# Back calculation to lumped RC values

Discussion in 'Math' started by terence1212, Nov 19, 2008.

1. ### terence1212 Thread Starter New Member

Nov 19, 2008
2
0
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

2. ### KL7AJ AAC Fanatic!

Nov 4, 2008
2,047
295
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

3. ### terence1212 Thread Starter New Member

Nov 19, 2008
2
0
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

4. ### markmain Member

Dec 28, 2008
13
0
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.