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A printed bias resistor connected to an external NTC temperature sensor sets more than the nominal divider voltage. Its value determines how the available voltage span is distributed across temperature and how much electrical power heats the sensing element. The best value for a wide measurement interval need not be the best value near a control threshold. Specify the objective before selecting the resistor, and retain the sensor's actual resistance-temperature data throughout the calculation.
Key design decisions
- Use the resistance range of the selected external sensor, not a generic NTC label.
- Choose between endpoint span and sensitivity at a particular temperature before optimizing bias resistance.
- Check sensor dissipation and bias-resistor error after selecting a nominal transfer function.
Define the two components and their separate responsibilities
Let a fixed bias resistor Rb connect the excitation voltage Vb to the measurement node, and let the external NTC resistance Rt connect the node to return. Output is measured across the NTC. As its resistance falls with increasing temperature, this output falls. Reversing the two elements reverses that direction and changes which node the receiver observes.
The bias element may be part of a reviewed printed resistor network, while the thermistor has its own material, package and calibration information. Do not substitute an ordinary printed resistor's temperature coefficient for the thermistor curve. The design package should identify both elements independently, including where each is located thermally and which component's temperature the measurement is intended to report.
Keep the resistance-to-voltage transformation explicit
With negligible receiver loading, the normalized output y is Rt divided by Rb plus Rt. Recovering resistance from a measured ratio requires Rt equal to Rb y divided by one minus y. The conversion becomes sensitive to small ratio errors near either rail, so the useful temperature resolution is not established by the ADC bit count alone.
If excitation and converter reference share the same actual voltage at the relevant terminals, their common scale factor can cancel in the ideal normalized reading. That does not cancel bias-resistor variation, thermistor curve error or terminal-temperature differences. Retain those separate inputs rather than treating a ratiometric arrangement as a complete calibration.
y = Vout/Vb = Rt/(Rb + Rt); Rt = Rb y/(1 - y)
- Rt is the external NTC resistance in ohms at the sensing-element temperature.
- Rb is the actual bias resistance in ohms at its own operating temperature.
- y is the dimensionless settled output ratio, strictly between zero and one for finite positive resistances.
An ideal excitation and negligible receiver current. The real receiver's loading, leakage and sampling behavior must be assessed separately.
Maximize the span between two specified sensor resistances
Let Rc be the larger sensor resistance at the cold end and Rh the smaller resistance at the hot end, both positive. The normalized endpoint span is Rc/(Rb + Rc) minus Rh/(Rb + Rh). Differentiating with respect to the positive bias resistance gives the maximum at Rb equal to the square root of Rc times Rh. This is the geometric mean of the endpoint resistances, not their arithmetic mean and not necessarily the resistance at the middle temperature.
This result optimizes only the difference between two endpoint voltages. It says nothing about how evenly voltage changes between them. Two sensor curves with identical endpoint resistances but different shapes have the same span-optimal bias value while producing different temperature sensitivity inside the interval. Keep the intermediate resistance data when evaluating either one.
Compare three bias choices using the same endpoints
Assume an external sensor has endpoint resistances of 100 kilohms and 10 kilohms for the temperature interval under review. With 3.3 V excitation, the span-optimal bias is approximately 31.623 kilohms. Its cold and hot outputs are approximately 2.50716 V and 0.79284 V, giving approximately 1.71433 V total span.
A 10 kilohm bias instead gives 3.000 V and 1.650 V, a 1.350 V span. A 100 kilohm bias gives 1.650 V and 0.300 V, also 1.350 V. The equal spans of these two choices do not mean their useful temperature regions are identical. One places more voltage change toward one end of the resistance range and the other toward the opposite end. The assumed endpoints are not an NTC grade specification.
| Bias resistance | Cold-end output | Hot-end output | Endpoint span |
|---|---|---|---|
| 10 kilohms | 3.000 V | 1.650 V | 1.350 V |
| 31.623 kilohms | 2.50716 V | 0.79284 V | 1.71433 V |
| 100 kilohms | 1.650 V | 0.300 V | 1.350 V |
Optimize a threshold temperature with a different criterion
At a selected temperature, the voltage slope is Vb Rb times dRt/dT divided by the square of Rb plus Rt. With the local sensor resistance and slope held fixed, its magnitude is greatest when Rb equals Rt at that temperature. Thus maximum local sensitivity and maximum endpoint span are distinct optimization problems.
For an illustrative local sensor resistance of 20 kilohms and slope of minus 800 ohms per kelvin, a 20 kilohm bias at 3.3 V gives minus 33.000 mV/K. The 31.623 kilohm span-optimal bias gives approximately minus 31.327 mV/K. Either may be appropriate depending on whether the measurement prioritizes a narrow threshold region or the broader interval. Neither choice makes the whole voltage-temperature curve linear.
Check the sensor's highest electrical dissipation
The sensor power is Vb squared times Rt divided by the square of Rb plus Rt. For a fixed bias and excitation, this expression reaches its maximum when Rt equals Rb, provided that resistance occurs inside the operating interval. The maximum is Vb squared divided by four Rb. If the interval excludes that point, compare the interval boundaries instead.
At 3.3 V, a 31.623 kilohm bias permits a modeled maximum sensor dissipation of approximately 86.093 microwatts, while a 10 kilohm bias gives 272.25 microwatts at its matching point. These numbers alone do not determine temperature rise. The installed sensor's thermal coupling and excitation timing are required to evaluate self-heating. Do not transfer a dissipation constant from a bare sensor in another medium to an attached heater assembly.
Propagate actual bias resistance into inferred temperature
Suppose software uses nominal bias Rb,nom while the circuit contains actual bias Rb,actual. Under the ideal divider model, the inferred sensor resistance equals the true sensor resistance multiplied by Rb,nom divided by Rb,actual. A bias resistance one percent above nominal therefore makes the inferred resistance approximately 0.990099 times its true value. For an NTC, that generally creates an apparent increase in temperature.
At the illustrative 20 kilohm point, the inferred resistance becomes approximately 19.80198 kilohms. Dividing the minus 198.02 ohm error by the local minus 800 ohm/K slope gives about plus 0.2475 K as a local estimate. Over a wider displacement, use the actual inverse sensor curve rather than extending the tangent indefinitely. Include the bias element's installed temperature when evaluating its resistance departure.
Deliver an objective-specific bias selection
Compare endpoint voltages, local slope and sensor power over the complete resistance-temperature table for each candidate bias value. Identify whether the limiting point is a temperature endpoint, a control threshold, a matching-resistance power maximum or a receiver constraint. A single nominal voltage at room temperature does not capture these different limits.
The resulting drawing review should specify the bias value and its permitted variation, sensor identity, relevant temperature interval, excitation method and conversion data. Verify the connected receiver separately, including its sampling conditions. This keeps the printed resistor requirement tied to a measurable sensor-interface function without implying that a larger ADC span automatically delivers the best temperature accuracy.
Provide the external sensor curve and bias-selection objective
Include the actual sensor data so the printed bias element can be reviewed against the intended measurement function.
- External NTC identity and resistance-temperature table with tolerance information over the intended interval.
- Circuit orientation, excitation and ADC-reference connections, receiver loading and sampling conditions.
- Whether the priority is endpoint span, a specific control-threshold sensitivity or another defined error objective.
- Bias-resistor nominal range, temperature environment and resistance-error allocation.
- Sensor mounting and medium, allowed self-heating contribution and excitation duty or settling requirements.
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