Hybridization of full-wave FDTD solver with a Multilevel Multiconductor Transmission Line solver with ngspice Interconnections

Hybridization of full-wave FDTD solver with a Multilevel Multiconductor Transmission Line solver with ngspice Interconnections Alberto Gasc´ on Bravo#1, Luis D. Angulo#2, Carlos J. Ramos-Salas#3 Ferran Silva$4, Marcos Qu´ ılez$5, Salvador G. Garc´ ıa#6, Enrique Pascual∗7 #Group of Electromagnetism of Granada, University of Granada, Granada, Spain $Group of Electromagnetic Compatibility, Universitat Polit` ecnica de Catalunya, Barcelona, Spain ∗Airbus DS, Madrid, Spain {1agascon, 2lmdiazangulo ,3cramoss, 6salva}@ugr.es, {4ferran.silva, 5marcos.quilez}@upc.es [email protected] Abstract—This article presents the hybridization of a 3D full-wave Finite-Difference Time-Domain (FDTD) solver with a multilevel multiconductor transmission line (MTL) solver, using the circuit simulator ngspice to treat TL interconnections. This schema is intended to study the electromagnetic response of 3D systems where networks of shielded cables are present. The MTL solver can simulate networks of shielded cable bundles with arbitrary topology. Bundles interconnects and terminations are described using ngspice. Interaction between different levels in the bundle is described by means of the transfer impedance of the cable shields. I. INTRODUCTION THE finite-difference time-domain (FDTD) method [1], [2] is one of the most widely used methods in electromagnetic analysis. It is customarily used to analyze transient fields in a three-dimensional domain and to obtain the time-domain response of non-linear complex systems, such as the bulk structures of aircraft. FDTD is also used to obtain the time-domain response of wiring networks, modeled as one-dimensional multiconductor transmission lines (MTLs) [3], However, the scale of the aeronautical and automotive system and of their wiring networks greatly differ. The time step of the FDTD method is proportional to the space step and if the latter is reduced enough to adequately describe the wires, the former will decrease accordingly, leading to extremely long simulation times. The are some methods to include sub-cell structures in FDTD without reducing the space step [4], [5], [6], [7], [8]. However, none of them can handle all the characteristics of multiconductor networks: large number of nested, multiconductor cable bundles, and complex connections between bundles and between bundles and structures. In MTL solvers based on the FDTD method the connections and terminations of MTLs are usually limited to resistive elements [9] and to combinations of non-linear elements [10], [11], [12]. In the latter the implementation depends on the particular connection topology, not being possible to generalize them to describe arbitrary connections between bundles. Dispersive elements can be included [13] adding auxiliary equations. The simulation of interconnections where electronic components are present is not possible in any of the methods above. We propose the hybridization of opensemba/fdtd, a state-of-the-art open-source 3D full-wave FDTD solver [14] with a comprehensive MTL-FDTD solver that can simulate networks of shielded cable bundles. Each shielded MTL can have an arbitrary number of nested levels. The system can be excited by internal sources and/or external plane waves. The coupling of the shields with the conductors therein is described using a bidirectional transfer impedance. The interconnections and the terminations of the MTL are treated using a SPICE simulator of electric and electronic circuits, ngspice [15]. There other FDTD-SPICE hybridization works in the literature (see for example [16], [17]), where the full-wave model is used to compute the excitation fields interacting with the lines, and it is then converted into a equivalent circuit, which interacts with the circuit of the transmission line. This work differs from that approach and in that offers some advantages. The most important, that the coupling is bidirectional: the currents in the wire bundles result in a modification of the field outside them. In addition, online sources, or interferences reaching the bundle from another line can result in radiated fields. In addition, the FDTD modeling of inhomogenous transmission line allows for a degree of detail larger than that of transmission lines in circuit simulators, having tranmission lines whose per unit length parameters change along the line or incorporating transmission lines connectors. II. MATHEMATICAL MODEL To hybridize the full-wave domain, using the Maxwell’s Equations, with the MTL domain, using the Telegraphist’s Equations, both set of equations have to coupled. Consider a shielded MTL with mlevels inside a field domain. m= 1 corresponds to the outermost level, the exterior shield that interacts with the external field. niis the number of conductors of the i-th level, and nc=Pm i=1 niis the total number of conductors. The set of equations that describes the interaction between the fields and the TL is: ∇ × E=−µ∂H ∂t (1a) ∇ × H=ε∂E ∂t +σE+Js(1b) ∂z{V}=−[L]∂t{I} − [R]{I}+ [Z]∗ {I}+{ET}(1c) ∂z{I}=−[C]∂t{V} − [G]{V}+ [Y]∗ {V}(1d) where E=E(x, y, z, t)and H=H(x, y, z, t)are the electric and magnetic vector fields, Js=Js(x, y, z, t)is the source current density and µ,εand σare, respectively, the permeability, permittivity and conductivity of the medium outside the MTL. {V}(z, t) = hV1 1, V 2 2. . . V 2 nj. . . V m nk. . . V m nciT , where V1 1is the in-cell voltage of the outermost shield and Vi jis the voltage of the j-th conductor (belonging to the i-th level) with respect to its reference conductor, with i∈ {2, . . . , m},j∈ {2, . . . , nc}are the voltages of the rest of the conductors with respect to their reference conductor. {I}(z, t) = hI1 1, I2 2. . . I2 nj. . . Im nk. . . Im nciT is the vector of currents on those conductors. Fig. 1 shows a schematic view of a bundle in a FDTD cell, with its levels and conductors numbered. In this example, there are three levels (l1 to l3), with eight conductors (1to 8) distributed among these levels. For instance, the voltage value will have the structure {V}(z, t) = V1 1, V 2 2, V 2 3, V 2 4, V 3 5, V 3 6, V 3 7, V 3 8T 1 2 3 4 6 78 5 Fig. 1. Schematic view of a cable bundle inside a FDTD cell. There are three levels (labeled l1to l3) and 8 conductors (labeled 1to 8). [L]and [C]are the per-unit-length (p.u.l.) inductance and capacitance between all conductors and [R]and [G]are the per-unit-length losses: resistance (along conductors) and conductance (between conductors). [L]and [C]are nc×ncblock square matrices, each block a ni×nimatrix, where niis the number of conductors of the i-th level. In particular, the first blocks [L]1=L11,[C]1=C11 are, respectively, the in-cell p.u.l. inductance and capacitance which can be computed as [4], [6]: L11 =µ 2πRR∆k1∆k2,r>rsh ln(r/rsh)dk1dk2 RR∆k1∆k2,r>rsh dk1dk2 (2a) C11 =µε/L1(2b) where rsh is the radius of the external shield of the bundle, ∆k1 and ∆k2are the spatial steps transversal to the direction kof the bundle, and µand εare the permeability and permittivity of the medium surrounding the cable. The rest of blocks, [L]iand [C]i, with i∈ {2, . . . , m}, are the p.u.l. inductance and capacitance of the i-th level with respect to the reference conductor of that level. These can be computed using the theoretical expressions for shielded multiconductors [3] or using an electrostatic solver [18]. [R]has the same structure as [L]and [C]but is a diagonal matrix. (3) shows the structure of [L]for the bundle shown in figure Fig. 1. [L] =             L10000000 0L2 2,2L2 2,3L2 2,40 0 0 0 0L2 3,2L2 3,3L2 3,40 0 0 0 0L2 4,2L2 4,3L2 4,40 0 0 0 0 0 0 0 L3 5,5L3 5,6L3 5,7L3 5,8 0 0 0 0 L3 6,5L3 6,6L3 6,7L3 6,8 0 0 0 0 L3 7,5L3 7,6L3 7,7L3 7,8 0 0 0 0 L3 8,5L3 8,6L3 8,7L3 8,8            (3) [Z]and [Y]are, respectively, the shield transfer impedance and admittance. These are frequency dependent quantities and the ∗operator indicates that these terms are time-domain convolutions of frequency dependent quantities. The transfer impedance describes the voltage that appears inside an imperfect shield when a current circulates through it, whereas the transfer admittance describes the current that appears on the inside a the shield that has a voltage difference with respect to its reference [19]. Except for very large braid apertures [20], [21] or large termination impedances of the external line [19] the contribution of the admittance can be neglected. Each level is potentially coupled to the level above and the level below through the transfer impedance and admittance describing the shields. We will consider that the coupling in the shields is bidirectional. The transfer impedance matrix is a block matrix where the only non-zero components are those Zij and Zji such that j correspond to the conductors shielded by conductor i. (4) shows the structure of [Z]for the bundle shown in figure Fig. 1. [Z] =              0Z1,2 1,2Z1,2 1,3Z1,2 1,30000 Z2,1 2,1000Z2,3 2,5Z2,3 2,60 0 Z2,1 3,100000Z2,3 3,7Z2,3 3,8 Z2,1 4,10000000 0Z3,2 5,2000000 0Z3,2 6,2000000 0 0 Z3,2 7,300000 0 0 Z3,2 8,300000              (4) The cross-coupling between levels created by the bidirectional transfer impedance prevents the mequations stemming from (1c) and (1d) from being solved independently for the TL that represents each level, and all levels have to be solved as a single matrix equation. {ET}(z, t) = E1 T,10. . . 0is the vector of external tangential fields. Only the external shield is subject to the direct influence of the field, hence only the first component is non-zero. We will assume that the lines are uniform and homogeneous, and that their per-unit-length properties are frequency independent. The coupling between the full-wave domain and the MTL domain is mediated by the interaction between (1b) and (1c) through the electric field and current terms. A. Discretization Cables are introduced as elements in the full-wave domain, and discretized along the grid used in the 3D-FDTD method, each section of the cable belonging to a cell of the grid. The equations in (1) are discretized using the FDTD method described in [3]. The [Z]∗ {I}term represents the time-domain convolution of a frequency dependent transfer impedance with a current vector. Following [22], [13], if [Z]is substituted by a rational approximation the Piecewise Linear Recursive Formulations (PRLC) can be used to express the convolution in the time domain. If the frequency-dependent transfer impedance is approximated by a rational approximation, the convolution term can be written as: [Z]∗ {I}=[d]I+ [e]∂tI+X i [r]iZt 0 e[p]i·tI(z, t −τ)dτ (5) where [d],[e],[r]and [p]are the matrices of terms of the rational approximation. If (5) is discretized, the summation term takes the form X i [r]iZt 0 e[p]i·t{I}(z, t −τ) = X i [φ]n i(6) with [φ]n i= [q1]i{I}n+1/2 k+ [q2]i{I}n−1/2 k+ [q3]i[φ]n i Details of the implementation of the method and the derivation of the terms [φ],[q1],[q2]and [q3]can be found in [23], where it was implemented to describe frequency-dependent lumped impedances. The final discretized field, voltage and current equations are: Hn+1/2=Hn−1/2−δt µ∇ × En+1 (7a) En+1 =ε−σ∆t/2 ε+σ∆t/2En +∆t ε+σ∆t/2∇ × Hn−1/2−Jn−1/2 s(7b) {V}n+1 k=F−1 V+[FV−{V}n k−({I}n−1/2 k− {I}n−1/2 k−1)] (7c) {I}n+1/2 k=F−1 I+FI−{I}n−1/2 k−({V}n+1 k+1 − {V}n+1 k) +ET−∆zX i [q3]i[φ]n−1 i(7d) where FV±=∆z ∆t[C]±∆z 2[G](8a) FI+= ∆z ∆t[L] + ∆z 2[R] + ∆z 2[d] + ∆z ∆t[e]+∆zX i [q1]i! (8b) FI−= ∆z ∆t[L]−∆z 2[R]−∆z 2[d] + ∆z ∆t[e]−∆zX i [q2]i! (8c) B. Terminations and interconnections The equations above describe a shielded cable bundle but not the elements on its extremes. The problem in incorporating MTL terminal conditions in FDTD is that the voltages and currents of the line are not collocated neither in space or time, whereas the terminal conditions relate the current and voltage at the same point and the same time. The procedure to implement the terminal conditions is explained in detail in [24]. We use the circuit solver ngspice to treat the terminations of the MTLs. Since ngpsice is a lumped circuit solver, we first find a circuit equivalent of the transmission line, shown in Sec. II-B. C′=1 2C∆zand R′= (1 2G∆z)−1, where Cand Gare the p.u.l capacitance and conductance, respectively. ZSand ZLrepresent the impedance of the terminations connected at source and load ends of the line, respectively. At each time step, the value of the current sources (ˆ I1and ˆ IN+1) are set to the value of the corresponding current segment from the MTL-FDTD simulation. Then, the circuit is evolved to next time step, producing the voltages corresponding to the line extremes, which are read and passed to the MTL-FDTD algorithm (V1and VNDZ+1). C. Algorithm With the discretization of the field equations and the MTL equations in (7), and the procedure to use ngspice to advance Fig. 2. Lumped circuit model of the extremes of the transmission line. the voltage of the extremes of the MTLs, a combined FDTD solution can be constructed. The iterative procedure to update fields, currents and voltages is: 1) advance the electric field En+1 using (7b) and Jn−1/2=In−1/2· S−1k, where S= ∆i∆jis the cell area transversal to direction k, the direction of E. 2) advance the voltage of each TL at all nodes except those on the extremes {V}n+1 k(k= 2 . . . N) using (7c); 3) advance the voltage on the extremes of each TL {V}n+1 1and {V}n+1 N+1, advancing all ngspice circuits: •for each TL attached to a junction, update the corresponding current source ˆ Ii,k, using the current of the transmission line segment attached to that node, In−1/2 i,k , with k= 1, k =N, and icorresponding to the TL; •advance the transient ngspice simulation by ∆t; •assign each ˆ Vi,k to the corresponding voltage on the TL Vn+1 i,k (k= 1 or k=N). 4) advance the current of each TL {I}n+1/2 k(k= 1 . . . N) using (7d); 5) advance the magnetic field Hn+1/2using (7a) III. VALIDATION The proposed hybridized solver is validated comparing simulations with measurements performed at the laboratory of the GCEM-UPC[25]. The validation uses two different methods: a plane wave incident on a shielded wire using a EUROTEM TEM cell [26] (Fig. 4) and a wire panel, where we measure the cross-talk in a two-conductor transmission line between the excited line and the receptor line, terminated with an operational amplifier (op-amp, from now on). The validation using the wire panel can be found in [24] A. Code implementation The full-wave FDTD solver and the MTL-FDTD solver are written in Fortran and are part of a state-of-the-art general-purpose time-domain simulator developed by the authors, it has been validated for several EMC-related applications [27], [28], [23], the code is published under MIT license and available at [29]. Moreover, ngspice is an open-source spice simulator for electric and electronic circuits. It is SPICE compatible, and PSPICE or LTSPICE device model parameters and netlists can be used for simulating discrete circuits [15]. For this work, all the simulated cases are prepared using the Electromagnetics workbench [30] implemented in FreeCAD [31]. This tool allows defining problems with a significant complexity and generating all the input files necessary to run the simulation. A case prepared in the workbench can be seen in Fig. 3. Fig. 3. Overview of the Electromagnetics Workbench implemented in FreeCAD used to prepare the test cases. B. Validation with a TEM cell 1) Measurement setup: The TEM cell consists of four stripline antennas housed inside an absorbent-lined enclosure. According to the manufacturer, inside the cell there is a volume of 20x20x20 cm3in which the field can be assumed to be a plane wave. The antennas can be excited to propagate verticallyor horizontally-polarized TEM waves, the direction of propagation being the longest dimension of the cell. To test the plane wave assumption we measured the electric field intensity at the extremes of the position where the wire would be positioned. The deviation from the plane wave assumption should not relevant as long the field along the wire remains constant. We measured the field using an electric field probe FL7006[32]. According to the manufacturer, this has an uncertainty of ±0.8 dB in the frequency range 0.1 GHz to 1 GHz. The function generator was a HAMEG 8134 [33]. The difference between the fields measured at the extremes of the wire for the ydirection was negligible (1.6%). 2) Wire excited by plane wave: We used the TEM cell to excite a bare and a coaxial cable. Fig. 5 shows a simplified view of the measurement setups. The cable entered the cell along the direction of propagation of the plane wave through a metallic cylinder in ohmic contact with the enclosure. At the center of the volume where the excitation can be considered a plane wave a segment of 7 cm was bent into the vertical position. We used a vertically polarized plane wave. Thus, only the vertical segment of the cable should be excited by the wave. In the case with the bare wire, the shielding was removed in this portion of the cable. In the case of the shielded wire, the BNC connector of the extreme of the coaxial cable inside the cell was removed and terminated on a 50 Ω resistor connecting the shield and the Fig. 4. Outside view of the TEM cell Fig. 5. Scheme of the measured setups. Top: plane wave on a bare wire. Outside the cell, the wire is connected to an oscilloscope. Bottom: plane wave on a shielded wire. The end of the coaxial inside the TEM cell is connected with a 50 Ωresistor to the shield. Outside the cell, the coaxial wire and shield are connected to the oscilloscope channel and ground (through the oscilloscope). The input signal is amplified. inner wire. In both cases the rest of the cable was left intact. According to the manufacturer, the transfer impedance of the coaxial cable can be modeled as ZT(ω) = RT+jωLT, with RT=0.01 Ω m−1and LT=10 nH m−1. This value is extremely sensitive to the manufacturing process and to the proper use and storage of the cable. According to [34], the uncertainty in the transfer impedance can be as high as 25%. The shielded conductor was modeled as a nested transmission line with two levels and two conductors, one corresponding to the shield and the other corresponding to the inner wire. A vertically polarized plane wave was injected equal to E(t) = E0sin(2πft), with E0=1.85 V/m and E0=42.29 V/mfor the bare and shielded wire, respectively, and f=550 kHz in both cases. When measuring using the coaxial cable the voltage of the function generator was amplified using a B1080M-10 amplifier to obtain a larger field intensity. Otherwise, the field inside the coaxial would be below the sensitivity of our instruments. The frequency was chosen in a frequency range where the suppression of the longitudinal and transversal field components is maximal according to the manufacturer of the TEM cell. The simulation ran for 20 ns with a time step ∆t=2.5 fs Fig. 6 shows the validation results: the voltage induced on the bare wire (Fig. 6a) and the voltage at the end of the inner wire of the coaxial cable (Fig. 6b), comparing the simulation to the measured results. The uncertainty in the measurement corresponds to the uncertainty of the oscilloscope (1 division in the scale 1 mV/div). To account for the uncertainty in the simulation we repeated the simulation varying the shield transfer impedance by a ±25%, and the field intensity by a ±0.8 dB. The uncertainty in the simulation corresponds to sum in quadrature of the uncertainties obtained in each case. (a) Voltage induced on a wire excited by an incident plane wave. (b) Voltage induced on the inner conductor of a shielded wire excited by an incident plane wave. Fig. 6. Voltage induced on a wire (bare and shielded) excited by a plane wave inside the TEM cell. IV. CONCLUSIONS We have presented a hybridization of a FDTD full-wave solver with a MTL-FDTD solver to treat networks of cable bundles whose interconnections are treated using the circuit simulator ngspice. The MTL can simulate shielded cable bundles of arbitrary topology, with bidirectional shields, allowing to study both the emission and the susceptibility of realistic cable networks in complex electromagnetic environments. Using a circuit simulator to treat the connections in the network allows to simulate connections where non-linear and electronic components are present. The method has been validated comparing with two laboratory setups that test the features and integration levels of the hybridized solver, with very good agreement between the simulations and the measurements. ACKNOWLEDGMENTS This work was funded in part by the Spanish National Project PID2022-137495OB-C31 (ESAMA) and by the European Union under GA no 101101961 - HECATE. 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