Minimum-Inductance Distribution of Current

In preparation for writing my new text, "High-Speed Signal Propagation", I took the time to read some classic works in electrical engineering. One particular work stood out as a magnificent tribute to man's ability to solve complex problems by intuition and analogy. "A Treatise on Electricity and Magnetism", by James Clerk Maxwell, is available from in a modern, reprinted edition. This is a two-volume set, vol. I-ISBN 0486606368 and vol II-ISBN 0486606376.

This newsletter tries to impart a sense of the intuitive power of Maxwell's approach to field calculations. If you want to experience the full effect, GET HIS BOOK. Although Maxwell's electrical vocabulary (from the 1880's) can at times be difficult for people in the modern world to interpret, the clarity of his vision is stunning. What particularly appeals to me is the fact that he developed his entire theory without access to any computing resources more powerful than a pencil. He did not have Mathematica, or field-calculation software, or a three-dimensional visualization package. Everything happened in his head. I like that kind of engineering. That's the kind of intuitive understanding I teach in class.

The following exchange took place during a protracted discussion on the si-list about the distribution of current on a reference plane underneath a pcb trace. Sainath is trying to grasp the relation between the inductance of a microstrip trace distribution of current on the reference plane. He wonders if the phrase "least inductance distribution of current" implies that perhaps there exist other distributions of current with other values of inductance, and in the quote below, he questions whether you get equivalent results integrating the total magnetic flux surrounding a microstrip by taking the region between the trace and its reference plane, or the region above the trace.

Those of you who, like I, have struggled a lot with magnetic field theory may find Sainath's questions very familiar.

Minimum-Inductance Distribution of Current

Sainath Nimmagadda writes to the si-list:

[Responding to a message from Andrew Byers]   ...I get the themes that inductance is a one number affair and current returns through the least inductance path. [But,] is there a contradiction in these themes?

Let me borrow the following from your previous mail.

"If you were to put your integrating surface on the other side of the trace, extending up from the top of the trace, you theoretically would have to make the area of the surface extend to infinity to "catch" all the field lines."

[Does the mean that] the inductance of the microstrip is going to be infinity because of the infinitely-large surface of integration? Or any other value? Or does the inductance remain the same as it was when integrating [the total flux on the] surface below the trace?

The mysteries of magnetic-field integration are indeed sometimes difficult to comprehend. In answer to your question about the surface of integration, the best mental image for this appears in the famous work by James Clerk Maxwell, 

"A Treatise on Electricity and Magnetism". The first volume of this work (Electricity) is available on in the form of a modern reprint of an old Dover version, circa 1954.

From the preface of that book, here is the key idea that renders sensible this whole business of integration of magnetic field intensity over a surface: "Faraday, in his mind's eye, saw lines of force traversing all space".

The "lines of force" concept underlies all of electromagnetism. What you need to know about Faraday's "lines of force" idea, in the context of your problem having to do with evaluating the inductance of your trace, is that magnetic lines of force form continuous loops having no beginning and no end. The total number of lines extant is a measure of the total magnetic flux produced by a magnetized structure.

Of course you can re-normalize any magnetic field picture to produce a different number of lines by declaring each line to represent a different quantity of flux, for example 1/10th the original amount would produce 10x the number of lines, etc. Presumably you have scaled the flux represented in your (mental) magnetic field picture in such a way as to produce a manageable number of lines that is at once enough to represent accurately the pattern of field intensity and also not too many to clutter the image. Keep in mind, however, that regardless of the number of lines, there are a finite number of them and each is a continuous entity forming a complete, unbroken loop.

In Maxwell's view, integrating the magnetic flux passing through a surface is simply a matter of simply COUNTING how many lines pass through it.

For example, consider a closed surface (a sphere) in space. Any particular line that enters the ball must, since it cannot end within the sphere, exit at some other point. Therefore, when counting the number of lines penetrating the surface, since each line must both enter (a positive count) and exit (a negative count), the sum of entrances and exits penetrating the sphere must be zero. From this simple idea, Maxwell derives the idea that the integral of flux over any closed surface (of any shape) must be zero.

[Mathematical aside--you may be familiar with certain complications having to do with the integration of field vectors penetrating a surface whereby you have to dot product the field intensity direction vector with a vector normal to the surface--these difficulties disappear when you simply "count lines", which is the beauty of Faraday's brilliant intuitive approach. When the surface is tilted so that the lines intersect the surface at an oblique angle, the number of lines penetrating each square area of surface is naturally reduced. This reduction is precisely accounted for, in multidimensional vector calculus, by the dot-product operator.]

Now let's apply the line-counting analogy to Sainath's trace-inductance problem. Imagine a certain finite number of magnetic lines of force wrapped around a microstrip trace. [I'll assume the reference plane extends infinitely in the x and y directions. The plane is located at z=0, and the trace is at z=1. Since the plane is infinite and perfectly conducting, no magnetic lines of force exist below z=0.]

Assume I hook up my inductance meter to one end of the trace. Connect the other end of the trace to the reference plane. Now stretch an imaginary "soap bubble" in the region between the trace and the reference plane. Beginning at my end of the trace, the edge of the bubble touches the trace all along the length of the trace, following along the shorting connection at the end down to the reference plane, and returning along the plane back to my meter. For completeness, also consider how the edge of the bubble also must track along the ground lead of my inductance meter all the way up to the actual instrument and then back down the signal lead of the instrument to the beginning of the trace. We'll assume the meter is tiny compared to the size of the trace so we don't have to worry too much about the shape of the bubble at the source end (this is a serious real-life complication in the measurement of tiny inductances, that we will ignore).

Next step: apply 1 amp of current to the trace, and count the number of field lines penetrating the soap bubble. Since the bubble is an "open" shape (i.e., it is bounded at its edge in such a way that it does not enclose any space), a careful count will reveal some non-zero amount of magnetic flux penetrating the bubble. NOW comes the cute part of this mental experiment. I want you to blow on the bubble, stretching it. It's still anchored at the edges, but no longer a flat sheet. The remarkable thing that happens is that the number of magnetic field lines penetrating the bubble does not change. It doesn't matter how you stretch or modify the shape of the bubble, or how far you blow it out of position, as long as you don't change where the bubble is anchored around the edges, you haven't changed the number of lines penetrating it. That property (of the total flux not changing regardless of the exact shape of the surface of integration used) is essential to understanding how to calculate inductance.

To prove that distorting the bubble doesn't change the total flux, Maxwell imagines two surfaces, A and B, both anchored to the trace and plane just like your soap bubble. When connected together, these two surfaces A and B form a single closed surface. Therefore, using our earlier reasoning about the sphere, the total number of lines penetrating the combined object A+B (that is, coming into A and leaving through B) must equal zero--from which you may correctly deduce that when measured separately the total flux passing through surface A must precisely equal the total flux passing through surface B.

In a minute, I'm going to directly address Sainath's question about making "the area of the surface extend to infinity to catch all the field lines", but first I need to go over one more detail. That detail has to do with how a 2-dimensional surface with infinite extent acts kinds of like a closed surface, in that it partitions space into two regions. Instead of the regions being "inside" and "outside" as they are for an ordinary closed surface, the regions are "this side" and the "other side", but the partition exists just the same. I bring this up because the partition idea helps you see why the total flux penetrating any infinite plane must equal zero. Just like with the sphere, any line of flux that passes through the infinite sheet to the other side (a positive count) must eventually make its way back (a negative count), making the total number of crossings equal zero. I'm now going to apply this idea (finally) to the problem at hand.

I want you to turn your mental picture so you are looking at the side of the trace (a broadside view of your soap bubble). Color the bubble pink. Now, pick some particular line of magnetic flux that penetrates the pink region. If it passes through the pink region then there are two possibilities for how it returns to its source (completing the loop): either it comes back through the pink region, in which case it cancels itself out contributing nothing to the total count of flux penetrating the pink region, or it comes back SOME OTHER WAY. The only other way back is through the "white space" that you see above, below, and to the sides of the apparatus. Therefore, if you erect a white curtain above, below, and to the sides of the apparatus, covering all the space you see that isn't already pink (looking from your perspective like a photographic negative of the pink region), and if the sheet is anchored at its edges along the trace and plane precisely coincident with the edges of the pink soap bubble, then you may rightly conclude that any flux that contributes to the total flux count in the pink region must also, on its return voyage, penetrate the white sheet. In other words, you can count the flux passing through the pink region, or count the flux passing through the white sheet, either way you get the same answer. This property directly relates to the discussion above about the infinite plane partitioning space. As long as the pink and white surfaces, when combined, form a complete partition of all space, the total flux through that partition must be zero, ergo; the flux through the pink and white surfaces must be the same. That is what I think Andrew was talking about when he said that if you extended the area of integration to infinity you could catch all the flux.

The total flux passing through the pink region in reaction to a current on the trace of 1 amp is defined as the inductance of the circuit formed by the trace and its associated reference plane.

I hope my rather lengthy discussion helps you sort out some of the paradoxes associated with magnetic-field integration.

Buried in the definition of inductance is the assumption that current always assumes minimum-inductance distribution. We say, "Current always follows the path of least inductance", or more precisely, "Current at high frequencies, if not altered by significant amounts of resistance, always assumes a distribution that minimizes the inductance of the loop formed by the signal and return paths". If you put something in the way of your current that alters the distribution of current on the return path (like a hole in the reference plane), then the current assumes some alternate distribution that must necessarily raise the inductance of the configuration (moving to any distribution other than the minimum-inductance distribution must necessarily raise the inductance).

Regarding the subject of visualizing the distribution of current in a "least-inductance" configuration, let me propose an analogy that I find quite helpful in working through that problem.

First, replace your dielectric medium (the space between the trace and reference plane) with a slightly resistive material. I like to imagine salt water occupying that space. Leave the trace open-circuited at both ends, and apply 1-V DC to the trace. A certain pattern of current will flow through the salt water to the reference plane. I'll bet you could draw a picture showing the pattern of current flow in this situation. Start with a cross-sectional view of the trace. Suppose you use 100 lines for the picture, each line representing a certain fraction of the total current. Each line emanates from the trace and terminates on the plane (unlike magnetic lines of force these current density lines have beginnings and endings). A great density of lines will flow directly between the trace and plane, with the lines feathering out to lower and lower densities as you work your way further from the trace. The lines always leave the surface of the trace in a direction perpendicular to the surface of the trace, and land perpendicular to the reference plane.

Here's why I like this exercise: Your picture of the DC current flow exactly mimics the picture of lines of electric flux in a dielectric medium operated at high frequency. I find many people have no difficulty imagining how DC currents would behave in salt water, which turns out to be precisely the same problem as figuring out how AC currents behave in a dielectric medium.

Now we get to the part of this discussion about the density of current in the reference plane. Your electric-field picture shows a great density of current flowing from trace to plane at a position directly underneath the trace, and less and less density of current flowing to positions on the plane remote from the trace. This picture shows precisely how the current gets from trace to plane (i.e., it flows as displacement current through the parasitic capacitance that exists between the trace and the plane). If you assume that current arriving on the plane then flows parallel to the trace (making the cross-sectional picture the same at each position along the trace, as required by symmetry), then you have a mental picture that shows the density of current flowing on the plane as a function of position. Most of the current flows on the reference plane right under the trace, with less and less as you move away from the trace (it happens to fall off approximately quadratically for microstrips, even faster for striplines).

Of course, you are going to want to know "why" current should behave in such a manner. The principle in question here is the "minimum energy" principle. My understanding of Maxwell's equations is that the distributions of charge and current in a simple statics problem with no internal sources will always fall into a pattern that satisfies all the boundary conditions around the edges of the region of interest, has a Laplacian of zero in the middle, and also just happens to store the minimum amount of energy in the interior fields. In other words, you never get huge, unexplained, spurious magnetic fields in the middle of an otherwise quiet region (unless you believe in vacuum fluctuations, which is a different subject entirely...).

The stored energy for inductive problems equals: E=(1/2)LI2, where L is the system inductance and I2 is the total current squared. As you can see, stored magnetic energy E and inductance L vary in direct proportion to one another. Therefore, the distribution of current on the reference plane that minimizes the total stored magnetic energy and the distribution of current that minimizes the inductance are one and the same.

In answer to what might logically be your next question, "Why do electromagnetic fields tend towards the minimum-stored-energy distribution?" I can only say that I'm not sure anyone really knows-we just observe that this is the way nature seems to operate. Perhaps someone more well-versed in electromagnetic theory can provide an answer.

By assuming that current does not adopt a minimum-energy distribution you can demonstrate the existence of a mode of current that, once started, leads rapidly to a lower-energy state. Presumably, such a mode would start up immediately, making any state but the lowest-energy state unstable. This demonstration would convince you of the absurdity of the non-minimum energy situation only if you also intuitively believe that nature is not absurd. Further discussion of that issue is probably best left to physicist-philosophers.

I hope this discussion is helpful to you, and doesn't just stir up other doubts.

For further reading, try the following articles: "High-Speed Return Signals", "Return Current in a Plane", "Proximity Effect", "Proximity Effect II", "Proximity Effect III", and "Rainy-Day Fun".

Best Regards,
Dr. Howard Johnson