Back calculation to lumped RC values

terence1212

Joined Nov 19, 2008
2
Hi all,

I have two sets time aligned of data. The first we can call the "step" the second the "step response". The step in reality represents the applied change in temperature of a hotplate, from ambient to a higher value over a few minutes - therefore an fairly "soft" step. The step response is the decreasing resistance of a calibrated NTC thermister on a PCB on top of the hotplate. I want to model the step response as an RC circuit in order characterize the thermal conduction properties of the PCB. How do I compute these two data sets to give me estimates of R and C?

Any help will be much appreciated. I'm a very poor mathematician. Is there a program that will churn the numbers and pop out the right answer!? (well...I can hope).

Terence

KL7AJ

Joined Nov 4, 2008
2,208
You can probably come pretty close by substituting the thermal mass with an RC time constant. 1 farad discharging through a resistance of 1 ohm will decay to 63% of its original value every second. Since thermal systems are also exponential (asymptotic) you can probably get by with the natural log function, as a first approximation. To get a "endpoint" to this, you need to find out how long the thing takes to get to 99% of its "settled" value after you change the input.

Eric

terence1212

Joined Nov 19, 2008
2
Thank-you, Eric. I was was starting to worry that my question was too dumb for consideration. You have provided a useful stating point!

-Terence

markmain

Joined Dec 28, 2008
13
For an RC circuit, I wanted a simple formula to calculate how many seconds it will take to charge a capacitor starting from an initial Low Voltage mark (L) to a target High Voltage mark (H); and then to do the reverse by discharging from H to L.

I was not able to find something that laid this out easily and so I came up with this:

**Note: "LN" in the formula represents the Excel Function "LN", which returns the Natural Logarithm of a number; it's the inverse of the EXP function that is used for calculating e raised to an exponent--which is used in the Universal Time Constant Formula that we often see

V=Source Voltage
L=Low Voltage Level (initial value on charge and target value on discharge)
H=High Voltage Level (target value on charge and initial value on discharge)
R=Resistor Ohms
C=Total Capacitor Farads if Fully Charged
S=Seconds To Charge from Zero to Reach the Target Voltage (the Charge Time)
D=Seconds to Discharge from Target Voltage to Zero (the Discharge Time)

S=R*C*LN(1/(1-((H-L)/(L-V))))
D=R*C*LN(1/((H-L)/(L-V)))

If the Low Voltage Level (L) is going to be Zero, then the formula can be simplified to:

S=R*C*LN(1/(1-(H/V)))
D=R*C*LN(1/(H/V))

Using the source voltage (V), charging from Zero volts for (S) seconds, the voltage level of the capacitor will be (X) volts; here is the formula that I used to determine what the X volts would be (and I used it to confirm my formulas above:

X=V*(1-EXP(-S/(C*R)))

For discharging, starting from full capacity (C), and discharging for (S) seconds, I used this formula to determine what the (X) volts would be after the discharge:

X=V*EXP(-S/(C*R))

If anyone spots an error in my formulas let me know, but I believe these look good. I just wasn't finding something written down in a format that I could easily transfer to Microsoft Excel.

This link was a big help to me.