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Time-Domain Coupling to a Device on Printed Circuit Board Inside a Cavity

Chatrpol Lertsirimit, David R. Jackson and Donald R. Wilton

Applied Electromagnetics Laboratory Department of Electrical Engineering, University of Houston

Houston, Texas 77204-4005

Abstract Time-domain coupling from an incident plane-wave pulse to a device on a printed circuit board inside of a metallic cavity enclosure is calculated and studied, using an efficient hybrid method. The cavity has an exterior feed wire that penetrates through an aperture and makes direct contact with the printed-circuit board trace that leads to the device. The signal level at the input port of the device is calculated and studied. The incident electromagnetic field is assumed to be a time-domain plane wave in the form of a pulse, and two pulse shapes (a Gaussian pulse and an exponentially damped sinusoidal pulse) are studied. Results show how different pulse characteristics produce different types of signals at the input to the device. The time-domain results are validated by comparing with simple expressions based on the resonant frequencies and Q of the cavity.

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1. Introduction

The electromagnetic coupling from an exterior field to a circuit component on a printed circuit

board (PCB) inside a conducting cavity (shield) via a direct connection from a wire or cable that

penetrates an aperture in the cavity can be a very important mechanism in determining the signal

levels at the device on the PCB due to the coupling from an exterior field; indeed, it may be the

dominant coupling mechanism. An accurate and efficient analysis of this important EMC

problem requires the combination of PCB analysis with the analysis of field penetration into a

cavity, two analyses that are on very different size scales, making this a difficult problem.

Both PCB analyses and cavity analyses have received significant attention. For example, Ji et.

al. [1999] used FEM/MoM method to analyze the radiation from a PCB. The partial element

equivalent circuit approach (PEEC), which employs quasi-static calculations, has also been

widely used to analyze PCB signals (e.g. [Archambeault et. al., 2001] and [Ji et. al., 2001]). For

cavity analyses, many studies have investigated coupling of electromagnetic waves to a simple

element, such as a wire, inside the cavity enclosure. For example, Carpes et al. [2002] used

FEM to analyze the coupling of an incident wave to a wire inside a cavity. Lecointe et al. [1992]

analyzed a similar problem using the MoM. However, the analysis of a complete system,

containing a metallic cavity enclosure and a PCB inside the cavity with a trace on the PCB

leading to a device, remains a difficult problem.

Recently, Lertsirimit et. al. introduced an efficient hybrid method for calculating the frequency-

domain coupling from an exterior wave to a device on a PCB for the type of structure mentioned

above. The method is efficient since it does not require the complete discretization of the entire

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system, as would be the case in a completely numerical solution (using, e. g., the method of

moments). A complete numerical solution would require many unknowns, and furthermore, the

level of discretization would be very different throughout the system, since the conductor trace

on the PCB is at a very different feature size than the cavity or the feed wire that penetrates the

aperture are. This would lead to a very significant computation time, and also a potential loss of

accuracy. The hybrid method, however, analyzes the PCB trace and the cavity/feed-wire system

separately, reducing the number of unknowns as well as avoiding a discretization of the PCB

trace when analyzing the cavity and feed wire. The method allows for a calculation of the

Thévenin equivalent circuit for the entire system leading up to the input port on the digital

device.

In this paper, the hybrid method is used to calculate the time-domain voltage at the input port of

the digital device due to an incident time-domain plane wave pulse that is incident on the system.

The calculation is done by using the frequency-domain hybrid method together with a Fourier

transform in time. Because the inverse Fourier transform that is used to calculate the port voltage

at the device requires many frequency-domain calculations to obtain an accurate result

(especially for short pulses), the benefits of the hybrid method are very significant.

A brief summary of the frequency-domain hybrid method is discussed in Section 2 for

completeness, although the reader is referred to [Lertsirimit et. al.] for a more complete

discussion.

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In Section 3, the time-domain analysis is presented, and a brief discussion of signal transmission

through the system is given. Simple approximate formulas for the time-domain signal at the

device port are also presented, based on the assumption of a high-Q cavity. These simple

formulas are very useful for validation. The concepts of phase and group delay are also briefly

discussed, as these aid in the physical understanding of the signal transmission through the

system.

The formulation is applied to two different pulse shapes. The first is a sinusoidal signal

modulated by a Gaussian envelope. The second is a sinusoidal signal that begins at t = 0 and is

modulated by a damped exponential.

In Section 4 results are presented for the two different pulses, with varying center frequency and

bandwidth. A physical interpretation of the results is given, based on system theory. The

calculations also reveal the level of signal voltage at the device port that can be expected due to

representative time-domain incident fields, so that practical issues such as the potential for

device upset can be determined.

In Section 5 conclusions are given, including a summary of the important physical properties of

pulse propagation through the system to the device port.

2. Hybrid method

The hybrid method introduced in [Lertsirimit et. al.] is summarized here for convenience. This

method provides an efficient means for calculating the frequency-domain voltage at the input

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port of the digital device due to an exterior incident field. A generic picture of the system under

consideration in shown in Fig. 1. A feed wire penetrates an aperture in a conducting cavity (box)

and then makes contact with a PCB trace. The PCB trace leads to a device of interest on the

PCB, and the trace may also lead to other (linear) loads as well. For convenience, it is assumed

here that the input to the digital device is a high impedance load, so that the device may be

modeled as an open-circuit port at which the voltage is to be obtained. (If this is not the case,

then the open-circuit voltage may still be calculated, in which case it is regarded as the Thévenin

voltage for the entire system leading to the port. A Thévenin impedance would then need to be

calculated, and this could be done by calculating the short-circuit current at the port by following

similar steps as outlined below, using a short-circuit load instead of an open-circuit load at the

port.) The key calculation steps in the hybrid method are outlined below.

Step 1

The interior part of problem is first considered. The input impedance ZinAP seen by the feed wire

looking through the aperture into the interior of the system is calculated. This is done by first

shorting the aperture and replacing it by a 1 [V] source between the feed wire and the cavity

wall. Next, assuming that the PCB substrate is thin compared to a wavelength (typical in

practice), transmission line (TL) theory can be used to obtain the frequency-dependent input

impedance ZL seen by the feed wire looking down at the contact point with the PCB trace. In this

step the trace may be fairly complicated, although the calculation involves only TL analysis. The

interior problem is thus reduced to a closed cavity with an interior feed wire that begins at the

cavity wall with a 1V source and terminates at the bottom of the cavity with a load ZL. A full-

wave solver such as the method of moments is used to solve this interior problem, to find the

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current on the feed wire at any given frequency. The input impedance ZinAP is then calculated

directly by the ratio of the source voltage (1 [V]) and the current on the feed wire at the voltage

source location. The full-wave solution is relatively efficient, since only the cavity and the feed

wire need to be discretized, and not the PCB trace.

Step 2

The exterior problem is now considered. The goal in this step is to find the voltage at the

aperture between the feed wire and the surrounding cavity wall, due to the incident field that

impinges on the system (e.g., a plane wave). The aperture is shorted and the impedance ZinAP is

used to connect the exterior part of the feed wire to the cavity. An exterior problem is thus

created, consisting of the cavity (with no aperture), and an exterior feed wire connected to the

cavity via a load impedance at the base. This problem is illuminated by the incident field. This

scattering problem is solved using a full-wave solver such as the method of moments. The

voltage VAP across the load ZinAP is obtained directly from the solution of the exterior problem.

Step 3

The current on the feed wire inside of the cavity due to the incident field is then obtained by

taking the solution for the feed wire current obtained in step 1 (using a 1V source at the aperture

location) and scaling the feed-wire current by the factor VAP.

Step 4

Once the current on the feed wire in known, the value of the feed-wire current IJ at the junction

with the PCB trace may be determined. Transmission-line theory is then used to determine the

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voltage at any point on the PCB trace, using the known current source IJ as an ideal parallel

current source in the TL model. The open-circuit port voltage at the location of the digital device

on the PCB is thus determined.

Figure 2 shows a canonical structure that is used to obtain results. In this structure the PCB has

an air substrate for simplicity, and thus the PCB trace is actually a wire that is at a height of 1

mm above the bottom of the box. This simplification does not significantly affect the solution

time in the hybrid method (which uses transmission-line theory to model the substrate), nor does

it affect any of the qualitative conclusions obtained later. However, this simplification does allow

a complete numerical solution of the entire problem (used to provide validation of the hybrid

method) to run much faster, since it is not necessary to discretize the substrate. The canonical

problem exercises all of the main features of more realistic problems, including a cavity, an

aperture, a feed wire, and a transmission line wire that makes contact with the feed wire.

Figure 3 (taken from Lertsirimit et. al.) shows frequency-domain results for the voltage at the

device port, due to a unit-amplitude incident plane wave as shown in the figure. The agreement

between the port voltages calculated using the hybrid method and using the numerically-exact

moment-method solution of the entire structure, is quite good. The results also demonstrate that

increased coupling to the device port may occur for several reasons, including exterior

resonances of the structure, interior resonances of the feed wire, and most importantly, interior

cavity resonances (a few of which have been labeled in the figure).

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3. Time domain analysis

1. General Formulation

An incident time-varying electromagnetic signal (pulse) is now assumed to illuminate the

structure. The incident pulse and the output voltage at the device port are related by a linear,

time-invariant, continuous-time system. In this case, the system consists of the feed wire, the

box, the PCB trace, and any linear loads that the PCB trace terminates in before arriving at the

device port of interest (which is assumed to be open-circuited). Figure 1 shows once again the

general system, although results will be presented for the structure of Fig. 2. The incident

electromagnetic field is now assumed to be a time-varying plane wave of the form

0ˆPW

inc r rE E p tc

⎛ ⎞⋅= −⎜ ⎟

⎝ ⎠ (1)

where p(t) is an arbitrary time-varying signal (two specific pulse shapes will be considered for

the results presented later) and ˆPWr is the direction of propagation of the plane wave. The real-

valued unit vector E0 gives the polarization of the incident wave. The voltage signal v(t) at the

device port can be written as [Lathi, 1998]

( ) ( ) ( )v t p t h t= ∗ (2)

where ( )h t is the unit impulse response (the response when ( ) ( )p t tδ= ) and ‘*’ refers to

convolution. In the frequency domain, the relationships can be written as

( ) ( ) ( )V P Hω ω ω= (3)

where ( )P ω is the Fourier transform of the input pulse, ( )V ω is the Fourier transform of the port

voltage, and ( )H ω is the Fourier transform of the unit impulse response of the system. The

transform definitions used here are

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( ) ( ) j tA a t e dtωω∞ −

−∞= ∫ (4)

( ) ( )12

j ta t A e dωω ωπ

∞

−∞= ∫ . (5)

Because the voltage signal v(t) is real valued, the output signal can be obtained from its Fourier

transform as

( ) ( )0

1Re j tv t V e dωω ωπ

∞= ∫ . (6)

Hence,

( ) ( ) ( )0

1Re j tv t P H e dωω ω ωπ

∞= ∫ . (7)

The incident signal can be represented in terms of its Fourier transform as

( )0 0 0

ˆ 1RePW

inc jk r j tr rE E p t E P e e dc

ωω ωπ

∞ − ⋅⎛ ⎞⋅= − =⎜ ⎟

⎝ ⎠∫ (8)

where 0 ˆPWk k r= .

Comparing (7) and (8), it is concluded that ( )H ω in (7) physically represents the frequency-

domain voltage at the device port due to a unit-amplitude incident plane wave at a radian

frequency ω , of the form

( ) 0inc j k rE E eω − ⋅= . (9)

Therefore, the transfer function ( )H ω of the system can be calculated by the hybrid method

with a unit-amplitude incident plane wave excitation. The magnitude of this transfer function is

shown in Fig. 3 for the canonical structure in Fig. 2.

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2. Special Cases

There are two special cases that are useful to consider, since they aid in the physical

interpretation of the results presented later. For the results presented later, the pulse will be in the

form of a modulated carrier wave (either a sin or cosine) of radian frequency sω , having the

form

( ) ( ) ( )( )

sincos

s

s

tp t s t

tωω

⎡ ⎤= ⎢ ⎥

⎣ ⎦ (10)

where ( )s t is modulating envelope function.

In the first case, it is assumed that the system response has a linear phase response over the

bandwidth of the pulse, the system response in the frequency domain can be approximated as

( ) ( )0j BH Ae φ ωω +=

where A is a real constant. The output signal in this case is in the form [Couch, 2001]

( ) ( )( )( )( )( )

sin

cos

s p

g

s p

t Tv t As t T

t T

ω

ω

⎡ ⎤−⎢ ⎥= −⎢ ⎥−⎣ ⎦

(11)

where /gT d d Bφ ω= = is the group delay of the system response, and 0 /pT φ ω= is the phase

delay of the system response.

In the second case, the system response is assumed to be dominated by a single high-Q resonant

type response, in which case the system responses in the time and frequency domains are

approximated as

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( ) ( ) ( )( )2 cos sinc

ct

Qc ch t e b t c t

ω

ω ω−

= + (12)

( )1 2 1c

cc c

b jcHj Q

Q

ωω ω

ω

−=

⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

, (13)

where cQ is the Q of the cavity resonance. In this case the port voltage at late time will be

dominated by the exponentially decaying type of sinusoidal response shown in (12).

3. Pulse Shapes

Two particular pulse shapes are used for the numerical results. The pulse shapes and their

Fourier transforms are listed below.

A. Modulated Gaussian Pulse

The modulated Gaussian pulse is of the form

( ) ( )2

22 cost

sp t e tσ ω−

= (14)

( ) ( ) ( )2 2

2 2

2 21 22

s sP e eσ σω ω ω ω

ω σ π− + − −⎛ ⎞

= +⎜ ⎟⎜ ⎟⎝ ⎠

(15)

where sω is the frequency of the carrier and σ determines the pulse width, which is inversely

related to the bandwidth of the pulse in the frequency domain. The absolute bandwidth of the

pulse in the frequency domain is defined from the frequency limits ω+ and ω− where the

amplitude of the pulse spectrum in decreased by a factor of 1e− , and is given approximately as

2 2 /ABW ω ω σ+ −= − = . Figure 4a shows this pulse.

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B. Damped Sinusoidal Pulse

The damped sinusoidal pulse is of the form

( ) ( ) ( )2 sins

st

Qsp t e t u t

ω

ω−

= (16)

( ) ( )2 22

11 1

2

c s

s scs

s

QP jjQ

jQ

ωωω ωω ω ω

⎡ ⎤⎢ ⎥= ≈ −− −⎢ ⎥⎛ ⎞ ⎣ ⎦+ +⎜ ⎟

⎝ ⎠

(17)

/ sω ω ω= , u(t) is the unit step function, cω is the frequency of the carrier, and sQ is the quality

factor of the signal (this type of signal would originate from a resonator circuit that has this value

of Q). The approximate form in (17) is accurate for a large sQ . The Qs of the signal is inversely

related to the bandwidth of the pulse. The relative bandwidth in this case in the frequency

domain is defined from the -10dB bandwidth limits ω+ and ω− , and is given approximately as

( )/ / 3 /R s s sBW Qω ω ω ω ω+ −= ∆ = − = . Figure 4b shows this pulse.

4. Results and Discussion

Each of the two pulse types discussed previously is applied in three different regions in the

frequency domain, as shown in Fig. 5. The pulses are labeled according to the type (Gaussian or

exponential) and their region (1, 2, or 3). The polarization is chosen as 0 ˆE x= , the incident

wavevector direction is ˆ ˆPWr z= − , and the origin is at the corner of the box (see Fig. 2.)

Pulse 1 in Fig. 5 is chosen so that the center frequency sω is located in a region of the frequency-

domain response where the magnitude of the transfer function ( )H ω is relatively flat with

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frequency. Furthermore, the phase of the transfer functions is relatively linear over the fairly

small bandwidth of the pulse.

Pulse 2 is chosen so that the center frequency sω is centered at a cavity resonance of the system

(the (011) mode of the cavity). The bandwidth of the pulse is fairly narrow, so that mainly a

single cavity resonance is expected to be excited. For this particular resonance, the parameters in

(12) and (13) for this cavity mode are Qc = 128, b = -1.082E7, and c = 2.125E6.

Pulse 3 has a center frequency sω that is roughly in the center of the frequency range plotted,

and has a very large bandwidth, covering several resonances.

A. Gaussian Pulse

In this subsection, the Gaussian pulse with different characteristics (1, 2, and 3) as shown in Fig.

5 are applied to the system.

Gaussian Pulse 1

The pulse has the center frequency of 0.35 GHz. The decay parameter σ is chosen as 9.66E-9 s

and the corresponding bandwidth is 0.28 GHz. The open-circuit port voltage is calculated

numerically using (7). Figure 6 shows the output (port) voltage and the input (signal) voltage for

comparison. Also shown is an envelope that is obtained by shifting the envelope of the input

signal by the group delay. As expected, the output signal has the form of (11). That is, the output

signal is a scaled version of the input signal, with the carrier shifted by the phase delay and the

envelope shifted by the group delay. A calculation shows that the group delay of the output

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signal is about -7.027E-10 s so that the envelope of the output signal is advanced by 7.027E-10 s,

while the phase delay of the output signal is about 8.347E-10 s. Figure 6 shows good agreement

with these calculations, although the shift in the envelope is difficult to see on the scale of the

plot. An expanded scale (not shown here) verifies that the envelope of the output signal does

match well with the shifted envelope of the input signal.

Gaussian Pulse 2

Pulse 2 is centered about the (011) cavity mode resonance of the system response. The pulse has

a center frequency at 0.51 GHz. The decay parameter σ is chosen as 5.6825E-8 s and the

corresponding bandwidth is 0.05 GHz. The output port voltage that is calculated numerically

using (7) is shown in Fig. 7. The envelope plotted in Fig. 7 is a plot of the exponentially

decaying sinusoidal response function in (12). The response function in (12) would be expected

to be a good approximation to the system response to the Gaussian pulse provided that the pulse

has a sufficiently narrow bandwidth that primarily one only cavity resonance is excited.

Examination of Fig. 5 shows that this is the case for pulse 2. Figure 7 shows that the output

signal has the same form as the input pulse in the early time. After some time, around 4σ, the

input signal has decayed sufficiently to have an insignificant effect on the output signal. At this

point the output signal begins to behave like the cavity response.

Gaussian Pulse 3

Gaussian pulse 3 has a broad frequency spectrum that covers almost the entire range of the

system response shown. The pulse has the center frequency of 0.4 GHz. The decay parameter σ

is chosen as 1.21E-9 s and the corresponding bandwidth is 2.33 GHz. The output port voltage is

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calculated numerically using (7) and is shown in Fig. 8. As for pulse 2, the input pulse has a

strong effect on the output signal in early time. However, this effect dies off quickly and a rather

complicated ringing is observed, which is due to the superposition of several cavity resonance

responses, similar to (12), arising from the different system resonances that are excited. For

later time the output response appears to be mainly a beating between dominant resonances. For

still later time (beyond the scale of the plot) the response would be dominated by a single cavity

response, corresponding the system resonance that has the highest Qc factor.

B. Damped Sinusoidal Pulse

In this subsection, results are shown for the exponentially damped sinusoidal pulse in (16).

Results are again shown for three different pulses, having different center frequencies and

bandwidths (pulses 1, 2, and 3) as shown in Fig. 5.

Sinusoidal Pulse 1

Pulse 1 has a center frequency at 0.35 GHz and a signal Q factor of 525sQ = . The corresponding

relative bandwidth is 0.0057. The output port voltage is calculated numerically using (7) and is

shown in Fig. 9. Also shown for comparison is an envelope that is obtained by shifting the

envelope of the incident pulse p(t) by the group delay -7.027E-10 s (so that the envelope is

shifted to the left). The output signal has an envelope that matches well with the predicted

envelope based on the group delay.

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Sinusoidal Pulse 2

Pulse 2 is centered about the (011) cavity mode resonance. The pulse has the center frequency of

0.51 GHz, the same as the center frequency of the cavity resonance. Two different values of the

signal quality factor sQ are used. One value is greater than the cQ value of this cavity mode and

the other is smaller. The two values of sQ are 510 and 51. Recall that the value of cQ of this

cavity mode is 128. Figures 10 and 11 show the output signal as well as the envelope functions

corresponding to the input signal and the cavity response. The envelope for the input signal is

given by the exponential term in (16). The envelope of the cavity response is obtained from the

exponential term in (12), using (13) to obtain the parameters (b, c, ωc, Qc) by fitting with the

actual response ( )H ω near the resonance peak.

Figure 10 shows results for the high-Q signal. Both the input signal and the cavity response are

in the form of exponentially decaying sinusoidal waves. However, because the Qs of the signal is

much larger than the Qc of the cavity, the late-time response of the system is dominated by the Qs

of the signal. Hence, for late time it is seen that the envelope of the output signal matches quite

well with the envelope of the input signal.

Figure 11 shows results for the low-Q signal. Because the Qc of the cavity is now larger than the

Qs of the signal, the envelope of the late-time system response is approximated by the envelope

of the response for the (011) cavity mode. However, because the Qs of the signal is now fairly

low, the signal excites additional cavity modes to some extent. Therefore, in late time the output

signal is not a pure sinusoidal wave modulated by an envelope (as would be expected if a single

cavity response dominated) but is a beating between a couple of different cavity responses. A

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single cavity-mode envelope matches quite well with the output signal for most of the time

shown, however.

Sinusoidal Pulse 3

Pulse 3 has a broad spectrum that covers almost the entire frequency range of the system

response (see Fig. 5). The pulse has the center frequency of 0.4 GHz. The signal quality factor

is 12.1sQ = and the corresponding bandwidth is 0.25. Figure 12 shows the output port voltage.

As for the case of the broad-spectrum Gaussian pulse (Fig. 8), the output signal is rather

complicated, and exhibits interference between several cavity-mode resonances. For later time,

the interference is mainly a beating between two cavity resonances, and eventually (beyond the

scale of the plot) a single cavity resonance response would dominate.

5. Conclusions

The coupling of a time-domain plane wave pulse to a digital device on a printed-circuit board

(PCB) inside a cavity has been formulated. The cavity has an aperture though which an exterior

feed wire passes, with the feed wire making contact with the PCB trace at some point. This type

of direct-contact coupling can be one of the most important contributors to the EMC coupling to

devices on circuit boards inside of a metallic enclosure. The formulation uses a “hybrid” method

that decouples the analysis of the PCB trace with the analysis of the cavity and the feed wire to

obtain an efficient and accurate solution. Results were obtained for a canonical structure in

which the PCB was replaced by an air substrate for simplicity, although this simplification

should not affect any of the conclusions.

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Results were presented for two different types of pulses. One is a sinusoidal signal modulated by

a Gaussian envelope function. The second is an exponentially decaying sinusoidal signal that

starts at t = 0. Furthermore, for each type of pulse, three different pulses were used, with

different center frequencies and bandwidths. One pulse (pulse 1) was centered at a frequency for

which the transfer function of the system response was fairly slowly varying. The bandwidth of

the pulse was small enough so that the transfer function of the system could be approximated as

having a constant magnitude and a linear phase over the bandwidth of the pulse. The second

pulse (pulse 2) was a narrow-band pulse that was centered at one of the cavity resonance of the

system. The third pulse (pulse 3) was a very broadband pulse, which excited several cavity

modes. The results show that the characteristics of the pulses are important parameters in

determining the signal at the device on the PCB.

For pulse 1, the output signal is a scaled version of the input signal, with the envelope and

sinusoidal carrier each shifted by the group delay and phase delay, respectively. For pulse 2, the

output signal behaves like a single cavity response in late time when the input signal is the

Gaussian pulse. When the input signal is a damped sinusoid, the late-time response is also in the

form of a damped sinusoid, but the character of the output signal depends on whether the signal

Q is less than or greater than the cavity Q. When the signal Q is higher, the late-time response is

dominated by the signal itself. When the cavity Q is higher, the late-time response is dominated

by the cavity response. For a broadband pulse that excites more than one cavity mode, the output

signal is rather complicated, and even for relatively late time, there is a beating between two or

more cavity responses that is observed.

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References

Archambeault, B. & Ruehli, A. E., Analysis of power/ground-plane EMI decoupling performance using the partial-element equivalent circuit technique, IEEE Transactions on Electromagnetic Compatibility, vol. 43, No. 4, pp. 437-445, Nov., 2001.

Carpes Jr., W. P., Pichon, L. & Razek, A., Analysis of the coupling of an incident wave with a wire inside a cavity using an FEM in frequency and time domains, IEEE Transactions on Electromagnetic Compatibility, vol. 44, No. 3, pp. 470-475, Aug., 2002.

Couch, L. W. II, Digital and Analog Communication Systems, 6th edition, Prentice Hall, pp. 242-244, 2001.

Ji, Y., Chen, J., Hubing, T. H. & Drewniak, J. L., Application of a hybrid FEM/MOM method to a canonical PCB problem, 1999 IEEE International Symposium on Electromagnetic Compatibility, vol. 1, pp. 91-96, Aug. 2-6, 1999.

Ji, Y., Archambeault, B. & Hubing, T. H., Applying the method of moments and the partial element equivalent circuit modeling techniques to a special challenge problem of a PC board with long wires attached, 2001 IEEE International Symposium on Electromagnetic Compatibility, vol. 2, pp. 1322-1326, Aug. 13-17, 2001.

Lathi, B. P., Modern digital and analog communication systems, 3rd edition, Oxford University Press, chapter 3, 1998.

Lecointe, D., Tabbara, W. & Lasserre, J. L., Aperture coupling of electromagnetic energy to a wire inside a rectangular metallic cavity, Dig. IEEE AP-S Antennas Propagat. Soc. Int. Symp., vol. 3, pp. 1571-1574, 1992.

Lertsirimit, C., Jackson, D. R., Wilton, D. R., An efficient hybrid method for calculating the EMC coupling to a device on a printed circuit board inside a cavity, Electromagnetics, (submitted).

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Figure 1. A pictorial illustration of a feed wire that penetrates an aperture in a conducting cavity enclosure,and then makes contact with a conducting trace on a printed circuit board (PCB). The PCB trace leads to the input port of a device, such as an integrated circuit (IC). The voltage at the aperture and the current at thejunction with the PCB trace are shown.

Figure 2. Canonical structure used to obtain results. A feed wire penetrates an aperture in aconducting cavity and makes contact with a transmission-line wire. The aperture is square withdimensions 6x6 cm and is centered horizontally on the left side of the box. The center of theaperture is 0.15 m above the bottom of the box. The feed wire has a radius a2 = 0.25 mm, andprotrudes a distance of 12 cm outside the box. The feed wire extends 20 cm inside the box beforebending down to make a contact with the transmission-line wire. The transmission-line wire has aradius a1 = 0.383 mm and is 1 mm above the bottom of the box. The transmission-line wire has atotal length of 26 cm, and extends 10 cm to the left to reach the open-circuit port, and 16 cm to theright to reach the 50 Ω load. The incident plane wave is traveling vertically downward and ispolarized with the electric field parallel to the feed wire.

IC

E inc

IJ

device port VAP+

-

2a2

2a1

TL wire (Z0 = 100 Ω)

feed wire

device port

cavity

0.4 m

0.36 m

0.6 m

h=1mm

0.15 m load (50) Ω

x

z

21

Figure 3. Magnitude of the open-circuit (Thévenin) port voltage due to a unit-amplitude incidentplane wave. The result from the hybrid method is compared with that obtained from a rigorous full-wave simulation (“complete EM”). The first resonance around 0.15 GHz is due to an exterior boxresonance. The second resonance at about 0.46 GHz is due to a resonance of the feed wire in the box.The next three resonances are due to cavity-mode resonances (the first two of which are labeled).

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8frequency [GHz]

volts

mag (hybrid method)mag (complete EM)

011110

box wire

cavity resonances

22

Figure 4. Pulses under investigation. (a) Gaussian pulse. (b) Damped sinusoidal pulse.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

frequency [GHz]

volts

System response

1

23

Figure 5. Magnitude of the system response versus frequency. Shown on the plot are sketches of thespectral content for the three different pulses that are used. Pulse 1 is in a region where the amplitude ofthe system response is approximately constant. Pulse 2 is centered around a cavity mode (the 011 mode). Pulse 3 has a large bandwidth that spans several resonances.

4σ

1

1 ?

g t ( )

55 ? t 4a

t

p(t)

4b

t

p(t)

23

Figure 7. The open-circuit port voltage excited by Gaussian pulse 2.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-3.0E-07 -1.0E-07 1.0E-07 3.0E-07 5.0E-07

[s]

volts

envelope (single resonance)output signal

4σ

Figure 6. The open-circuit port voltage excited by Gaussian pulse 1.

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

-2.0E-08 -1.0E-08 0.0E+00 1.0E-08 2.0E-08 3.0E-08

[s]

volts

output signalenvelopeinput pulse

24

Figure 8. The open-circuit port voltage excited by Gaussian pulse 3.

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

-2.0E-08 2.0E-08 6.0E-08 1.0E-07 1.4E-07 1.8E-07 2.2E-07 2.6E-07 3.0E-07

time [s]

volts

4σ

Figure 9. The open-circuit port voltage excited by damped sinusoidal pulse 1.

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.0E+00 5.0E-08 1.0E-07 1.5E-07 2.0E-07 2.5E-07 3.0E-07 3.5E-07 4.0E-07

[s]

volts

output signalenvelope

25

Figure 10. The open-circuit port voltage excited by damped sinusoidal pulse 2 with sQ =510.

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

1.0E-07 2.0E-07 3.0E-07 4.0E-07 5.0E-07 6.0E-07 7.0E-07 8.0E-07 9.0E-07

[s]

volts

output signalenvelope of cavity responseenvelope of input signal

Figure 11. The open-circuit port voltage excited by damped sinusoidal pulse 2 with sQ =51.

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

1.0E-07 1.5E-07 2.0E-07 2.5E-07 3.0E-07 3.5E-07 4.0E-07

[s]

volts

output signalenvelope of cavity responseenvelope of input signal

26

-0.025-0.02

-0.015-0.01

-0.0050

0.0050.01

0.0150.02

0.025

0.0E+00 4.0E-08 8.0E-08 1.2E-07 1.6E-07 2.0E-07 2.4E-07 2.8E-07

[s]

volts

Figure 12. The open-circuit port voltage excited by damped sinusoidal pulse 3.