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 in the "least-inductance" configuration, let me propose an analogy that I find quite helpful in working through that 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 perpindicular to the surface of the trace, and land perpindicular 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., tel +1 509-997-0505, howiej@xxxxxxxxxx http:\\sigcon.com -- High-Speed Digital Design articles, books, tools, and seminars -----Original Message----- From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of Sainath Nimmagadda Sent: Thursday, July 17, 2003 11:44 PM To: andrew.c.byers@xxxxxxxxxxxxxx Cc: si-list@xxxxxxxxxxxxx Subject: [SI-LIST] Re: si-list Digest V3 #194 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 ---------Included Message---------- >Date: Thu, 17 Jul 2003 17:37:12 -0700 >From: <andrew.c.byers@xxxxxxxxxxxxxx> >Reply-To: <andrew.c.byers@xxxxxxxxxxxxxx> >To: <gigabit@xxxxxxxxxx> >Cc: <si-list@xxxxxxxxxxxxx> >Subject: RE: [SI-LIST] Re: si-list Digest V3 #194 > >Hello Sainath, > >Clearing up some terminology here. > >"Least inductance" refers to the path that the current will travel because >it has the least inductance of all possible paths in the system. Current >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 > > > >-----Original Message----- >From: Sainath Nimmagadda [mailto:gigabit@xxxxxxxxxx] >Sent: Thursday, July 17, 2003 4:01 PM >To: Byers, Andrew C >Cc: si-list@xxxxxxxxxxxxx >Subject: RE: [SI-LIST] Re: si-list Digest V3 #194 > > >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 > >---------Included Message---------- >>Date: Thu, 17 Jul 2003 10:02:49 -0700 >>From: <andrew.c.byers@xxxxxxxxxxxxxx> >>Reply-To: <andrew.c.byers@xxxxxxxxxxxxxx> >>To: <gigabit@xxxxxxxxxx>, <beneken@xxxxxxxxxxxx> >>Cc: <si-list@xxxxxxxxxxxxx> >>Subject: RE: [SI-LIST] Re: si-list Digest V3 #194 >> >>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 >> >>-----Original Message----- >>From: Sainath Nimmagadda [mailto:gigabit@xxxxxxxxxx] >>Sent: Thursday, July 17, 2003 9:25 AM >>To: beneken@xxxxxxxxxxxx >>Cc: si-list@xxxxxxxxxxxxx; gigabit@xxxxxxxxxx >>Subject: [SI-LIST] Re: si-list Digest V3 #194 >> >> >>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 >> >>---------Included Message---------- >>>Date: Thu, 17 Jul 2003 12:04:57 +0200 >>>From: "Thomas Beneken" <beneken@xxxxxxxxxxxx> >>>Reply-To: <beneken@xxxxxxxxxxxx> >>>To: <si-list@xxxxxxxxxxxxx> >>>Subject: [SI-LIST] Re: si-list Digest V3 #194 >>> >>>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 >>> >>>> Msg: #12 in digest >>>> Date: Wed, 16 Jul 2003 11:55:35 -0800 >>>> From: "Sainath Nimmagadda" <gigabit@xxxxxxxxxx> >>>> Subject: [SI-LIST] Microstrip Inductance >>>> >>>> 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 >>> >>> >>>--------------------------------------------------------- --------- >>>To unsubscribe from si-list: >>>si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field >>> >>>or to administer your membership from a web page, go to: >>>//www.freelists.org/webpage/si-list >>> >>>For help: >>>si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field >>> >>>List archives are viewable at: >>> //www.freelists.org/archives/si-list >>>or at our remote archives: >>> http://groups.yahoo.com/group/si-list/messages >>>Old (prior to June 6, 2001) list archives are viewable at: >>> http://www.qsl.net/wb6tpu >>> >>> >>> >>---------End of Included Message---------- >>__________________________________________________________ ___ >> >> >>---------------------------------------------------------- -------- >>To unsubscribe from si-list: >>si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field >> >>or to administer your membership from a web page, go to: >>//www.freelists.org/webpage/si-list >> >>For help: >>si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field >> >>List archives are viewable at: >> //www.freelists.org/archives/si-list >>or at our remote archives: >> http://groups.yahoo.com/group/si-list/messages >>Old (prior to June 6, 2001) list archives are viewable at: >> http://www.qsl.net/wb6tpu >> >> >---------End of Included Message---------- >___________________________________________________________ __ > > ---------End of Included Message---------- ____________________________________________________________ _ ------------------------------------------------------------ ------ To unsubscribe from si-list: si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field or to administer your membership from a web page, go to: //www.freelists.org/webpage/si-list For help: si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field List archives are viewable at: //www.freelists.org/archives/si-list or at our remote archives: http://groups.yahoo.com/group/si-list/messages Old (prior to June 6, 2001) list archives are viewable at: http://www.qsl.net/wb6tpu ------------------------------------------------------------------ To unsubscribe from si-list: si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field or to administer your membership from a web page, go to: //www.freelists.org/webpage/si-list For help: si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field List archives are viewable at: //www.freelists.org/archives/si-list or at our remote archives: http://groups.yahoo.com/group/si-list/messages Old (prior to June 6, 2001) list archives are viewable at: http://www.qsl.net/wb6tpu