1. EM Solvers Theory Basics
Basic theory review 1

2. 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

3. 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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

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

18. 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

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

20. 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

21. 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

22. 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

23. Internal impedance (2) If we define the internal impedance per
unit length as:
we can then write:
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24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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 , 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

46. 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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