EM Solvers Theory Basics Basic theory review 1

The goal Derive, from Maxwell field equations, a circuital relationship bounding voltages and currents Give to the single terms of the circuit equation the meaning of capacitance, inductance and resistance Explain the partial inductance concept Summarize the high frequency effects We will summarize some electromagnetic concepts useful to understand the basics under the hood of quasistatic em solvers and the related circuit theory. This is only a review; we will not dig in deep details into the formulae, but we will try to give an intuitive view of the involved mechanisms. So, after reviewing the relations between the Maxwell equations and the circuit parameters (inductance, capacitance and resistance) we’ll deal with the partial inductance concept, of which we will give an accurate definition. This concept is very important to give the right meaning to the data obtained from em solvers, since it allows to give an unique meaning to the inductance of an open loop. In the end, we will briefly discuss some high frequency effects taking place in the conductors, again useful to understand the physics under the data coming from the em solvers. 2

Maxwell equations Differential form of Maxwell equations: Our aim is now to derive, from Maxwell field equations, a circuital relationship which bounds voltages and currents. In doing so, we will mainly follow the approach of [1]. 3

The vector potential By math, the divergence of the curl of a vector is always zero: Since then we can write where is a magnetic vector potential becomes 4

The scalar potential implies that there exists a scalar function Φ such that We can rewrite it as 5

The electric field (1) In the electric field is expressed in terms of a scalar potential and of a vector potential Note that is not uniquely defined; but provided we are consistent with a gauge choice, it does not matter here The A vector is not uniquely defined since its divergence has not been specified. We could choose the Lorenz or the Coulomb gauge, for instance, but the following discussion is not dependent on this choice. 6

The electric field (2) Assume now that Ohm’s law is valid inside the conductors: Let’s split in , induced by the charges and currents in the system, and , applied by an external generator Note that the total field to be used in the Ohm’s law is the sum of the applied field and of the induced field. The field we considered up to now, the E vector, was only the induced field. 7

The electric field (3) Remembering and , can be rewritten as: This equation is what we were seeking: it allows to derive a circuital relationship bounding voltages and currents. Let’s work on it a bit more. The applied field is therefore split into an ohmic term and terms due to the charges and currents in the systems 8

The circuit equation Consider a conductor mesh. For any point along the path, is valid To obtain a circuit equation, let’s integrate this equation on the path: Other Authors at this point make different assumptions, to be able for example to consider an integral formulation which allows for the conductor volume discretization. In this way is possible to derive a discrete model suitable to be implemented in computer program. See, for example, [2] and [3]. 9

Kirkhoff’s second law If is defined as voltage applied to the circuit and as voltage drops along the circuit, then has the form of the Kirkhoff’s second law 10

Voltage drop terms For low frequencies, i.e. small circuit dimensions vs. wavelength (quasistatic assumption), we can name the terms as: internal applied impedance voltage voltage drop For high frequencies these terms loose their usual meaning, but for low frequencies we recognize in them the usual terms of the circuit theory. ’inductive’ ’capacitive’ voltage drop voltage drop 11

Capacitance (1) Reacting to the applied field, free charges distribute along the circuit Since all charges are regarded as field sources, we can account for the conductor through its effect on the field distribution and assume that the charges are in free space We are going now to dig deeper into the ‘capacitive’ voltage drop term introduced with the circuit equation. To start with, we have to find the relationship between the scalar potential field and the charges. To this purpose, is useful to use an equivalent free-space formulation, where all conductore are removed. Their presence is accounted for through its effect on the field distribution. The generated field in this equivalent free-space formulation is found thorugh the above integration over a suitable volume of homogeneous dielectric. So the generated field is: 12

Capacitance (2) Assume to have a capacitor along the chosen conductive path, as in the figure: So along the path there is a discontinuity (i.e. on the capacitor plates). Therefore: In this discussion, we neglect the stray capacitance and assume that the only capacitor on the path is the one shown in the figure. The intergral of the gradient of the scalar potential along the path ‘c’ is therefore equal to the circuitation along the path (as if it were closed, i.e. no discontinuity) minus the integral on the discontinuity. 13

Capacitance (3) Let’s work on the terms: (always zero) Therefore, The first term on the left is always zero because, by math, the integral of a gradient computed on any closed path (circuitation) is zero. The second term is calculated by considering that the gradient of Φ along the direction of the path is the partial derivative of Φ evaluated in the path direction. Therefore, 14

Capacitance (4) By hypotesis, all charge Q is lumped on the discontinuity Φ is proportional to Q (remember ) so we can write where C is constant 15

Capacitance (5) Considering the current continuity equation , the charge on the discontinuity is related with the current flowing towards it. is the usual capacitive term of the circuit theory. 16

Inductance (1) The vector potential depends in general on the current density . For example, assuming , it follows: 17

Inductance (2) The current density can be expressed as the total current I multiplied by a suitable coordinate-dependent vector function along the conductor section. Because of the quasistatic assumption, I is constant along the circuit 18

Inductance (3) We can therefore define a coefficient, L, depending on the circuit geometry but not on the total current: 19

Inductance (4) Using L, the ‘inductive’ voltage drop becomes: Note that the partial derivative has been replaced with a normal derivative, since we are considering only non-moving circuits. This term has the usual form of inductive voltage drop in the circuit theory 20

Inductance (5) As an alternative, consider a closed circuit, and using Stoke’s theorem and the fact that we can define Ψ as: So Ψ is the magnetic flux linked to the circuit; this gives the usual inductance definition: Remark: S is the surface enclosed in the integration path chosen for the vector potential A. 21

Internal impedance (1) In general, the current is not homogeneously distributed on a conductor section. Therefore, the term in depends strictly on the mesh integration path. If we choose a path on the conductor surface, gives the electrical field on the surface, . 22

Internal impedance (2) If we define the internal impedance per unit length as: we can then write: Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved 23

Internal impedance (3) In sinusoidal steady state, has a real and an imaginary part, since the surface field is not in phase with the total current in the conductor, because of the magnetic flux distribution inside the conductor. 24

Internal impedance (4) The real part of gives the resistance of the conductor at a certain frequency. The imaginary part of gives the internal reactance, that is the part of the reactance generated by the magnetic flux inside the conductor 25

Internal impedance (5) The choice of the integration path was arbitrary; but in this way, the magnetic flux Ψ in can be considered as the flux linked with the internal of the mesh but not with the conductor. Other choices could be made but with this one there is a distinct advantage 26

Internal impedance (6) With this integration path choice: L = Ψ / I can be called ‘external inductance’, since is bounded to the magnetic flux external to the conductor The inductive term of the internal impedance (the imaginary part) is linked to the flux inside the conductor and can be called ‘internal inductance’. 27

Partial inductance (1) Inductance is a property of a closed conductive path. However, it is possible to define a partial inductance, applicable to open paths Suppose to divide a path into small, rectilinear segments; the idea is to distribute the total inductance to these segments in a unique way The concept works for any path, provided that we can choose composing segments as small as needed. For a formal definition of partial inductance, please see [4]. 28

Partial inductance (2) Let’s work on a simple example. Consider the rectangular path in figure: The interruption in the path in which the current generator is inserted can be thought small at will. Neglect the fact that the path is not closed; assume also a constant current density 29

Partial inductance (3) Remembering the definition of magnetic flux, we can define Ψ1, Ψ2, Ψ3 and Ψ4 as integrals along the 4 segments: This subdivision suggests that can be written as the sum of four terms 30

Partial inductance (4) So, L=L1+ L2+ L3+ L4 where The magnetic potential vector can be split in its turn into the sum of the potential vectors generated by the currents in the four sides Assuming that the segments are infinitesimally thin, we can now define the partial inductances. 31

Partial inductance (5) The partial inductances Lij are: where is the vector potential along the i segment caused by the current Ij along the j segment 32

Partial inductance (6) The total inductance of the closed path is therefore: The definition can be extended to non-infinitesimally thin segments with non-uniform current densities 33

Partial inductance (7) Let’s consider an equivalent formula: where Si is the area enclosed between segment i, the infinite and two lines passing along the endpoints and perpendicular to segment j Let’s now consider another formulation of the same concept, as done in [4]. The advantage of this equation is that can be easily relates to the problem geometry, giving an intuitive idea of the concept of partial inductance. 34

Partial inductance (8) For instance, if i = j, the area enclosed in the path to be used in the integral is shown in the figure: 35

Partial inductance (9) Let’s derive the alternative equation. Using Stoke’s theorem on the path ci enclosing Si, we have: This demonstration is valid only for parallel, infintesimally thin segments, but it can be generalized to non- infintesimally thin segments with arbitrary orientation, see [4]. 36

Partial inductance (10) has the following properties: is parallel to the source current increasing the distance from the source, tends to zero By construction, lateral sides of Si are perpendicular to source segment j By construction, the last side of Si lies at infinite distance 37

Partial inductance (11) Therefore, the only contribution to Lij along ci is given by the i segment, i.e.: Please note that there are infinite ways to attribute partial inductances to the sides of a closed loop, see for instance [5] and [6]. However, only the presented choice has the advantage of an intuitive meaning and, moreover, is easily incorporated in a program for the automatic calculation of the inductance, see [4]. The equivalence has thus been proved. 38

High frequency effects (1) In this context, high freqency means the maximum frequency at which we can still use the quasistatic assumption (i.e. geometrical dimensions << wavelength) Three main effects: Skin effect Edge effect Proximity effect 39

High frequency effects (2) At low frequencies, the current is uniform on the conductor section. Increasing the frequency, the distribution changes and the three effects take place. The effects are not independent, but for sake of simplicity are treated on their own For reference, see [7]. We will provide an intuitive explanation only 40

High frequency effects (3) The edge effect is the tendency of the current to crowd on the conductor edges. The proximity effect, especially visible on ground planes, is the tendency of the current to crowd under signal carrying conductors. The skin effect is the tendency of the current to crowd on a thin layer (‘skin’) on the conductor surface 41

High frequency effects (4) The skin thickness is usually taken as δ, the depth at which the field is only 1/e of the surface field, in case of a plane conductor The skin thickness formula is: 42

High frequency effects (5) The proximity effect can be thought also as the skin effect of a group of conductors At low frequencies, L and R are almost constants. However, increasing the frequency, the current crowds on the surface: For more information on the proximity effect, see also [8]. The resistance increases The external inductance slightly decreases The internal inductance decreases 43

High frequency effects (6) At frequencies even higher, the R’ increases with the square root of the frequency and the L’ tends to a constant, which can be expressed as: Note that here we are considering the unit length parameters. That is why we can relate the inductance to the capacitance through the speed of light. where C0’ is the capacitance per unit length when the dielectrics are subsituted with void 44

References (1) [1] S. Ramo, J. R. Whinnery, T. Van Duzer. Fields and Waves in Communication Electronics. John Wiley & Sons Inc., 1993. [2] A. E. Ruehli. Equivalent Circuit Models for Three-dimensional Multiconductor Systems. IEEE Trans. on Microwave Theory and Techniques, vol. 22, no. 3, March 1974. [3] M. Kamon. Efficient Techniques for Inductance Extraction of Complex 3-D Geometries. Master’s Thesis, Depart. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 1994. [4] A. E. Ruehli. Inductance Calculations in a Complex Integrated Circuit Environment. IBM J. Res. Develop., vol. 16, pp. 470-481, September 1972. [5] E. Hallen. Electromagnetic Theory. Chapman & Hall, London, 1962. [6] B. Young. Return Path Inductance in Measurements of Package Inductance Matrixes. IEEE Trans. on Components, Packaging and Manufacturing Technology, part B, vol. 20, no. 1, February 1997. 45

References (2) [7] A. R. Djordjevic, T. K. Sarkar. Closed-Form Formulas for Frequency- Dependent Resistance and Inductance per Unit Length of Microstrip and Strip Transmission Lines. IEEE Trans. on Microwave Theory and Techniques, vol. 42, no. 2, February 1994. [8] S. S. Attwood. Electric and Magnetic Fields. John Wiley & Sons Inc., 1949. 46

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