On 20.06.2014, at 11:11, Tim Goetze <t...@quitte.de> wrote:
> I
> mistakenly thought you were proposing nodal analysis including also
> the nonlinear aspects of the circuit including valves and output
> transformer (which without being too familiar with the method I
> believe to lead to a system of equations that's a lot more complicated
> to solve).
Well, it might not be as straight forward, but there are various methods to
solve non-linear systems of equations.
The main problem arises when a non-linear element occurrs on both sides of an
equation, such as
Vout = g * ( tanh( Vin ) - tanh( Vout ) ) + iceq
As one can't solve this for Vout, there's a few different approaches to deal
with it computationally.
The most common one is to introduce a unit delay (aka the "naive method")
Vout[ n ] = g * ( tanh( Vin[ n ] ) - tanh( Vout[ n - 1 ] ) ) + iceq
This however leads to mangled phase/frequency response and worse time invariant
properties, e.g. when sweeping g
A quick and easy method to resolve this is to only apply the unit delay to the
effect of the non-linearity itself, but keep the system delay-less for its
linear components:
Vout = g * ( tanh( Vin ) - f[ n-1 ] * Vout ) + iceq
f[ n ] = tanh( Vout )/Vout;
This way the equation can be solved for Vout and the non-linearity becomes a
factor that's applied with one sample delay.
Similar methods apply an offset and a tangent to linearize the non-linearity
for a moment in time.
So far this is computationally easy going, but it's also a compromise in
accuracy. To become numerically accurate one typically uses iterative methods
to "look into the future". A very simple approach is to estimate the value in
question, calculate the equation and compare the estimate with the result. From
there on one can refine the estimate until it converges with the result in any
desired accuracy, such as
while ( abs( Vout_result - Vout_estimate ) > 0.0001 )
{
Vout_estimate = rootFindingAlgorithmOfChoice( Vout_result );
Vout_result = g * ( tanh( Vin ) - tanh( Vout_estimate ) ) + iceq
}
There you go. It's very basic and it can be optimised to run on a modern CPU.
Cheers,
- Urs
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