RMS Power

[sreid]

I need some conformtion of a power calculation.

If a resistor [R] is droven by a sinusoidal current source that at time zero is

i = Asin[wt]

and "A" decreases to zero linearly, what is the RMS Power.

This is not a home work assignment, this is an approximation of the current waveform I'm driving into a Voice Coil.

 

[VEBill]

Forgot the ;-) !!

 

[electricpete]

There's a duplicate in there - that only counts once!

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[sreid]

Thank you for all your help and suggestions. I'm going to be very conservative since while "Speed is King," burning up Voice Coils is verboten. 250 Watts continous gets me to an average winding temperature of 155 deg C. This was determined by injecting a fixed DC current into the voice coil and measuring the voltage drop across the coil. At 155 deg C, the voltage increases 50% [or the resistance increases by 50%]. Roughly 6 amps RMS at 7.5 Ohms [5 Ohms cold].

But I wanted to know a "Fudge Factor" I might use based on a linear current taper from OD to ID if management wanted "Just a little more" which, of course, they always do.

 

[electricpete]

250 Watts continuous gets me to an average winding temperature of 155 deg C.

IF we assume
1 - this is the temperature in a steady state temperautre
2 - your ambient temperature is 40C
THEN we can calculate Rth = (155-40) C/ 250 watts = 0.46 degC / watt
(hey - I wasn't too far off with 0.2 degC / watt!)

But we have no idea your time constant. If you apply DC current as a step increase from 0, how much time does it take for temperature to get 63% of the way from it's initial value to final-steady-state value? That would be the time constant. That info along with the ramp time, frequency, initial temperature (before the ramp-down) and ambient temperature would complete the inputs required for the spreadsheet.

To sharpen the pencil some more, one could also program the temperature coefficient of resistance into the model.

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[electricpete]

Here is revised spreadsheet that includes correction of resistance for temperature and a few miscellaneous format items cleaned up as shown in the new "instructions" tab.

We start with three thermal unknowns :Rth, Cth, Tau.
We have two equations:
1 - Tau = Rth, Cth
2 – Rth = 0.46 degC

We need one more equation or unknown value to solve the thermal parameters. Two possibilities to get there:

1 - As discussed above , Tau could be from step change in current (or as close as you can get with an inductor).
or...
2 - What might be easier is to estimate Cth. We can guesstimate that if you tell us the mass of materials: how much mass of copper (and iron?...not sure if voice coil has a core) heat up to this temperature.


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[Skogsgurra]

I need to clarify a few things, see posting 20 Apr 09 0:16.

I do not skim to find something to criticize. I take part in threads when I think that I have something to contribute. I think that long term members of Eng-Tips realize that.

The two "crimes" that I seem to have committed are that I used the words "RMS Power" and that I wrote "High" instead of "Low".

The first crime is an adaption to the sloppy usage of the term "RMS Power" in the thread heading and to the words used in the question I was answering. If hat is a terrible crime, I think it would have been fair to point that out long before I happened to use the words.

The second crime - well, I guess I am guilty. Slip of mind. I actually realized that just after posting, but didn't bother to post a correction. I guess I should have done that. It is a habit that I perhaps should pick up.

Gunnar Englund

http://www.gke.org

--------------------------------------
100 % recycled posting: Electrons, ideas, finger-tips have been used over and over again...

 

[sreid]

The heat input is extremely complicated. The cutting RPM may be limited by the max Velocity, the max Acceleration and the max Jerk of the fast tool [voice coil driven]. And the lenses [limits] are in random order.

There are "N" Rough Cuts [done with a fast feed rate] followed by the slow finish cut.

There ia a dead time between [load-unload] on the lenses

From a cold start with the maximum current that gets me to 155 deg C, it takes 1.5 minutes to get to 63% of the final coil voltage. I did not test the cool down time as I already knew I was going to "Play it safe."

 

[electricpete]

I assume that your current waveform is something like an amplitude modulated waveform. The carrier frequency is at 50hz or higher. The envelope is slowly varying - steady in places, step change in places, ramps in places.

Since your thermal time constant (90 sec) is so much higher than the period of the carrier frequency (1/50hz or shorter), we can simplify the problem by ignoring the carrier frequency component and replacing the waveform with envelope (divided by sqrt2 to convert to 1-cycle-rms). That is of course the similar to what you already did to begin with when you replaced the ramp sinusoid with a ramp to calculate the factor 1/3. This would simplify any analytical solution and I think would also reduce the number of iterations required for numerical solution (numerical which would be my preference since it easily allows us to accmodate non-linearities such as temperature coefficient of resistance.... could also easily be adjusted to account for non-linearities in heat transfer characteristic if they are known).

If I get a chance this weekend, I will adjust the spreadsheet to add a tab where the user can easily specify an arbitrary input current waveform (which could be the full am waveform or the envelope /sqrt2).

There is now enough info to solve the thermal model (simply plug 90 sec into Tau). If you wanted to do some rough validation of the thermal model there are some some double-checks which can be done:
1 - provide the info on mass of copper and steel and calculate thermal capacitance from that to see if it matches the thermal capacitance provided from Cth = Tau/Rth
2 - if you happen to have plot of temperature vs time during your dc test (could be constructed from resistance vs time accounting for temp coefficient), it should be roughly a straight line on a log-log scale if you have a 1-degree of freedom thermal system (one mass and one temperature). If 2dof system, we might see two different slopes in different parts of the curve.
3 - Multiple dc tests of course provide more detailed info on the thermal model. The form of heat transfer out of the device is most likely Ith = dT^m / Rth
where Ith is heat transfer out (watts)
dT is temperature rise above ambinet
R is thermal parameter
m is unknown coefficient. would be near 1 for conduction heat transfer but can be in the range 0.1 - 0.2 for radiation heat transfer near room temperature, approximately 0.25 for laminar convective heat transfer and 0.33 turbulent convective heat transfer. If multiple heat transfer mechanisms exist thatn the overall effective value of m would be something like a weighted average of those numbers Comparing the final equilibrium temperautres for different values of heat input = heat output = Ith would allow us to estimate the exponent m.



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[electricpete]

I have been assuming only I^2*R losses are relevant. If it is an iron core device (is it?), then core losses may play a role also.

We don't know the frequency or wire size - I assume conductors are small enough and frequency low enough that skin effect / proximity effect are not significant.

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[electricpete]

Attached is a new version of the spreadsheet that allows you to put in an arbitrary current waveform.

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[electricpete]

The "max step" parameter was set to small in the last spreadsheet - makes the program do many more calculations than required. I suggest setting it up much higher (0.02) for slowly varying envelopes like the one shown.

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