Open source cable models for EMI simulations

This paper describes the progress of work towards an Open Source software toolset suitable for developing Spice based multi-conductor cable models. The issues related to creating a transmission line model for implementation in Spice which include the frequency dependent properties of real cables are presented and the viability of spice cable models is demonstrated through application to a three conductor crosstalk model. Development of the techniques to include models of shielded cables and incident field excitation has been demonstrated.


INTRODUCTION
There has been a significant amount of progress in the development of software based simulation tools that focused on advanced models of cables and multiconductor transmission lines (MTL) [1,2]. These models can include the effects of frequency dependent cable properties, imperfect shielding and external field to cable coupling mechanisms. However advanced cable models are either not readily available to the wider EMC community or come at significant cost in terms of both procurement and effort required to learn how to use these tools effectively. These issues are being addressed by the development of sophisticated Spice based multi-conductor transmission line models that implement the state-of-the-art in cable simulation techniques in a form of a toolset that will be made available through an open source license.
From the outset the development has been governed by the requirement for an accurate, robust, numerically efficient model with the following features:  Time domain and frequency domain simulation capabilities  Ability to describe frequency dependent cable properties  Models of shielded cables, with transfer impedance coupling, including multiple levels of shielding  Twisted pair cable model  Capability for the inclusion of coupling onto cables from external electromagnetic field sources  Portability between Ngspice, LTSpice and PSpice

METHODOLOGY
A MTL is assumed to comprise multiple cables that may or may not feature over-shields and a ground plane. Individual cables within the bundle will have frequency dependent characteristics and each of these individual cables may themselves include shielded conductors as illustrated in Fig. 1. In developing the Spice model the multi-conductor propagation model based on modal decomposition is used [1]. The frequency dependence of mode propagation on the multi-conductor system is accounted for through the method of characteristics with the inclusion of a propagation correction. The propagation within shields is further dealt with by domain decomposition and coupling across imperfect shields and is accounted for via a weak form of the transfer impedance coupling [3]. External incident field excitation is assumed to be in the form of a plane wave.
If we assume that we wish to model a single mode transmission line whose properties are described by frequency dependent inductance and resistance, frequency independent capacitance and zero admittance. The frequency dependence of the inductance and the resistance is assumed to be due to the penetration of the field into the conductors due to their finite electrical conductivity. The total inductance may be separated into two components, the first due to the magnetic field outside the conductors which is assumed to be frequency independent and the second due to the magnetic field within the conductors. Similarly the resistance is separated into a d.c. resistance and a frequency dependent resistance. Eq. 1.
The internal impedance for a cylindrical conductor, assuming a uniform current distribution around the conductor circumference may be calculated as, Eq. 2.
For a test case of a single cable where L0=1.727 μH/m C0=6.440 pF/m, Rdc=0.558Ω/m on each conductor and Z0=518Ω, the calculated internal impedance is shown in Fig. 2.  Figure 3 shows the characteristic impedance of the transmission line given by Eq. 4 and the high frequency limit of the impedance given by Eq. 5. Therefore a proposed propagation model which has the correct low frequency behaviour and includes frequency dependent loss is shown in Fig. 4.   Lossless model: The series resistance and internal inductance are assumed to be zero. Fig.  4 . shows that the lossless model is inaccurate at low frequency and at high frequency.  Model with d.c. resistance added: The d.c.
resistance of the conductors is added at the terminations. No frequency dependent resistance or inductance is included. Figure 2.8 and Fig. 5 shows that including the d.c. resistance corrects the low frequency results however the high frequency results remain inaccurate.  Frequency dependent method of characteristics solution. The model includes the d.c. resistance and the frequency dependent propagation correction. Fig. 6 shows that the combination of the d.c. resistance and the propagation correction in the model gives results which are accurate across the whole frequency range of analysis.

RESULTS
An implementation of the proposed model in Ngspice is demonstrated here so as to give confidence that it is capable of modelling frequency dependent propagation on multi-conductor transmission lines. The frequency dependence is included in the model as a s-domain impedance function, Zint Eq. 6, which appears in series with the inductive impedance of the transmission line.
To test and validate the approach a three conductor configuration, Fig. 9, with frequency dependent loss was used. The model includes a 'modal transformer' in order to perform a modal decomposition. The complete specification for the cables, including the filter parameters used in Eq. 6, is given in Tab. 1.  The near end crosstalk is calculated for this configuration the results of the frequency dependent model described here are plotted against the analytic solution. The results show the correct behaviour at low frequency, <100MHz, Fig. 10.  Next a test configuration consisting of three parallel cylindrical conductors, Fig. 12, illuminated by an external E field which is polarised along the transmission line and the k vector is in the plane of the transmission line, perpendicular to the line. The incident field excitation model is based on that published in [4]. The cable is parameters for this configuration are given in Table 2. The results are presented in Fig. 13.

Figure 12.
Three wire incident field configuration.

Cylindrical conductors with no dielectric coating
Conductor radius 1mm conductor separation, s 1cm transmission line length 1.0m No frequency dependent loss present   The crosstalk voltage as a function of frequency shows distinct peaks where the crosstalk is enhanced by resonance in the circuit formed by the two shields, short circuited at either end. The agreement between the results is reasonable, but we note that this may be improved by implementation of a more rigorous model for coupling through two levels of shielding.

CONCLUSION
The requirements for modelling multi-conductor cable bundles for EMI, in particular for space applications, have been identified and suitable assumptions have been established that lead to viable Spice cable bundle model. During this work the incorporation of frequency dependent cable characteristics into the models has been found to be significant and one which most developers of Spice cable models have neglected in their models to date.