Claus Turtur's proposal for a capacitor with plates connected by springs and resonantly driven is presented. It is found that a mistake was made in Turtur's model, where the charge on the capacitor was not updated with each time step, only the initial value was used. When the correct time-varying value is used in a revised model, no energy is generated by the system.

1. Sudy of Prof. Turtur’s resonant circuit simulation
Reference:
http://www.ostfalia.de/export/sites/default/de/pws/turtur/DownloadVerzeichnis/Series-english-5Articles.pdf
On pdf-pages 48-68 of the referenced document, a system is described where an LCR circuit includes a capacitor
whose plates are allowed to vibrate on a spring which separates the plates. It is claimed that this can be arranged
to produce an excess of energy which may be endlessly extracted to a load resistor.
However, there is a critical mistake in the simulation code on pdf-page 68
Fc:=-Q[0]*Q[0]/4/pi/epo/(2*x[i-1])/(2*x[i-1]); {Coulomb- force}
This calculates the force between the capacitor plates using only the initial charge Q[0], and does not update the
charge at each step. At the very least, the program should be modified to read
Fc:=-Q[i-1]*Q[i-1]/4/pi/epo/(2*x[i-1])/(2*x[i-1]); {Coulomb- force}
The following presentation shows the implications of these modifications.

2. Replication of main result
Fc:=-Q[0]*Q[0]/4/pi/epo/(2*x[i-1])/(2*x[i-1]); {Coulomb- force}
Original figure, copied from pdf-page 56 Figure calculated using my implementation for the same
parameters
Using just the initial charge on the plates to calculate the force between the plates does indeed
give a non-conservative result. But this cannot be correct, because the charge on the plates
varies in time.

3. Updating the charge on the capacitor plates
Fc:=-Q[i-1]*Q[i-1]/4/pi/epo/(2*x[i-1])/(2*x[i-1]); {Coulomb- force}
Turtur’s finite difference implementation My finite difference implementation
If the charge on the plates is allowed to change, the behaviour is very different. For the same initial
conditions, the displacement of the capacitor plates increases until the plates touch.
There is a slight difference between the two implementations, but I would consider this ‘minor’
compared to the difference between letting the charge vary or keeping the charge fixed.
I will show where this discrepancy comes from in the next two slides.

4. Error analysis
Turtur’s finite difference implementation My finite difference implementation
Once a solution is simulated, we can check how good that solution is. There are two homogeneous equations that need to be
checked, ‘L d2Q/dt2 + R dQ/dt + 1/C Q = 0’ and ‘m d2x/dt2 + D*(x-CD/2) + (1/4*pi*epo)*Q^2/((2*x)^2) = 0’.
The solution found for each of the three terms of the equations are plotted above, and the sum, which should be zero, is
plotted as the thick black line.
Both implementations have the thick black line close to zero in comparison to the individual terms, so both solutions seem
good approximations.

5. Error analysis
In these plots, just the errors are plotted (the thick black lines on previous plots).
Blue lines are for errors in Prof. Turtur’s implementation, green are for errors in my implementation.
We can see that in the charge homogeneous equation that Turtur has a residual value in the solution. This appears
mathematically as if there is a driving voltage of this amplitude on the circuit, and so will alter the currents and charges
away from the ideal solution. Therefore I estimate that my implementation is more accurate. However, this is not the main
issue of this document.

6. Choosing values of spring constant
By ‘playing’ with the parameters, one can get many varied and interesting trajectories of position and
charge over time, but I couldn’t find any which give ‘free’ energy. Whenever one amplitude increases,
the other one decreases.
Is there a ‘sweet’ set of conditions which can be found to increase both amplitudes together???

7. Other typos in the paper
There were several other minor issues with the paper, but none as serious as the main issue outlined above.
Some of these were:
where the sign of the Coulomb term should be negative.
In the equation
and the previous similar equations, the inductive term should be positive. However, both of the above ‘typos’ had the
correct sign in the simulation code.
Comparison of the capacitance equation and the Coulomb force equation shows that the capacitance includes
relative permittivity but the Coulomb force doesn’t. The simplest solution here is to have a relative permittivity equal
to unity, as it is not trival to calculate the Coulomb force when some of the space contains dielectric and some
doesn’t.
In the equation
only half the spring length and extension are used. This doesn’t change the physics, but it does change the
apparent value of spring constant ‘D’ if one wants to compare to a real-life construction, even though the plate
movements are symmetrical about the origin.
In the implementation of the finite difference equations, residual errors can creep in due to the way the terms feed-
forward. For an alternative implementation, see the code in ‘BRrescapsim_v2p0.m’ for example.

8. Summary
The main conclusion of the paper is found to be in error. This is due to usage of Q[0] as a fixed
charge on the capacitor plates which does not get updated as the simulation proceeds.
Correcting this to Q[i-1] gives a simulation which behaves like standard coupled oscillators.
Can such a situation be ‘tweaked’ to give the free energy that the author suggests? I would like to see
how!