# Equation to calculate voltage divider resistor value for an input of op-amp

I found this guide for a temperature sensing circuit design with NTC thermistor by Texas Instruments. According to my opinion, optimum R1 value is calculated so that the voltage divider can provide the widest voltage variation at its output for the given resistance range of the NTC thermistor.

In the Design Step 1 they have used this equation to calculate R1 of the voltage divider. Here minimum and maximum values of the resistance range of the NTC thermistor are used.

Can this equation always be used as a general equation when designing voltage dividers for similar applications? Is this equation is a standard one and how is it derived, at what situations this equation should not be used?

• This equation will give you the maximum voltage change (swing) at the output of a voltage divider. – G36 May 6 '20 at 19:31
• Interesting! I'm trying to think how to prove it. – Transistor May 6 '20 at 19:47
• This may not be optimum if self-heating errors are considered. Maybe a couple °C error with the given circuit (assuming still air). – Spehro Pefhany May 6 '20 at 19:51
• Are you sure that equation isn't to give you a centred swing between the temperatures of 25 and 50? Normally with NTCs, the trick is to have the series resistor equal to the NTC at the middle of the range you want to measure, that gives you maximum swing for your range. Due to the fortunately compensating non-linearities of the NTC and the voltage divider, +/- 5C range will be essentially linear voltage with temperature, and +/- 10C will still be usably close to linear. – Neil_UK May 6 '20 at 19:53

You can prove this by writing an equation for the difference of the two voltages and setting the derivative to zero.

$$\\frac{V_{DIFF}}{V_{CC}}=\frac{R_X}{R_1+R_X} - \frac{R_X}{R_2+R_X} \$$

Differentiate (and set to zero) and you get: $$\\frac{R_1}{(R_1+R_X)^2} - \frac{R_2}{(R_2+R_X)^2} = 0 \$$

Simplify and you get Rx = $$\\sqrt{R_1 \cdot R_2}\$$

Given the fixed percentage change of thermistors per degree this will also maximize the voltage change per degree at the extremes (making it equally bad at each end, compared to best which occurs at the center of the resistance range- when the thermistor resistance equals the series resistor).

Edit: The principle behind setting the derivative to zero is that this will detect maxima (or minima). You have to apply some thought to ensure you have a maxima and to eliminate the constraints (if any) as potential points. This is a standard technique.

As to the calculation, the quotient rule makes easy work of differentiating wrt Rx and the rest is just algebra.

• It took me some time to figure out the calculation. Further explanation is welcome for the sake of fast digestion. Anyway, great answer. – Yasindu May 7 '20 at 5:54