Recently I have been simulating the time domain response of the Celani replication cells that are being tested by the MFMP and am getting excellent results. My purpose is to detect any excess power that may emanate from the cells as the temperature sweeps upwards due to power increases. I suspect that the time domain behavior will be dramatically different if additional power is effectively added to the drive as a result of excess power generation.
My hypothesis is yet to be confirmed and I felt like it was time to reveal my process for others to share the fun and help with the large amount of data that is coming available. I hope to publish several posts as time permits that will allow other interested parties to duplicate my system and perhaps find answers to a few questions that remain. For this particular description I am using Excel, in others I have used LTSpice and the results agree to a reasonable level of confidence. I also, generated a simple numerical integration procedure which also matched the others. Three different techniques that demonstrate very similar results must be good for something. And, they fall closely to the calibration data on the MFMP site. The technique to be described is the solution to a non linear differential equation of the following form: dx/(a*x^2+b*x+c)=dt . The closed form solution to this equation is in the form of an exponential series similar to this: x(t)=k1 + k2*(e^(w*t+c1) + (e^(w*t+c1))^2 + (e^(w*t+c1))^3 +.....). This mess can be simplified to an equation of this sort: x(t)=k3 + k4*(e^(w*t+c1))/(1-e^(w*t+c1)) which is how I enter the information into my Excel chart. Notice that there are two pieces to the solution, one is a constant and the second one is the transient behavior. The constant is that actual final value that the temperature reaches if a very long time is applied to the system allowing the transient portion to approach zero. The second part fades away with time, and as a result will follow a certain path that can be carefully matched to the real world revealing any excess heating. I should mention that I have been forced to add a short lived transient to the equation solution to counter the initial edge delay which does not appear normally with the exponential expression. On some of my testing, I have just waited out the time required for this transient to pass so that I do not have to include it. The time constant is typically 30 to 40 seconds, but depends upon the cell design. It would be a good exercise for others to speculate as to what causes this delay. My current thoughts are that it is the time required for a pulse of heat added to the cell to mix within the gas and finally impact the outside glass. To replicate my file, you need to find values for the constants of the equation: P(T)=a*T^2+b*T+c. Where P(T) is power as a function of the Outside Glass Temperature(T) of the cell. These coefficients can be easily determined by analyzing a calibration run for the cell of interest. I obtain the data from the live feed of the MFMP site averaged over a 30 second window. They usually start with a step of very nearly 0 watts and the associated temperature at the outer glass wall and then step the input power upwards until they reach about 103 watts. This procedure yields several steps and thus pairs of values that can then be placed into a X-Y chart. I take my readings generally from the second to last constant input power point proceeding a power step upwards. I like to use the individual point plot for the X-Y plot so that it is clear where the trend line intercepts these for an indication of a good curve match. Open the chart modification box with the trend line options showing and select polynomial of second order. Choose to show the equation on your chart as well as the R^2 value for confirmation. I edit the trend line label so that my numbers are displayed with 6 or more decimal places. The values you desire are shown associated with the quadratic equation displayed on the chart. I call these a,b, and c which are modified by the next process that will be given on a following post. If there are questions please do not hesitate to post them as I will do my best to answer. Once you apply this technique, you will be amazed at the closeness with which the theory follows the real world results. On many tests, I find it difficult to detect an indication of the underlying transient curve that is many times greater than the noise surrounding the ideal response. Dave
