The data has been flooding in from the MFMP and I have been seeking a time domain model of the system behavior when power to the Celani replication device is modified. Most of my effort has been exerted by analyzing the rising edge of the time domain waveform when the drive power is stepped up by a significant amount. The temperature follows a certain path as it ramps up to the value required to balance the input and output power of the cell.
We have been fortunate in this particular case to find that the relationship between temperature and input power is well behaved and follows a second order curve to a remarkable degree. It is not uncommon to see a curve fit with R^2=.9999 or better in many independent test runs. I initially was expecting to see a power series that included a forth order term of significance due to the S-B radiation equation. This has not ever been dominate in any test and I still am trying to understand why this is true. For the time being I will accept this gift happily. A quick glance at the shape of the rising edge of the temperature curve suggests that it follows an exponential. I thus began my model by making that assumption and got fairly reasonable results. It was always evident that my curve fit contained holes, but a couple of degrees of error did not seem too excessive at that time. Being a perfectionist, I decided to improve the situation and to determine how well a model could match the real life test. I very soon added a second exponential to the mix and noticed that the fit improved remarkably. Also, I noticed that the second real frequency was close to the second harmonic of the first one determined by my earlier work. A light went off inside my head and I realized that this would be expected since the non linearity is mainly of second order in the relationship between variables. Now, I saw that the accuracy of my model was becoming very acceptable. There remained a short period of time at the initial power increase where the fit was not as good as I hoped. To fix this problem I added another exponential with an associated time constant of about 40 seconds. With this model, I could obtain an excellent match between my simulation and the real world data. I could have left it in this state, but it is hard to accept imperfection. To pursue the matter further I used a LTSpice model of the system. I guessed correctly in my first try with the model and was rewarded with a well behaved simulation that included the second order distortion effects. This model was used for a significant time as it matched the real world waveforms everywhere except for the initial short period that required another time constant to fix. Looking at my spice model gave me an interesting idea. I used a capacitor to represent storage of the incoming energy and the node it is connected to reads expected time domain temperature for the outside glass sensor. In parallel with the storage capacitor is a pair of current sources, one representing power applied to the cell, the other power being taken away by the various paths. The draining current source appears as a parallel conductance who's value depends upon the voltage at the temperature node. I, of course, was seeking verification of the time constant associated with the exponential rise waveforms and attempted to use the effective conductance value in parallel with my storage capacitor for a quick check. This lead to the non linear differential equation definition that works so well. It occurred to me that my model could be expressed in the form of a non linear differential equation with a little manipulation of the shape. Basically you have a parallel capacitor being driven by a current source that is paralleled by a non linear conductance. The non linear conductance is neatly defined by the second order equation derived from the calibration runs for the Celani cell. Now, all I had to do was to solve the non linear differential equation that I constructed and insert the initial conditions to define the temperature and power over any time frame. My first thought was yipes! I consulted our favorite source wikipedia to find the solution to unusual integrals. The one I needed to solve was in the form of: Integral dx/(a*x^2+b*x+c) with initial condition of the temperature of the steady state value just prior to the application of an increase in power. I transformed the time scale so that time = 0 was with this application of extra power. It turns out that there is an exact solution to such an equation which you can look up at your convenience to save time and space here. I had to perform some interesting series adjustments to get the curve within the desired temperature band, and I was a bit rusty at first. Finally, a perfect curve was being generated that matched the time domain data extremely well except for that nagging time region at the very start. I continue to have to include an additional exponentially rising pulse function with a time constant around 40 seconds at the application of the extra drive to get virtually perfect tracking to the real world data. Next, I included another drive current waveform of this nature to my spice model and it tweaks the start of the rising edge a tiny amount much like a delay. I am still seeking a good explanation for the necessity of this extra pulse source and I wonder if it can be traced to the IR effects or some other relatively large time constant such as the glass heating. The nature of the extra leading edge drive pulse can be described as a signal that begins at a certain level and decays exponentially to zero with a time constant of 40 seconds. The effective DC component of the waveform is taken out by the action of the non linear conductance. One interesting observation is that the calibration determined a,b, and c that constitute coefficients of the second order equation defining Power versus Outside Glass Temperature along with the input power uniquely determine the steady state temperature of the device. These four variables define operation over the entire range of input powers. My model also includes a capacitor that acts as the energy storage stand in. One good temperature rising transition allows me to choose the correct capacitor to enter into the model. The additional short time exponential must be determined by curve fitting within a short initial period typically 100 seconds. I have found this exercise interesting and educational. If a good explanation for that initial power pulse is obtained I can relax and fool with the incoming data. I am hoping that my contributions will enable us to discover any excess power that may occur by its signature outside of the normal that I now model and observe. Dave
