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Analog modeling of a diode clipper (3a): Simulation

Now that we have a few methods, let’s try to simulate them. For both circuits, I’ll use the forward Euler, then backward Euler and trapezoidal approximations, then I will show the results of changing the start estimate and then finish by the Newton Raphson optimization. I haven’t checked (yet?) algorithms that don’t use the derivative like the bisection or Brent algorithm.

All graphs are done with a x4 oversampling (although I also tried x8, x16 and x32).

First diode clipper

Let’s start with the original equation:

Analog modeling of a diode clipper (3a): Simulation

Forward Euler

Let’s now figure out what to do with the integral by deriving the equation:

Analog modeling of a diode clipper (3a): Simulation

So now we have the standard form that can used the usual way. For the derivative of the input, I’ll always use the trapezoidal approximation, and then for the output one, I’ll use the forward Euler which leads to the “simple” equation:

Analog modeling of a diode clipper (3a): Simulation

Backward Euler

For the backward Euler, I’ll start from the integral equation again and remove the time dependency:

Analog modeling of a diode clipper (3a): Simulation

Now the discretization becomes:

Analog modeling of a diode clipper (3a): Simulation

I didn’t use this equation for the Backward Euler because I would have had a dependency in the sinh term, so I would still have required the numerical methods to solve the equation.

Trapezoidal rule

Here, we just need to change the discretization for a trapezoidal one:

Analog modeling of a diode clipper (3a): Simulation

Starting estimates

Starting from the different rules, we need to replace sinh(x) :

  • for the pivotal by Analog modeling of a diode clipper (3a): Simulation
  • for the tangent rule by Analog modeling of a diode clipper (3a): Simulation

Graphs

Let’s see now how all these optimizers compare (with an estimate of the next element being the last optimized value):

Analog modeling of a diode clipper (3a): Simulation
Numerical optimization comparison

Obviously, the Forward Euler method is definitely not good. Although is on average 4 times lower, the accuracy is definitely not good enough. On the other end, the other two methods give similar results (probably because I try to achieve a convergence quite strong, with less that 10e-8 difference between two iterations).

Now, how does the original estimate impact the results? I tried the Backward Euler to start, and the results are identical:

Analog modeling of a diode clipper (3a): Simulation
Original estimates comparison

To have a better picture, let’s turn down the number of iterations to 1 for all the estimates:

Analog modeling of a diode clipper (3a): Simulation
One step comparison

So all the estimates give a similar result. By comparing the number of iterations with the three estimates, the pivotal method gives the worst results, whereas the affine estimates lowers the number of iterations by one. Of course, there is a price to pay in the computation.

So the obvious choice is to use trapezoidal approximation with affine starting point estimate, which is not my default choice in SimpleOverdriveFilter .

To be continued

The post is getting longer than I thought, so let’s keep it there for now and the next post on the subject will tackle the other diode clipper circuit.

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