EMI/EMC ANALYSIS OF ELECTRONIC SYSTEMS SUBJECT TO NEAR ZONE ILLUMINATIONS

Transcription

1 EMI/EMC ANALYSIS OF ELECTRONIC SYSTEMS SUBJECT TO NEAR ZONE ILLUMINATIONS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Zulfiqar A. Khan, M.S., B.S. * * * * * The Ohio State University 2007 Dissertation Committee: Approved by Professor John L. Volakis, Adviser Professor Prabhakar H. Pathak Dr. Chi-Chih Chen Adviser Graduate Program in Electrical and Computer Engineering

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3 c Copyright by Zulfiqar A. Khan 2007

4 ABSTRACT There is an increasing interest to evaluate performance of electronic systems subject to electromagnetic interference (EMI). To this end, an efficient technique for system level EMI/EMC analysis of electronic systems was recently proposed. This scheme, referred to as the hybrid S-matrix method, is based on introducing an additional port on the printed circuit board (PCB) and cable network to represent the plane wave excitation. The scattering matrix for the network (with the added port) is generated once the transmission line (TL) modes are extracted from the induced voltages. The resulting hybrid S-matrix allows for complete characterization of a microwave network in presence of enclosures or other scatterers. It can then be ported into a circuit solver for complex EMI/EMC analysis of electronic devices. This dissertation extends the hybrid S-matrix approach to near zone sources. A key aspect of this generalization is the use of the FFT rather than the GPOF to identify the TL modes in presence of the continuous spectrum of forced modes introduced by the near zone sources. Examples are shown to validate the generalized hybrid S-matrix method. These include a multi-layered PCB in the presence of nearby cables and when enclosed by the metallic structures with openings. The final chapter of the dissertation contains numerous experimental examples that provide for the first time a set of realistic EMI/EMC reference data. Specifically, field penetration through apertures and coupling of penetrating wires onto the printed circuit boards (PCBs) ii

5 enclosed by resonant structures are considered. In contrast to other measurements, we focus on multi-cavity enclosures in presence of cables and PCBs. The purpose of this experimental study is to provide accurate reference data for possible future validation of various computational tools and to test their accuracy and efficiency on realistic platforms. In particular, we validate the proposed hybrid S-matrix method. iii

6 Dedicated to Laila iv

7 ACKNOWLEDGMENTS I have so many reasons to thank Prof. Volakis that I can t mention all of them here. He has not only guided me in my technical work, encouraged me to take risks in my research and helped me in improving my communication skills but also he also helped me resolve many personal challenges that I had to face during my PhD. For all these reasons and more, I will always remain grateful to him. I also want to thank Dr. Yakup Bayram for his guidance throughout my work and helping me understand very important details of the hybrid S-matrix approach. I am grateful to Prof. Prabhakar Pathak, Dr. Chi-Chih Chen and Professor Patrick Roblin for their valuable suggestions and agreeing to be on my PhD committee. Throughout my stay in the Lab, I had a great time with fellow ESL students. I wish to thank all of them. Special thanks are to Brian Usner, Edwin Lim, Ming Lee, Ioannis Tzanidis, Tao Peng, Paul Chang, Salih Yarga, Gokhan Mumcu, Jae-Young Chung and Stylianos Dosopoulos; for the memorable time I spent with them. During these years, I had to face some very difficult challenges in my personal life. I could not have survived in these hardships if my wife, Laila, had not constantly supported me. I thank her for everything she did for me. Finally, I thank my parents who always supported and prayed for me. v

8 VITA February 24, Born - Sargodha, Pakistan December, B.Sc. Electrical Engineering University of Engg. & Technology, Lahore, Pakistan December, M.S. Electrical Engineering Oklahoma State University Stillwater, OK February, July, Assistant Manager R & D, Advanced Engineering Research Organization - Pakistan January, July, Graduate Research Assistant, Oklahoma State University Stillwater, OK. October, present Graduate Research Assistant, Ohio State University Columbus, OH. PUBLICATIONS Research Publications Z. A. Khan, C. F. Bunting and M. D. Deshpande, Shielding Effectiveness of Metallic Enclosures at Oblique Incidence and Arbitrary Polarizations. IEEE Trans. Electromagnet. Compat., vol. 47, no. 1, pp , Feb V. Rajamani, C. F. Bunting and M. D. Deshpande and Z. A. Khan, Validation of Modal/MoM in Shielding Effectiveness Studies of Rectangular Enclosures with Apertures and a Preliminary Investigation of the Input Impedance of a Wire located inside the Cavity for Varying Positions. IEEE Trans. Electromagnet. Compat., vol. 48, no. 2, pp , May vi

9 Z. A. Khan, Y. Bayram and J. L. Volakis, EMI/EMC Measurements and Simulations for Cables and PCBs enclosed within Metallic Enclosures, submitted to IEEE Trans. Electromagnet. Compat., Z. A. Khan, Y. Bayram and J. L. Volakis, Hybrid S-parameters Analysis of PCBs and Cables subject to Non-Plane Wave Illuminations. IEEE APS International Symposium, June Z. A. Khan, Y. Bayram and J. L. Volakis, An Integrated Hybrid Solver and Measurements for EMI/EMC Analysis of Cables and PCBs Enclosed within Metallic Structures. IEEE EMC International Symposium, vol. 2, pp , August Z. A. Khan, J. L. Volakis, Experimental and Analytical Study of EMC/EMI effects on PCBs and Cables enclosed within Metallic Structures. IEEE APS International Symposium, pp , 9-14 July Z. A. Khan, Y. Bayram and J. L. Volakis, Experimental Study of Electromagnetic Interference (EMI) on PCBs and Cables enclosed in Complex Structures. National Radio Science Meeting, 4-7 January Y. Bayram, Z. A. Khan and J. L. Volakis, Experimental and Theoretical Study of Digital Circuits Subject to Electromagnetic Interference, URSI Boulder Meeting, January 2005 Z. A. Khan, C. F. Bunting, Shielding Effectiveness Studies of Rectangular Enclosures with Apertures against EM Fields with Arbitrary Angles of Incidence and Polarizations, IEEE EMC International Symposium, vol. 1, pp , August Z. A. Khan, Shih-Pin Yu and C. F. Bunting, Statistical Shielding Effectiveness - a Modal/Moment Method Approach to Characterize the Average Shielding Effectiveness Over a Wide Frequency Range including Resonances, IEEE EMC International Symposium, vol. 2, pp , August vii

10 FIELDS OF STUDY Major Field: Electrical and Computer Engineering Studies in: EMI/EMC Analysis of Mixed Signal Circuits EMI/EMC Analysis of Shielding Enclosures Professor John L. Volakis Dr. Charles F. Bunting viii

11 TABLE OF CONTENTS Page Abstract Dedication Acknowledgments Vita ii iv v vi List of Figures xii Chapters: 1. EMI/EMC Analysis of Electronic Systems Introduction and motivation Challenges in EMI/EMC analysis of electronic systems Mixed signal EMI/EMC analysis of complex electronic systems using a decomposition approach Hybrid S-matrix approach for concurrent on/off board EMI/EMC analysis of electronic systems Contributions of this dissertation Dissertation layout Hybrid S-Parameters for External Plane Wave Excitation of Electronic Circuits Hybrid S-matrix Approach Derivation of the hybrid S-matrix Decomposition of the induced transmission line voltage Extracting V T L and V forced from V total ix

12 2.2.3 Incorporating V T L into S-parameters Calculating the hybrid S-parameters {HS} Algorithm for calculating the hybrid S-matrix Plane wave coupling to non-uniform transmission lines Plane wave coupling to PCBs with connecting planar wires Plane wave coupling to PCBs with connecting vertical wires Hybrid S-matrix for PCBs and wires enclosed by cavities Rectangular cavity enclosing a PCB with connecting planar wires PCB connected with vertical wires and enclosed by a rectangular cavity PCB connected to oblique wires and enclosed by a rectangular cavity Conclusions Hybrid S-matrix Approach for Non plane wave EMI Excitations A spectral domain approach for extracting TL and forced modes from the EMI induced voltages on a transmission line Bandwidth approximation of the induced forced modes Algorithm for extracting the hybrid S-matrix for near-zone EMI excitations Validation Study Multi-layered PCB exposed to EMI from a near zone wire Enclosed PCB exposed to EMI from a near zone source Conclusions Experimental Validation of the Hybrid S-matrix Approach for Complex Structures Measurement setup: Description of cavities, PCB and the field sensing probe EMI coupling onto enclosed cables via apertures and penetrating wires Field penetration through apertures into the empty multisection enclosure EMI coupling onto enclosed cables via apertures and penetrating wires: Validation of the scattering matrix approach for EMI/EMC analysis of PCBs enclosed by multi-cavity structures EMI coupling to PCB traces enclosed by multi-cavity structures Conclusions of the experimental study x

13 5. Conclusions and Future Work Appendices: A. Agrawal s Coupling Method Bibliography xi

14 LIST OF FIGURES Figure Page 2.1 Single layer PCB layout Hybrid circuit model of the PCB (shown in Fig. 2.1) subject to plane wave excitation Plane wave illumination of a PCB enclosed by a rectangular cavity Interconnect layout for the PCB employed in Fig Induced transmission line voltage at PCB ports (see Fig. 2.4) Comparison of the hybrid S-matrix method with Agrawal s coupling technique Single layer PCB layout with four traces PCB with connecting planar wires Hybrid ADS model of the PCB-wire structure (Fig. 2.8) Full wave HFSS simulation to calculate induced transmission line voltages Induced voltage on port P1 for the PCB with connecting planar wires (see Fig. 2.10) Induced voltage on port P3 for the PCB with connecting planar wires (see Fig. 2.10) Induced voltage on port P1 for the PCB connected to open circuit planar wires (see Fig. 2.10) xii

15 2.14 Induced voltage on port P3 for the PCB connected to open circuit planar wires (see Fig. 2.10) Full wave HFSS simulations for the PCB with connecting vertical wires Induced voltage on port P2 for the PCB-wire structure of Fig Induced voltage on port P8 for the PCB-wire structure of Fig PCB connected to planar wires enclosed by a rectangular cavity Full wave HFSS solution for the PCB-wire-cavity structure of Fig Induced voltage on port P4 for the for the PCB-wire-cavity structure of Fig Induced voltage on port P6 for the for the PCB-wire-cavity structure of Fig Induced voltage on port P1 for the for the PCB-wire-cavity structure of Fig with open circuit wires Induced voltage on port P5 for the for the PCB-wire-cavity structure of Fig with open circuit wires PCB with connecting vertical wires enclosed by a rectangular cavity Full wave HFSS simulations for the PCB-wire-cavity structure of Fig Induced voltage on port P5 for the PCB-wire-cavity of Fig Induced voltage on port P7 for the PCB-wire-cavity of Fig HFSS analysis of inclined wires connecting to a PCB enclosed by a rectangular cavity Induced voltage on port P5 for the PCB-wire-cavity structure of Fig Induced voltage on port P7 for the PCB-wire-cavity structure of Fig xiii

16 3.1 PCB trace exposed to radiation from a near zone source Two layered PCB structure exposed to a nearby radiating wire Load configuration at ports of PCB trace on Layer 2(see Fig. 3.2(b)) Fourier spectrum of the induced transmission line voltage along PCB trace on Layer 2 (see Fig. 3.2(b)) ADS model of the hybrid S-matrix for the PCB-cable structure given in Fig Two layered PCB exposed to nearby radiating wire, all enclosed by a cavity (see Fig. 3.2) Illustration of the cavities and the PCB used for the reported measurements Photographs of the measurement setup and actual fabricated Cavities and PCBs Balanced dipole probe used for field measurements Setup for field measurements inside the empty cavity (Fig. 4.1(a)) Field magnitude at the center (of either section) of the empty cavity (see Fig. 4.4 for the measurement setup Setup for measuring the effect of a penetrating wire on EMI coupling to the enclosed cables Field coupling to the enclosed cable for the setup given in Fig Measurement setup for a penetrating wire connected to a PCB inside small rectangular cavity (see Fig. 4.1 for dimensions) S 21 (db) measurements on the PCB for the setup given in Fig Measurement setup with a penetrating wire connected to a PCB at the inner cavity of a multi-cavity structure (all dimension are in meters; see Fig. 4.1) xiv

17 4.11 S 21 (db) Measurements and simulations on the PCB for the setup given in Fig A.1 EMI coupling to an open circuit wire enclosed by a rectangular cavity 80 A.2 EMI coupling to loads connected to a wire enclosed by a rectangular cavity xv

18 CHAPTER 1 EMI/EMC ANALYSIS OF ELECTRONIC SYSTEMS 1.1 Introduction and motivation There is an increasing interest to evaluate electronic systems for possible susceptibility to electromagnetic interference (EMI) and to ensure their electromagnetic compatibility (EMC) with nearby systems sharing the same EM environment. This is important because all electronic systems (including commonly used wireless devices, television stations as well as radar systems) act as potential sources of EMI which can affect proper operation of nearby systems, sometimes leading to severe consequences. EMI was a possible cause for the explosion of Trans World Airlines Boeing 747 Flight 800 (TWA-800) [1] and experimental studies have shown that High Power Microwave (HPM) can seriously damage electronic systems [2, 3]. With recent security threats, increasing clock speeds, crowded frequency spectrum and higher device densities on PCBs, EMI/EMC analysis of electronic systems has become very important in recent years. For these reasons, national and commercial bodies have established certain EMC regulations and standards for both commercial and military installations [4]. These regulations specify limits for EMI emissions from electronic devices as well as their 1

19 susceptibility to EMI generated from nearby systems. Moreover, detailed testing procedures have been defined in these regulations. For example, Mil-Std-461/462D requires semi-anechoic chambers for EMC compliance testing of all electronic devices. Failure of a product to pass these EMC tests results in tedious troubleshooting to locate the problem components. Consequently, post-design fixes and sometimes complete re-designing of the product are necessary. These factors delay market entry of the product and result in significant economic loss to the manufacturers. Therefore, it is desirable to have simulation tools which can efficiently and accurately carry out EMI/EMC analysis of real electronic systems. 1.2 Challenges in EMI/EMC analysis of electronic systems An efficient and accurate analysis of electronic systems, however, is quite challenging due to geometrical and electronic complexity of these systems. Important computational issues associated with major components of an electronic system can be listed as follows: 1. Surrounding structures enclosing the electronic systems are highly resonant giving rise to convergence issues for numerical solvers. Moreover, the presence of small apertures (for ventilation, penetrating wires and display panels) on large metallic enclosures results in huge mesh sizes leading to impractically large computational memory requirements. These factors make it difficult for available computational resources to efficiently handle these structures. 2. Cable bundles typically comprise of strongly coupled closely spaced wires, giving rise to convergence issues for numerical solvers. Moreover, non-uniform 2

20 construction (bent/twisted wires) of these cable bundles makes it difficult to model them in numerical simulations. 3. PCBs have complex multi-layer and highly dense interconnect layouts with linear and non-linear devices operating in active mode which are difficult to accurately model using EM solvers. 1.3 Mixed signal EMI/EMC analysis of complex electronic systems using a decomposition approach While accurate system level solvers are not available, there are many existing tools that can efficiently address a specific EMI/EMC problem. For example, regular metallic enclosures can be efficiently investigated using semi-analytical methods based on cavity Green s functions [5 10], small hole diffraction theory [11, 12], transmission line approximations [13 18], circuital approach [19] as well as Fourier Transform and mode matching techniques [20]. For more general structures, numerical methods based on Finite Element Method (FEM) [21 24], Finite Domain Time Domain (FDTD) method [25 27], Method of Moments (MoM) [19], Transmission Line Matrix (TLM) approach [28] as well as hybrid and Multi-level Fast Multipole Methods (MLFMM) [29,30] have been successfully employed. Similarly, EMI coupling to cable bundles is primarily calculated using multiconductor transmission line theory [31 36] for lower frequencies while iterative solutions of Telegrapher s equations (TICE) [37] are also available for higher frequencies. EMI coupling to printed circuit boards has been investigated using full wave solvers like MoM [38 44], FEM [45], FEM with TEM approx. [46] and FDTD [47] methods. Moreover, approximate methods like Partial 3

21 Element Equivalent Circuit (PEEC) [48 50] and multi-conductor transmission line theory [51] are also available for PCBs. Development of so many approximate methods for these relatively simple problems (despite availability of accurate full wave solutions) is due to the large memory and CPU requirements of rigorous solvers. It is also difficult to accurately model nonlinear devices in EM solvers. Therefore, rigorous EMI/EMC analysis of complex systems involving cavities, cables and PCBs with linear/nonlinear devices, is a formidable task and requires more efficient yet accurate alternative techniques. To this end, following EM Topology concepts [52], we propose to decompose the whole system into smaller and simpler EMI/EMC problems that can be separately solved using computational tools developed for cavities, cables and/or PCBs, thereby significantly reducing the computational overhead. Moreover, one may employ the most efficient and accurate computational method available for a specific problem. This means that while EMI coupling due to the large surrounding structures is calculated using EM solvers, the linear/nonlinear devices can be modeled using circuit solvers and methods like HSPICE [53 55] and Harmonic Balance Methods [56 59]. 1.4 Hybrid S-matrix approach for concurrent on/off board EMI/EMC analysis of electronic systems In the context of the proposed system decomposition approach, a hybrid scheme was recently proposed which integrates EM and circuit solvers in an efficient and accurate manner. Referred to as the hybrid S-matrix approach [60, 61], this scheme relies on the observation that plane wave coupling to a transmission line induces a distinct forced mode and a pair of natural TL modes on a transmission line [62]. 4

22 Forced modes are directly associated with the incident plane wave while the natural TL modes arise due to reflections from the line ends. The hybrid S-matrix approach employs full wave simulations to calculate induced transmission line voltages in the absence of circuit devices. Then the induced forced and natural TL modes are extracted using generalized pencil of function (GPOF) method [63]. The TL modes are incorporated into the S-matrix of the transmission line network by introducing an additional hybrid port, representing EMI coupling. The obtained hybrid S-matrix allows the whole PCB interconnect and cable layout to be transported to a circuit solver where the forced modes are also introduced as constant voltage sources at the ports. Now circuit devices are connected to the ports and their performance subject to EMI is investigated. Since full wave solvers are employed to account for the effects of large surrounding structures, PCB traces and cables while circuit devices are handled in circuit solvers, the hybrid S-matrix approach fully exploits the advantages of the EM and circuit solvers without making any quasi-static assumptions. As originally proposed, the hybrid S-matrix approach is limited to plane wave illuminations. However, in realistic environments, circuits and cables are usually exposed to EMI generated by near zone sources, such as antennas, cables and On/Off-board circuit devices. Therefore, implementing the hybrid S-matrix approach for near zone illuminations is of significant importance. Since near-zone source induce a continuous spectrum of forced modes, it is difficult to employ GPOF for accurately extracting the TL modes. For this purpose, in this dissertation, we propose a new spectral approach to extract the forced and natural TL modes induced on a transmission line subject to non plane wave excitations. We also study the implementation of the hybrid S-matrix 5

23 approach for cavity modes. In addition, we discuss results of an experimental study for validation of the hybrid S-matrix approach for complex EMI coupling problems. 1.5 Contributions of this dissertation Major contributions of this dissertation are: - Extension of the hybrid S-matrix method to non-plane wave EMI excitations - Demonstration of the validity of the hybrid S-matrix method for realistic EMI/EMC scenarios - Experimental validation of the hybrid S-matrix approach for complex EMI coupling scenarios including (a) multi-cavity structures, (b) cable to cavity interactions and (c) cable to PCB interactions - Providing accurate reference data for possible validation of future EMI/EMC numerical tools 1.6 Dissertation layout This dissertation is divided into five chapters. The next chapter, Chapter 2, presents formulation of the hybrid S-matrix approach for plane wave coupling to electronic systems. Some examples involving non-uniform transmission lines as well as PCBs enclosed by cavities are also discussed. In Chapter 3, we present a spectral method based on DFT to implement the hybrid S-matrix approach for near-zone EMI sources. Chapter 4 presents a series of measurements to validate the hybrid S-matrix approach for realistic EMI coupling scenarios. Finally, in Chapter 5, we discuss our conclusions based on the present study. 6

24 CHAPTER 2 HYBRID S-PARAMETERS FOR EXTERNAL PLANE WAVE EXCITATION OF ELECTRONIC CIRCUITS As mentioned in Chapter 1, electronic systems are typically composed of highly resonant cavities, strongly coupled non-uniform cables and very small PCBs with linear and nonlinear devices. These factors make system level EMI/EMC analysis of real electronic systems quite challenging. However, there are many available computational tools that can perform standalone EMI/EMC analysis of cables, PCBs, or cavities. Following EM topology concepts [52], one may combine these individual tools in an integrated fashion to solve problems of higher complexity. For example, it is common practice to employ Multiconductor Transmission Line Theory (MTLT) along with Agrawal s coupling equations to calculate EMI induced voltages and currents on PCBs and cables [31 35]. As expected, some computational tools are more accurate for circuit simulations, while others are more efficient in handling large EM structures. Therefore, employing selective computational tools in an integrated fashion makes system level EMI/EMC analysis not only computationally efficient but also allows for optimum use of the computational tools [36]. Such integration schemes, however, suffer from lower accuracy issues since combining different computational tools often leads to approximations. For example, the 7

25 aforementioned Agrawal s coupling technique is limited to lower frequencies since it assumes only quasi-static coupling among cable sections. Accuracy can be improved via iterations but at the cost of efficiency [37,64]. Full wave solutions such as FDTD, for concurrent EM and circuit analysis are accurate but have impractically large computational overhead [36]. To address issues of efficiency but still maintain accuracy, a new technique for integrating computational tools was recently presented for system level EMI/EMC analysis of electronic systems [61, 65]. This scheme, referred to hybrid S-matrix approach, is based on using the scattering parameters to integrate EM and circuit solvers by introducing a hybrid port to account for the EMI coupling to PCBs and cables. Introduction of this hybrid port allows fully exploiting the computational advantages of EM and circuit solvers without compromising accuracy. Specifically, EMI coupling to PCB and cable ports is calculated using full wave EM solvers which take into account the contributions from the large surrounding structures, PCB interconnects and cables but in the absence of circuit devices. This means no quasi-static assumptions are made. Then, modifying the system s S-matrix by introducing the contribution from EMI coupling, the entire system is modeled in a circuit solver where linear and nonlinear devices are introduced and their performance subject to EMI is evaluated. We will discuss this hybrid approach in this chapter with some validation examples. Section 2.1 gives an overall formulation of this hybrid approach while mathematical derivations are given in Section 2.2. Various EMI/EMC scenarios not studied before using this hybrid approach are discussed in Section 2.3 and Section 2.4. Finally, some conclusions based on important limitations of hybrid S-matrix approach are given in Section

26 2.1 Hybrid S-matrix Approach For illustration of the hybrid S-matrix approach, consider a typical PCB structure subject to plane wave illumination as shown in Fig. 2.1(a). As illustrated in Fig. 2.1(b), the PCB has N physical ports. According to microwave theory, this PCB can be characterized by a scattering matrix (or S-matrix) given as: {b} = [S]{a} (2.1) where [S] is an N N matrix, {a} is a N 1 vector and refers to the incoming voltage waves (or excitations) at ports while {b} is a N 1 vector referring to the outgoing or reflected voltage waves at ports. A plane wave impinging upon the PCB will induce voltages and currents on various PCB ports. To account for these external fields, (2.1) can be modified by noting that the total voltage induced on a transmission line can be decomposed into the forced and natural TL modes: V total = V forced + V ± T L (2.2) where V forced is directly associated with the illuminating plane wave and propagates on the transmission line with the wave number equal to the wave number of the incident plane wave. For an infinite transmission line, V forced is the only mode that exists due to the incident plane wave. For finite transmission lines, however, mismatched terminations at the ports give rise to additional forward and reverse propagating TL modes, V ± T L. The wave numbers of these TL modes are determined by the transmission line characteristics. 9

27 (a) PCB structure (b) PCB Interconnect layout with devices removed Figure 2.1: Single layer PCB layout 10

28 Figure 2.2: Hybrid circuit model of the PCB (shown in Fig. 2.1) subject to plane wave excitation To account for V ± T L, we introduce an additional hybrid port representing the external field excitation. This can be done by modifying (2.1) as: {b} = [S]{a} + {HS}a N+1 (2.3) where {HS} are the hybrid S-parameters with a N+1 being a coefficient associated with the external plane wave strength. Equation (2.3) implies that the entire PCB network illustrated in Fig. 2.1 can be characterized by an [(N + 1) (N + 1)] matrix. Once this matrix is defined, it can be ported to a circuit solver with the (N + 1) th -port representing the external EMI excitation. For example, the PCB structure of Fig. 2.1 can be modeled as an N + 1-port hybrid device in ADS as shown in Fig

29 2.2 Derivation of the hybrid S-matrix The hybrid S-matrix approach is based on identifying and extracting the forced and natural TL modes from the induced transmission line voltage. Once extracted, the TL modes are incorporated into the hybrid S-matrix which allows the microwave network to be ported to a circuit solver. Here, the forced modes are applied as constant voltage sources at the corresponding ports. The mathematical details of this formulation will be discussed in the following sections. Section explains the decomposition of the induced transmission line voltage into the forced and natural TL modes. Extraction of these modes using GPOF is discussed in Section and incorporation of the TL modes into the S-matrix is given in Section Section2.2.4 gives the derivation of the formulas for calculating the hybrid S-parameters. Finally, the series of calculations required to obtain the hybrid S-matrix is given in Section (We mention here that some material in this section is taken and paraphrased from Bayram s dissertation [66].) Decomposition of the induced transmission line voltage We start our analysis of plane wave illumination of PCB traces and cables by noting that the total electric field must satisfy the Dirichlet boundary condition on PEC surfaces: ˆn E total = 0 (2.4) where E total is composed of: E total = E inc + E scat (2.5) 12

30 In (2.5), E inc refers to the incident electric field and E scat is the electric field radiated by the induced currents on the PCB interconnects. To satisfy boundary conditions, (2.4) implies that the induced currents should match the wave number of the incident plane wave. For finite transmission lines, however, additional voltage/current waves are generated due to reflections from the ports. These additional waves propagate on the line with wave numbers depending on the line characteristics. To distinguish between the fields induced directly due to the incident plane wave and the fields arising due to the reflections from the ports, we refer the former as forced waves, E forced while the latter as the TL waves, E T L. Therefore, the total field given in (2.5) can also be represented as: E total = E forced + E T L (2.6) where, as implied by (2.4), E T L also satisfies the Dirichlet boundary conditions on PEC surfaces: ˆn E T L = 0 (2.7) Corresponding to (2.6), we can write the induced transmission line voltage as: V total = V forced + V T L (2.8) Depending on the frequency of the incident wave and line characteristics, a number of TL modes can possibly be excited on a transmission line. Therefore, we may express V T L as a modal sum of forward and reverse propagating waves: V T L = k A k e k e γ kz + k B k e k e γ kz (2.9) where γ k represents the wave number of the k th mode of V T L with A k and B k being the corresponding mode coefficients. 13

31 2.2.2 Extracting V T L and V forced from V total To extract V T L and V forced from V total, we note that any measured signal can be approximated in terms of complex exponentials: y(r) R i e sir ; 0 r L (2.10) i=1,m where R i is the residue or complex amplitude of the i th mode and s i are the complex poles of the data array. For an equally spaced discrete data sequence of length N, (2.10) can be written as: y(kδr) R i e sikδr ; k = 0, 1,..., N 1 (2.11) i=1,m where δr is the sampling interval. Several methods are available which can achieve the approximation given in (2.10) and (2.11). More popular among them are the Prony method [67 69], the pencil of function (POF) method [70 74] and the generalized pencil of function (GPOF) method [63]. Among all these methods, GPOF has the advantage of having no practical limit on the maximum number M of dominant modes and being more robust to the noise compared to the other linear techniques. Moreover, GPOF has the lowest variance in extracted poles as compared to other parametric estimation methods. For these reasons, throughout this chapter, we will employ GPOF for extracting the V T L and V forced from the total induced transmission line voltage. 14

32 2.2.3 Incorporating V T L into S-parameters Once extracted from the total field, V T L can be incorporated into the S-matrix, which is defined for an N port network (for the k th mode) as: b k S1,1 1. k... S1,N k a k 1 = (2.12) b k N SN,1 k... SN,N k where a k i refers to the incident wave and b k i refers to the reflected wave at the i th port. Also, a k N Sij k = bk i a k a k n =0 n j (2.13) j where a k n = 0 can be achieved by terminating n th port with its corresponding reference impedance Z refn. To account for V T L induced on the ports, we introduce an additional port representing plane wave excitation and modify the existing N-port S-matrix as following: b k S1,1 1. k... S1,N k HS1,N+1 k a k 1 b k = (2.14) N S k b k N,1... SN,N k HSN,N+1 k a k N+1 SN+1,1 k... SN+1,N k N HSk N+1,N+1 a k N+1 where the (N + 1) th port, characterized by the hybrid S-parameters HS k i,n+1, represents the plane wave coupling to the i th port for the k th TL mode. To understand the relation between the TL modes and the hybrid S-parameters HSi,N+1 k, we note that the port voltages and currents can be defined in terms of TL modes as following: V k i I k i = = C i C i E k dl (2.15) H k dl 15

33 where (V k, I k ) denote the TL voltages and currents at the i th port due to the k t h mode. Furthermore, the mathematical relation between the k th incident and reflected TL modes amplitudes (i.e. a k i and b k i respectively), is given as [75]: a k i = V i k + Z refi Ii k 2 (2.16) Z refi b k i = V i k Z refi Ii k 2 Z refi where Z refi is the reference impedance for the corresponding i th port. As may be noted, (2.15) and (2.16) serve as an interface between the circuit and RF solvers, where the TL modes extracted by GPOF are incorporated into the system S-matrix which can subsequently be modeled by a circuit solver Calculating the hybrid S-parameters {HS} To calculate the hybrid S-parameters, the inherent relations between the incident and reflected waves, and voltages and currents at the port is exploited. For this purpose, corresponding to the hybrid S-matrix defined in (2.14), we introduce the following impedance matrix for the (N + 1) port network: I V 1 Z 1,1... Z 1,N HZ 1 1,N+1. = VN k I Z N,1... Z N,N HZ N = [Z]{I} + {V T L,oc} (2.17) N,N+1 I N+1 where {V T L,oc } = {HZ}I N+1 refers to the open circuit TL voltage induced at the ports due to the incident plane wave. It may be noted that coupling to the (N + 1) th port, being of no interest in our analysis, has been excluded in the Z-matrix by setting the (N + 1) th row of the Z-matrix to zero. Employing (2.16) in (2.17) gives, [Z ref ]({a} + {b}) = [Z]( [Z ref ]) 1 ({a} {b}) + {V T L,oc } (2.18) 16

34 where {a} T = {a 1... a N } (2.19) and {b} T = {b 1... b N } (2.20) Z 1,1... Z 1,N [Z] =..... (2.21) Z N,1... Z N,N Zref1 [Z ref ] = ZrefN (2.22) where [Z] and [Z ref ] are already known matrices. Rearranging the terms, we find that the coefficients of the incident and reflected waves at the physical ports are given by, {b} = {[Z]( [Z ref ]) 1 + [Z ref ]} 1 {[Z]( +{[Z]( [Z ref ]) 1 + [Z ref ]) 1 [Z ref ]} 1 {a} (2.23) [Z ref ]} 1 {V T L,oc } Comparing (2.23) with (2.17) and setting a N+1 = to normalize the incident plane wave, we finally get, E 0 ZrefN+1 {HS} = ZrefN+1 E 0 {[Z]( [Z ref ]) 1 + [Z ref ]} 1 {V T L,oc } (2.24) where E 0 is the magnitude of the incident plane wave and {HS} = {HS 1... HS N }. The evaluation of [Z ij ] in (2.17) is done in the usual manner via open circuit analysis. Therefore, once V T L,oc are extracted, {HS} can be calculated via (2.24). Moreover, since V forced do not depend on the terminations, they can be directly exported to a circuit solver and applied as constant voltage sources at the ports. The details are given in the following Section

35 2.2.5 Algorithm for calculating the hybrid S-matrix To generate the hybrid S-matrix coefficients, a series of full wave and circuit simulations must be carried out. First, leaving the ports open, a full wave solver calculates the total induced voltage, V total, along PCB traces and cables. Then, GPOF is employed to extract V T L and V forced from V total. Subsequently, V T L are substituted in (2.24) to calculate the hybrid S-parameters, {HS}. Finally, using the calculated hybrid S-matrix, the cable/pcb microwave network is transported to a circuit solver for system performance evaluation. These step by step calculations are listed as under: 1. Remove all devices/loads from the PCB/cable ports and terminate them with some reference impedance, Z ref 2. Employ full wave simulations to calculate the S and Z matrices defined in (2.12) and (2.17). 3. With no device connected to the ports, employ full wave simulations to calculate equally spaced samples of the induced transmission line voltages, V total. 4. Using GPOF, extract V T L and V forced from V total for each port. 5. Substitute V T L into (2.24) to calculate the hybrid S-parameters {HS}. 6. Incorporate {HS} in (2.14) to obtain the hybrid S-matrix. 7. Model the transmission line network as an N port hybrid device, characterized by the hybrid S-matrix, in a circuit solver such as Agilent s ADS. 8. Apply V forced values as constant voltage sources at each port. 18

36 9. Connect a voltage source representing the incident plane wave to the (N +1) th hybrid port. 10. Connect circuit devices at the ports in a the circuit solver 11. Run circuit simulations to evaluate system performance subject to plane wave incidence. Figure 2.3: Plane wave illumination of a PCB enclosed by a rectangular cavity 19

37 For demonstration, consider the example illustrated in Fig As shown, a plane wave is incident normally on a rectangular cavity enclosing a PCB. Fig. 2.4 gives the load configuration on the PCB ports. Since there are four ports, the PCB layout can be characterized by a [4 4] S-matrix. Using the algorithm described above, we calculate the hybrid S-parameters for the cavity/pcb structure subject to the incident plane wave, obtaining a [5 5] hybrid S-matrix. With this hybrid S-matrix, the whole PCB layout is then modeled inside Agilent s ADS and the load configuration (shown in Fig. 2.4) is applied at the ports. The results are plotted in Fig. 2.5 and show a very good agreement with the full wave HFSS simulations. Figure 2.4: Interconnect layout for the PCB employed in Fig

38 (a) Voltage magnitude at port 1 (b) Voltage phase at port 1 (c) Voltage magnitude at port 2 (d) Voltage Phase at port 2 Figure 2.5: Induced transmission line voltage at PCB ports (see Fig. 2.4) 21

39 To show that the above hybrid approach is also valid for cables, we consider another example, shown in Fig. 2.6(a). As illustrated, a plane wave illuminates a rectangular cavity enclosing a cable terminated with lumped loads. We calculate the induced voltages using the hybrid S-matrix approach. For comparison, we also calculate the induced voltages using Agrawal s coupling solution [33] which is obtained using cavity Green s functions and a transmission line solver (via Onera a CRIPTE [76]). As seen in Fig. 2.6(b), the results show a very good agreement between the two integration schemes. 2.3 Plane wave coupling to non-uniform transmission lines The hybrid S-matrix approach with plane wave EMI illumination on PCBs and cables has already been validated [60, 66]. Specifically, the hybrid S-matrix approach was employed to calculate plane wave coupling to a pair of transmission lines. It was also used to characterize the performance of an inverter, an RF amplifier and an RF mixer. In all these examples, a continuous reference plane was assumed for the transmission lines. In practical situations, however, the discontinuities arise when different PCBs are connected via wires. As such, these discontinuities introduce reflections on the transmission line in addition to port reflections. Our goal, in this section, is to demonstrate the validity of the hybrid S-matrix approach for these relatively complex structures. For this purpose, in this section, following two variations of the PCB with connecting wires are discussed: 1. PCBs with connecting planar wires (Section 2.3.1) 2. PCBs with connecting vertical wires (Section 2.3.2) 22

40 (a) A loaded wire enclosed by a rectangular cavity (b) Voltage magnitude on port 2 Figure 2.6: Comparison of the hybrid S-matrix method with Agrawal s coupling technique 23

41 The PCB structure employed in these examples is illustrated in Fig.2.7. It is assumed that the PCB substrate is 2mm thick and has a dielectric constant of ɛ r = 2.2. Figure 2.7: Single layer PCB layout with four traces Plane wave coupling to PCBs with connecting planar wires The case when the PCB (illustrated in Fig. 2.7) is connected to the incoming planar wires is shown in Fig We assume that the PCB and wire structure is being illuminated by a plane wave impinging normally on the PCB-wire structure. Following the algorithm given in Section 2.2.5, we first calculate the [8 8] S-matrix for the PCB/wire ports using HFSS. Full wave HFSS simulations were also employed to sample induced transmission line voltages, V total along the PCB traces and wires with all ports open (no loads connected). Using GPOF, V T L and V forced are extracted from V total. Incorporating V T L in (2.24), we calculate the hybrid S-parameters {HS}, 24

42 thereby obtaining the [9 9] hybrid-s matrix. As shown in Fig. 2.9, the entire PCBwire structure was then ported as a 9-port hybrid device in ADS. As shown, a voltage source representing EMI to the additional 9 th -port. The loads are then connected at the ports and ADS simulations were carried out to calculate the induced voltages. For comparison, full wave solution was also obtained via HFSS (see Fig. 2.10). Fig and Fig plots the calculated voltages on ports P1 and P3 and show an excellent agreement between the hybrid S-matrix and HFSS. Figure 2.8: PCB with connecting planar wires We now demonstrate that once the hybrid S-matrix for a transmission line network is calculated, it can be re-used if the load configuration at the ports is changed. For this purpose, consider the case when the incoming wires are left open. As EM domain is not changed, we don t need to re-calculate the hybrid S-matrix. Only ADS simulations are required for new load configurations at the ports. This significantly 25

43 Figure 2.9: Hybrid ADS model of the PCB-wire structure (Fig. 2.8) Figure 2.10: Full wave HFSS simulation to calculate induced transmission line voltages 26

44 (a) Magnitude (b) Phase Figure 2.11: Induced voltage on port P1 for the PCB with connecting planar wires (see Fig. 2.10) (a) Magnitude (b) Phase Figure 2.12: Induced voltage on port P3 for the PCB with connecting planar wires (see Fig. 2.10) 27

45 reduces computational time. The results are plotted in Fig and Fig and agree well with the HFSS simulations. (a) Magnitude (b) Phase Figure 2.13: Induced voltage on port P1 for the PCB connected to open circuit planar wires (see Fig. 2.10) Plane wave coupling to PCBs with connecting vertical wires Next, we consider the case when the PCB is connected to vertical wires. Ports P1, P2, P7 and P8 are now redefined at points where PCB traces connect to the incoming wires. This means we have to re-calculate the S and Z parameters of the PCB-wire layout. As in the previous example, we modify the S-matrix by incorporating the hybrid S-parameters, thereby obtaining the [9 9] hybrid S-matrix. Similarly, using the hybrid S-matrix, we transport the network as a 9-port device to ADS with the additional 9 th -port acting as an input port to allow EMI coupling to the system 28

46 (see Fig. 2.9 for ADS model). The given load configuration is then applied at the ports and the induced voltages are calculated at the ports. For comparison, HFSS simulations(fig. 2.15) were also employed to calculate total induced voltages on the ports. As plotted in Fig and Fig. 2.17, we see that the hybrid S-matrix agree well with the full wave solution. 2.4 Hybrid S-matrix for PCBs and wires enclosed by cavities In following examples, hybrid S-matrix approach is employed to calculate EMI induced voltages on cable and PCB ports when they are enclosed by a rectangular cavity. Specifically, we consider the following cases: 1. Rectangular cavity enclosing a PCB with connecting planar wires (Section 2.4.1) 2. Rectangular cavity enclosing a PCB with connecting vertical wires (Section 2.4.2) 3. Rectangular cavity enclosing a PCB with connecting oblique wires (Section 2.4.3) Rectangular cavity enclosing a PCB with connecting planar wires Fig illustrates a PCB connected with planar wires and enclosed by a rectangular cavity. As shown, all PCB and cable ports are terminated with lumped loads (For load configuration at PCB ports as well as PCB layout, refer to Fig. 2.7). Following the algorithm given in Section 2.2.5, we calculate the S-parameters of the given network (PCB and wires within rectangular cavity) using Ansoft s HFSS obtaining an [8 8] S-matrix for the given frequency range. For calculation of the hybrid S- parameters {HS},the PCB and cable ports were left open and the induced voltages along the PCB and wires were calculated. From these induced voltages, we extracted 29

47 (a) Magnitude (b) Phase Figure 2.14: Induced voltage on port P3 for the PCB connected to open circuit planar wires (see Fig. 2.10) Figure 2.15: Full wave HFSS simulations for the PCB with connecting vertical wires 30

48 (a) Magnitude (b) Phase Figure 2.16: Induced voltage on port P2 for the PCB-wire structure of Fig (a) Magnitude (b) Phase Figure 2.17: Induced voltage on port P8 for the PCB-wire structure of Fig

49 Figure 2.18: PCB connected to planar wires enclosed by a rectangular cavity V T L terms and calculated {HS} using (2.24). Finally, we incorporated calculated hybrid S-parameters into the S-matrix of the network, thereby obtaining a [9 9] hybrid S-matrix for the given structure. Using the hybrid S-matrix, thus calculated, the network was ported as a 9 port device to ADS with the additional 9 th -port acting as an input port to allow EMI coupling to the system. As shown in Fig. 2.9, the given loads were connected to the ports and the ADS simulations were employed to calculate the induced voltages. For full wave solution, the whole network with the given load configuration was modeled in HFSS (see Fig. 2.19). For each port of the network, the magnitude and phase of the induced voltages are plotted in Fig and Fig As seen, the hybrid-s method compares well with HFSS. 32

50 Figure 2.19: Full wave HFSS solution for the PCB-wire-cavity structure of Fig (a) Magnitude (b) Phase Figure 2.20: Induced voltage on port P4 for the for the PCB-wire-cavity structure of Fig

51 (a) Magnitude (b) Phase Figure 2.21: Induced voltage on port P6 for the for the PCB-wire-cavity structure of Fig Next, we calculate the induced voltages at the cable and PCB ports when the wires connecting to the PCB are open at the ends aways from the PCB. Since only port conditions are changed and the EMI excitation and other setup remains the same, we don t need to re-calculate the hybrid S-matrix and employ the one that was calculated in the Section Therefore, all we need is to run the ADS simulations again for the new load conditions at the ports (i.e., with ports P1, P2, P7 and P8 left open). For validation, the HFSS simulations had to re-run for the new load configuration. This shows the computational efficiency of the hybrid S-matrix approach. Results are are plotted in Fig and Fig As seen, the hybrid-s method compares well with HFSS. 34

52 (a) Magnitude (b) Phase Figure 2.22: Induced voltage on port P1 for the for the PCB-wire-cavity structure of Fig with open circuit wires (a) Magnitude (b) Phase Figure 2.23: Induced voltage on port P5 for the for the PCB-wire-cavity structure of Fig with open circuit wires 35

53 Figure 2.24: PCB with connecting vertical wires enclosed by a rectangular cavity PCB connected with vertical wires and enclosed by a rectangular cavity Next, we consider the case when the PCB is connected to vertical wires and is enclosed by a rectangular cavity as illustrated in Fig Again, we calculate [9 9] hybrid S-matrix using the algorithm described in Section and model the PCBwire-cavity structure as a 9-port device in ADS (see Fig. 2.9 for ADS model). The loads are now connected to the ports and the induced voltages are calculated. Results are plotted in Fig and Fig and show a good agreement with HFSS. 36

54 Figure 2.25: Fig Full wave HFSS simulations for the PCB-wire-cavity structure of (a) Magnitude (b) Phase Figure 2.26: Induced voltage on port P5 for the PCB-wire-cavity of Fig

55 (a) Magnitude (b) Phase Figure 2.27: Induced voltage on port P7 for the PCB-wire-cavity of Fig PCB connected to oblique wires and enclosed by a rectangular cavity Finally, consider the case when the wires connect to the PCB at 45 degrees. Again, we calculate a [9 9] hybrid S-matrix and model the network as a 9-port device in ADS (see Fig. 2.9 for ADS model). The loads are connected to ports and the induced voltages at the ports are calculated. For comparison, full wave solution was also obtained via HFSS (see Fig. 2.28). Results are plotted in Fig and Fig. 2.30, and we see that the hybrid S-matrix agrees well with HFSS. 2.5 Conclusions As seen from the presented examples, the hybrid S-matrix approach is an efficient and accurate method for modeling electronic systems, particularly when the load configuration at the ports is changing. Moreover, since no quasi-static approximations 38

56 Figure 2.28: HFSS analysis of inclined wires connecting to a PCB enclosed by a rectangular cavity (a) Magnitude (b) Phase Figure 2.29: Induced voltage on port P5 for the PCB-wire-cavity structure of Fig

57 (a) Magnitude (b) Phase Figure 2.30: Induced voltage on port P7 for the PCB-wire-cavity structure of Fig are made in its formulation, this approach is more accurate than prevalent schemes like Agrawal s method or PEEC. A primary advantage of this approach, however, is that it allows separate EM and circuit domain analysis of electronic systems. As EM solvers can efficiently handle large surrounding structures and cables and circuit solvers can model nonlinear devices more accurately, therefore, the proposed decomposition is very useful for realistic EMI/EMC analysis of electronic systems. In formulating the hybrid S-matrix, some assumptions were made. For example, it was assumed that the dominant electric field is bound between the line and the ground. Also, scattering from the PCB devices was neglected. These are realistic assumptions as circuit devices are typically very small and are usually covered by 40

58 ground plane which is included in the EM solution. Also typical substrate thicknesses are around transmission lines. λ 100 to λ 50 implying that the TEM mode is the dominant mode in Formulation of the hybrid S-matrix assumes a plane wave coupling to the circuit devices. This is not a practical EMI coupling scenario as the circuits are usually exposed to EMI generated by near zone sources such as connecting cables, antennas and On/Off board circuit devices. Thus, for practical use of the hybrid S-matrix approach, we must adopt it to allow for near-zone EMI excitations. This is discussed in Chapter 3. 41

59 CHAPTER 3 HYBRID S-MATRIX APPROACH FOR NON PLANE WAVE EMI EXCITATIONS As discussed in Chapter 2, the described hybrid S-matrix approach relies on plane wave excitation i.e. it is not applicable to near zone sources such as antennas or cables illuminating electronic devices. A major challenge in modifying the hybrid S-matrix approach for near zone EMI sources is the determination of the forced and natural TL modes excited on the PCB traces and cables. As is well known, near zone sources generate a continuous spectrum of plane waves and this can be easily illustrated by noting the Weyl s identity, e jkr 4πr = j e jkz z e j(kxx+kyy) dk 8π 2 x dk y (3.1) k z That is, a spherical wave can be represented by a continuous spectrum of plane waves of the form, A(k x, k y )e jkz z. This means that an infinite number of modes are induced on a transmission line when exposed to near zone excitations. Since curve fitting approaches like the Prony s method or the GPOF can only approximate a limited number of distinct modes, they cannot be applied to extract the forced and 42

60 natural TL modes (induced on the PCB traces and cables due to near zone illuminations). Specifically, when dealing with a continuous spectrum of modes, GPOF has two major computational disadvantages: 1. As a parametric technique, GPOF is applicable only to the data composed of distinct set of modes. Therefore, GPOF is not suitable for extracting TL modes when the induced voltage contains a continuous spectrum of dominant forced modes [77]. 2. If number of modes is increased for better approximation of the continuous spectrum, the very large computational overhead makes GPOF impractical [78]. On the other hand, the accurate extraction of the forced and natural TL modes is very important in implementing the hybrid S-matrix approach (see Chapter 2). As the wave number of these modes is known, the Discrete Fourier Transform (DFT) can be a better alternative to GPOF. Unlike GPOF, which provides the best fit approximation of the measured data in the least square sense, the DFT decomposes a signal into a series of orthogonal basis functions. Thus each mode of the induced transmission line voltage can be extracted independently from the Fourier spectrum without interference from the other modes. In employing the hybrid S-matrix approach using DFT, we first carry out (full wave) simulations to calculate the induced voltages on an N-port transmission line network (with all ports open). Using the DFT, we then obtain the mode spectrum of the induced voltages and extract the complex amplitudes of the TL modes by inspection. Substituting these TL modes into (2.24), we subsequently obtain the hybrid S-matrix for the transmission line network. This hybrid S-matrix can then 43

61 be ported in a circuit solver as an (N + 1)-port device where (N + 1) th -port port represents the EMI source. The forced mode is actually included in the circuit solver by applying a constant voltage at the ports. The value of this constant voltage is obtained from the full wave solution of the finite transmission line after subtracting the natural TL mode contributions. Before proceeding to better explain our approach in detail, we like to mention the following major challenges associated with the DFT for data analysis. 1. The DFT algorithm requires large bandwidth to resolve two impulses that may represent reflections from the transmission line terminations. However, as we are only interested in extracting the complex conjugate TL modes, the bandwidth to resolve these modes can be relaxed as they are well separated. 2. The DFT entails an implicit periodicity. This introduces discontinuities at the edges of the measurement interval resulting into spectral leakage to nearby modes. However, as explained in Section 3.1.1, for transmission lines supported by a dielectric substrate such as the PCB traces (with effective permittivity ɛ de > ɛ 0 ), the TL modes are well separated from the forced mode spectrum. Therefore, spectral leakage is not of concern in our case. We discuss our approach in Section 3.1 with an explanation for the bandwidth of the forced modes in Section The algorithm for calculating the hybrid S-matrix is given in Section 3.1.2, followed by some validation examples in Section 3.2. Some concluding remarks are given in Section

62 3.1 A spectral domain approach for extracting TL and forced modes from the EMI induced voltages on a transmission line To better illustrate the mode extraction method, let us consider a PCB trace of length L illuminated by a near-zone EMI source (such as an on-board device, antenna or cable) as illustrated in Fig The thickness of the substrate is h and has a permittivity and permeability of ɛ d = ɛ 0 ɛ r and µ d = µ 0, respectively. Let (E i, H i ) denote the EMI fields incident upon the trace with (E, H) being the total fields (scattered and incident). The voltage induced at any point (x = x l ) along the PCB trace is then given by: V total (x l ) = h where E z is the z-component of the total electric field. 0 E z (x l, z)dz (3.2) As illustrated in Fig. 3.1, V + T L (x) and VT L (x) represent the natural TL modes propagating in the +x-axis and x-axis directions. Denoting the forced modes as V forced, V total (x) can be written in terms of forced and TL modes as: V total (x) = V + T L (x) + V T L (x) + V forced(x) (3.3) To extract the forced and TL modes from V total, we define the following Discrete Fourier Transform (DFT) pair: V (n) DF T Ṽ (k) (3.4) where V (n) = 1 N N 2 Ṽ (k)e j 2π N kn (3.5) k= N 2 45

63 (a) PCB structure (b) Cross-sectional View Figure 3.1: PCB trace exposed to radiation from a near zone source 46

64 and Ṽ (k) = 1 N N 2 V (n)e +j 2π N kn (3.6) n= N 2 where Ṽ represent the Fourier Transform of the voltage V. For a given EMI source, (E, H) can be calculated via a full wave simulation. Subsequently, using (3.2), equally spaced samples, [V ( N ), V ( N + 1), V ( N ),..., V ( N )] of V 2 total along the transmission line are collected. Substituting these voltage samples into (3.6), we obtain the mode spectrum of V total. As the TL mode wave numbers are known from (k d = ω µ d ɛ de ) (where ɛ de is the effective dielectric permittivity of the microstrip), their complex modal amplitudes, a + k d and a k d, can be extracted from Ṽ (k) in (3.6). On identifying the dominant TL mode pair, we can now write: V total (x) = a + k d e j 2π N k dx + a k d e j 2π N k dx + 1 N N 2 k= N 2, k d V (k)e j 2π N kx (3.7) implying, V forced = 1 N N 2 V (k)e j 2π N kx (3.8) k= N 2, k d For an arbitrary excitation source, we expect V forced to have a broad spectrum. However, we are only interested in the value of V forced at the ports, viz V forced (x = ± L) 2 for a simple transmission line. So there is no need to have accurate values for the modes and coefficients of each mode. Once dominant TL modes and V forced at the ports are known, the S-matrix can be calculated for porting to a circuit solver such as ADS. As already noted, the N - port transmission line network subject to EMI is modeled as an (N + 1) - port hybrid device, characterized by the hybrid S-matrix. Electronic devices can now be 47

65 connected to the ports and their performance subject to EMI subject to EMI can be evaluated (via circuit and SPICE like packages in ADS) Bandwidth approximation of the induced forced modes As can be understood, before implementing the DFT (defined in (3.5) and (3.6)), knowledge of the bandwidth of the mode spectrum of V forced and V ± T L is required. Although, no analytical way to determine this bandwidth exists, we can approximate it intuitively. For illustration, let us consider the microstrip structure illustrated in Fig Assuming that the EMI source is a radiating dipole, Chow [79] and Aksun [80, 81] derived closed form spectral-domain Green s functions associated with the dipole radiation in presence of the substrate, in the following form, G = [A(k x, k y )e jkzz + B(k x, k y, ź)e +jkzz ] (3.9) Here A and B are the complex amplitudes of the plane waves, propagating with wave numbers k z. Therefore, the corresponding forced modes induced on the transmission line must also have waves numbers close to k z. Since k z = k 0 cos(θ i ), where 0 θ i π, the values of k z lie in the range k 0 k z k 0, implying that the induced forced modes are band limited by 2k 0. Consequently, if PCB substrate has an effective permittivity ɛ de > ɛ 0, then the wave numbers k d associated with TL modes satisfy the condition k d > k 0. The above can be taken into account when discretizing V total (x) in (3.6). Specifically, the sampling interval, x = L,(where L is the length of the line, and N is the N 48

66 number of samples), must satisfy the relation, x = 1 2k d (3.10) implying that N 2k d L. Typically, setting N = 2k d L is sufficient to resolve TL and forced modes Algorithm for extracting the hybrid S-matrix for nearzone EMI excitations To calculate the hybrid S-matrix for near zone EMI sources, the following steps are needed: 1. Remove all devices/loads from ports and terminate these ports with some reference impedance, Z ref (e.g. 50 Ω). 2. Employ full wave simulations (e.g. Ansoft s HFSS) to calculate the [N N] S and Z matrices (defined in (2.12) and (2.17)) for the N port transmission line network. 3. Leave the ports open and employ full wave simulations to calculate total induced transmission line voltage, V total (as defined in (3.2)). 4. Apply the DFT to V total as in (3.5) and (3.6). 5. Identify the modes with wave numbers k d = ω ɛ de µ, where ɛ de is the effective dielectric permittivity of the substrate. These are the natural TL modes, V ± T L, and their complex amplitudes, a ± kd, correspond to the excitations a N+1 from the hybrid port representing EMI. 6. Using (3.3), calculate the V forced values at each port. 49

67 7. Substitute a ± kd values for each port in (2.24) to obtain hybrid S-parameters {HS}. 8. Expand the S-matrix to incorporate {HS} and obtain the (N + 1) (N + 1) hybrid S-matrix as given in (2.14). 9. Model the N port PCB and cable network in ADS as an (N +1) port hybrid device characterized with the hybrid S-matrix. 10. Apply V forced as a constant voltage source at each port. 11. Introduce circuit devices and loads at the ports. 12. Connect a voltage source to the (N + 1) th port as an EMI source. 13. Carryout the EMI/EMC analysis of the entire system and evaluate its performance subject to EMI. 3.2 Validation Study For validation of our proposed EMI/EMC analysis, we considered a typical EMI/EMC scenario in electronic systems where a PCB is exposed to near zone illuminations from interconnecting cables. Specifically, the following two cases are considered: 1. Multi-layered PCB exposed to EMI from a near zone wire (Section 3.2.1) 2. Multi-layered PCB exposed to EMI generated by a nearby wire, all enclosed by a cavity (Section 3.2.2) 50

68 3.2.1 Multi-layered PCB exposed to EMI from a near zone wire A: Problem Description Fig. 3.2 illustrates a multi-layered PCB structure exposed to radiation from a nearby wire. This wire may represent data and power cables, not usually shielded. As shown in Fig. 3.2(b), the PCB structure consists of two substrate layers. The bottom layer (Layer 2) is grounded by a PEC plane while the medium above top layer (Layer 1) is free space (µ 0, ɛ 0 ). We assume that both substrate layers are 2 mm thick and have a dielectric constant of ɛ r = 2.2. The PCB trace on Layer 1 is 250 mm long, while the trace on Layer 2 is 500 mm long. It is also noted (from Fig. 3.2(b)) that the PCB traces are laterally shuffled with respect to each other by 4 mm. Finally, we note that the radiating wire is connected to a voltage source (10 V) at one and to a resistive termination (5 kω) on the other. The load configuration on the ports of the PCB trace (on Layer 2) is given in Fig As shown, two linear lumped loads are connected on either port of the PCB trace. For the sake of our analysis, we assume that no loads are connected to the PCB trace on Layer 1. The reason for choosing linear loads was to make full wave HFSS simulations possible since we will use HFSS results for comparison with the hybrid S-matrix results. Otherwise, the proposed solution can also be applied to non-linear loads. S-parameters Extraction We start by calculating the S-matrix for the PCB structure of Fig Since we intend to calculate the induced voltages at the PCB trace on Layer 2 only, we limit our analysis to only two physical ports at the bottom trace, significantly reducing 51

69 (a) PCB-cable structure (b) PCB Dimensions Figure 3.2: Two layered PCB structure exposed to a nearby radiating wire 52

70 Figure 3.3: Load configuration at ports of PCB trace on Layer 2(see Fig. 3.2(b)) the computational overhead. Using HFSS, we calculate the following [2 x 2] S-matrix with respect to 50 Ω reference impedances, [ ] (3.11) Extracting the natural TL modes from the Fourier spectrum Having obtained the S-matrix for the PCB on Layer 2, we next proceed to extract the forced and the natural TL modes from the induced transmission line voltage. To do so, we remove the loads from the ports and employ HFSS simulations to calculate the induced voltage along the trace on Layer 2. Subsequently using (3.6), we obtain the Fourier spectrum of the transmission line voltage and plot it in Fig As seen, the TL modes are strongly excited whereas the forced modes are associated with small amplitude values. From the spectrum in Fig. 3.4, we can readily identify the wave number and complex amplitudes of the TL modes. The wave numbers of the extracted TL modes are 53

71 Figure 3.4: Fourier spectrum of the induced transmission line voltage along PCB trace on Layer 2 (see Fig. 3.2(b)) given in Table 3.1 along with the corresponding wave numbers extracted by GPOF. Comparing them with the theoretical values of the wave numbers, we see that the DFT results are very accurate as compared to those of GPOF. This clearly demonstrates the importance of our approach for near zone EMI/EMC analysis. Theoretical Proposed Method (DFT) GPOF Table 3.1: Comparison of TL modes extracted using DFT and GPOF 54

72 Hybrid S-matrix for the PCB structure We next proceed to substitute the complex amplitude of the extracted TL modes in (2.24) to calculate the {HS} parameters. Incorporating the {HS} parameters into the S-matrix given in (3.12), we obtain the hybrid S-matrix as following, (3.12) Finally, the forced modes are obtained by subtracting the TL modes at the ports from the total induced transmission line voltages as given in (3.3). 3-port hybrid model of the PCB structure subject to EMI Having obtained the hybrid S-matrix (3.13) and the forced mode voltages, we now transport the entire PCB structure and the radiating wire to ADS. As shown in Fig. 3.5, the PCB structure subject to EMI from the radiating wire is modeled as a 3-port hybrid device. Ports 1 and 2 represent the physical ports of the PCB trace whereas the third port is the represents the external EMI excitation. Additionally, the forced voltages are applied at each port as constant voltage sources. The results are given in Table 3.2, and are compared to the full wave simulations. HFSS Hybrid S-matrix Magnitude (dbmv) Phase (deg) United Table 3.2: Induced Voltages on Port 1 55

73 Figure 3.5: ADS model of the hybrid S-matrix for the PCB-cable structure given in Fig Enclosed PCB exposed to EMI from a near zone source We now proceed to a more complex EMI/EMC scenario. In this case, the PCB structure and the radiating wire (Fig. 3.2) are enclosed by a rectangular cavity as illustrated in Fig The aperture on the top wall of the cavity was chosen to represent a possible ventilation hole or aperture for penetrating cables. Using the open port analysis explained in Section 3.2.2, the obtained hybrid S- matrix for the PCB trace on Layer 2 is, (3.13) Next, we transport this scattering matrix to ADS, where (as before) loads are introduced and the induced port voltages are calculated. Table 3.3 gives the calculated port voltages using our hybrid S-matrix vs. full wave calculations via HFSS. It is seen that reasonably good agreement is obtained even for the voltage phase validating our approach. 56

74 Figure 3.6: Two layered PCB exposed to nearby radiating wire, all enclosed by a cavity (see Fig. 3.2) HFSS Hybrid S-matrix Magnitude (dbmv) Phase (deg) United Table 3.3: Induced Voltages on Port Conclusions A spectral method based on DFT was proposed in this chapter to generalize the hybrid S-matrix approach for near zone EMI illuminations. In contrast to GPOF (which is limited to estimating distinct modes), DFT gives better estimation of the continuous spectrum of the induced forced modes. Since only the TL modes are 57

75 extracted, only a few samples of the induced transmission line voltage are needed. This significantly reduces the required computational overhead. For validation, a multi-layered PCB structure exposed to a near zone radiating wire was considered. We also considered the case when this PCB structure and the radiating wire were enclosed by a cavity with aperture. In both cases, the induced port voltages were calculated using the proposed method and compared with the HFSS simulations. An excellent agreement was seen. In Chapter 4, we will show the accuracy of the hybrid S-matrix approach for more complicated and realistic EMI/EMC scenarios. 58

76 CHAPTER 4 EXPERIMENTAL VALIDATION OF THE HYBRID S-MATRIX APPROACH FOR COMPLEX STRUCTURES An important issue in the development of new system level EMI/EMC tools is their validation against accurate reference data. Some validations can certainly be carried out using well-established numerical solvers. However, for realistic system characterizations (with large enclosures, cables, PCBs and digital circuits) carefully designed measurements can better serve as validation examples. Towards this end, the IEEE EMC Society TC-9 Committee and the Applied Computational Electromagnetic Society (ACES) have recently proposed a set of simple yet challenging problems for measurements and validation of computational tools [82]. These include a perforated enclosure, PCB cards inside a rectangular cavity, different seam shapes of enclosures, and some complex connectors among others. Though these are important EMI/EMC problems, they do not consider multi-component coupling scenarios which are often quite important and more close to real systems. For example, penetrating wires coupling to enclosed cables or EMI coupling to PCBs within cylindrical and multi-cavity structures are very common EMI/EMC scenarios and computational tools should be able to investigate these important problems. 59

77 Therefore, to validate our hybrid S-matrix approach for non-plane wave EMI excitation of electronic systems, we carried out an experimental study involving multiple EMI coupling scenarios, frequently encountered in real systems. New measurement setups were designed and measurements were made inside compact range of Electro- Science Lab (ESL). Various common EMI/EMC scenarios including EMI coupling via apertures and penetrating wires onto the internal wiring and enclosed PCBs as well as the effects of increasing the shielding levels were investigated. The data obtained through these measurements was employed as reference for validation of hybrid S-matrix approach. This chapter presents this series of measurements to study various EMI/EMC coupling scenarios frequently encountered in electronic systems. Specifically, field penetration through apertures and coupling of penetrating wires onto cables and printed circuit boards (PCBs) enclosed by resonant structures are considered. In contrast to other measurements, here we focus on multi-cavity enclosures in the presence of cables and PCBs. The measurements (controlled environment) allowed for two or more subcomponent interactions such as (a) cables with a cavity, (b) PCB with cables in a single cavity, and (c) cables with PCBs within a multi-aperture and multi-cavity configuration. In the following sections, results of these measurements with validations of hybrid S-parameters approach are discussed. Section 4.1 gives a description of the measured components including the field probing technique. Section 4.2 discusses the measurements for field penetration via apertures as well as field coupling to enclosed cables. Finally, Section 4.3 presents experimental validation of the hybrid S-matrix approach for a multi-cavity structure enclosing a PCB with cables. 60

78 4.1 Measurement setup: Description of cavities, PCB and the field sensing probe Fig. 4.1 shows two small cavities and one large multi-section enclosure to be used in measurements. The smaller cavities in Fig. 4.1(b) and Fig. 4.1(c) can be placed inside the large cascaded enclosure, shown in Fig. 4.1(a), for multi-cavity arrangements. The aperture at the center plate dividing the large enclosure ensures cross coupling between the two sections. We also note that the PCB, shown in Fig. 4.1(d), is a matching microwave circuit transitiong a 50 Ω transmission line to a 100 Ω termination (not connected). All dimensions shown in Fig. 4.1 are for the interior cavity sizes and in meters. Photographs of the actual cavities and the PCB are shown in Fig These cavities were constructed using aluminum sheets of thickness mm except for the middle plate that was 6.35 mm thick. This was necessary since the metal sheets were joined together using fine metal screws, i.e., no welding was done (right angled flanges were used for joining the outer perimeter of the cavity with the screws). For the cylindrical cavity, the top plates were attached to the hollow cylinder (formed by a 6.35 mm thick aluminum sheet) using metal screws. The PCB was attached to the bottom plate of the small rectangular cavity using a conductive glue. As shown in Fig. 4.2(d), all measurements were made inside an anechoic chamber. Fig. 4.3 illustrates the design of the probe used for measuring the fields inside the cavity. The probe was constructed by joining (via soldering) the outer ground conductors of the two semi-rigid coaxial cables. One end of the probe was then tapered to achieve impedance matching to free space. As illustrated, the inner conductors of both coaxial cables are extended to form a 0.05 m long balanced dipole. The other 61

79 (a) Two-section rectangular cavity of size (0.8 x 0.6 x 0.5) (b) Small rectangular cavity of size (0.2 x meters. All apertures are (0.1 x 0.1) meters x 0.2) meters with aperture of size (0.08 x 0.05) meters. (c) Small cylindrical cavity with height 0.2 (d) PCB used in measurements, L=length and W=width of meter high and 0.12 meter in diameter. The the strip. circular aperture has diameter of 0.05 meter. Figure 4.1: Illustration of the cavities and the PCB used for the reported measurements. 62

80 (a) Photograph of the cavities. (b) Arrangement of smaller cavities placed inside large enclosure for multi-cavity measurements. (c) Photograph of the PCB with attached wire. (d) Measurement setup inside the compact range. Figure 4.2: Photographs of the measurement setup and actual fabricated Cavities and PCBs. 63

81 end of the probe connects to a Network Analyzer through a 3-dB, 180 degree hybrid. Therefore, the return currents in both coaxial cables, flowing in opposite directions, cancel each other, suppressing re-radiation. This reduces the scattered field by the probe, especially at lower frequencies, where the differential mode currents can be dominant. Figure 4.3: Balanced dipole probe used for field measurements. 4.2 EMI coupling onto enclosed cables via apertures and penetrating wires In this section, we discuss our measurements for estimating field penetration (through apertures) into cavities and the effect of the penetrating wires on EMI coupling to the enclosed cables. 64

82 4.2.1 Field penetration through apertures into the empty multi-section enclosure As a first step, we evaluated field penetration through apertures into the large empty cavity (Fig. 4.1(a)). For this purpose, the large two-section cavity was illuminated by a plane wave and the interior fields were measured at the center of either section of the cascaded cavity using the probe described in the previous section. This probe was inserted into the cavity via a small hole at the bottom plate. For calibration purposes, fields were also measured in the absence of the cavity. This latter set of measurements was subtracted from the first to suppress cable and probe effects. Measurement setup is illustrated in Fig. 4.4 and the measurements are plotted in Fig. 4.5 where we also include rigorous simulations obtained using Ansoft s HFSS. In general, the agreement is very good and shows the accuracy of the measurement setup, even at locations where the cavity is resonant (corresponding to field spikes in Fig. 4.5) EMI coupling onto enclosed cables via apertures and penetrating wires: In real systems, external EMI may couple onto the internal wiring due to small cracks in cable shields. This scenario becomes more of concern in presence of penetrating wires that may couple external EMI into the cavity. To investigate this scenario, we employed the measurement setup shown in Fig As illustrated, the cylindrical cavity encloses a coaxial cable inserted from the bottom. There is also a wire penetrating the cylindrical cavity via the top slot. We note that the coax has a small (17 mm long) section of its outer conductor removed. This should couple 65

83 Figure 4.4: Setup for field measurements inside the empty cavity (Fig. 4.1(a)). 66

84 Figure 4.5: Field magnitude at the center (of either section) of the empty cavity (see Fig. 4.4 for the measurement setup. 67

85 the cavity fields to the interior conductor of the coax. As shown, one end of the enclosed coaxial cable inside the cavity was left open while the other was connected to the Network Analyzer ensuring a constant 50 Ω termination for the entire frequency range. The penetrating conductor (from the top) was a simple copper wire, 2 mm in diameter. As illustrated, a small length (5 cm) of the penetrating wire is bent horizontally for maximum field coupling. Figure 4.6: Setup for measuring the effect of a penetrating wire on EMI coupling to the enclosed cables. 68

86 To induce EMI coupling, the cylindrical cavity was illuminated by a radiating horn inside the compact range, ensuring a plane wave illumination (see Fig. 4.6). First, the measurements were carried out without a penetrating wire and the slot was the only opening to cause coupling. Subsequently, the wire was also added. In both cases, we subtracted the S 21 values obtained in the absence of cavity to normalize the corresponding S 21 data of the cavity system. Comparison of the measurements with simulations (using Ansoft s HFSS) is given in Fig As seen, a good agreement is observed verifying the measurements. Further, we observe that the cavity shielding is significantly deteriorated due to the wire presence. This is particularly true at lower frequencies as already noted in [29]. Figure 4.7: Field coupling to the enclosed cable for the setup given in Fig

87 4.3 Validation of the scattering matrix approach for EMI/EMC analysis of PCBs enclosed by multi-cavity structures Having obtained confidence with the above simpler structures, we now proceed with a more complex structure. Our goal with this experiment is to validate the hybrid S-matrix approach for concurrent EMI/EMC analysis of circuit devices and EM structures. We begin with small cavity enclosing a PCB and then we will employ a complex multi-cavity structure for our measurements EMI coupling to PCB traces enclosed by multi-cavity structures Consider the setup described in Fig. 4.8, where a PCB is placed inside a small rectangular cavity. A penetrating wire through the slot was directly connected to the microstrip circuit on the PCB (see Fig. 4.1 for dimensions). One end of the PCB interconnect is open while the other is connected to the Network Analyzer (port 2) via an SMA connector. The setup was measured in a compact range (plane wave illumination) and S 21 measurements were recorded. For calibration purposes, another set of S 21 measurements was made without the rectangular cavity and subtracted (in db) from the actual measurements. For simulations, a plane wave excitation was assumed and the induced voltages along the PCB traces were calculated. Following the algorithm described in Section 3.1.2, V modal was extracted using GPOF and (2.24) was employed to calculate the hybrid S-parameters. From Fig. 4.9, we observe good agreement between measurements and simulations over the entire frequency range. A more complex measurement setup is illustrated in Fig where the large cascaded enclosure integrates the setups of Fig. 4.6 and Fig Again, plane wave 70

88 Figure 4.8: Measurement setup for a penetrating wire connected to a PCB inside small rectangular cavity (see Fig. 4.1 for dimensions). 71

89 Figure 4.9: S 21 (db) measurements on the PCB for the setup given in Fig

90 illumination was assumed and the coupling results are shown in Fig The comparison between measurements and calculations is truly remarkable, given the cavity complexity. More importantly, the hybrid S-matrix approach picks up all the resonance spikes over the entire bandwidth. Figure 4.10: Measurement setup with a penetrating wire connected to a PCB at the inner cavity of a multi-cavity structure (all dimension are in meters; see Fig. 4.1). It is interesting to point out that a comparison between the data in Fig. 4.9 and Fig shows more resonances due to the cavity complexity. However, the level of EMI coupling remained about the same, meaning no overall improvement due to 73

91 multi-cavity shielding. This was expected, since the major source of coupling is the penetrating wire, directly connected to the PCB trace. Figure 4.11: S 21 (db) Measurements and simulations on the PCB for the setup given in Fig Conclusions of the experimental study A series of measurements were presented in this chapter to investigate EMI coupling onto PCBs enclosed by multi-cavity structures with cable penetrations. We observe that the penetrating cables can seriously compromise shielding capacity of metallic enclosures. The primary objective of this experimental work was to gather 74

92 reference data for validation of EMI/EMC computational tools for realistic multicavity enclosures. Particularly, we employed the collected data to validate our hybrid S-matrix approach for non-plane wave EMI excitations. As the results show, hybrid S-matrix approach works very well for complex enclosures, verifying our assumptions made in Chapter 3. 75

93 CHAPTER 5 CONCLUSIONS AND FUTURE WORK The discussions in the previous chapters showed that the hybrid S-matrix method is an efficient and accurate approach for system level EMI/EMC analysis of electronic structures. Our several numerical examples and measurements involving transmission lines with discontinuous reference planes and complex structures with resonant cavities demonstrated our claim. Based on this work, we can summarize the following: 1. Efficient EMI/EMC analysis of large electronic structures at the system level can be achieved by decomposing the system into EM and circuit domains. 2. Using several numerical examples in Chapter 2, it was demonstrated that the hybrid S-matrix method can be employed for realistic EMI/EMC platforms. Specifically, we analyzed PCBs with connecting cables in the presence of resonant structures subject to plane wave illuminations and confirmed the accuracy of the hybrid S-matrix approach. 3. For near zone EMI illuminations of electronic circuits, a spectral method based on DFT was introduced and successfully employed to calculate EMI coupling onto PCBs in presence of resonant structures. 76

94 4. In the context of our method, it was found that only the TL modes need be resolved accurately. The forced modes need not be resolved accurately as we only require their total induced voltages at the ports. 5. The accuracy of the hybrid S-matrix approach was also confirmed against measurements on carefully designed multi-cavity resonant structures (with penetrating wires) enclosing PCBs. Having confirmed the validity and accuracy of the hybrid S-matrix approach against complex electronic structures, several realistic EMI/EMC scenarios can now be studied. For example, digital and non-linear analog devices can be employed at the ports and their performance subject to EMI illuminations be evaluated. Since circuit and EM domains are solved separately, surrounding structures and devices can be individually optimized for a robust EMI/EMC design. Therefore, the hybrid S-matrix approach may also be applied for design of the enclosing structures for optimum performance subject to external EMI. Moreover, PCB layouts and port locations can be efficiently optimized for a given EMI/EMC scenario. 77

95 APPENDIX A AGRAWAL S COUPLING METHOD In chapter 2, we gave a comparison of the hybrid S-matrix approach with Agrawal s coupling equations. Since, Agrawal s coupling equations are widely used for calculating EMI coupling to cables and PCBs, here we give a brief description of this hybrid approach. In real systems, cable bundles are typically comprised of strongly coupled closely spaced wires. Moreover, due to bending and twisting of wires, these cables are of nonuniform construction. These factors make it difficult for full wave solvers to model these cable bundles and efficiently carryout their analysis. However, Telegrapher s equations have been shown to efficiently handle such multi-conductor transmission line problems [52]. These TL equations can also be combined with full wave simulations to calculate EMI coupling to cable bundles. One such approach, known as Agrawal s coupling equations [33], is discussed in the following lines. The total induced voltage on the line can be expressed in terms of scattered and incident voltage terms: V total (x, y, z) = V s (x, y, z) + V inc (x, y, z) (A.1) 78

96 where V s (x, y, z) is a forced terms induced by tangential component of the incident field, Ex inc (x, y, z) (assuming that line is oriented along x-axis) and can be calculated using the following set of modified Telegrapher s equations: dv s (x, y, z) dx + jwli(x, y, z) = E inc x (x, y, h) (A.2) di(x, y, z) dx + jwcv s (x, y, z) = 0 (A.3) where L and C, are per unit inductance and capacitance of the line, respectively, and h is the height of the line. In (A.1), V inc (x, y, z), is a forced term induced due to the vertical component of the incident electric field Ez inc (x, y, z) (assuming that z-axis is the vertical axis) and can be calculated using the following equation: V inc (x, y, z) = h 0 E inc z (x, y, z)dz (A.4) The fact that only incident field are required to calculate the total induced voltage on the line allows one to combine full wave solvers with transmission line solvers. Moreover, since fields are calculated in the absence of wires, a significant reduction in computational overhead is possible. We demonstrate this approach by calculating EMI coupling to a wire enclosed by a rectangular cavity, as illustrated in Figure A.1(a). We calculate the cavity fields in the absence of the wire using a semi-analytical full wave solver, namely Modal MoM [7]. These fields are then substituted in (A.2) and (A.3) which are solved using a commercial TL software (Onera s CRIPTE[31]). The calculated current along the wire is plotted in Figure A.1(b). As shown, the results agree well with the full wave moment method solution (EMCAR [29]). 79

97 (a) Plane wave illuminating a rectangular cavity enclosing an open circuit wire (b) Magnitude of the induced current on the wire Figure A.1: EMI coupling to an open circuit wire enclosed by a rectangular cavity 80

98 One important limitation of the above approach, however, is that transmission line (TL) solution being a quasi-static approximation of Maxwell s equations is valid only for lower frequencies. To understand this limitation, we consider the problem illustrated in Figure A.2(a). As shown, a plane wave illuminates a cavity enclosing a wire terminated with lumped loads (R = 50 Ω,L = 50mH). Once again, we calculate the EMI induced voltage at the loads using Modal MoM/CRIPTE integrated solution. As shown in Figure A.2(b), the results agree well with full wave solution (obtained using Ansoft s HFSS). However, as expected, the agreement is not good at higher frequencies. (This limitation can be overcome by using an iterative solution of the Telegrapher s Equations [37], but at an additional computational cost.) 81

99 (a) Cable with lumped loads inside a rectangular enclosure (b) Voltage induced on lumped loads attached to the cable Figure A.2: EMI coupling to loads connected to a wire enclosed by a rectangular cavity 82