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Tue Oct 21 09:35:13 EDT 2014

Circuit sim, a-symmetry of static NL transformation

Basic idea: add inertia to a static nonlinearity to make simulation
simpler.


                 Ic>
             /---||---\
             |    C   |
       L     |        |
V1 ---OOOO---o--|>|---o--- V0
    I>       Vx   Id>

I = Id + Ic
L DI = V1 - V0
Ic = C DVx
Id = f(Vx-V0)

6 variables, 4 equations -> 2 deg freedom

Compare to 3, 1 -> 2  (V1,V0,I) when L,C absent



So this transformation shields the diode from voltage transients on
V1,V0.

BUT

This introduces a-symmetry: the resulting 2-terminal can no longer be
driven by a current source.

Is that a problem?
And why is this a-symmetry choice so natural?


( There's a deeper philosophical questions: why this a-symmetry in
thinking about impedance?  V and I are dual.  Why do we prefer voltage
sources? )


When making one side of the V/I dual, one needs to ask which is the
other.  What is the V/I dual circuit of this? Is it relevant?

How does this relate to I=f(V) V=f^-1(I) duality for the nonlinear
element?


Start with the linear approximation of Id = Is e(Vx/VT)

   Id/Is = e(Vx0/VT) (1 + [Vx-Vx0]/VT)

Here Vx0 is the value from the previous time step.  This approximation
is based on the assumption that Vx0 ~ Vx over the range of the update,
which for a diode is the right thing.

This also gives a retrofitted reason for a-symmetry:
at small time scales, a diode behaves as a voltage source.


PROBLEM SETTING:

Solving a circuit differential equation can be separated in "space"
and "time" steps.

- Space step satisfies static relations at a specific time, based on
  the current state of energy storage devices (modeled as current or
  voltage sources).

- The time step approximates computes time evolution of energy storage
  state based on the static update calculated in the previous step.

In this approach, the time update step is just function evalutation,
i.e. it is "local", while the temporal step involves the solution of a
system of equations, i.e. it is "global".

Nonlinearities that include in the time (local) step are easy to
handle, but those that occur in the spatial (global) step require a
separate solution strategy: from linear to non-linear sets of
equations.

The idea then is to transform non-linearities from the spatial to the
temporal domain by surrounding them by energy storage devices,
essentially "low-pass filtering" the approximation errors.

This is done in such a way that the functioning of the system is not
compromised, i.e. the time constants added would be present in the
circuit anyway in parasitic amounts.


CONCLUSION:

( intuitive.. quantify this )

- a nonliearity can be "low-pass shielded", but this intruduces
  a-symmetry: is inertia added to voltage or current?

- express the static non-linearity in terms of the "slowed down"
  variable.  ( in the case above it would be Vx )




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