EM Solvers Theory Basics Basic theory review
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved The goal Derive, from Maxwell field equations, a circuital relationship bounding voltages and currents Explain the partial inductance concept Give to the single terms of the circuit equation the meaning of capacitance, inductance and resistance Summarize the high frequency effects
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Maxwell equations Differential form of Maxwell equations:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved The vector potential Since then we can write where is a magnetic vector potential becomes By math, the divergence of the curl of a vector is always zero:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved The scalar potential implies that there exists a scalar function Φ such that We can rewrite it as
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved The electric field (1) In Note that is not uniquely defined; but provided we are consistent with a gauge choice, it does not matter here the electric field is expressed in terms of a scalar potential and of a vector potential
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Assume now that Ohm’s law is valid inside the conductors: The electric field (2) Let’s split in, induced by the charges and currents in the system, and, applied by an external generator
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved The electric field (3) Remembering The applied field is therefore split into an ohmic term and terms due to the charges and currents in the systems and can be rewritten as:,
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved The circuit equation Consider a conductor mesh. For any point To obtain a circuit equation, let’s integrate this equation on the path: is validalong the path,
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Kirkhoff’s second law If is defined as voltage applied has the form of the Kirkhoff’s second law as voltage drops along the circuit, then to the circuit and
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Voltage drop terms For low frequencies, i.e. small circuit dimensions vs. wavelength (quasistatic assumption), we can name the terms as: applied voltage internal impedance voltage drop ’inductive’ voltage drop ’capacitive’ voltage drop
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Capacitance (3) Therefore, Let’s work on the terms: (always zero)
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Φ is proportional to Q (remember ) so we can write where C is constant Capacitance (4) By hypotesis, all charge Q is lumped on the discontinuity
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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.
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved For example, assuming, it follows: Inductance (1) The vector potential depends in general on the current density.
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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 Inductance (2)
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Inductance (3) We can therefore define a coefficient, L, depending on the circuit geometry but not on the total current:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Inductance (4) Using L, the ‘inductive’ voltage drop becomes: This term has the usual form of inductive voltage drop in the circuit theory
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Therefore, the term in depends strictly on the mesh integration path. Internal impedance (1) In general, the current is not homogeneously distributed on a conductor section. If we choose a path on the conductor surface, gives the electrical field on the surface,.
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved we can then write: Internal impedance (2) If we define the internal impedance per unit length as:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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. Internal impedance (3)
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved The real part of gives the resistance of the conductor at a certain frequency. Internal impedance (4) The imaginary part of gives the internal reactance, that is the part of the reactance generated by the magnetic flux inside the conductor
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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. Internal impedance (5) Other choices could be made but with this one there is a distinct advantage
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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’.
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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 written as the sum of four terms
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Partial inductance (4) So, L=L 1 + L 2 + L 3 + L 4 where Assuming that the segments are infinitesimally thin, we can now define the partial inductances. 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
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Partial inductance (5) The partial inductances L ij are: where is the vector potential along the i segment caused by the current I j along the j segment
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Partial inductance (7) Let’s consider an equivalent formula: where S i is the area enclosed between segment i, the infinite and two lines passing along the endpoints and perpendicular to segment j
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Partial inductance (9) Let’s derive the alternative equation. Using Stoke’s theorem on the path c i enclosing S i, we have:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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 S i are perpendicular to source segment j By construction, the last side of S i lies at infinite distance
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved Partial inductance (11) Therefore, the only contribution to L ij along c i is given by the i segment, i.e.: The equivalence has thus been proved.
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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: Proximity effect Edge effect Skin effect
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved High frequency effects (3) The edge effect is the tendency of the current to crowd on the conductor edges. The skin effect is the tendency of the current to crowd on a thin layer (‘skin’) on the conductor surface The proximity effect, especially visible on ground planes, is the tendency of the current to crowd under signal carrying conductors.
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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:
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved At low frequencies, L and R are almost constants. However, increasing the frequency, the current crowds on the surface: High frequency effects (5) The proximity effect can be thought also as the skin effect of a group of conductors The resistance increases The external inductance slightly decreases The internal inductance decreases
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved References (1) [1] S. Ramo, J. R. Whinnery, T. Van Duzer. Fields and Waves in Communication Electronics. John Wiley & Sons Inc., [2] A. E. Ruehli. Equivalent Circuit Models for Three-dimensional Multiconductor Systems. IEEE Trans. on Microwave Theory and Techniques, vol. 22, no. 3, March [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, [4] A. E. Ruehli. Inductance Calculations in a Complex Integrated Circuit Environment. IBM J. Res. Develop., vol. 16, pp , September [5] E. Hallen. Electromagnetic Theory. Chapman & Hall, London, [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.
Copyright 2012 Enrico Di Lorenzo, All Rights Reserved 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 [8] S. S. Attwood. Electric and Magnetic Fields. John Wiley & Sons Inc., 1949.