Fast Field Solvers EM Solvers Theory Basics Basic

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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 2

Maxwell equations Differential form of Maxwell equations: May 30 th, 2017 Copyright 2017 Fast.

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 May 30 th, 2017 becomes Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 4

The scalar potential implies that there exists a scalar function Φ such that We can rewrite it as May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 7

The electric field (3) Remembering and , can be rewritten as: The applied field is therefore split into an ohmic term and terms due to the charges and currents in the systems May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 10

Voltage drop terms For low frequencies, i. e. small circuit dimensions vs. wavelength (quasistatic assumption), we can name the terms as: applied voltage ’inductive’ voltage drop May 30 th, 2017 internal impedance voltage drop ’capacitive’ voltage drop Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 So the generated field is: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 13

Capacitance (3) Let’s work on the terms: (always zero) Therefore, May 30 th, 2017

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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved ) Slide 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. May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 16

Inductance (1) The vector potential depends in general on the current density. For example, assuming May 30 th, 2017 , it follows: Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 18

Inductance (3) We can therefore define a coefficient, L, depending on the circuit geometry but not on the total current: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 19

Inductance (4) Using L, the ‘inductive’ voltage drop becomes: This term has the usual form of inductive voltage drop in the circuit theory May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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, . May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 22

Internal impedance (2) If we define the internal impedance unit length as: per we can then write: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 23

Internal impedance (3) May 30 th, 2017 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. Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 24

Internal impedance (4) The real part of gives the resistance of the conductor at a certain frequency. May 30 th, 2017 The imaginary part of gives the internal reactance, that is the part of the reactance generated by the magnetic flux inside the conductor Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 The inductive term of the internal impedance (the imaginary part) is linked to the flux inside the conductor and can be called ‘internal inductance’. Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 27

Partial inductance (1) Inductance is a property of a closed conductive path. May 30 th, 2017 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 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 28

Partial inductance (2) Let’s work on a simple example. Consider the rectangular path in figure: Neglect the fact that the path is not closed; assume also a constant current density May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 29

Partial inductance (3) Remembering the definition of magnetic flux, we can define Ψ 1, Ψ 2, Ψ 3 and Ψ 4 as integrals along the 4 segments: This subdivision suggests that can be written as the sum of four terms May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 30

Partial inductance (4) So, L=L 1+ L 2+ L 3+ L 4 where May 30 th, 2017 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. Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 35

Partial inductance (9) Let’s derive the alternative equation. Using Stoke’s theorem on the path ci enclosing Si, we have: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 37

Partial inductance (11) Therefore, the only contribution to Lij along ci is given by the i segment, i. e. : The equivalence has thus been proved. May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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: May 30 th, 2017 Skin effect Edge effect Proximity effect Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 We will provide an intuitive explanation only May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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 May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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: May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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: May 30 th, 2017 The resistance increases The external inductance slightly decreases The internal inductance decreases Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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: where C 0’ is the capacitance per unit length when the dielectrics are subsituted with void May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 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. May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 45

References (2) [7] A. R. Djordjevic, T. K. Sarkar. Closed-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. May 30 th, 2017 Copyright 2017 Fast. Field. Solvers S. R. L. - https: //www. Fast. Field. Solvers. com - All Rights Reserved Slide 46