Electronic – NTC thermistor series/parallel resistor calculation for MCP73833 linear Li-Ion charger

chargerthermistor

I am using MCP73833 which has the following datasheet
https://ww1.microchip.com/downloads/en/DeviceDoc/22005a.pdf

Internal diagram as follows:
enter image description here

They says that "An internal 50 μA current source provides the bias for
most common 10 kΩ negative-temperature coefficient
thermistors (NTC). The MCP73833/4 compares the
voltage at the THERM pin to factory set thersholds of
1.20V and 0.25V, typically"

Now the formulas shows that
enter image description here

For example, by utilizing a 10 kΩ at 25C NTC thermistor with a sensitivity index, β, of 3892, the charge temperature range can be set to 0C – 50C by placing a 1.54 kΩ resistor in series (RT1), and a 69.8 kΩ resistor in parallel (RT2) with the thermistor as depicted in Figure 6-1.

enter image description here

Now I just can't figure out how to properly calculate the resistors required series/parallel with NTC for desired window set, a more generic way based on RHOT and RCOLD (ie. for other range)

The best I came up with and I am stuck. Appreciate a little help.

1           1           1
----    =  ------ + --------
24-RT1      RT2       rcold  

1           1           1
----    =  ------ + --------
5-RT1      RT2       rhot

Best Answer

You're a good part of the way there. Simply subtract the two equations and you get

\$R_{T1}^2 -29R_{T1}+m = 0\$

where \$ m = 24\cdot5 - \frac{19}{1/R_{HOT}-1/{R_{COLD}}}\$ (units of k\$\Omega^2\$)

So \$R_{T1}= \frac{29 \pm \sqrt{29^2 -4m}}{2}\$ (standard solutions to a quadratic)

Pick the correct answer for RT1 (minus) and you can then subsitute RT1 back into either of the first two equations to get (say)

\$R_{T2} = \frac{1}{\frac{1}{5K-R_{T1}}- \frac{1}{R_{HOT}}}\$

(If you end with a negative answer for either resistor, you picked the wrong answer- assuming there is a solution for that thermistor).

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