Path of least resistance should really be path of least inductance.

[William Gaatjes]

A couple of weeks ago my smart colleagues and myself where sharing thoughts about how electric currents flow through copper planes and copper traces and how several different currents towards different switching loads cause EMI/EMC noise. I had done some reading and thought i had found some possible interesting similarities between electric currents flowing in plasma and currents flowing in multilayer copper planes and traces of printed circuit boards. When it comes to EMI and EMC, it is best that the return path of the current is of the same length as is the providing path. The AC (or pulsed DC) electric current always seems to have the desire to follow the path of least inductance. Although it is only shallow related, it made me think of the filament currents of a plasma where the filament is created by the resulting forces of the magnetic fields of the currents itself . But this is just as a side note.

We started thinking how planes work better then traces because parallel copper planes also function sort of as capacitor plates but with low capacitance. But really planes work, because of the wide path. The inductance of the plane is low with respect to a copper trace because of the eddy (Foucault) currents countering the generated magnetic field by producing their own fields where most eddy current generated magnetic fields are not aligned and actually weaken the primary generated magnetic field. We talked about it and if i understand correctly, when an electric current starts flowing, it seeks actually the path of least inductance and when the inductance is saturated the path of least resistance comes in effect. It makes intuitively very much sense.

But as most people i heard the phrase : "The path of least resistance "but i never really gave it much thought. I had to analyze together with my colleagues some switch mode power supplies with strange random failures (the smps was just bad designed). But the really interesting part was that how everything was connected and how power was supplied, prevented the smps to burn itself up initially. After our recommendations to the existing design, random components started to over heat and burn up because finally there was enough electric power that the loads(with transient behavior) could draw power and overloaded varies parts of the smps to the point of failure(crash and burn i might say D: ).

To return to the topic of the desired path of an electric current :
When i thought about it, it is only true for dc always. But for AC signals (or pulsed dc) it is only true after the "inductor" of the path is saturated.

Am i wrong about this assumption ?
It makes sense to me when i visualize it in very small time steps. I have to admit, i have smart colleagues.

 

[TuxDave]

Inductors has a high impedance at high frequencies so for high frequency signals, path of least impedance is the path of least inductance (assuming all else equal). Of course if we're doing that, may as well say it's the path of most capacitance (assuming all else equal) as well.

Or you can just nerd if up and just settle for path of least impedance.

 

[William Gaatjes]

That is true what you write about impedance. But impedance is very different from Ohmic resistance. These get confused a lot and should not be. And that was what my post is about.

With an inductor it is called inductive reactance because the change in the "flow" of electrons creates a magnetic field. The faster the change, the more the "flow" experiences a counter force. You cannot compare the alignment of electrons (the magnetic field build up until saturation is reached) with the scattering of electrons(Ohmic resistance). IIRC DC current flows. AC currents only pass energy on.

 

[Biftheunderstudy]

Except, Ohmic resistance is a subset of impedance. Yes, its only true in the case of DC that current follows the path of least resistance, but thats because at DC, resistance == impedance.

Obviously, to generalize to AC we need to use a more general term. As inductance, capacitance and resistance are all subsets of impedance in an AC circuit, then "current follows the path of least impedance" would be the most correct way of saying it.

 

[William Gaatjes]

Except, Ohmic resistance is a subset of impedance. Yes, its only true in the case of DC that current follows the path of least resistance, but thats because at DC, resistance == impedance.

Obviously, to generalize to AC we need to use a more general term. As inductance, capacitance and resistance are all subsets of impedance in an AC circuit, then "current follows the path of least impedance" would be the most correct way of saying it.

Indeed this is correct as you and Tuxdave mentioned. But the most interesting thing is that as long as the inductor is not saturated, it is the impedance that is dominant. After saturation, the inductance is zero or near zero and the Ohmic resistance is the dominant one. Of course the inductance for inductors and the capacitance for capacitors is frequency depended (the rate at which the electric current changes in size or in size and direction). Why is this interesting ? I wonder how superconductors respond to this situation. Is the inductance of an inductor different during superconduction when compared to room temperature ? I should look it up if the inductance changes with temperature. When thinking about, it makes sense that the inductance is higher when the temperature is colder because materials shrink. But that are just numbers behind the comma when thinking of windings and core material.

What you write for DC current is true but :
For DC you could say that at the moment the dc current starts flowing, the current experiences impedance and not resistance. Experiencing resistance only happens after the inductance is saturated. Pulsed DC is similar as AC without the reversal of the direction of the current(in reality some series inductance may produce some EMK that has of course the opposite polarity) .

 

[Denbo1991]

Experiencing resistance only happens after the inductance is saturated.

This doesn't make very much sense, because the impedance of a resistor isn't zero even during the switch on for a DC source. It is simply dwarfed in comparison to the voltage drop over the inductor during the short period when the voltage source is turned on.

If what you are saying was true, then only the inductance value could change the time it takes for LR circuit to reach steady state, whereas in the real world both the inductance of the inductor and the resistance of the resistor affect the time it takes to reach steady state.

I dont see how a system could possibly just ignore the resistive elements, regardless of the type of source (AC, DC, pulsed DC, what have you)

 

[William Gaatjes]

This doesn't make very much sense, because the impedance of a resistor isn't zero even during the switch on for a DC source. It is simply dwarfed in comparison to the voltage drop over the inductor during the short period when the voltage source is turned on.

If what you are saying was true, then only the inductance value could change the time it takes for LR circuit to reach steady state, whereas in the real world both the inductance of the inductor and the resistance of the resistor affect the time it takes to reach steady state.

I dont see how a system could possibly just ignore the resistive elements, regardless of the type of source (AC, DC, pulsed DC, what have you)

You are right when reading it back, i typed it wrong.
The bold part of your text is what i wanted to write.

 

[Mark R]

The current does not seek the path of least impedance, as you suggest. Instead, the current density is distributed in inverse proportion to the impedance. Current will flow via all paths, but the quantity may be markedly disparate.

Apropos planes and eddy currents. You are correct that the contribution of eddy currents is related to the aspect ratio of the conductor. In a plane, there is a reduced loop area for the currents leading to an effectively higher resistance for the EMF generated by the leakage inductance. The eddy currents, however, provide a net effect of resistance - not inductance (the impedance they provide does not lie in the complex plane, and they dissipate real power).

 

[William Gaatjes]

The current does not seek the path of least impedance, as you suggest. Instead, the current density is distributed in inverse proportion to the impedance. Current will flow via all paths, but the quantity may be markedly disparate.

This is true. But IIRC when thinking of generating EMI or EMC one can state that the optimum path is the path of least inductance and interference with other currents.

Apropos planes and eddy currents. You are correct that the contribution of eddy currents is related to the aspect ratio of the conductor. In a plane, there is a reduced loop area for the currents leading to an effectively higher resistance for the EMF generated by the leakage inductance. The eddy currents, however, provide a net effect of resistance - not inductance (the impedance they provide does not lie in the complex plane, and they dissipate real power).

That is the counter intuitive part of eddy currents. Because a copper plane has a large cross section area, the resistance should be lower when compared to a thin trace. I assume that in effect it still is even when the eddy currents consume some of the energy. I do know that there is also the skin effect to be taken into account which the eddy currents are the cause of it seems. These eddy currents force the main ac current to flow in the area near the surface, repel the main current to the surface. Thus a copper plane compared to a thin wire has more surface area and thus the main current encounter less eddy currents and thus less inductance. (The eddy currents in a typical thin wire like trace would just counter the the main ac current and the eddy currents themselves because of being shorted emk loops turn the energy into heat because of indeed resistance effects.) What is strange is that all these eddy currents are not aligned naturally. Must be maybe because of the constant outer bombardment of EM waves ?

At the moment i am working so i cannot really think that much about it. I have to think the theory over when i can.

 

[William Gaatjes]

I forgot to add, that it is not really the eddy currents forcing the main current to flow closer to the outer surface... I think it is more that the main current and the combined effects of the eddy currents(and the eddy currents repelling and attracting each other) try to avoid each other because they repel each other because of the resulting opposing magnetic fields. It may be very well similar as the way electrical currents flow in a liquid or a plasma.

 

[William Gaatjes]

Although my former statement about current flow only holds up when the lattice of the atoms does not restrict the movement of electrons. Which IIRC is the case in a solid and less to none in a liquid and not at all in a plasma.

 

[Onceler]

I always thought since electrons repel each other that the path of least resistance becomes the path of most resistance (freak lightning stikes where a woman gets struck 100 feet away from a flagpole).

 

[William Gaatjes]

I think it is a matter of dimension. Electrons do repel each other. But when a strong enough electric field is present, then the repulsion of the electrons can be overruled. The polywell, an experimental fusion reactor (by the late Robert Bussard and group) that functions on the principle that a large clustering of electrons will attract positive ions. The polywell creates a virtual (negative) cathode in the center of the polywell by forcing electrons to the center by use of strong focused electric fields and magnetic fields. This cathode then attracts positive ions. And these ions can when all variables meet fuse together.

 

[William Gaatjes]

Another side note is :

IIRC :
Copper has lower resistance then aluminum. But aluminum has more free electrons than copper. Copper has more atoms in the lattice then aluminum.

That explains why copper is more heavy then aluminum. But because the electrons in copper encounter less repulsion effects from other free electrons when compared to aluminum, the electrons can flow more freely in copper. Here the balance is between the lattice shape, the amount of free (valence ?) electrons and the amount of atoms for a given dimension. Dimension meaning here for example 1mm * 1mm * 1mm.

 

[disappoint]

I always thought since electrons repel each other that the path of least resistance becomes the path of most resistance (freak lightning stikes where a woman gets struck 100 feet away from a flagpole).

At such high potential differences even the air is essentially a conductor.

 

[disappoint]

IIRC DC current flows. AC currents only pass energy on.

I always understood that both "pass energy on" as you put it. In other words, the electrons themselves move very slowly even in a DC circuit. It's the electric field that "passes the energy on". So the current is really the electric field.

 

[William Gaatjes]

I always understood that both "pass energy on" as you put it. In other words, the electrons themselves move very slowly even in a DC circuit. It's the electric field that "passes the energy on". So the current is really the electric field.

That is interesting. Especially when assuming as Mark R mentioned, the current flows through all possible paths more here , less there (The current density). Only some paths carry more "current" and others less. Thus locally inside the conductor there are many different electric fields. And to add to the confusion also magnetic fields. The most interesting part is then again, what happens prior before the current flows ? What happens at the moment the current starts flowing. And what happens when the current finally flows ?

Which makes me wonder about the skin effect, the depth or the current distribution in the conductor and at what speed a DC current is passed on from the outside of the conductor to the inside. This would all depend on local electric fields and magnetic fields inside the conductor. And determines the depth the signal can penetrate at a given frequency.

It also made me think of something else. When i have to shield something from magnetic fields or field lines, i cannot block them. I have to create another route to make sure the magnetic field lines do not penetrate for example an electronic circuit that would be influenced by those field lines...