Re: [Phys-L] inductors cutset and condenser loops, previous and initial conditions.

[John Denker <jsd@av8n.com>]

Executive summary:  Skip to "constructive suggestion" below.

Longer version: on 08/04/2015 05:28 PM, Diego Saravia wrote:

> If you say in previous conditions, that in an inductor cutset the sum of
> currents is different from zero, you will be in contradiction to the
> current Kirchoff law.

That's an overly-complicated yet less-than-precise way of
expressing a simple idea.

For starters, keep in mind that Kirchoff's «laws» are not
exact;  they describe the asymptotic behavior in certain
limits, subject to certain idealizations.  When in doubt,
rely on the Maxwell equations instead.

Conservation of charge can be seen as a corollary of the 
Maxwell equations ... and/or as a fundamental principle
unto itself.

   Homework:  Prove the corollary.

For any given circuit, suppose you divide the circuit 
into two pieces.  Sum(*) up all the current crossing the
boundary.  If this is nonzero, then charge is accumulating
on one side of the boundary (and opposite charge is
accumulating on the other side).  That's the basic
fact.  Some corollaries follow:
  1) As a matter of principle, this *can* happen.
   Charge can flow, and it can accumulate.  Evidence:
     https://www.youtube.com/watch?v=9tzga6qAaBA&t=0m52s
  2) As a matter of practical engineering, if you 
   assume the self-capacitance of the circuit is 
   tiny, this leads to a transient that is so brief
   that you might not notice it.  For a typical capacitor,
   the self-capacitance of each plate is tiny compared
   to the mutual capacitance between the plates ...
   but this is true because you engineered it to be so,
   not because of any law of nature.
  3) According to Kirchhoff's «law», it is simply not
   possible to have current flowing with no return path.
   This approximates reality in the DC limit.

This cannot easily be expressed in terms of a cut-set,
because the aforementioned sum(*) requires you to keep 
track of the /direction/ of current in each contribution
to the sum, whereas simply "cutting" a current-path does 
not care about direction.
  http://mathworld.wolfram.com/CutSet.html

===========================

> My problem is with "previous" and "initial" conditions.

> Perhaps the solution is as the simulation program suggests:  undetermined,

That's a completely different question AFAICT.

When designing a circuit, especially after cutting a 
circuit, it is possible to create some /floating nodes/.
The simplest example is an ideal capacitor with leg A
tied to ground and leg B open-circuited.  The amount 
of charge on the B side is entirely determined by the
initial conditions.  As another way of saying the same
thing, there is a "startup transient" that lasts forever.

In the real world, there will always be some leakage
from A to B.  On the other hand, the parasitic RC time
constant could be on the order of years.

In the real world, there do exist specialized circuits
with floating nodes.  They can be used as memory circuits.
This soon becomes a subject for the advanced engineering 
course, not the introductory physics course.

Ordinary less-specialized circuits are designed to not 
have floating nodes, and more generally to not have
ultra-long startup transients.  Simulation programs 
have a choice:  They can
 a) provide mechanisms to specify the initial conditions
   on each floating node, or
 b) assume such things do not exist, or
 c) notice such things and complain about them, on the 
  grounds that they are probably a mistake.

Constructive suggestion:
Assuming we are not in situation (a), the time-honored
way to proceed is to provide your *own* initialization
mechanisms.  That is, tie a switch and a voltage source
to each floating node.  Now it's not floating anymore.
Some time after the simulation has started, open the
switch.

   This reflects real-world engineering anyway:  In
   practice there should be some way of initializing
   the floating node, even if the initialization was
   years in the past.

Of course duality applies:  Anything you can say about
floating nodes with charge on them can also be said
about isolated loops with current in them.

==========

If you have read this far, you might also be interested in:

 *) Basic notions of what we mean by "voltage"
   https://www.av8n.com/physics/voltage-intro.htm

 *) A non-nonsense discussion of Kirchhoff's «laws»
   https://www.av8n.com/physics/kirchhoff-circuit-laws.htm

 *) The definition of capacitance, including the 
  capacitance matrix for multi-terminal capacitors:
   https://www.av8n.com/physics/capacitance.htm

 *) Spreadsheet methods for solving Laplace's equation,
  including the capacitance of oddly-shaped multi-terminal
  capacitors:
    https://www.av8n.com/physics/laplace.htm

 *) A discussion of the notorious Two-Capacitor Problem:
   https://www.av8n.com/physics/capacitor-transfer.htm

 *) A modern view of the Maxwell equations:
   https://www.av8n.com/physics/maxwell-ga.htm