## 1. Introduction

[2] 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.* [1995] 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.

[3] 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.

[4] Both PCB analyses and cavity analyses have received significant attention. For example, *Ji et al.* [1999] 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.* [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.

[5] Recently, *Lertsirimit et al.* [2005] 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.

[6] In this paper, the frequency domain hybrid method developed by *Lertsirimit et al.* [2005] 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.* [2005] are very significant. Although the frequency domain hybrid method was developed and thoroughly studied by *Lertsirimit et al.* [2005], a brief summary of it is given here for completeness in section 2.

[7] 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.

[8] 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.

[9] 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.