star ground for power tubes and ct, or soldered to chassis?

[Tooboob]

No insult intended, some folks (certainly none of you fine posters) need to know this and don't know that they don't know it:

Ground is just a concept, and Ohm's law isn't. It helps to be the electrons.

Ground routing is certainly key, but don't forget the caps. Let's say that you have a triode amplifying a signal. It has a "ground" reference, and variable current through the triode. The current comes from the wall, through the PT, through the bridge, over the caps, through dropping/filter resistors/capacitors to form the supply for the triode, and goes to the "ground", where it runs back to the PT and back to the wall, right?

Not really. There's a big cap on that triode's supply, and it is the current source for signal related AC. One end of the cap sources AC current to the load (the triode), the other end of the cap sucks it out of "ground", and by the miracle of electronics, it's about the same AC current that the load draws. If you put the cap way over by the power transformer, HV caps and dropping power resistors, and put the triode elsewhere where there's more room, or the tube will be easy to get to, you make a big loop, and ground current goes a long way, creating an IR drop (V=IR, so that's bad).

Caps at sources of AC (signal related, not 120VAC) current (mostly tubes in this case) can pretty much eliminate AC from the chassis. Supply source resistance can work with these local caps to reduce this AC current from the supply, helping the caps localize the current loops. The cap-powered AC small loops also help fight hum, since the AC loop is smaller, so it's less susceptible to EMI from the PT, florescents, etc. If the filtering capacitance results in a supply that's too tight, stick a resistance between the cap and the load, and maybe another cap, to yield the proper response. and maintain the small loop. If you can tie that loop to your ground network at one point, the other circuits will never know that there are monstrous current disturbances going on, and disturbances from radiation picked up by the small loop will stay in the loop.

You don't want too much supply sag, or the wrong response time, and you probably don't want your tubes to see supplies that are too tight, so it's still an art, but at least you're working on the right painting.

Most designs have the caps. They should go near the load (the tubes), not the cap's input source (the PT and stuff). Don't forget that the OT is in the loop for the power tubes. That's unavoidable, and you want to minimize that loop too.

Don't think like the power company, and think that the AC signal currents in the amp necessarily come from and go back to the PT. Instead, make sure they stay local. Draw the AC loops on your schematic, minimize them in your layout, and try not to use the shared ground network to do it.

[NateS]

^-- Much more detailed version of what I was trying to say, but I was trying to boil it down to something pithy.

(Note to self: If you're trying to be pithy, make sure numbered bullet lists are sequential, and don't use the teeny tiny screen on your phone. Particularly on boards that have random and capricious edit policies.)

[Alan0354]

There is nothing pithy about grounding. It is a big subject and it is not easy to understand. Whether you guys buy it or not, it is all electromagnetics. This is a subject that is much better studied in the last 15 years as system getting bigger, more complicated, more analog/digital mixed signal circuits co-exist in a tight space. As technology goes towards surface mount and VLSI, high speed, high current circuits are packed tighter and tighter and noise coupling becomes more and more of a problem.

I work many years in grounding of systems, I was contracted to KLA Tencor to layout their critical 4 GBits CCD ( as in Giga, not Mega) 10 years ago. The many drivers have to drive high current into the CCD and signal coming out has to be free of induced noise. And I was in charge of passing the CE test that mainly deal with noise emission and susceptibility. These are all grounding issues. They even have a special tittle for this kind of engineer call "Signal Integrity" engineer that specialized just on the grounding.

I gave the link to my brief explanation. You guys should look into this. There are books on this topics of EMC. Even though they mainly talk about high speed digital and mixed signals, but EVERYTHING applies to what you are talking. I explained in theory why Fender old SF and BF amp has the best grounding scheme, there is no black magic, just electromagnetics................that current runs in a very predictable path in the ground plane( chassis) and can be tightly manipulated by design. It is not like current travel every which way. In electromagnetics, current travels in the path of least impedance, not path of least resistance like Ohm's law like to let you believe. In fact, Ohm's law does not really work if you look deeper just like Newton's law collapsed under microscope and Quantum theory takes over. Electromagnetic wave theory take over in AC signal over Ohm's law.

I know this is very hard to swallow and some of you are going to say I am reinventing the wheel and get a rise. But it's up to you to go do some research into signal integrity. The biggest revelation for me is to realize signal does not travel as electrons and currents. Signal travels as EM wave, current and voltage you see is ONLY the consequence of the boundary condition of the EM wave confined in the transmission structure( chassis and wire combo). Again, there is no easy explanation for this, there is nothing pithy about this. That's the reason most electronics is explained in Ohm's law like most mechanics are explained by Newton's Law, not Quantum Mechanics even people know those really do not work for all cases.




FYI. Ohm's Law, Kirchhoff Law......ALL FAILED under electromagnetic theory, these are well proven and you can even find on youtube the professor Lewin of MIT disproved Kirchhoff Law of current and voltage loop.
https://www.youtube.com/watch?v=eqjl-qRy71w
https://www.youtube.com/watch?v=1bUWcy8HwpM
I argued passionately against it.......I lost!!! I spent the whole Christmas time off on this, my user name is Yungman there:
http://www.physicsforums.com/showthr...t=Walter+Lewin

[R.G.]

...Whether you guys buy it or not, it is all electromagnetics....there is no black magic, just electromagnetics................that current runs in a very predictable path in the ground plane( chassis) and can be tightly manipulated by design. It is not like current travel every which way. In electromagnetics, current travels in the path of least impedance, not path of least resistance like Ohm's law like to let you believe. In fact, Ohm's law does not really work if you look deeper just like Newton's law collapsed under microscope and Quantum theory takes over....
I know this is very hard to swallow and some of you are going to say I am reinventing the wheel and get a rise. But it's up to you to go do some research into signal integrity.... That's the reason most electronics is explained in Ohm's law like most mechanics are explained by Newton's Law, not Quantum Mechanics even people know those really do not work for all cases.

Before we get off into arguments about How The Universe Works, it's worth looking at the Correspondence Principle.
Here's a brief, but useful description of the Correspondence Principle, using Newton's Laws and Einstein's Relativity:

The general rule is what physicists call the correspondence principle or the Classical limit. This states that any new theory must give the same answers as the old theory where the old theory has been confirmed by experiment. Newton's laws and Einstein's Relativity give the same answers in ordinary conditions, they only give different answers in extreme conditions such as near the speed of light, refining the accuracy of the GPS system, or calculating the orbit of Mercury (none of which Newton could confirm by experiment).

(from the "Atomic Rockets" web site - good reading!"
Electromagnetics did supplant the earlier, simpler observations of electrical phenomena that things like Ohm's Law was based on, but under extremes of frequency. However, the Correspondence Principle held: under non-extreme conditions, Maxwell's Equations reduce to things like Ohm's and Kirchoff's and the others. So at low frequencies, electromagnetics (that's Maxwell) and the earlier, simpler rules say the same thing.

Audio - and power line frequencies! - are so close to DC that the differences between electromagnetic theory and the simpler laws is so minute as to make messing about with the differences require huge efforts of refinement to even detect differences.

So before we get off into having to run a solution to Maxwell's Equations for figuring cathode resistors, it's useful to note that it's quite difficult to tell the difference in that and Ohm's law in most situations. And before we go haring off into the tall grass putting ground planes on every ground return, it's worth noting that isolated return wires and determining where the currents go is much simpler with individual wires at frequencies where this works.

There's an other principle at work here. Einstein is reputed to have said one version of it - everything should be as simple as possible, but no simpler. That's a good form of Occam's Razor.

[Alan0354]

Electromagnetics did supplant the earlier, simpler observations of electrical phenomena that things like Ohm's Law was based on, but under extremes of frequency. However, the Correspondence Principle held: under non-extreme conditions, Maxwell's Equations reduce to things like Ohm's and Kirchoff's and the others. So at low frequencies, electromagnetics (that's Maxwell) and the earlier, simpler rules say the same thing.

Audio - and power line frequencies! - are so close to DC that the differences between electromagnetic theory and the simpler laws is so minute as to make messing about with the differences require huge efforts of refinement to even detect differences.

So before we get off into having to run a solution to Maxwell's Equations for figuring cathode resistors, it's useful to note that it's quite difficult to tell the difference in that and Ohm's law in most situations. And before we go haring off into the tall grass putting ground planes on every ground return, it's worth noting that isolated return wires and determining where the currents go is much simpler with individual wires at frequencies where this works.

There's an other principle at work here. Einstein is reputed to have said one version of it - everything should be as simple as possible, but no simpler. That's a good form of Occam's Razor.

See that is what the conventional believe is. But this is not true. current and voltage does not magically transform when frequency goes up, it is still governed by the same basic principle. You cannot use electron or current to explain why you turn on a light even through a 10 feet cable and the light turn on instantaneously. If you reference the speed of the electron or "hole" movement, it is governed by the velocity which is inverse proportional to the conductivity. In copper or steal, the velocity is in mm/sec., not 3X10EE8 m/sec. of light speed. The reason a 60Hz signal travels at light speed through the Tx line in air is because it is travel in form of EM wave. And please don't use the ornament of a few metal balls where you hit on one side, the ball at the other end bounce out. That has no scientific relevance at all. And the water flow in a host does not explain either.

Back to the relevance on the subject. The ground current in a plane does not travel every which way, it absolutely follow the path of least impedance. If you have a wire on top of the chassis, the ground current will try to follow under the wire to a certain extend. You are right a low frequency, it is not as exact. BUT the distribution of current density still follow the wire. This mean the current is localized. What fender did is have the most sensitive preamp tubes on one side and propagates down. So the ground "image" current is localized even if you don't use a star. Fender also put the filter caps in the way it bus the power and ground of each cap to the power and ground where the stages are to make sure all current are confined to the local region. You cross one of the ground return wire from the capacitor box, you'll be in trouble!!!

Fender put the cathode of the power tubes to the chassis but there is no ill effect, because that part of the chassis is confined to only having the current from the power tubes. The layout is ingenious and it works. The only thing I would change is I put one star right in the middle between the two power tubes, that's where I put the cathode, the ground of the reservoir cap and the screen cap on. All the high current of the power stage does not even touch the chassis.

I have my grounding on my amp all grounded onto the chassis, I have 3 stages in the high gain stage. The main noise is from the grid stop resistor at the input and the AC filament of the first tube. I remove the grid stop and just simply rectified to DC filament cut 90% of the noise. I think I can cut a lot more with regulated DC. Ground noise is not even an issue. Fender's layout work and is brilliant.

If people playing with star and one point ground, they are playing with fire. they better be very very careful to account for every single ground current return, or it will be trouble. I remember I helped a guy here like two years ago in a hum issue. He build a Twin from a kit and had hum noise. I had to look at the picture for hours to find he mistakenly put the ground wire of the PI stage to one of the ground star of the preamp stage. He wanted to be fancy, thinking star ground is better, he missed one and that caused him all the grief. I advised him to ripped the star grounding and copy Fender, worked like a champ.


Point to note: wire over ground plane ( chassis) is just microstrip. You can use the fomulas of microstrip. There are formula to show the current density on the ground plane under the wire that is proportion to the distance of the wire from the ground and the frequency. yes, low frequency make it not as obvious and the current spread out, but it is still localized. That's the reason you tag the wire close to the ground to the ground plane to localize the current on the plane and less loop area to serve as a loop antenna.

Noise induce in circuit mainly from ground noise couple and loop antenna effect, both can be explained only be electromagnetics. If you know the mechanism of cross talk, then you know how to avoid noise coupling. This is just as important at audio frequency.

[Tooboob]

Yeah, but sometimes you have to design a circuit. You don't need a 3-D solver to do a guitar amp, and you'll save $50,000 and $10,000/year in support. I send around lots of 5 GHz signals, the fastest scope I use is 350 MHz, and I pass CE. You just have to see what's going on, and return currents and loop management are a very big thing.

Where do you get your EM waves if current and voltage are only a consequence? It's sort of a chicken and egg thing. You can get all quantum on it, but there's safety in numbers, and if you play rock through an amp, the old-school models work quite nicely. You've got a bunch of "stages", with single-digit milliamps flowing between them, trying to hit 20kHz. Is it on yet?

It's a freaking guitar amp. It makes a seven transistor radio look complicated (the radio's got an antenna and RF tuner). You can relax now. Identify the big problem and solve it efficiently. That's what engineers do. Unlike what you've been doing for years, the really big things are all subjective. You know where I get stuck? How many knobs, and what they should do. Everything else is just implementation.

Shot noise in a grid leak resistor? The guy's playing a Strat!

Enjoy the electrons hitting a quarter of the speed of light heading from the grid to the plate (driven by an EM field, generated by driving a current of electrons througha capacitor, creating a voltage). Enjoy the high voltage. Enjoy the tradition and achievement of the pioneers. Enjoy the lack of an SOIC solution that reduces you to an ME and power supply designer, and puts you at the mercy of bit bangers. Enjoy the music. Anything that isn't fingers, strings and wood is frosting!

You do appear to be a wizard. You should make magic.

[Alan0354]

Yeah, but sometimes you have to design a circuit. You don't need a 3-D solver to do a guitar amp, and you'll save $50,000 and $10,000/year in support. I send around lots of 5 GHz signals, the fastest scope I use is 350 MHz, and I pass CE. You just have to see what's going on, and return currents and loop management are a very big thing.

Where do you get your EM waves if current and voltage are only a consequence? It's sort of a chicken and egg thing. You can get all quantum on it, but there's safety in numbers, and if you play rock through an amp, the old-school models work quite nicely. You've got a bunch of "stages", with single-digit milliamps flowing between them, trying to hit 20kHz. Is it on yet?

You don't need simulation. If you know the principle, just layout with that in mind and it's all good.

It is a chicken and egg thing. You control the tube/transistor. The current and voltage launch the EM wave.

Yes, it is not obvious for low frequency, BUT the effect is the same, current is localize and seemingly an inferior ground using the chassis turn out to be much better taking into effect of the current distribution from EM theory.

Old school theory cannot quite explain why the Fender's way of grounding is far superior than star grounding. It is a lot safer, as long as you layout the component in progressive order, you are quite safe. Problem arise only when people don't understand this and think they can do better by doing one point or star ground. You do CE, you have to study signal integrity to a certain extend, you should know this work like a champ. If it is so old school, why people keep running into noise problem?

[R.G.]

Sigh.

[Dave H]

Sigh.

What was that 'pig' quotation again?

[Alan0354]

OK OK, I got carry away, I'll stop.

[R.G.]

OK, I can be calm now.

See that is what the conventional believe is.

There's a reason for that.

current and voltage does not magically transform when frequency goes up, it is still governed by the same basic principle.

I believe that's what I said.

You cannot use electron or current to explain why you turn on a light even through a 10 feet cable and the light turn on instantaneously.

Actually, I can. And in any case, the light doesn't come on instantly. The speed of the field trying to move current through it only goes through the wires at the speed of EM fields in copper.

If you reference the speed of the electron or "hole" movement, it is governed by the velocity which is inverse proportional to the conductivity. In copper or steal, the velocity is in mm/sec., not 3X10EE8 m/sec. of light speed.

Yep. We had long philosophical discussions of the discrepancies of hole and electron flow/drift velocity versus the speed of the field in the material back in semiconductor physics class. And the speed of an EM field (not light, light won't pass through copper ) in copper is not the approximately 3x10^8 meters/sec of light in a vacuum. Light (and other EM) travels at different speeds in different mediums. One consequence of which is Cherenkov radiation. Fun stuff to read about.

The reason a 60Hz signal travels at light speed through the Tx line in air is because it is travel in form of EM wave.

Yes, it's an EM field. That's fine. As I said, Maxwell's Equations reduce to the low frequency forms ... at low frequencies. That doesn't say they're not different in nature.

Quick - what's the speed of a DC signal? Careful now...

And please don't use the ornament of a few metal balls where you hit on one side, the ball at the other end bounce out. That has no scientific relevance at all. And the water flow in a host does not explain either.

Good. I didn't use either one.

The ground current in a plane does not travel every which way, it absolutely follow the path of least impedance.

Quick - what's the impedance of a wire at 0.000001Hz? [Hint - it's really, really, really close to its DC resistance.]

And what path does a DC signal follow?

If you have a wire on top of the chassis, the ground current will try to follow under the wire to a certain extend.

Yep. To the extent that frequency effects can affect the current in the chassis. Ooops. There is that word "frequency" again.

You are right a low frequency, it is not as exact.

Oh, OK, good. You scared me there for a minute. I thought I had the universe all wrong.

BUT the distribution of current density still follow the wire. This mean the current is localized.

You left off a bit. It follows the wires **to the extent that frequency effects and resistance force it to**. When frequency is low compared to the impedances involved, the inductive and capacitive effects are asymptotic to zero. Never exactly zero, but asymptotic.

Our professors had a favorite question - show how the current in an infinite sheet of 1 ohm resistors approximates the current distribution in an infinite sheet of resistive material. Think about that one.

What fender did is have the most sensitive preamp tubes on one side and propagates down. So the ground "image" current is localized even if you don't use a star. Fender also put the filter caps in the way it bus the power and ground of each cap to the power and ground where the stages are to make sure all current are confined to the local region. You cross one of the ground return wire from the capacitor box, you'll be in trouble!!!

Yep. Get a ground wrong, it makes noise.

Fender put the cathode of the power tubes to the chassis but there is no ill effect, because that part of the chassis is confined to only having the current from the power tubes. The layout is ingenious and it works. The only thing I would change is I put one star right in the middle between the two power tubes, that's where I put the cathode, the ground of the reservoir cap and the screen cap on. All the high current of the power stage does not even touch the chassis.

I have my grounding on my amp all grounded onto the chassis, I have 3 stages in the high gain stage. The main noise is from the grid stop resistor at the input and the AC filament of the first tube. I remove the grid stop and just simply rectified to DC filament cut 90% of the noise. I think I can cut a lot more with regulated DC. Ground noise is not even an issue. Fender's layout work and is brilliant.

If people playing with star and one point ground, they are playing with fire. they better be very very careful to account for every single ground current return, or it will be trouble. I remember I helped a guy here like two years ago in a hum issue. He build a Twin from a kit and had hum noise. I had to look at the picture for hours to find he mistakenly put the ground wire of the PI stage to one of the ground star of the preamp stage. He wanted to be fancy, thinking star ground is better, he missed one and that caused him all the grief. I advised him to ripped the star grounding and copy Fender, worked like a champ.

]
As I often point out, there are an infinite number of ways to get grounding to be quiet. But star grounding is the only one that you know ahead of time. And you're right - leave off a star wire and things are bad. OK. Leave any ground off a ground plane, and I suspect it's equally bad.

Point to note: wire over ground plane ( chassis) is just microstrip. You can use the fomulas of microstrip.

Quick - is there a frequency term in the microstrip impedance calculations?

There are formula to show the current density on the ground plane under the wire that is proportion to the distance of the wire from the ground and the frequency. yes, low frequency make it not as obvious and the current spread out, but it is still localized. That's the reason you tag the wire close to the ground to the ground plane to localize the current on the plane and less loop area to serve as a loop antenna.

OK, how big a loop do you need to have a quarter wave at 60Hz?

[Alan0354]

OK, I can be calm now.


There's a reason for that.


I believe that's what I said.

Actually, I can. And in any case, the light doesn't come on instantly. The speed of the field trying to move current through it only goes through the wires at the speed of EM fields in copper.


Yep. We had long philosophical discussions of the discrepancies of hole and electron flow/drift velocity versus the speed of the field in the material back in semiconductor physics class. And the speed of an EM field (not light, light won't pass through copper ) in copper is not the approximately 3x10^8 meters/sec of light in a vacuum. Light (and other EM) travels at different speeds in different mediums. One consequence of which is Cherenkov radiation. Fun stuff to read about.


Yes, it's an EM field. That's fine. As I said, Maxwell's Equations reduce to the low frequency forms ... at low frequencies. That doesn't say they're not different in nature.

Quick - what's the speed of a DC signal? Careful now...

When it is stable for a long time, there is not EM field as E and H are decoupled in static case. But the moment you change at one end, change of voltage cause change of E and dE/dT cause curl of B. You launch the EM wave. We gone through that before.

Good. I didn't use either one.


Quick - what's the impedance of a wire at 0.000001Hz? [Hint - it's really, really, really close to its DC resistance.]

And what path does a DC signal follow?

I look at it as the current density spread away from the wire in microstrip. In high frequency and trace close to the plane, current density drop off close to zero at about 3 to 5 wire width from the wire. At lower frequency, it spread much wider, but still is going to be highest right under the wire, not however which way.


Yep. To the extent that frequency effects can affect the current in the chassis. Ooops. There is that word "frequency" again.
Don't keep thinking at low frequency, it does not matter, the current is more spread out, but it still follow the general path. The current is still localize if you just use common sense in the layout. So don't just imply it does not matter at audio frequency.

Oh, OK, good. You scared me there for a minute. I thought I had the universe all wrong.

You left off a bit. It follows the wires **to the extent that frequency effects and resistance force it to**. When frequency is low compared to the impedances involved, the inductive and capacitive effects are asymptotic to zero. Never exactly zero, but asymptotic.
Characteristic impedance does not depend on frequency. Yes, at low frequency, the image current does not follow exactly the wire on top, but still it is concentrate under and around the wire. This gives the proximity property that is very useful in layout and grounding when using a ground plane. That is the very point I am trying to drive, that with a progressive layout from preamp to power amp, you can safely use the chassis as ground and the image current will stay local and not affecting each other. The theory is more complicate, but the application is very easy, just layout in a logical order and ground on the plane like Fender and you get the best result. I never said you need to use simulation and all. BUT ALL THESE ARE BASED ON ELECTROMAGNETIC THEORY, not by ohm's law.

Our professors had a favorite question - show how the current in an infinite sheet of 1 ohm resistors approximates the current distribution in an infinite sheet of resistive material. Think about that one.
For DC, it just follow the path of least resistance. For AC, it is a different animal, it follow the path of least impedance.

Yep. Get a ground wrong, it makes noise.

]
As I often point out, there are an infinite number of ways to get grounding to be quiet. But star grounding is the only one that you know ahead of time. And you're right - leave off a star wire and things are bad. OK. Leave any ground off a ground plane, and I suspect it's equally bad.

This is where I am differ, I believe you can predict the current using the concept of EM theory and you know exactly where the current flow. Yes, for a star ground, you know ahead of time, but if you make a mistake, you'll be in trouble. Maybe you can sort out the stars, BUT obviously a lot of people cannot even though they think they know. Or else, there won't be a continuous discussion of this subject...Right? This is why I keep talking, people think they know but they don't!!!! It is safer to use chassis and use progressive layout and the current will sort it out by itself.

Quick - is there a frequency term in the microstrip impedance calculations?
microstrip has a characteristic impedance no exact a function of frequency. When you design a 50ohm line, it is 50 ohm regardless of frequency until it gets beyond a certain frequency.
OK, how big a loop do you need to have a quarter wave at 60Hz?
You don't need quarter wave, read the small loop, all it said is EMF induced by the TOTAL magnetic field through the loop. Yes, a small loop is not efficient, but guitar amp has a lot of gain, it does not take much to make noise.

I thought you want me to stop talking about EM. But yes, we can go through it if you want. I did spend years studying EM trying to understand it. Only problem is this site does not support Latex equation writing and EM can only be explained by calculus. As you know Maxswell did not use calculus to explain his theory, people invent multi variable calculus to explain Maxwell's equation. This is very important to know and there is NO common sense explanation of EM that I know of.

Yes, for a star ground, you know ahead of time, but if you make a mistake, you'll be in trouble. Maybe you can sort out the stars, BUT obviously a lot of people cannot even though they think they know. Or else, there won't be a continuous discussion of this subject...Right? This is why I keep talking. It is safer to use chassis as ground and use progressive layout and the current will sort it out by itself. Just like people don't think and start cutting ground and power plane in pcb thinking they can cut noise by separating the analog and digital ground. But if they don't know better, you put a digital trace crossing the analog ground, you get burned!!! That's the reason, unless you really know what to do ( not think), it is safer to use one continuous ground. That you can take it to the bank and I am speaking with many years of experience in system grounding and pcb layout for much more critical circuits. AND I fixed too many of other people's( experts) problem to say this.

[Jazz P Bass]

"OK, how big a loop do you need to have a quarter wave at 60Hz? "

Really small?

[Alan0354]

You don't need to be quarter wave loop to pickup noise. You can be 1/100 or smaller and still pickup noise of audio frequency. That's the point that you cannot discount by saying it's audio!!!

If it is not big deal, why are we even talking about it? Obviously a lot of people screwed up and have problem that this subject keep coming up in the pass, now and will be in the future!!!! Learn from Fender, they did it right from the 60s, and I can assure you that there is sounded EM theory behind what Leo Fender did whether he knew it at the time. This, to me, is call geneous. And we common people have to try to use EM theory to justify what he did. It works. Is that the whole point of this thread....why Fender works.

this is like Maxwell came up with the 4 equations that is so ingenious but without good explanation, that people had to invent multi variable calculus to help explain Maxwell's equation, not the other way around. This is science to you!!! Like the one of the biggest discovery of chemistry.....the Benzene ring!!! The people cannot fit all the theory together for the compound. One night, he dream of a snake biting it's own tail, and it goes round and round.............That's how the Benzene ring was discovered and it changed the chemistry world into cyclic compound.

[R.G.]

I wonder if we're in some kind of electronic/net epidemic.