from (1),(2) and (3) as,

r� ðm�1 ��m½ ��1 � r� EÞ � o2e ��e½ �E ¼ �joJ ð4ÞMultiplying the vector testing function T on both sides of(4) leads to the weak form integral over each cell of the finiteelement mesh:Z

V

1

m ��m½ � � r� E

� �� r� Tð ÞdV �

ZS

T� 1

m ��m½ � r�E

� �dS

�Z

VT � o2e ��e½ �EdV ¼ �jo

ZV

T � J i dV ð5Þ

The vector testing function T is chosen to be the sub-domain finite-element basis function defined for eachelement. Merging the weak-form integrals for each cellleads to the volume integral over the complete computa-tional domain, whereas the surface integral terms reduce toan integral over the outer boundary of the computationaldomain only. This is because of the continuity of thetangential electric fields on the inter-element boundaries.The boundary integral vanishes when the computationaldomain is enclosed by the conductor-backed PML. Withedge element vector bases, the electric field vector withineach cell can be expanded in terms of its values on the edges,which is

E ¼XN

j¼1N jej ¼

XN

j¼1ejðNj

x ;Njy ;N

jzÞ ð6Þ

The testing vector T can also be expressed in its componentform, consisting of the same set of bases:

T ¼ ðN ix;N

iy ;N

izÞ ð7Þ

Substituting (6) and (7) into (5) yields:

� o2ejQxsysz

sxþ ejPx

sx

sysz� o2ejQy

sxsz

syþ ejPy

sy

sxsz

� o2ejQzsysx

szþ ejPz

sz

sxsy¼ �jo

Zv

N i � J i dv ð8Þ

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where

Qx ¼Z

veNj

xN ix dv Qy ¼

ZveN j

yN iy dv

Qz ¼Z

veNj

z N iz dv Px ¼

Zv

1

mðr�N jÞxðr�N jÞx dv

Py ¼Z

v

1

mðr�N jÞyðr�N jÞy dv

Pz ¼Z

v

1

mðr�N jÞzðr�N jÞz dv ð9Þ

To solve (8), three auxiliary variables need to be furtherdefined, which are:

Fx ¼sx

syszei;Fy ¼

sy

sxszei;Fz ¼

sz

sxsyei ð10Þ

Thus, (8) is reduced to:

�o2FzQxs2y þ PxFx � o2FxQys2z þ PyFy

� o2FyQzs2x þ PzFz ¼ �joZ

vN i � J i dv ð11Þ

Note that both (8) and (11) are frequency-domain formulasand they will be transformed into the envelope domain todescribe the time variation of the signal envelope. Thecomplex signal envelopes for the above quoted variables aredefined as:

ejðtÞ ¼ ujðtÞejoct

FjðtÞ ¼ cjðtÞejoct

JiðtÞ ¼ jiðtÞejoct

ð12Þ

where oc is the carrier frequency. To generate the fieldequations in the envelope domain, the above frequency-domain formulas can be first transformed into the time-domain through Fourier transform, then further convertedto the envelope domain through (12). An alternative is todirectly apply the following frequency domain to theenvelope-domain transform,

jo$ @

@tþ joc � o2 $ @2

@t2þ 2joc

@

@t� o2

c ð13Þ

Substituting (3) into (11) and applying (13), the weak-formwave equation in the envelope domain becomes:

Qxd2cz

dt2þ S1½ �

dcz

dtþ S2½ �cz þ Qy

d2cx

dt2þ S3½ �

dcx

dtþ S4½ �cx

þ Qzd2cy

dt2þ S5½ �

dcy

dtþ S6½ �cy ¼ �

@fi

@tþ jocfi

� �

ð14Þwhere

S1 ¼ 2 joc þsy

e0

� �Qx S2 ¼ joc þ

sy

e0

� �2

Qx þ Pz

S3 ¼ 2 joc þsz

e0

� �Qy S4 ¼ joc þ

sz

e0

� �2

Qy þ Px

S5 ¼ 2 joc þsx

e0

� �Qz S6 ¼ joc þ

sx

e0

� �2

Qz þ Py

fi ¼Z

vN i � J i dv

ð15Þ

To solve (14) by computer, the Newmark-beta formulation[3] is used to discretise the differential equation in the time


domain, which yields:

R1cnþ1z þ R2c

nz þ R3c

n�1z þ R4c

nþ1x

þ R5cnx þ R6c

n�1x þ R7c

nþ1y þ R8c

ny þ R9c

n�1y

¼ � f nþ1i � f n�1

i

2Dtþ joc

f nþ1i þ 2f n

i þ f n�1i

4

� �ð16Þ

where

R1 ¼Qx

Dt2þ S1

2Dtþ S2

4

� �

R2 ¼ �2 Qx

Dt2þ S2

2

� �

R3 ¼Qx

Dt2� S1

2Dtþ S2

4

� �

R4 ¼Qy

Dt2þ S3

2Dtþ S4

4

� �

R5 ¼ �2 Qy

Dt2þ S4

2

� �

R6 ¼Qy

Dt2� S3

2Dtþ S4

4

� �

R7 ¼Qz

Dt2þ S5

2Dtþ S6

4

� �

R8 ¼ �2 Qz

Dt2þ S6

2

� �

R9 ¼Qz

Dt2� S5

2Dtþ S6

4

� �

ð17Þ

and n is the time-step number.Similarly, (10) has to undergo the same frequency to

envelope transform given by (13), which leads to

cnþ1x ¼ ax5c

nx þ ax6c

n�1x þ ax7unþ1 þ ax8un þ ax9un�1

ð18Þ

The coefficients are defined as:

ax1 ¼ 2joc þsx�1 þ sx�2

e0

ax2 ¼sx�1sx�2

e20þ joc

sx�1 þ sx�2e0

� o2c

ax3 ¼ 2joc þsxe0

ax4 ¼� o2c þ

sxe0

joc

ax5 ¼2

Dt2� ax2

21

Dt2þ ax1

2Dtþ ax2

4

ax6 ¼�1

Dt2� ax1

2Dtþ ax2

41

Dt2þ ax1

2Dtþ ax2

4


ax7 ¼1

Dt2þ ax3

2Dtþ ax4

41

Dt2þ ax1

2Dtþ ax2

4

ax8 ¼� 2

Dt2þ ax4

21

Dt2þ ax1

2Dtþ ax2

4

ax9 ¼1

Dt2� ax3

2Dtþ ax4

41

Dt2þ ax1

2Dtþ ax2

4

ð19Þ

To solve the time-dependent complex signal envelopes, oneneeds to associate (16) with (18). A simple way to solvethem jointly is to substitute (18) into (16), which yields atime-marching form of the field variables:

az7R1 þ ax7R4 þ ay7R7

� �unþ1

j þ az8R1 þ ax8R4 þ ay8R7

� �un

j

þ az9R1 þ ax9R4 þ ay9R7

� �un�1

j þ ax5R4 þ R5ð Þcnx

þ ax6R4 þ R6ð Þcn�1x þ ay5R7 þ R8

� �cn

y

þ ay6R7þR9

� �cn�1

y þ az5R1þR2ð Þcnzþ az6R1 þ R3ð Þcn�1

z

¼ � f nþ1i � f n�1

i

2Dtþ joc

f nþ1i þ 2f n

i þ f n�1i

4

� �ð20Þ

The time iteration can then start with (20), relying on (18) tosupply all the known variables for the next time step. Theinitial conditions to start the time iterations are:

u0j ; u�1j ;c0

j ;c�1j ¼ 0 ð21Þ

The time stepping is carried out as follows:

Step 0: Solve for the inverse of the matrix coefficient of unþ1j

in (20) using sparse matrix techniques. This only needs to bedone once.

Step 1: Use (20) to calculate the envelope vector U n+1 withthe signal envelope vectors U n, U n� 1 and wn, wn� 1 inprevious time steps.

Step 2: Use (18) to calculate the envelope vector wn+1 withthe signal envelope vectors U n+1 U n, U n� 1 and wn, wn� 1.

Step 3: n¼ n+1, and go back to step 1.

With the above iterative scheme and the sparse matrixtechniques, the complex signal envelope vectors for eachtime step u¼ [u1,u2,y,uN] and W¼ [C1,C2,y,CN] canthus be solved.

The computational efficiency of the EVFE technique issuperior to that of the conventional FETD method,especially when only a small fraction of the bandwidth isinterested. If the fraction of the bandwidth is defined asBW¼ ( fhigh� flow)/fc, where fhigh, flow are, respectively, thehighest and lowest frequencies in the band and fc is thecentre frequency. The total number of time steps requiredshould be less than in the conventional FETD method by afactor of BW/2, which results in the same proportion ofcomputational time saving if an iterative solver is used toinverse the finite-element matrix. Moreover, the dispersionerror caused by the time discretisation is also significantlylower than that in the FETD method in the interestedfrequency band [13]. The overall computational efficiency ofthe EVFE technique can not be directly compared to theFDTD method, because it differs in terms of the

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implementation of the matrix solver and the meshingscheme. However, the number of time steps needed for theEVFE technique is usually far less than in the FDTDmethod, since there is no stability condition which limits themaximum number of time-sampling intervals. The EVFEtechnique also offers the standard advantages of finite-element methods over the FDTD method in terms of itsflexible meshing schemes, which can well represent the finedetails of the physical structure.

3 PML stability and performance test

The reflection of a non-perfect PMLmainly comes from thespatial discretisation error. To minimise this error, aspatially variant conductivity along the normal axis of thewave absorption should be chosen [10]:

szðzÞ ¼smax z� z0j jmffiffiffiffi

erp

dm ð22Þ

where z0 is the interface between the PML region and thenon-PML region, d is the depth of the PML and m is theorder of the polynomial variation. Order m¼ 2 is chosen forthe numerical test here.

Two numerical experiments are performed to validate theEVFE-PML formulation and to examine its absorptionperformance. The first numerical example is a 3-D emptyrectangular waveguide terminated with a PML absorber, asshown in Fig. 1. The cross-section of the waveguide is10.16mm (x direction) by 22.86mm (y direction). Theminimum mesh sizes in the x, y directions are, respectively,Dxmin¼ 1.693mm and Dymin¼ 1.95mm. The mesh size inthe z direction, which is the propagation direction, isuniform with Dz¼ 2mm. The second example, see Fig. 2, isa shielded microstrip line printed on a substrate with adielectric constant of 9.8 and thickness of 7.4mil. Theindividual cell dimensions are Dxmin¼ 7.4mil, Dymin¼1.85mil and Dz¼ 10mil, respectively [1mil¼ 0.001 inch�0.0254mm].

Figure 3 shows the PML absorption performance againstdifferent PML depths for the rectangular waveguide. Theexcitation’s carrier frequency is about 10GHz and thebandwidth is about 4GHz. The time step is Dt¼ 5ps andthe parameter smax changes according to different PMLdepths. The excitation plane is set at the planar metalcontacts (PMC) boundary at z¼ 0, whereas the observationpoint is set at z¼ 10mm. The result shows that only fourlayers of PML can provide about 40dB absorption whenthe smax is carefully set. Figure 4 shows the PMLperformance against different smax for the second example,i.e. the shielded microstrip line. Again, the excitation planeis set at the PMC boundary underneath the strip line atz¼ 0 and the observation point is set at z¼ 100mil. In this

Fig. 1 Empty waveguide with PML, a¼ 10.16 mm, b¼ 22.86 mmand c¼ 300 mm

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case, the PML depth is fixed to 200mil but different smax

values are used. It shows that an optimal value of smax

exists, which can maximise the PML absorption. In general,the optimal value of smax can be estimated through thefollowing formula [10]:

smax �ðmþ 1Þ150pDz

ð23Þ

where m is the order of variation in s in (22). The thicknessof the PML should be at least two times the skin depth ofthe PML at the lowest frequency for sufficient absorption.

Fig. 2 Shielded microstrip line with PML, a¼ 74 mil, h¼ 7.4 mil,b¼ 100 mil and c¼ 500 mil (er¼ 9.8)1mil¼ 0.001 inch� 0.0254mm

Fig. 3 PML absorption the rectangular waveguide for differentnumbers of layers of PML

Fig. 4 PML absorption versus different smax values for theshielded microstrip line


Fig. 5 Normalised time-domain waveforms at z¼ 10 mm with different time steps Dt¼ 5Dt0, 20Dt0, 50Dt0, 100Dt0 for the waveguide example

A good approximation of the minimum thickness d is givenas follows:

d ¼ 1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif � 10�7 � smax

p ð24Þ

Figure 5 shows the stability of the EVFE algorithm with thePML boundary condition. For the above waveguideexample, the maximum time step satisfying the CFLcriterion is Dt0¼ 3.85ps, which serves as a reference. Fourdifferent time-step sizes: Dt¼ 5Dt0, 20Dt0, 50Dt0, 100Dt0 arethen separately used to run the EVFE simulation. Thewaveforms at the z¼ 10mm are plotted in Fig. 5. It showsthat the algorithm remains stable as the time-step sizeincreases. However, the time dispersion increases as well.

4 Anaylisis of microwave integrated circuits

Passive structures in microwave integrated circuits, such asMMIC interconnects and other on-chip parasitic compo-nents, require accurate electromagnetic modelling. Some ofthe components, such as high-Q chip inductors, may exhibitstrong frequency dependency at their resonant frequency. Itmay take a long time to precisely predict the frequency-domain response of such structures when a conventionaltime-domain simulator is used. For example, a microstriplow-pass filter, with a dielectric constant of 2.2 is analysedand the physical structure is shown in Fig. 6. In thisexample, four layers of the PML are set in each direction,except on the ground plane. The minimum mesh sizesin each direction are: Dxmin¼ 31mil, Dymin¼ 23.75mil andDzmin¼ 25mil. The excitation is a Gaussian pulse modulat-ing on a carrier frequency of 5GHz, with a total bandwidthof 8GHz The time step used in the simulation is 10ps,which is about 6 times that required by the Courant–Friedrichs–Lewy (CFL) stability criterion. Figure 7 showsthe time-domain waveforms of the electric field and Fig. 8shows the magnitudes of the S parameters. The resultscorrespond well with those calculated using the FDTDmethod. In fact, if the interested frequency band is just inthe vicinity of the resonant frequency, more computationsaving can be achieved by just cutting down the excitationbandwidth and increasing the time step accordingly.

The second example is a vertical metallic via hole in thedielectric substrate, which is widely used in modern MMIClayouts. It provides a short circuit to the different layers of


the MMIC or a feeding point for the RF excitation signals.Figure 9 shows a chip-level grounded via in a microstrip lineprinted on a GaAs substrate (er¼ 12.9). The sizes of thisstructure are: h1¼ 250mm, h2¼ 600mm, w2¼ 600mm,w3¼ 1785mm. Here we assume that the via has a squarecross-section with an edge length of 85mm. The two sidesand the bottom and top surfaces are perfect electric

Fig. 6 Geometry of a microstrip low-pass filter1mil¼ 0.001 inch� 0.0254mm

Fig. 7 Time-domain envelopes of the electric field

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conductor (PEC) walls. The front and end are truncated byPML absorbers. In this example, we want to show theEVFE-PML technique’s capability of predicating broad-band performance ( from 0 to 30GHz); thus we set thecarrier frequency as 15GHz with the excitation’s bandwidthas 15GHz. The time step is 2ps, which is about 4 timessparser than that required by the regular FETD. The PMLdepth is about 5000mm. The magnitudes of S11 and S21 areplotted in Fig. 10, and the results have good agreement withYook’s in [14].

Fig. 8 Magnitudes of S parameters

Fig. 9 Geometry of a microstrip grounded via

Fig. 10 Magnitude of S11 and S21

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The third example is two microstrip lines joined by an airor dielectric bridge, as depicted in Fig. 11. The microstripline has a width a and a thickness 0.2a, which is againprinted on the GaAs substrate (er1¼ 12.9). The bridge has alength 12.7a with the dielectric constant er2. Like theprevious example, the structure is enclosed by PEC walls,except the front and end which are terminated by PMLabsorbers. In EVFE formulations, we set the centrefrequency to k0a¼ 0.24 and the bandwidth to twice thecentre frequency, where k0 is the wave number in free space.The PML depth is about 4a. The magnitude of thesimulated S11 is shown in Fig. 12 against different values ofer2. The results are also compared to those of the frequency-domain finite-difference method [15] in Fig. 12. A goodcoincidence between the two sets of results can be observed.

5 Conclusion

In this paper, an anisotropic PML formulation has beendeveloped and implemented into a 3-D EVFE solver.Numerical examples of an empty rectangle waveguide and ashielded microstrip line have been presented to validate thePML’s performance; more than 40dB absorption has beenachieved when a 4-layer absorber is set. The performance ofthe EVFE-PML technique is validated through thesimulation of three microwave integrated circuit structures.It is expected that the EVFE-PML solver can be easilyextended for modelling of other types of 3-D electromag-netic problems.

1rε0.2a

a

2a

d

1rε

5a

a

2rε

Fig. 11 Geometry of an air or dielectric bridge in MMICs

Fig. 12 Magnitude of simulated S11


6 References

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9 Sacks, Z.S., Kingsland, D.M., Lee, R., and Lee, J.-F.: ‘A perfectlymatched anisotropic absorber for use as an absorbingboundary condition’, IEEE Trans. Antennas Propag., 1995, 43,pp. 1460–1463

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13 Yao, W., andWang, Y.: ‘Numeric dispersion analysis of 3D envelope-finite element (EVFE) method’. Proc. IEEE Antennas and Propaga-tion Society Int. Symp., 2003, Vol. 1, pp. 706–709

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3-d envelope finite element (evfe) solver with pml boundary conditions for microwave integrated...

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