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 the quality factor of the cavity.
 Electromagnetic interference (EMI) has become an important subject of interest since the 1950s [Montrose, 1999]. The source of the interference may come from nature or be artificially created. In the late 1970s, the problem of electromagnetic compatibility (EMC) among electronic devices became eminent. Analog devices were once believed to be more susceptible to EMI than digital devices. However, digital devices, which are now ubiquitous, are also vulnerable to EMI. For example, Laurin et al.  showed the susceptibility of a complementary metal oxide semiconductor device to radiated EMI. In addition, both device types can also be sources of EMI. To minimize the level of EMI in the environment, EMC considerations are now an important factor in the design of electronic products.
 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.  used finite element method (FEM)/method of moments (MOM) 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 and Ruehli, 2001; 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.  used FEM to analyze the coupling of an incident wave to a wire inside a cavity. Lecointe et al.  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 system, as would be the case in a completely numerical solution (using, for example, 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 area. 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 frequency domain hybrid method developed by Lertsirimit et al.  is used to calculate the time domain voltage at the input port of the digital device due to a time domain plane wave pulse that is incident on the system. The time domain analysis and characterization of various signals that might appear at such a port are analyzed with a view toward assessing their potential for device upset or failure. The device port is assumed to be open-circuited (many digital devices have a high impedance) or terminated in a load that is linear, at least up to the time of device failure (so that the system is linear) [Baum, 1983]. 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 frequency domain hybrid method developed by Lertsirimit et al.  are very significant. Although the frequency domain hybrid method was developed and thoroughly studied by Lertsirimit et al. , a brief summary of it is given here for completeness in section 2.
 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 quality factor (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 on the basis of system theory. The calculations also reveal the level of signal voltage at the device port that can be expected because of 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 frequency domain hybrid method introduced by Lertsirimit et al.  is summarized here for convenience. This method provides an efficient means for calculating the frequency domain voltage at the input port of the digital device due to an exterior incident field. A generic picture of the system under consideration in shown in Figure 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 fundamental assumptions in the hybrid method are that the substrate is thin compared with a wavelength (so that it may be modeled using transmission-line theory) and that the aperture size is fairly small compared with a wavelength (so that it may be replaced by a lumped voltage source in the interior problem and a lumped load in the exterior problem). The key calculation steps in the hybrid method are outlined below.
2.1. Step 1
 The interior part of the 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 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.
2.2. 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.
2.3. 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.
2.4. 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 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 Figure 3. 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 except at the (110) resonance, where the numerically exact solution loses accuracy because of ill conditioning. 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 Figure 3).
3. Time Domain Analysis
3.1. 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 Figure 2. The incident electromagnetic field is now assumed to be a time-varying plane wave of the form
where p(t) is an arbitrary time-varying signal (two specific pulse shapes will be considered for the results presented later), PW is a unit vector in the direction of propagation of the plane wave and r is the vector to the observation point. 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]
where h(t) is the unit impulse response (the response when p(t) = δ(t)) and ‘*’ refers to convolution. In the frequency domain, the relationships can be written as
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
Because the voltage signal v(t) is real valued, the output signal can be obtained from its Fourier transform as
The incident signal can be represented in terms of its Fourier transform as
where k = k0PW and k0 = ω/c, the propagation constant in free space, and c is the speed of light.
 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
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 Figure 3 for the canonical structure in Figure 2.
3.2. 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 sine or cosine) of radian frequency ωs, having the form
where s(t) is modulating envelope function.
 In the first case, where it is assumed that the system response has a constant amplitude and linear phase response over the bandwidth of the pulse, the system response in the frequency domain can be approximated as
where A = ∣H(ω0)∣ is a real constant and the phase of H(ω) near ω0 has been approximated as ϕ ≈ ϕ0 + B(ω − ω0). The output signal in this case is in the form [Couch, 2001]
where Tg = −dϕ/dω = −B is the group delay of the system response, and Tp = −ϕ0/ωs 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
where ωc is the resonant frequency of the cavity mode and Qc is the Q of the cavity resonance. (The constant c in this equation does not denote the speed of light.) In this case the port voltage at late time will be dominated by the exponentially decaying type of sinusoidal response shown in (12), at least for those cases where the response function in (12) decays slower than does the input pulse.
3.3. Pulse Shapes
 Two particular pulse shapes are used for the numerical results. The pulse shapes and their Fourier transforms are listed below.
3.3.1. Modulated Gaussian Pulse
 The modulated Gaussian pulse is of the form
where ωs is the radian 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 is decreased by a factor of e−1, and is given approximately as BWA = ω+ − ω− = 2/σ. Figure 4a shows this pulse.
3.3.2. Damped Sinusoidal Pulse
 The damped sinusoidal pulse is of the form
where = ω/ωs, u(t) is the unit step function, ωs is the frequency of the signal carrier, and Qs 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 Qs and ≈ 1. 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 −10 dB bandwidth limits ω+ and ω−, and is given approximately as BWR = Δω/ωs = (ω+ − ω−)/ωs = 3/Qs. 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 Figure 5. The pulses are labeled according to the type (Gaussian or exponential) and their region of extent (1, 2, or 3).
 Pulse 1 in Figure 5 is chosen so that the center frequency ωs is located in a region of the frequency domain response where the magnitude of the transfer function H(ω) is relatively flat with frequency. Furthermore, the phase of the transfer functions is approximately linear over the fairly small bandwidth of the pulse.
 Pulse 2 is chosen so that the center frequency ωs 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 ωc = 3.2E+9 rad/s, Qc = 128, b = −1.082E7, and c = 2.125E6. (The units of b and c are in meters.)
 Pulse 3 has a center frequency ωs that is roughly in the center of the frequency range plotted, and has a very large bandwidth, covering several resonances.
4.1. Gaussian Pulse
 In this subsection, the Gaussian pulse in (14) with different bandwidth extents (1, 2, and 3) as shown in Figure 5 are applied to the system.
4.1.1. Gaussian Pulse 1
 The pulse has a center frequency of 0.35 GHz. The decay parameter σ is chosen as 9.66E-9 s and the corresponding bandwidth is 0.29 Grad/s. 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 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. Plotting the signal on an expanded scale (not shown here) permits verification that the envelope of the output signal does match well with the shifted envelope of the input signal.
4.1.2. Gaussian Pulse 2
 Pulse 2 is centered about the (011) cavity mode resonance of the system response. The pulse has a center frequency of 0.51 GHz. The decay parameter σ is chosen as 5.6825E-8 s and the corresponding bandwidth is 0.05 Grad/s. The output port voltage that is calculated numerically using (7) is shown in Figure 7. The envelope plotted in Figure 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 late-time 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 Figure 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.
4.1.3. 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 a center frequency of 0.4 GHz. The decay parameter σ is chosen as 1.21E-9 s and the corresponding bandwidth is 2.33 Grad/s. The output port voltage is calculated numerically using (7) and is shown in Figure 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.
4.2. 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 Figure 5.
4.2.1. Sinusoidal Pulse 1
 Pulse 1 has a center frequency of 0.35 GHz and a signal Q factor of Qs = 525. The corresponding relative bandwidth is 0.0057. The output port voltage is calculated numerically using (7) and is shown in Figure 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 on the basis of the group delay.
4.2.2. Sinusoidal Pulse 2
 Pulse 2 is centered about the (011) cavity mode resonance. The pulse has a center frequency of 0.51 GHz, the same as the center frequency of the cavity resonance. Two different values of the signal quality factor Qs are used. One value is greater than Qc for the cavity mode and the other is smaller. The two values of Qs are 510 and 51. Recall that the value of Qc 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. Note that for this high-Q signal, the peak output voltage is large compared with that from pulse type 1 and the output response extends to very late time.
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 single cavity-mode envelope matches quite well with the output signal for most of the time shown, however.
4.2.3. Sinusoidal Pulse 3
 Pulse 3 has a broad spectrum that covers almost the entire frequency range of the system response (see Figure 5). The pulse has a center frequency of 0.4 GHz. The signal quality factor is Qs = 12.1 and the corresponding relative bandwidth is 0.25. Figure 12 shows the output port voltage. As for the case of the broad-spectrum Gaussian pulse (Figure 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.
 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.
 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 resonances 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.
 Support for this project was provided from the Air Force Office of Scientific Research MURI award F49620-01-1-0436, “Analysis and design of ultrawide-band and high-power microwave pulse interactions with electronic circuits and systems.”