trace and the ground plane also a new discretization of parts of the dielectric layer .... for the infinite ground plane in the computation without the dielectric the ...
Efficient Computation of Radiated Fields from Finite-Size Printed Circuit Boards Including the Effect of Dielectric Layer Marco Leone, Hermann Singer Technical University Hamburg-Harburg 2 1079 Hamburg - Germany
Abstract -
The rigorous analysis of a finite-size printed circuit board by the Method of Moments based on a full discretization of the whole three-dimensional structure requires a high numerical and modeling effort. A suitable simplification which drastically reduces the computation time is to use an equivalent-wire model for the traces situated within an homogeneous medium with an effective dielectric constant to account for the dielectric layer. For the subsequent determination of the radiated fields the dielectric layer is commonly ignored. In this paper we show in which cases this can lead to considerable prediction errors, and present a method based on polarization currents to include the effect of the dielectric layer, retaining the numerical efficiency of this approach. Examples are given to demonstrate the importance of the proposed extension.
INTRODUCTION
The radiated-emission analysis of a complex printed circuit board (PCB) with a full-wave approach such as the Method of Moments (MOM), based on a full discretization of the structure, represents a challenging task with regard to computational resources and modeling effort. The discretization of traces, ground plane and dielectric layer, generally leads to huge linear systems to be solved, with thousands of unknowns, resulting in very long computation times. For highly accurate results, which can in principle be obtained by a full discretization, the three-dimensional structure has to be carefully modeled, considering the required strong variation of the discretization density, e.g. nearby traces. The difficulties in handling such very large systems quickly arise if a modification, e.g. a change of a trace is necessary. Besides the trace and the ground plane also a new discretization of parts of the dielectric layer would be needed, which additionally increases the costs for simulation during a product design phase. Therefore, the question arises how to reduce the computational effort. A suitable approach is to use an equivalent-wire model, replacing the traces by wires with an equivalent radius, situated in an effective dielectric medium. The discretization of the dielectric substrate is avoided and the fastidious transverse discretization of the traces to account for the edge singularities of current and charge is not necessary. Only a longitudinal segmentation of the equivalent wires is required, according to the signal wavelength. A trace section
0-7803-5057-X/99/$10.00 © 1999 IEEE
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may be simply defined by its start and end coordinates. The resulting drastic reduction of complexity enables the practical simulation of a finite-size PCB, providing a reasonable compromise between numerical effort and accuracy. This was demonstrated for the infinite structure by a detailed applicability study [l]. But as shown e.g. in [2], this approach applies also well for arbitrarily shaped finite-size PCBs. The remaining two-dimensional discretization of the finitesize ground plane yields the dominant part of the total number of unknowns. A suitable strategy to minimize the discretization effort is demonstrated in this paper. The dielectric layer which is considered in the computation of the currents on traces and ground plane must be also taken into account for a subsequent complete field computation. This cannot be done in the same way as for the infinite structure [ 11, for which appropriate far-field Greens functions exist. The order of the resulting error when the dielectric is ignored is clarified by a comparative example. To include the effect of the dielectric on radiated fields also for finite size PCBs we present a new method based on an equivalent polarization: current model.
COMPUTATIONOFCURRENTS, ONTRACESANDGROUNDPLANE
The basic idea of the equivalent-wire approach is to replace each trace section by a wire line with equal propagation characteristic, i. e. equal per-unit length transmission line parameters. As sketched in Figure 1, this is accomplished by placing the whole structure within a homogeneous medium with an effective dielectric permittivity r&rr and choosing a suitable equivalent-wire radius req. The unknown currents on the traces and on the ground plane are determined by the MOM, which is applied on the whole equivalent structure.
Figure 1. Equivalent wire model For finite-size PCB, the dielectric layer is accounted for by the effective permittivity E,.,,~.
The obtainable reduction of the number of unknowns comparedto a full surface discretizationof tmces and dielectriclayercanbe roughlyestimatedto be at leasta factor of 113,if we only considerthe additionaldiscretizationof the upper and lower plane of the dielectric layer. Assuminga standardLU solutiontechniquefor the linearequationsystem, with a growthof computationtime.whichis proportionalto the third power of the numberof unknowns,we can expect a computational speedup of > 27.
(2)
Er,& =
z, =
The Equivalent Wire Model
The electromagnetic propagationalonga traceon a multilayer PCB with a groundplanecanbe well described by its per-unitlengthtransmissionline parameterswithin a wide frequency band.
-Weff+1.393+ 0.66; 120x51 h $z
we7 h=h+
w
1.25rth
; I?r,l h 2n’
(4)
n
!r>l. h‘ -211
Figure2. A PCBhate with mtcmstripcross-section Within this quasi-TEMrange.[4] the cross-sectional electric Therequiredradiusfor the equivalentwire line rq is foundby and magneticfield distributionscorrespondto the staticcase. a characteristic impedance matchgivenby It is common to refer to the secondarytransmission-line parameters phasevelocityandcharacteristic impedance. Since the substrateis assumedto be nonmagnetic (Jo,= 1) the phase res= 2hexp (3 velocity is usuallygiven by an effectiverelativepermittivity E,~ of a fictitious homogeneous medium.The dielectricloss tangentfactorsas well as the conductorresistivitiesof typical As wasfoundin [l] relation(5) impliesa maximumw/h ratio PCBsarerelativelysmallso that lossescan be neglectedwith of about6. This valueis only slightly dependent on common regardto usualtrace lengths[3]. With increasingfrequency sbip thicknesses. thequasi-TEMassumption graduallybreaksdown.Theregion of transitionto a more.complicateddispersivebehavioris For thecaseof a net with tracesof differentwidthstheuseof a relatively wide so that only an approximatefrequencylimit mean-valueeffective pennittivity is proposed.Using the mean-valueeffective pennittivity in (5) the individual f b,,mcanbe given[4] characteristicimpedanceof eachtrace can still be matched. For the phase-propagation constantsa small error remains, f&ml =m (1) which lies within a few percent,if the w rangeis not too large [Il. The validity of the equivalent-wireapproachwith regardto The evaluationof (1) with typical PCB dimensions(trace trace discontinuitiesand mutual trace coupling was also width w and dielectric thicknessh) yields valuesfor f8,sminvestigatedin [ 11.The observeddeviationswere found to ranging from 1 to lOGI&. Thereforethe whole range of have only a minor influence on the overall radiation practicalinterestis coveredby the quasi-TEManalysis. prediction. The transmissionline parameters,expressedas effective This results we also attributed to the applicationof the dielectric constant E+, and charactaistic impedance&, equivalent-wireapproachto finite-sizePCBs.However,with dependon the substratedielecaicconstantE, and the ratiosof increasingfrequencywhen the size of the ground plane strip width and strip thicknessto substrateheightw/h and t/h, becomescomparable to the wavelengththe currentdistribution respectively. z, andE,.~~canbe calculatedby variousclosed- will certainlyshowanerror,whichis dueto the approximation form expressions of differentaccuracygivenin the microwave by the effective homogeneous medium.It would be worth literahue,e.g. [4], [5]. A simpleand accuratesetof equations clarifyinghow the overallradiationpredictionis influencedby whichincludesthe stripthicknessf is givenby [5] this error.
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Because the tracewidth w is generallymuchsmallerthanthe wavelength L a simplecurrent-filament approximation applies. After all currentsin the conductingstmctureshave been The filamentscarry the total tracecurrentsas determined by detemGned asdescribed in the preceding sectiontheresulting thepreceding equivalent-wire M O Mcomputation. radiatedfields are.computedin a secondstep.For a rigorous field computation the intluenceof the dielectriclayershould To calculatethe far-field EL,,, the explicitknowledgeof J,, be takeninto account.For electricallylargePCBswith a solid is not necessary.Due to w, h cc h and the fact that the groundplanethis can be accuratelydoneusingclosed-form polarization currentis mainlyconcentrated within a few trace Green’sfunctionsfor currentsaboveand within an infinite widthsthe total per-unit-length polarizationcurrentflowing grounded dielectricslab(e.g.vias)[l]. For the finite structure no suchGreen’sfunctionsexistandtheeffectof thedielectric from the traceto the groundmay be lumpedto a vertically orientedper-unit-length Hertziandipole.momentdI,+i&e, layeris commonlyignored. Due to symmetry.the contributionof the horizontalcurrent component Jpol,Y is cancelled (seeFigure4). COMPUTATTONOF RADIATED FIELDS
The Polarivrtion Cwrent Model
Figure3 showshow the radiatedfield E’ of a PCB can be decomposed into two components. Onecomponent denotedas Eb is due to the currentsflowing in the tracesand in the groundplane,without the dielectric.The other component EL1 accountsfor the effectof the dielectricwhich is repw sentedby an equivalentpolarizationcurrentdensityJ,I [7] inducedby theelectricfield E in thedielectriclayer
J por= joeo(&r -l)E
.
\,
.
Figure4. Per-unit-lengthHertzian dipole-moment far-field approximation for polarization currents
(6)
The total radiatedfield is thereforethe sum of the two The unknownper-unit-length polarizationcurrentd1,dd.xis contributions givenby the integralover the verticalcomponent J,I,z of the polarization currentdensityin thedielectric E’ = E; +Erw, (7) (8)
Bothcurrentcomponents radiatein freespace. Inserting(6) into (8) yields dIpd_. --~oe~(~,-l) dx
(9)
Usingthedivergence theorem
andexpressing the per-unit-length chargeon the trace.dQ/& by thecontinuityequation 0 I
dQ_ dx
1 dI jodx
(11)
Figure3. Structuredecomposition: Contribution of currentsI in conductors (left), effectof dielectricrepresented by equivalent we finally obtainthesimplerelation polarization current.Jw(right). Theradiatedfield E& canbc directlycomputed by integration overtheknowncurrentsin the tracesandin the groundplane.
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(12)
The per-unit-length polarization current is therefore simply given by the derivative of the trace current I with respect to the trace coordinate x. It can be seen from (12) that for & + 1 (vacuum) the polarization current vanishes. The radiation contribution of the polarization current can be determined together with the integration over the horizontal trace current using the free-space Green’s functions of a vertical dipole [6].
infinite grounded dielectric slab. The same configuration as described above is used. Due to the infinite ground plane used in this example the field contribution of the polarization currents has to be calculated using image theory. Figure 6 shows the resulting vertical radiation diagrams Es@), as sketched by the insert of Figure 5. The frequency was set to 800 Ml& because of the maximum influence of the dielectric observed above for this configuration. The results are in excellent agreement.
EXAMPLES To demonstrate the potential prediction error when neglecting the dielectric we compare the radiated electric field of a single trace with and without the dielectric. The investigation is carried out first for the infinite structure using the exact Green’s functions for the reference computation. To account for the infinite ground plane in the computation without the dielectric the image principle [6] is applied to the trace current. Both, the horizontal as well as the vertical current contributions (vias) are considered. Figure 5 shows the frequency-dependent maximum deviation for Ea in the vertical plane, within /0l< 90”, (see insert of Figure 5). The following parameters were chosen: E,= 4.7, h = 1.5 mm, w = 1 mm, t = 35 pm (see Figure 2). The trace has a length of 10 cm and is connected by short vertical wires (vias) to the ground plane. It is excited at one end by a voltage source with an internal impedance of 50 LJ. The other end of the trace is simply short circuited to the ground. As can be seen from the resulting error curve considerable deviations occur with increasing frequency, i.e. with increasing electrical length of the trace. The difference has a maximum of about 22 dB in the range of 800 MHz. This frequency approximately corresponds to the half-wavelength resonance of the trace. For longer traces this maximum is even shifted towards lower frequencies.30 ,
1
I
I
I
252015lo5-
Figure 5. Maximum deviation of radiated electric field from a 10 cm trace on an infinite PCB when the dielectric layer is ignored (trace current correctly computed with dielectric). To validate the proposed polarization-current method we compare the radiation diagram of a single trace on an infinite PCB with the result obtained using the Green’s function of the
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-
0
Equiv. polarization currents ----------Green’siimctions Figure 6. Vertical radiation diagram with polarization currents compared to exact Green’s functions result (infinite structure). In the next example we present the radiated emission prediction of the same trace on a finite-size PCB with 20 cm length and 15 cm width using the proposed polarizationcurrent method. All other parameters are the same as in the previous example. The trace is centered to the middle of the ground plane running parallel to the longer side. The discretization of the ground plane was accomplished by triangular and rectangular patches (Figure 7). The computation were performed with CONCEPT [8], using the part of the MOM code which is based on the Electric Field Integral Equation (EFIE) for metallic wires and plates. The optimized mesh was created using the CAD tools included in the program. A total number of 692 unknowns was obtained. Very small patch sizes were used only where absolutely needed, e.g. under the wire to account for the strong transverse variation of the return current. This is illustrated by a zoom on the region near the left wire connection to the ground plane (see Figure 7). Also near the borders the discretization is finer to account for the edge currents. Figure 8 shows the vertical radiation diagram in the x-z plane for the frequency of 800 MHz. For comparison the radiation diagram without the dielectric is included. It should be noted that both diagrams are based on the same conductor currents, as determined by a preceding equivalent-wire MOM computation with the dielectric included. The observed maximum deviation at jet= 90” even exceeds the estimation of the infinite case in Figure 5. The results clearly demonstrate that the dielectric layer can have a considerable suppressing effect on radiated emissions. Further, the diagram reveals that at some frequencies the overall radiated field amplitude can be considerably underestimated if only the usual observation
point perpendicularabove the PCB (0 = 0”) is considered.It is interestingto note that the field in the shadowregion under the PCB is of the sameorder of magnitudeas above the PCB.
~,=4.7, h=0.32nnn, w=O.13 mm, r=50pm (see FignreZ). Both terminals of the trace are connected to the output and input of digital components,which are representedby linear equivalent circuits as depicted in Figure 10. The excitation represented by the voltage source V(t) has a trapezoidal waveform with 5 V amplitude and a period time of 30 ns. The duty cycle is 50 W and the rise and fall time is each 1.4 ns.
T”
cm
Figure 9. Exampleof B trace running &se to the edgeof a PCB (exampletaken from [9]).
Figure 7. Discretizationof the ground plane (20 cm x 15 cm), the areaaround the left wire connectionis zoomedout.
e
Figure 10.Linear equivalentcircuits to representthe output Oeft) and the input (right) of logic gatesconnectedto the trace The resulting spectraldistribution of the electric field strength for the first 15 harmonicsat the distanceof 3m above the PCB (x = y = 0, z = 3m) is shown in Figure.11.
Figweft. Vertical radiation diagram of a ftnitestze PCB Figure 11.Spectraldistribution of the electricfield at the distance (20cmx 15cm) with a 10cm trace compared to the result of3 mabovethecenterdthePCB without dielectric(currentscorrectly computedwith dielectric). The spatial variation of the.electric field for the 15& harmonic As a last example the radiated emission of a trace placed near (f= 966.7 MHz) is demonstratedby the vertical and horizontal the edge of the PCB is investigated. The configuration was radiation diagram in Figure 12 and Figure 13, respectively. taken from the benchmarkexample No. 12 in [91. A sketch of Both diagrams were.calculated for a distance of 3 m from the the geometry is given in Figure 9. The further parametersare: center of the PCB. The vertical diagram again shows that the
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overall maximum field strength can be higher than the value at the usual observation point at 3 m above the center of the PCB.
CONCLUSION The Method of Moment in conjunction with an equivalentwire configuration provides an efficient tool for the radiatedemission analysis of finite-size printed circuit boards with arbitrarily shaped ground planes. The drastic savings in the number of unknowns, as a consequence of the reduction of complexity, leads to a reasonable modeling effort and computation time. A suitable strategy to keep the number of unknowns to a minimum has been presented, using an optimized meshing of the ground plane. Although including the presence of the dielectric in the computation of the conductor currents it has been found that considerable prediction errors occur if the dielectric is ignored in the subsequent field computation. Therefore a method based on equivalent-polarization currents has been developed to account for the effects of the dielectric, which is especially important for the study of the spatial variation of radiated emissions.
REFERENCES
Cl1 M. Leone, H. Brtins, H. Singer, “Fast EMC Analysis for Printed Circuit Boards Using an Equivalent-Wire Method of Moments”,
Figure 12. Vertical diagram of the electric field in the x-z plane at 3 m distance for the lSa’ harmonic (966.7 MHz).
Intern. Symp. on EM298 Roma, pp. 7-12, 1998
PI M. Bucker, S. Ging, “Simulator Coupling Technique for Complex PCB Structures”, IEEE 1998 Intern. Symp. on Electromagn. Compat., Denver USA, pp. 656-661, 1998. 131 R. Perez(Ed.), Handbook
of Electromagnetic
Compatibility,
Academic Press,1995.
[41 R. K. Hoffmann, Handbook of Microwave Integrated Circuits,
Artech House Inc., 1987.
151 K. C. Gupta, R. Garg, R. Chada, Computer-Aided Design of Microwave Circuits, Artech House Inc., 1981. Advanced Wiley & Sons, Inc., 1989
WI C. A. Balanis,
Engineering
Electromagnetics,
“Radiation Pattern Computation of Microstrip Antennas on Finite Size Ground Planes”, IEE PROCEEDINGS-H, vol. 139, no. 3, pp. 278-286, June 1990.
171 S. A. Bokhari, J. R. Mosig, F. E. Gardiol,
PI Th. Mader, Berechnung
elektromagnetischer Felderscheinungen in abschnittsweise homogenen Medien mit Oberjltichenstromsimulation, Ph. D. thesis,Technical University
Hamburg-Harburg, 1992. PI Figure 13. Horizontal diagram of the electric field in the x-y plane (z = 0) at 3 m distance for the 15* harmonic (966.7 MHz).
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V. Schulz, G. Mrozynsky, “A BenchmarkCatalog for Numerical Field Calculations in the Area of EMC”, 13’h Intern. Zurich Symp. on Electromagn. Compat., Tutorial Lectures, 8T2, pp. 115-121,February 16-18, 1999
