FastFieldSolvers EM Solvers Theory Basics Basic theory review Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved 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 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 2 Maxwell equations Differential form of Maxwell equations:
May 30th, 2017 D B E t B 0 D H J t Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 3 The vector potential By math, the divergence of the curl of a vector is alwayszero: A 0 Since B 0 then we can write B A
whereA is a magnetic vector potential B A E becomes( E ) 0 t t May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 4 The scalar potential ( E A ) 0 implies that there t exists a scalar function such that
A E t A We can rewrite it as E t May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 5 The electric field (1) In E A the electric field t is expressed in terms of a scalar potential and of a vector potential
Note thatA is not uniquely defined; but provided we are consistent with a gauge choice, it does not matter here May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 6 The electric field (2) Assume now that Ohms law is valid inside J Etot the conductors: Lets splitE in , induced by the E tot charges E0
and currents in the system, and , applied by an externalEtot E E0 generator May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 7 The electric field (3) A Remembering E an J Etot , t d Etot E E0 can be rewritten as: J A E0
t May 30th, 2017 The applied field is therefore split E0 into an ohmic term and terms due to the charges Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 8 The circuit equation Consider a conductor mesh. For any point J A along the is valid E0 t
path, To obtain a circuit equation, lets integrate this equation on the path: J A E0 dl dl dl dl t c c c c May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 9 Kirkhoffs second law If E0 dl is defined as voltage
applied c J A dl dl dl to the circuit t c c c and as voltage drops along the circuit, then J A
E0 dl dl dl dl t c c c c has the form of the Kirkhoffs second law May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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 E0 dl voltage c
internal J impedance dl voltage drop c A inductive capacitive dl dl
voltage drop voltage drop t c c May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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 dV
So generated field is: are in free andthe assume that the charges V 4r space May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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). 2 Therefore: dl dl dl c May 30th, 2017 1 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 13 Capacitance (3) Lets work on the terms: 2
dl dl dl c 1 dl dl 1 2 1 1 l 2 dl 0 (always zero) Therefore, May 30th, 2017 2 dl c 1
2 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 14 Capacitance (4) By hypotesis, all charge Q is lumped on the discontinuity dV is proportional to Q (remember V 4r ) so we can write Q 1 2 C where C is constant
May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 15 Capacitance (5) Considering the current continuity equation Q Idt , the charge on the discontinuity Q 2 with the current flowing is Q 1 related C dl towards it. c
dl 1 2 C c Q dl c C term May 30th, 2017 is the usual capacitive of the circuit theory. Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved
Slide 16 Inductance (1) TheA vector potential depends in general J on the current density . For example, assuming , it A 0 follows: JdV A V 4r May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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: A dl L c I
May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 19 Inductance (4) Using L, the inductive voltage drop becomes: A d d dI c t dl dt c A dl dt ( LI ) L dt This term has the usual form of inductive voltage drop in the circuit theory May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 20
Inductance (5) As an alternative, consider a closed circuit, and using Stokes theorem and the fact B A that A dl B ds we can define as: S So is the magnetic flux linked to the circuit; this gives the usual inductance B ds
definition: A dl S L May 30th, 2017 I I I Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved
Slide 21 Internal impedance (1) In general, the current is not homogeneously distributed on a conductor section. J Therefore, the termin dl J c depends strictly on the mesh integration path. If we choose a path on the conductor surface, gives the electrical field J / on the surface, ES . May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved
Slide 22 Internal impedance (2) If we define the internal impedance Zi per unit length as: ES Zi we can then write: I ES J dl I dl I Z i dl IZ C I C c May 30th, 2017
Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 23 Internal impedance (3) In sinusoidal steady state, has a real Zi 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. May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 24 Internal impedance (4) The real part of
gives the resistance Zi of the conductor at a certain frequency. The imaginary part Z ofi gives the internal reactance, that is the part of the reactance generated by the magnetic flux inside the conductor May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 25 Internal impedance (5) The choice of the integration path was arbitrary; but in this way, the magnetic flux in A dl B ds can be S 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 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 26 Internal impedance (6) With this integration path choice: May 30th, 2017 L = / 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 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 27 Partial inductance (1) Inductance is a property of a closed conductive path. However, it is possible to define a partial inductance, applicable to into opensmall, paths Suppose to divide a path May 30th, 2017 rectilinear segments; the idea is to
distribute the total inductance to these segments in a unique way Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 28 Partial inductance (2) Lets work on a simple example. Consider the rectangular path in figure: May 30th, 2017 Neglect the fact that the path is not closed; assume also a constant current density Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 29 Partial inductance (3) Remembering the definition of magnetic flux,
B ds A dl S we can define 1, 2, 4 4 3 and 4 as integrals along the 4 A dl i segi segments: i 1 i 1 This subdivision suggests that L A dl / I c can be written as the sum of four terms May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved
Slide 30 Partial inductance (4) So, L=L1+ L2+ L3+ L4 where Li A dl / I segi May 30th, 2017 The magnetic potential vector can be A split in its turn into the sum of the potential vectors generated by the currents in the Assuming that the segments are four sides infinitesimally thin, we can now define the partial inductances. Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 31
Partial inductance (5) The partial inductances Lij are: Lij May 30th, 2017 segi Aij dli Ij whereAij is the vector potential along the i segment caused by the current Ij along Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 32 Partial inductance (6)
The total inductance of the closed path is 4 4 therefore: L Lij i 1 j 1 The definition can be extended to non-infinitesimally thin segments with non-uniform current densities May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 33 Partial inductance (7) Lets consider an equivalent formula: Lij B
Si ij dsi Ij where Si is the area enclosed between segment i, the infinite and two lines passing along the endpoints and perpendicular to segment j May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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 30th, 2017
Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 35 Partial inductance (9) Lets derive the alternative equation. Using Stokes theorem on the path ci enclosing Si, we have: Lij May 30th, 2017 B Si ij Ij dsi
A ci ij dli Ij Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 36 Partial inductance (10) May 30th, 2017 A 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 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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.: Lij B Si ij Ij
dsi A ci ij Ij dli segi Aij dli Ij The equivalence has thus been proved. May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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 30th, 2017 Skin effect Edge effect Proximity effect Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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 Thethe effects are not independent, but for sake simplicity treated on their threeofeffects takeare place. own We will provide an intuitive explanation only May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 40 High frequency effects (3) May 30th, 2017
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 The skin effect is the tendency of the to crowd under signal carrying current conductors. to crowd on a thin layer (skin) on the conductor surface Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 41
High frequency effects (4) The skin thickness is usually taken as , 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: 1 0 f May 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 42 High frequency effects (5) May 30th, 2017 The proximity effect can be thought also as
the skin effect of L a and group of almost At low frequencies, R are conductors constants. However, increasing the frequency, the current crowds on the surface: The resistance increases The external inductance slightly decreases The internal inductance decreases Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.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 0 0 expressed as: L ' C 0 May 30th, 2017 where C0 is the capacitance per unit length when the dielectrics are subsituted Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 44 References (1) May 30th, 2017 [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. Masters 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 Copyright 2017 FastFieldSolvers - https://www.FastFieldSolvers.com - All Rights Reserved Matrixes. IEEE Trans. onS.R.L. Components, Packaging
and Slide 45 References (2) [7] A. R. Djordjevic, T. K. Sarkar. Closed-Form Formulas for FrequencyDependent 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 30th, 2017 Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved Slide 46 FastFieldSolvers Copyright 2017 FastFieldSolvers S.R.L. - https://www.FastFieldSolvers.com - All Rights Reserved
