I received these in my emails from si-list during July 2003.
Hello experts:
For a microstrip, we know the magnetic field distribution(for example, Fig. 2.3 Stephen Hall’s book) and current density distribution(Fig. 4.5 same book). Given these, how would you obtain the inductance distribution?
Thanks in advance,
Sainath
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Hello Sainath,
inductance is the proportional factor between the current and the magnetic flux. So far Your idea is ok. But calculating magnetic flux from magnetic field requires an integration across a closed surface surrounding the conductor carrying the current. So - as You see - You will not get a inductance distribution over conductor length but only an integral value for the conductor enclosed in the chosen sphere.
Sorry,
Thomas
Thomas,
Thank you. I agree, you get one value of inductance for one integration. If you repeat this for a number of ‘concentric spheres’, you will get a number of inductances- ranging from minimum to maximum. Does that make sense?
Sainath
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Sainath,
As Thomas pointed out, inductance is the ratio of magnetic flux to current in the conductor. Magnetic flux is the integral of B dot dA, or the magnetic field [dot product] the surface you are integrating over. The “dot product” is the same as multiplying the B-field by the area by the cosine of the angle between the B-vector and the normal to the area. So if the B-vector is perpendicular to the area surface, then the B-vector is parallel to the unit normal vector of the area surface, cosine of this zero degree angle is 1, and you simply multiply B*area. Here’s an example to illustrate.
You have a rectangular metal trace over a ground plane, length in the z-direction, height in the y, width in the x. Stretch a rectangle in the yz plane between the trace and the ground plane. Make it any length (smaller if you are simulating with EM tool). If we assume perfect conductors (ie no internal-conductor magnetic fields), then all of the magnetic field associated with that signal trace will pass through this rectangle. It is
kind of like a net. Magnetic field lines always have to end up in the same place they started, completing the circle. Also, in this configuration, all your field lines are perpendicular to the integrating rectangle. So
inductance is flux/I = B*A/I. In this case, you will actually have inductance per unit length because your net had a specific z-length.
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. By placing it between the signal line and the return path, you capture all the field lines. So you have one number for inductance if you account for all the B field lines. An inductance “distribution” would indicate that you are not catching all the magnetic field lines with your integrating surface.
This might open up a talk about internal inductance, when you have magnetic field lines (ie current) INSIDE the conductors. As frequency increases, the current crowds to the surface, and the internal inductance diminishes. But at lower or intermediate frequencies, this internal inductance can be a contributing factor. For PCB’s, this is typically in the low MHz range. But for square conductors on silicon, measuring a few microns wide and a few microns high, the internal inductance might have to be considered up to
several GHz. Does this affect you? Do you electrical models consider this effect? How about internal inductance of the ground plane? Interesting stuff here.
Salud,
Andy Byers
Andy,
Thanks. I appreciate the extra effort to explain detail of integration. In short, you’ve explained the current loop formed by a signal path on trace and signal return path beneath the trace and on the ground plane.
Such a return path, with its minimum loop area, is widely known to provide the path of “least” inductance for high-frequency currents(for example, Black Magic book). If inductance is thought of as one number,
what does “least inductance” refer to? Which is the path of “most” inductance for the microstrip? No doubt, I’m missing somethig.
Sainath
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Hello Sainath,
Clearing up some terminology here.
“Least inductance” refers to the path that the current will travel because will never choose an alternate path of “most inductance”. BUT you can have a different design in which the “path of least inductance” is longer. For example a two wire line with no ground plane where the wires are extremely far apart. Huge loop, huge inductance. But still the smallest loop for that system. For a microstrip, a path of More Inductance would be if there were a gap in the ground plane under the microstrip line. The current would be forced to diverge around the gap. This path would be more inductive than a solid ground plane, but the current would still be following the path of least inductance for that particular case.
The main challenge in most systems I’ve dealt with is making sure that return current paths have the least inductance possible. The simplest way to do this is go differential. Then you carry your virtual ground with you everywhere. If single ended, then be very conscious about where the return currents flow and try to provide a short path. Plenty of threads on this list about that.
Not sure if this clears up your last question, hope it helps though.
- Andy
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Hi Andy,
Thanks again. I get the themes that inductance is a one number affair and current returns through the least inductance path. 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.”
For this case, is the inductance of the microstrip going to be infinity(because of infinite surface)? or any other value? remains same as what it was for the integrating surface below the trace?
Sainath
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As someone previously stated, inductance is defined as the ratio of the magnetic field to the current. BUT both of those are vector quantities, not single numbers. And there is a different quantity for each point in a field. So “single values” for inductance are obviously simplifications. My interpretation of “the path of least inductance” would be the set of connected points for which the value of inductance is least.
Art Porter
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Inductance is the ratio of magetic flux (not field) to current. Flux is not a vector, it is a scalar. So is the magnitude of the current in a wire (closed integral of H dot dl). So you will get single inductance number for
a specific interconnect cross section.
See pg. 81-83 of “Fields and Waves in Communications Electronics (3rd ed)”, Ramo,Whinnery,and Van Duzer.
As you progress down the interconnect, the current will want to flow wherever this inductance in the smallest. The path that the current follows will be this path of “least inductance”.
Happy Weekend!
Andy
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Andy,
Yes, the inductance value should remain the same for both cases. Also, we are capturing all the magnetic flux lines in both cases.
Now comes the real question. When you capture all the flux lines, is the inductance going to be maximum? or minimum?
Sainath
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Sainath,
First of all, with your surface, either above the microstrip or below, you are capturing magnetic field lines, not “flux lines”. You integrate these field lines over the area of the surface to produce a scalar number which is your magnetic flux. A lot of times people get Flux and Field confused. Flux is a scalar number, while field is a vector.
So, like you say, if you capture all the field lines on your surface, you should calculate the true flux and therefore the correct inductance. Calling it a “maximum” or “minimum” does not really fit here. If you were to use a surface where you did not account for all the field lines, the inductance you calculate would indeed be smaller than the correct value. But it would be wrong. I guess you could say that “maximum” inductance calculation is correct, and “minimum” inductance calculation would be zero (you capture
none of the field lines).
Any 2D cross section of an interconnect system should have one correct inductance value. As you move along in the 3D direction of propagation, the 2D cross sections will change and your inductance at that point might change too. Once again this is assuming no internal inductance and a single mode.
With internal inductance, your total inductance becomes frequency dependent. The Ramo, Whinnery, Van Duzer book points this out as well.
Andy
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Andy,
Calling it a “maximum” or “minimum” does not really fit here.
C’mon! Let me explain why calling it a “maximum” or “minimum” does really fit here.
1) No other integration surface (other than the two we considered)can give a more inductance value. Right? So, obviously it is the maximum inductance value.
2) Conventional notion is that return current takes the path of minimum inductance and hence flows directly under the trace on the ground plane. Contrary to this notion, through this thread, we observed that such path provides maximum inductance.
Unless I’m missing something big time…
In conclusion, the return current, when it flows directly under the trace on the ground plane, is in fact taking the path of maximum inductance. In other words, the path directly under the trace on the ground plane is not the path of minimum inductance.
Sainath
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Wow - This is getting interesting. Heres my take on it. One can say that inductance is proportional to
field lines. SO more field lines (or flux) the higher the inductance. In thace case of two parallel wires, Flemings RHR shows that the field BETWEEN the lines in enhanced - maybe leadind one to think maximum inductance. However the fields on the *outside* of the wires actually cancels leading to
a reduction in overall inductance of the loop.
Best Regards
Charles Grasso
Senior Compliance Engineer
Echostar Communications Corp.
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Andy,
I disagree with your correction(about integrating magnetic flux lines). Please do a simple dimensional check.
Yes, there is this correct inductance value which we get in the limiting case when we capture all the flux. This is also the maximum inductance. Lower inductance values are possible depending on the chosen surface and the minimum can go as low as zero, like you said. So, there is a distribution ranging from zero to the correct value. I believe the significance of this and its SI application opens up new directions…
For SI application involving return current paths, I wonder how the idea of minimum(zero) inductance path stuck around so long.
Sainath
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Sainath,
The integral (maximum or minimal) depends on the loop of the surface edge, not the surface itself. Given a fixed loop, the integral will not vary on various surface. Its principle comes from the physics law that tells us the integral on a closed surface is always ZERO.
Fred
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Fred,
We’ve been talking about magnetic flux which is the surface integral of the normal component of flux density vector B. Right? Given that, please check your statements.
Sainath
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Sainath,
You are getting confused between the calculation of the inductance for a given current distribution and the variation of inductance caused by a variation in current distribution.
When you are calculating the inductance value for a given current distribution, you must integrate the normal of the B field over a surface area which captures ALL of the field lines surrounding (external
inductance) and within the current distribution (internal inductance). This is not the maximum inductance or the path of maximum inductance, it is simply the correct inductance. Any calculation which uses a surface
area which fails to have all of the field lines passing through it is wrong. Inductance (not partial inductance) is defined as the ratio of the amount of magnetic flux coupled through and created by a given
closed path current distribution to that current distribution. The irrelevant fact that performing the calculation while ignoring some of the field lines happens to give a lesser inductance value does not make
the correct calculation the maximum inductance value. By your logic, if I could find a different but equally wrong way of calculating the inductance and it happened to come out larger than the correct calculation, then the correct calculation should henceforth be known as the minimum inductance value.
If I were to integrate the electric field lines passing out of a closed surface and decided to ignore part of the surface, I would get a value for the charge within that surface which was smaller than the correct
value. Should I then refer to the charge within that surface as the maximum charge value?
The path of maximum inductance within the conductor would be the current distribution which maximizes the open surface area required to couple all of the B field. The path of minimum inductance within the conductor would be the current distribution which minimizes the open surface area required to couple all of the B field. The change in inductance is linked to the variation in loop size caused by the variation in current
distribution.
Additionally, as has been stated on this thread, the current will distribute itself on the path of minimum impedance or referring to the principle of least action, the path of least energy; depending on frequency this is not necessarily the path of minimum inductance.
Thanks,
Michael Smith
iZ Technology Corp.
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Michael Smith,
>By your logic, if I could find a different but equally wrong way of calculating the inductance and it happened to come out larger than the correct >calculation, then the correct calculation should henceforth be known as the minimum inductance value.
- That is what I need. Please give me a way to find an inductance value
that is larger than the correct value.
>The path of maximum inductance within the conductor would be the current distribution which maximizes the open surface area required to couple all of the B field. The path of minimum inductance within the
conductor would be the current distribution which minimizes the open surface area equired to couple all of the B field. The change in inductance is linked to the variation in loop size caused by the variation in current distribution.
- I don’t quite follow this technical language. Is there a reference you could suggest me on this?
>Additionally, as has been stated on this thread, the current will distribute itself on the path of minimum impedance or referring to the principle of least action, the path of least energy; depending on frequency this is not necessarily the path of minimum inductance.
- We all seem to agree that high-frequency currents need not necessarily
follow the path of minimum inductance.
Sainath
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From: “john lipsius”
To all pursuers of the maximum/minimum false dichotomy and the “path of maximum annoyance” 
Any further contributions to this thread that adhere to that confusion will, it seems, just confuse novices that subscribe to this list. Any further help from the experts is, unfortunately, wasted I believe.
Please pick up a physics or microwave text to get it straight and look at the illustrations. In short, it’s necessary to dot-product one’s interest with a little homework, whereupon the path of maximum edification shall reveal itself in all its glory and thence one shall go forth in peace and confidence.
A review of andrew’s and michael’s replies on this thread should suffice, below.
Basically, claiming there’s an inductance “distribution” is confusing these two:
1. a mathematical definition of flux that relies on an abstract surface chosen by you
2. the flux itself, which is constant for constant current, frequency, material and geometry.
-enough said
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Those of you interested in magnetic field theory may find Sainath’s questions about the integration of magnetic flux a fascinating subject; others may find this a good time to step out for a cup of tea…
Dear Sainath,
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 www.amazon.com as a modern reprint of an old Dover version, circa 1954. I read
a copy of the work in preparation for writing my latest book, “High-Speed Signal Propagation”, and found it most enlightening.
From the preface of Maxwell’s 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”.
It’s the “lines of force” concept that makes everything work. 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 also 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 dissappear 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.]
Now let’s apply the line-counting analogy to your trace-inductance problem. Imagine a certain finite number of magnetic lines of force wrapped around your trace. [I'll assume the reference plane is infinite in the x-y
directions. The plane is located at z=0, and the trace is at z=1. Since the plane is infinite, no 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 edges of the bubble touch the trace all along its length, following along at the end down to the reference plane, returning along the plane to the source. For completeness, let’s also consider how at the source the edges of the bubble also must track along the ground lead of my inductance meter up to the instrument and then back down the signal lead of the instrument to the beginning of the trace. We’ll assume the meter is really tiny compared to the size of the trace so we don’t have to worry too much about the shape of the source end of the bubble (this is a serious real-life complication in the measurement of tiny inductances).
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 the edges in such a way that it does not enclose any space), you will record some non-zero amount of flux penetrating the bubble. NOW comes the really 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 A must precisely equal the total flux passing through B.
In a minute I’m going to directly address your 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 an 2-dimensional surface with infinite extent acts kinds of like a closed surface, in that it partitiions 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 your problem.
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 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 errect 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 anchored at its edges along the trace and plane precisely coincident with the edges of the pink soap bubble, you may rightly conclude that any flux that contributes to the total flux count in the pink region must also 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 an infinite partition of space, the total flux through that partition must be zero, ergo, the flux through the pink and white surfaces must be the same. This is what I think Andy 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 this 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 which must necessarily raise the inductance of the configuration (moving to any distribution other than the minimum-inductance distribution must
necessarily raise the inductance).
Regarding your interest in the exact distribution of current problem. This analogy I’ve developed in the course of making up laboratory demonstrations for my new class on Advanced High-Speed Signal Propagation.
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 magetic 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–and it’s the same problem 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 through the parasitic capacitance between trace and plane). If you assume that once the current arrive on the plane it
flows parallel to the trace (making the cross-sectional picture the same at each position along the trace, as
required by symmetry), then you can see that the picture also 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 recollection of Maxwell’s equations (specifically I *think* it’s the ones that say the Laplacian of both electric and magnetic fields are zero within source-free regions) is that the distributions of charge and current in a statics problem fall into a pattern that satisfies all the boundary conditions around the edges of the region of interest, satisfies the Laplacian conditions in the middle, AND ALSO just happens to store the *minimum* amount of energy in the interior fields. In other words, you aren’t going to get huge, unexplained, spurrious magnetic fields in the middle of an otherwise quiet region (unless you believe in vaccuum fluctuations, which is a different subject entirely…).
The stored energy for inductive problems is: E = (1/2)*L*(I^^2), where where L is the system inductance and
I^^2 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 the current is *NOT* in the minimum-energy distribution you can demonstrate the existance of a mode of current that leads to a lower-energy state, but that 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 a lot of other doubts.
For further reading, try the following articles: “High-Speed Return Signals”, “Return Current in Plane”, “Proximity Effect”, “Proximity Effect II”, “Proximity Effect III”, and “Rainy-Day Fun”, (see http:\\sigcon.com, under “archives”, look for the alphabetical index).
Best regards,
Dr. Howard Johnson, Signal Consulting Inc.
As they say, interesting things happen when you are away from your mail. For a moment, I wondered if the list administrator has changed!
John,
You know, Lawrence is a good friend of mine. He never mentioned about your impressive language skills. Readers might be wondering why I am talking about Lawrence instead of Henry. Simple. John and I worked at a company called Cognigine and John used to report to Lawrence. Enough said. Oh, let me make sure, are you the same John Lipsius?
You seem to agree that there is some confusion. Novice or expert, it is important to sort out any confusion. If I am not looking at issues correctly, I better get the right perspective and this list is a good place. We all know about blindspots.
I consider myself a novice and tomorrow SI depends on today novices. So, novices should not be intimidated by confusion. For those who find this thread confusing or annoying, there is the delete button.
There is some useful contribution you can make(unless you think it is a wasted effort). Please give that physics or microwave text and illustrations. I will do my dot product and perhaps some cognitive integration also.
BTW, are we not concerned about changing currents?
Sainath
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From: “john lipsius
> BTW, are we not concerned about changing currents?
thank you, my point is thus proven (as I gambled it would be, by following that hard to
find but always-present “path of minimum energy”
…amazing)
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Dear Howard,
I appreciate your participation, time and views. I have some issues but let us consider the most important one. I’ve reproduced portions of your
text(in quotes) for discussion convenience. Any text without quotes is mine. It’s unusual and uncomfortable to read, but please bear with me.
“The stored energy for inductive problems is: E =(1/2)*L*(I^^2), where where L is the system inductance and
I^^2 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.”
For a microstrip, “Most of the current flows on the reference plane right under the trace, with less and less as
you move away from the trace” as shown in Fig. 5.3 in Black Magic book.
Using above formula and figure, “the distribution of current on the reference plane that minimizes the total stored magnetic
energy” occurs near the tail portions(away from the trace). This distribution is “one and the same” as “distribution of current that
minimizes the inductance”.
So, it appears to me that the path of least inductance for the return current is away from the trace where the current is minimum.
However, it is generally believed that the path of least inductance for the return current is on the reference plane right under the trace.
Using above formula and figure and the fact that “stored magnetic energy E and inductance L vary in direct proportion to one another”, right
under the trace, current is maximum => stored energy is maximum => inductance is maximum.
What’s wrong with my line of reasoning?
Sainath
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John,
What did you want us to understand from this message? It looks plain English. But, what does it convey?
I didn’t know that you gambled with technical issues.
Sainath
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Dear Sainath,
Let’s work for a minute on your concept of the “path of least inductance”.
I think a better wording here would be the “distribution of Imagine you have a long, straight pcb trace carrying a certain amount of signal current.
Underneath that trace I want you to construct not a plane, but an array of wires (kind like a ribbon cable). Place the wires on a very fine spacing so that (in the limit) they approximate a plane, but keep them as individual wires.
Now connect an individual current generator to each of the return wires. With such an arrangement you can FORCE any distribution of return current that you want. The current (in the return wires) always flows parallel to the signal trace, as it would (by symmetry) in a solid-plane situation.
[NOTE: here I'm just thinking about current in the return plane for a long, straight trace configuration. Towards the ends of the trace there are surely some deviations from straight-line flow which for now I will simply ignore].
Now I want you to adjust the current generators to energize only one of the return wires at a time. In each case, record the total magnetic flux (i.e., inductance) of the configuration using just that one return wire. You will find that the particular return wire that generates the least inductance is the one directly underneath the trace. Concentrating all the return current on that one wire, however, does not create the “path of least-inductance”.
To find the path of least inductance (actually, the return-current “distribution of least inductance”) you must
try ALL POSSIBLE distributions of return currents (with the constraint that the sum of return currents must equal the signal current in each case), and record the total magnetic flux for each distribution. You will find that the particular distribution of return current that generates the least inductance looks like the curve shown in Figure 5.3, page 191, of “High-Speed Digital Design: A Handbook of Black Magic”.
If you choose any distribution of current OTHER than the “distribution of least inductance” then I can always choose another distribution that has less overall inductance. How do I do this, you may wonder? There are several ways. The way nature does it is to make use of a theorem that says that if you are NOT using the distribution of least inductance then magnetic lines of flux must exist that penetrate the return plane (in your case, these lines penetrate your dense carpet of return wires). These magnetic field lines create circulating currents (eddy currents) in your wires that tend to modify your distribution of current, always making it more like the “distribution of least inductance”. If you use individual current sources on each wire you can prevent the eddy currents from having an effect, but if you connect the wires together at the ends of the ribbon the eddy currents will rapidly cause the wires to attain the “distribution of least inductance”. That’s what happens in a solid highly-conductive plane.
If, on the other hand your plane is not so conductive (perhaps it has some significant resistance) then the
resistance of the plane interferes with the eddy currents in such a way that the eddy currents may not be sufficiently powerful to form the “distribution of least inductance”. That happens in DC power distribution problems, where I’m sure you’ve heard the old adage “current follows the path of least resistance”. In the context of current flowing in a solid plane the “path of least resistance” is not be simply a straight line of infinitely thin width directly from source to load. Instead, the current spreads out, taking maximum advantage of all the available copper in the plane.
The distribution of current in the solid plane that minimizes the power dissipated in the plane is called the
“path of least resistance” (actually, it should be more properly called the “distribution of least resistance”).
In DC resistively-dominated problems the force that casues redistribution of current in the plane is the electric-field potential. In magnetically-dominated problems the force that causes redistribution of current in the plane is the magnetic-field potential (associated with eddy currents). In both cases the current spreads out, assuming not just one path but a distribution of pathways. For typical pcb geometries using 1/2 oz. copper planes the frequency at which the inductive (magnetic) effects supersedes the resistive (DC) effects is on the order of approximately 1 MHz. It is possible using matrix mathematics along with the concept of mutual inductance to derive the distribution of least inductance, but that’s not how nature
does it. In a natural setting the current re-distributes itself according to the action of localized eddy currents.
You are indeed correct that the “distribution of least inductance” includes the current flowing at positions remote from the trace.
I can provide one analogy that may help you understand the cooperative behavior of current both underneath and remote from the trace. Suppose I solder together two resistors in parallel. Let one of them be 1000 ohms, and the other be 1,000,000 ohms. The effective parallel impedance will be 999.000999000999000… ohms. What in this case would you call the “path of least resistance”? Surely it is true that MOST of the current flows through the 1000-ohms resistor, but PART of the current also flows through the 1,000,000 ohms resistor. If you force a current of 1 mA through the parallel combination you will develop a voltage of 0.999000999000… volts, and observe currents of 0.999000999000… mA and 0.000999000999000… mA in the two resistors, respectively. Give a name to this distribution of current, calling it distribution “A”.
Suppose now you connect individual current sources to the individual resistors so you can adjust the current separtely in each. I claim that, of all possible distributions of current adding up to 1 mA, the particular distribution “A” is the one that minimizes the total dissipated power. Distribution “A” also happens to have the property of generating the SAME voltage across both resistors. This example is highly analogous to the “distribution of least inductance”, where the distribution of least inductance has the property of minimizing the total magnetic flux (i.e., minimizing inductance) AND ALSO has the property of generating the same voltages at the ends of every wire (technically, this is the same as ensuring that no magnetic flux penetrates the solid reference plane).
The same general reasoning, by the way, applies to the distribution of current around the periphery of the pcb trace itself, and at every point on all other conductive objects near your trace.
I do wish we could have a chance to talk this over in person, as I sense you have many questions. I’ll be in San Jose right after labor day. Write to me at howiej@xxxxxxxxxx if you would like to arrange a time to
meet.
The mathematics associated with finding the path of least inductance are discussed further in my articles, “High-Speed Return Signals”, “Return Current in Plane”, “Proximity Effect”, “Proximity Effect II”, “Proximity Effect III”, and “Rainy-Day Fun”, (see http:\\sigcon.com, under “archives”, look for the alphabetical index).
Best regards,
Dr. Howard Johnson, Signal Consulting Inc.,
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