HP-SDI12-NANO heat-pulse sensor · background chapter · September 2026

How the heat-pulse sensor works

An introductory chapter on the electronics and the heat-transfer model behind the digital twin, written for a first-year electrical-engineering student and for anyone coming from agronomy or soil science.

Andres Patrignani, Garrett Peterson, Gerard Kluitenberg, Eduardo Santos, and Bala Natarajan

AI disclosure. This chapter was drafted with Claude Opus 5.5 (Anthropic), an AI assistant, in September 2026. Working from the authors’ instructions, peer-reviewed articles provided by the authors, team meeting notes, correspondence between team members, the sensor design files, the digital twin, and its own general knowledge, it wrote the text, wrote the Python code that draws the figures, and ran the calculations. Most calculations were also implemented in a Jupyter notebook for reproducibility and scrutiny by the authors. The authors directed the work, reviewed and edited the content, and are responsible for it.

What you will learn

• How a heat-pulse sensor measures soil thermal properties
• Basic electricity: voltage, current, resistance and capacitors
• How the sensor turns temperature into a number
• How a heat-flow model estimates thermal properties and water content
• How design choices affect the measurement

All numbers come from the current sensor design and its digital twin.

Contents

1. The measurement in one page

2. Electricity for soil scientists

  2.1 Voltage, current, resistance

  2.2 Ohm's law and power, with the heater

  2.3 Series resistors and the voltage divider

  2.4 Capacitors and the RC time constant

3. The thermistor circuit, part by part

  3.1 The thermistor

  3.2 Why a reference resistor

  3.3 From voltage to a number: the ADC

  3.4 Why the thermistors are switched on only to read

  3.5 Why R11

  3.6 Why C13 and C21–C24

  3.7 One reading, millisecond by millisecond

  3.8 Where the parts sit on the board

4. Heat in soil

5. The line-source model

  5.1 The heat equation

  5.2 Assumptions and boundary conditions

  5.3 Continuous and pulsed line source

  5.4 Reading soil properties from the peak

  5.5 Worked example

  5.6 From heat capacity to water content

6. The digital twin's model

  6.1 Seventeen point sources

  6.2 "Grid" resolution and time steps

  6.3 Why the estimate differs from the soil

  6.4 A real KD2 Pro curve

7. What if...? Design choices, one at a time

  7.1 How to read this section

  7.2 Two rules of thumb

  7.3 Needle spacing

  7.4 Heating time

  7.5 Longer needles

  7.6 How much of the curve to fit

  7.7 All cases in two tables

  7.8 Choosing settings

8. The whole chain in one table

9. Glossary

10. References

Appendix A. Parts list

1. The measurement in one page

The sensor has a small circuit board (the "head") and three thin needles that go into the soil. The center needle is a heater: seventeen tiny 3.3 Ω resistors in a row. The two side needles carry thermometers (thermistors TH1 and TH3), 8 mm away from the heater. A third thermistor, TH2, sits at the tip of the heater needle, and TH4 sits on the board to report the board's own temperature.

A data logger asks for each measurement over a single data wire, using SDI-12, a standard way for environmental sensors to talk to loggers. One measurement goes like this (Figure 1 shows where each part sits):

1. Background (10 s). Record the soil temperature before heating, so we can subtract it later.
2. Heating (15 s). Push about 2 W through the heater. That is roughly the power of a small flashlight bulb, spread along 44 mm of needle.
3. Cooling (150 s). Keep recording while the heat spreads outward. The side needles warm up by about half a degree, peak about 23 s after the heater switches off, and slowly cool.
4. Analysis. How high the side-needle curve rises tells us how much heat the soil can store (its heat capacity). How fast the peak arrives tells us how fast a temperature change spreads through it (its diffusivity). Together they give the thermal conductivity, and the heat capacity gives the water content, because water stores far more heat than soil minerals or air.

Analogy. Drop a spoonful of dye into still water and watch a spot 8 mm away. In thick syrup the color arrives late and stays concentrated; in water it arrives quickly and fades fast. The heat pulse is the dye, the soil is the liquid, and the side thermistor is your eye. The one difference: we know exactly how much "dye" (energy, in joules) we released, because we measure the heater power.

Two pieces of engineering make this work: an electronic circuit that measures temperature to a few thousandths of a degree (Sections 2 and 3), and a physical model that converts the temperature curve into soil properties (Sections 4 to 6).

2. Electricity for soil scientists

2.1 Voltage, current, resistance

Three quantities describe almost everything on this board:

Figure 2 pictures these quantities with water. Handy prefixes: 1 mA = 0.001 A, 1 µA = 0.000001 A, 1 kΩ = 1000 Ω, 1 mW = 0.001 W. Ground (GND) is simply the point we call 0 V; every other voltage on the board is measured from it, just as elevations are measured from sea level.

2.2 Ohm's law and power, with the heater

For a resistor, current is proportional to voltage:

and the power it turns into heat is

Worked example: the heater. Seventeen 3.3 Ω resistors in a row (in series, one after the other) add up: \(R_\text{heater} = 17 \times 3.3 = 56.1\ \Omega\). If the full 12 V battery sat across the heater, the current would be \(12/56.1 = 0.214\) A. In reality a few other things are in the same loop, and each takes its share:

So the heater gets \(P = I^2R = 0.2045^2 \times 56.1 = 2.35\) W while it is on, and each little resistor gets \(0.2045^2 \times 3.3 = 0.138\) W, well under its 0.33 W rating.

2.3 Series resistors and the voltage divider

Put two resistors in series between a voltage \(V_\text{top}\) and ground. The same current flows through both, \(I = V_\text{top}/(R_1+R_2)\), so the voltage at the point between them is

This is the voltage divider, and it is the heart of the temperature measurement: if \(R_2\) changes with temperature, so does \(V_\text{mid}\). If \(R_1 = R_2\), the midpoint sits at exactly half of \(V_\text{top}\).

2.4 Capacitors and the RC time constant

A capacitor stores electric charge, like a small bucket. Its size, the capacitance \(C\), is measured in farads (F); ours are 100 nF (0.0000001 F) and 1 µF. The charge it holds is \(Q = C\,V\): a bigger bucket, or a fuller one, holds more.

A capacitor does two useful jobs: it smooths (it absorbs sudden spikes and fills in brief dips, like a water tower on a town's water line), and together with a resistor it sets a delay. If you fill a capacitor through a resistor \(R\) from a voltage \(V_0\), it charges along

The time constant \(\tau\) is the whole story: after one \(\tau\) the capacitor is 63% full, after 5\(\tau\) it is 99.3% full, and after 10\(\tau\) it is within 0.005% of its final value. Example: 1.5 kΩ and 1 µF give \(\tau = 1500 \times 0.000001 = 1.5\) ms.

3. The thermistor circuit, part by part

3.1 The thermistor

A thermistor is a resistor made to change its resistance with temperature. Ours are "NTC" (negative temperature coefficient): warmer means lower resistance. At 25 °C it is nominally 10 kΩ (±1%). The standard "beta" equation describes it:

with \(T\) in kelvin (°C + 273.15). Near room temperature the resistance changes by about \(\beta/T^2 \approx 3.8\%\) per degree, which is large; that is why thermistors are good at sensing small temperature changes (Figure 4, left).

3.2 Why a reference resistor (R21–R24)?

The processor cannot measure resistance. It can only measure voltage. The reference resistor turns a resistance into a voltage by forming a divider with the thermistor (Section 2.3; Figure 3):

Example at 22 °C: \(R_\text{NTC} = 11.22\) kΩ, \(R_\text{ref} = 10\) kΩ, \(V_\text{REF} = 3.53\) V, so the current is \(3.53/21.22\ \text{k}\Omega = 166\ \mu\text{A}\) and \(V_\text{node} = 3.53 \times 11.22/21.22 = 1.87\) V. Warm the soil and the thermistor's resistance drops, so the node voltage drops.

Why 10 kΩ? Choosing the reference equal to the thermistor's nominal value puts the node near half of VREF, where the divider is most sensitive to small changes. Why 0.1% precision and 25 ppm/°C? Any error in the reference resistor looks exactly like a temperature error. A 1% resistor would shift the reading by about 1%/3.8% ≈ 0.26 °C; the 0.1% part keeps that below 0.03 °C, and its low temperature coefficient keeps it from drifting as the board warms or cools. For the soil properties, what matters is the temperature rise, and a fixed offset cancels when the background is subtracted: a 1% reference error changes the rise by only about 0.2%. The thermistor's \(\beta\) value is different: an error in \(\beta\) scales every rise, and therefore \(\lambda\) and \(C\), one for one.

Does the divider linearize the thermistor? Only as a side effect. The right panel of Figure 4 is not an approximation of the left one: it is the exact result, code = 16384 × R / (R + 10 kΩ). The fraction R / (R + R_ref) flattens toward 1 when the thermistor resistance is large and toward 0 when it is small, which counteracts the thermistor’s steep exponential curve. The result is an S-shaped curve that is nearly straight around the temperature where the thermistor equals the reference resistor (25 °C here) and bends more toward cold and hot temperatures. The processor never uses a straight-line approximation: it inverts the exact divider and beta equations (Section 3.3), which costs only a few lines of arithmetic. What the divider’s shape buys is nearly even resolution across the range of soil temperatures:

Near-surface soil can reach 50–60 °C in summer, especially while the heater is on. Even at 60 °C one code is about 0.011 °C, about 45 times finer than the half-degree signal. A smaller reference resistor would move the flattest part of the curve to warmer temperatures: with 6.8 kΩ, the value matching the thermistor at about 35 °C, resolution at 60 °C improves by about 20% but is about 13% worse at 10 °C, and the dividers draw more current, which lowers VREF and adds a little self-heating. Because noise, not the number of ADC steps, will limit the measurement, 10 kΩ remains a sound choice for the whole 0–60 °C range.

3.3 From voltage to a number: the ADC

The processor's analog-to-digital converter (ADC) compares the input voltage with a reference voltage and reports a whole number, the code. With 14 bits there are \(2^{14} = 16384\) steps:

Here is the trick in this design: the ADC's reference (AREF) is connected to the same VREF that feeds the dividers. Substitute the divider equation and VREF cancels:

Worked example. At 22 °C: code = 16384 × 11.22/21.22 = 8663. Going back: \(R = 10\ \text{k}\Omega \times 8663/(16384-8663) = 11.22\) kΩ, and inverting the beta equation,

How fine is the measurement? Near 22 °C the code changes by about 158 codes per °C, so one code is about 0.006 °C. The planned firmware averages 64 readings per sample (the twin assumes this), which reduces random noise further. The signal we want, the side-needle rise, is about 0.5 °C, or roughly 80 codes. The real limits are noise, thermistor calibration and the reference resistor, not the number of steps.

3.4 Why the thermistors are switched on only to read

Current through a thermistor heats it. This self-heating is a real problem when the signal is half a degree. During a reading the thermistor carries 166 µA with 1.87 V across it:

The maker quotes a "dissipation constant" of about 1 mW/°C in still air, so 0.31 mW would warm the thermistor by about 0.3 °C; in epoxy and soil heat escapes more easily, so the real warming is smaller. Left on all the time, that would be a steady warming comparable to the signal. A perfectly steady offset would cancel when the background is subtracted, but this one would not be steady: it depends on how well the thermistor is coupled to the soil, which changes with moisture, and it warms the very spot being measured. So pin D4 powers the whole thermistor network only 20 ms out of every second (2%). The average heating power drops to about 6 µW, and the self-heating to a few thousandths of a degree. It also saves battery.

3.5 Why R11 (1.5 kΩ)?

Pin D4 is a processor output, and processor pins are delicate: the RA4M1 data sheet lists 4 mA per pin for this port (Arduino's Nano R4 documentation quotes up to 8 mA; the design uses the more conservative data-sheet value). When D4 switches on, C13 is empty and behaves for an instant like a short circuit to ground. Without a resistor the pin would try to deliver a large current. R11 limits it by Ohm's law:

(An earlier version used 330 Ω, which would have allowed about 14 mA; that is why it was changed.) Once C13 is charged, R11 carries the steady current of the four dividers plus the ADC reference, about 0.8 mA. That current makes a voltage drop across R11, \(1500\ \Omega \times 0.8\ \text{mA} \approx 1.2\) V, which is why VREF settles near 3.5 V rather than 4.75 V. Thanks to the ratiometric trick, that lower value does not matter.

3.6 Why C13 and C21–C24?

C13 (1 µF) holds VREF steady. The ADC draws small, quick sips of current from its reference while it converts. Without a local "bucket," each sip would dent VREF through R11. C13 supplies the sips and refills between them. With R11 it charges with a time constant of about 1.1 ms (R11 in parallel with the divider load, times 1 µF).

C21–C24 (100 nF each) hold each thermistor node steady. Three reasons:

1. The ADC takes a gulp. Inside the ADC, a tiny capacitor is connected to the input for a moment and must be charged to the input voltage. Through 5 kΩ of divider resistance, that gulp would pull the voltage down briefly. The 100 nF capacitor is thousands of times larger than the ADC's internal one, so it supplies the gulp with almost no change in voltage.
2. The needle wires are antennas. The thermistor wires run 30 to 55 mm down the needles, right next to a heater that switches 100 times per second. They pick up electrical noise. The capacitor, together with the divider resistance, forms a filter that smooths out fast disturbances while passing the slow temperature signal.
3. A known, short settling time. Each node settles with \(\tau = (10\ \text{k}\Omega \parallel R_\text{NTC}) \times 100\ \text{nF} \approx 5.3\ \text{k}\Omega \times 100\ \text{nF} = 0.53\) ms. The planned firmware waits 10 ms before reading. That is nearly 19 of the node's own time constants, but the node also follows VREF, which settles more slowly (about 9 of its time constants in 10 ms). The node-to-VREF ratio is within about one ADC step at the first reading and closer on average, and the small remainder is the same every second, so it cancels when the background is subtracted.

3.7 One reading, millisecond by millisecond

This is the planned sequence that the twin simulates (Figure 5); the firmware is still to be written.

3.8 Where the parts sit on the board

Everything in Sections 2 and 3 is a real part you can find on the board. Figure 6 shows the whole sensor as KiCad renders it, turned the way the website draws it: cable end up, needles down, seen from the top face (so TH1 is on the right). Figure 7 zooms in on the head. Every part is marked with a dot and its name, each color is one job, and the text beside each group says in a sentence what it does and which section explains it. Appendix A lists every part with its manufacturer, part number and specifications.

4. Heat in soil

Three properties describe how a material handles heat:

Soil is a mix of minerals, organic matter, water and air, and the mix sets the properties:

The key fact for water content: water has twice the heat capacity of minerals and three thousand times that of air. When a soil gains water, the water replaces air, and the heat capacity climbs linearly with the water content. Measure \(C\), and you can back out the water content (Section 5.6).

5. The line-source model

5.1 The heat equation

Heat flows from warm to cold at a rate proportional to the temperature gradient (Fourier's law), and energy is conserved. Combining the two gives the heat equation:

In words: a spot warms up if it is colder than the average of its surroundings, and the rate is set by the diffusivity. For a long, straight heater it is natural to use distance from the heater, \(r\), as the coordinate; if nothing changes along the heater's length, the equation becomes

The extra \(1/r\) term appears because heat spreads over ever-larger rings: a ring twice as far away has twice the circumference to warm.

5.2 Assumptions and boundary conditions

To solve the equation we must say what the world looks like (Figure 8). The classic dual-probe analysis (Campbell et al. 1991, who treated the heat pulse as instantaneous; Kluitenberg et al. 1993 and Bristow et al. 1994, who included its finite duration) assumes:

• Geometry: the heater is an infinitely long, infinitely thin line.
• Medium: the soil is uniform (homogeneous and the same in every direction) and extends forever in every direction.
• Initial condition: everything starts at the same temperature; we work with the rise \(\Delta T = T - T_0\), which starts at zero everywhere.
• Far-field boundary condition: far away the soil never feels the heater, \(\Delta T \to 0\) as \(r \to \infty\).
• Source condition: the line releases a constant power per meter, \(q'\), from time 0 to time \(t_0\), then stops.
• Everything else is ignored: the needles' own size and heat storage, the epoxy, contact resistance between needle and soil, the soil surface, water moving in response to heat, and properties changing with temperature.

5.3 Continuous and pulsed line source

If the line heats forever, the solution (Carslaw and Jaeger 1959; Blackwell 1954) is

where \(E_1\) is the exponential integral, \(E_1(x) = \int_x^\infty \frac{e^{-u}}{u}\,du\). You do not need to compute it by hand: it is built into scientific libraries (for example scipy.special.exp1 in Python or expint in MATLAB), but not into common spreadsheets or data-logger function libraries; Knight and Kluitenberg (2004) give a simple polynomial approximation for those cases. All you need to know is that it is large for small arguments (long times, short distances) and falls quickly to zero for large ones. The group \(r^2/(4\alpha t)\) is a natural "clock": the heat front reaches a distance \(r\) at a time of order \(r^2/(4\alpha)\), which for our 8 mm and a typical soil is about 30 s.

Switching off: superposition. The heat equation is linear, so solutions can be added. Turning the heater off at \(t_0\) is the same as leaving it on and adding a second, negative heater that starts at \(t_0\):

Figure 9 shows how the two pieces combine.

5.4 Reading soil properties from the peak

The side-needle curve rises, peaks at a time \(t_m\) with height \(\Delta T_m\), and falls. Setting the time derivative of the pulsed solution to zero gives a formula for the diffusivity from the peak time alone (Bristow et al. 1994):

Then the peak height gives the conductivity, and the ratio gives the heat capacity:

Read it like this: the pulse of energy spreads into a cylinder of soil of roughly radius \(r\) (cross-section \(\pi r^2\)); the temperature rise is energy divided by (volume × heat capacity), and the factor \(1/e\) (with \(e \approx 2.718\)) is how far down the bell-shaped temperature profile the thermistor sits at that moment: at \(t_m\) the profile has \(\exp(-r^2/4\alpha t_m) = e^{-1}\) at distance \(r\). A soil that stores more heat (wetter) shows a smaller peak. The formula also shows why the spacing matters so much: \(C\) depends on \(r^2\), so a 1% error in spacing becomes a 2% error in heat capacity.

5.5 Worked example: Hodgeman 20 cm, 33 kPa

This is the twin's default soil, a silty clay loam core from the Kansas Mesonet soil database (Parker et al. 2022) whose KD2 Pro readings are \(\lambda = 1.283\) W m⁻¹ K⁻¹ and \(C = 2.345\) MJ m⁻³ K⁻¹ (so \(\alpha = 0.547\) mm² s⁻¹).

Now pretend we measured that curve and want the soil back. With \(t_m = 38\) s, \(t_0 = 15\) s and \(r = 0.008\) m:

The two \(E_1\) arguments are \(r^2/(4\alpha t_m) = 0.770\) and \(r^2/(4\alpha(t_m - t_0)) = 1.272\), with \(E_1\) values 0.3281 and 0.1415:

We recover the soil to within rounding, as we should when the data follow the model perfectly. The short-pulse shortcut gives \(C = 677/(2.718 \times \pi \times 0.008^2 \times 0.5221) = 2.37\) MJ m⁻³ K⁻¹, only 1% off, a good sanity check you can do on a calculator.

5.6 From heat capacity to water content

Heat capacity adds up by volume (de Vries 1963):

with bulk density \(\rho_b\) (Mg m⁻³), mineral specific heat \(c_s \approx 0.75\) MJ Mg⁻¹ K⁻¹, water \(C_w = 4.18\) MJ m⁻³ K⁻¹ and volumetric water content \(\theta\) (m³ m⁻³). Air adds essentially nothing. For the Hodgeman core, \(\rho_b = 1.363\):

The core's measured water content is 0.39. The gap shows that the de Vries relation has its own scatter (organic matter, mineral mix, the KD2 reading itself); in the Kansas dataset's wetter states (saturated, 33 and 70 kPa), measured heat capacities average about 6–9% below this line, with a scatter of about ±11–14%, and drier states scatter more. This core sits 12% below it (2.345 measured against 2.665 predicted). A sensor-specific calibration against measured water contents would remove much of this.

6. The digital twin's model

6.1 Seventeen point sources instead of a line

The real heater is not an infinite line. It is made of 17 separate resistors 2.6 mm apart (a 44.2 mm effective length), and it starts about 9 mm below the board. To predict what the real sensor would see, the twin treats each resistor as a point source in uniform soil. For a point releasing power \(P\) from time 0, the rise at distance \(d\) is (Carslaw and Jaeger 1959)

where erfc is the complementary error function (another standard function). The twin switches each source off at \(t_0\) by superposition, as in Section 5.3, and adds up all 17:

with the second term only after \(t_0\), \(P_i = 1.994/17 = 0.117\) W, and \(z_i = 8.8 + 2.6\,(i-1)\) mm the position of resistor \(i\) along the needle. TH1 and TH3 sit at \(z = 29.8\) mm, the middle of the heated section, and \(r = 8\) mm. TH2 sits on the heater needle, 3.68 mm past the last resistor. Figure 10 shows the resulting temperature field.

6.2 "Grid" resolution, time steps and boundary conditions in the twin

Many soil models divide the soil into a grid and step forward in time (finite differences or finite elements), and then grid spacing and time step control the accuracy. The twin has no grid. Both the line-source and the point-source solutions are exact formulas: the temperature at any point and time is computed directly, independently of every other point. So there is no numerical diffusion and no stability limit. What the twin does choose:

Boundary and initial conditions are those of Section 5.2, applied in three dimensions: uniform soil in every direction (no soil surface, no board or epoxy), uniform starting temperature, \(\Delta T \to 0\) far away, and zero-size needles that store no heat and touch the soil perfectly. The heater power is the PWM average; the 100 Hz flicker is not simulated (8 mm of soil smooths it out almost completely). TH4 on the board is held at the starting temperature because heat flow into the board is not modeled.

6.3 Why the twin's estimate differs from the soil values

The twin runs the measurement in two directions, like a real experiment:

1. Forward (the "truth"): the 17-point-source soil of Section 6.1, plus the electronics: 14-bit codes, averaging, one sample per second, subtracting the background.
2. Inverse (the "analysis"): what simple firmware would do: fit the ideal pulsed line source (Section 5.3) to TH1's rise, with the nominal 8 mm spacing.

Two analysis choices decide how close the answer comes. Take first a simple analysis that divides the power by the resistor-end-to-end length, 43.2 mm, and fits the whole 105 s record of a 90 s cooling phase. For the Hodgeman soil it reports \(\lambda = 1.331\) W m⁻¹ K⁻¹ (+3.7%) and \(C = 2.406\) MJ m⁻³ K⁻¹ (+2.6%). Three separate effects add up to that, and it is worth keeping them apart because they have different cures:

The effects multiply: \(1.023 \times 1.015 \times 0.999 \approx 1.037\) for \(\lambda\) and \(1.023 \times 1.004 \times 0.999 \approx 1.026\) for \(C\), the +3.7% and +2.6% above. The website uses the better choices: the effective length and a fit from heater-on to 1.5 times the measured peak time. It reports \(\lambda = 1.285\) (+0.1%), \(C = 2.344\) (−0.0%) and \(\alpha = 0.548\) mm² s⁻¹ for this soil, from 57 one-second readings (Sections 7.6–7.8).

A real sensor adds more differences that the twin deliberately leaves out, because they need measurements to quantify: the needles' own radius and heat capacity, which shift and rescale the peak (earlier and higher, or later and lower, depending on whether the needles store less or more heat per volume than the soil; Knight et al. 2012); epoxy, contact resistance, and air gaps around the needles; the exact spacing after assembly and insertion; heater resistance changing slightly as it warms; and the soil not being uniform. That is why heat-pulse sensors are usually calibrated: measure a material with known properties (water stabilized with agar is common), and adjust the effective spacing until the sensor reads correctly. That absorbs a constant bias in \(C\), including the heated-length bookkeeping above, but not in \(\lambda\): in the line-source analysis \(\lambda\) does not depend on \(r\) at all. Getting \(\lambda\) right needs the correct \(q'\), or a second check against a material of known conductivity (water, about 0.6 W m⁻¹ K⁻¹).

6.4 A real KD2 Pro curve

The Kansas dataset stores the raw temperature rise that the KD2 Pro recorded for each reading. The SH-1 probe has two 30 mm needles 6 mm apart and heated this core for 60 s at 18.3 W m⁻¹.

Figure 12 shows the curve. One point first: the KD2 computed its reported \(\lambda\) and \(C\) from this same curve with its own analysis, which we do not know in detail. So any gap between those values and our fit is first a difference between two analyses of identical data. With that in mind, here is what the curve teaches, with numbers checked by simulation:

• The finite length matters more with a long pulse on a short probe. Simulate the SH-1 geometry (30 mm heater, 6 mm spacing, 60 s at 18.3 W m⁻¹) in the reported soil and fit the ideal line to that noise-free, finite-heater curve over the same 118 s: the fit returns \(\lambda = 1.407\) (+9.7%) and \(C = 2.342\) (−0.1%). The heat-capacity estimate is almost untouched; the conductivity is not.
• Finite length does not explain everything. Fitting the ideal line to the measured curve gives \(\lambda \approx 1.56\) and \(C \approx 2.27\). The finite length accounts for about half of the conductivity difference and none of the heat-capacity difference. The rest comes from things neither ideal model includes, such as the needles' own radius and heat capacity (Knight et al. 2012), contact between needle and soil, and the instrument's own processing; this one curve cannot separate them.
• The peak comes only 4–6 s after the heater stops, and the readings are 2 s apart, so single-point peak methods are fragile here: moving \(t_m\) from 64 s to 62 or 66 s moves \(\alpha\) by +67% or −25%.

7. What if...? Design choices, one at a time

With the equations in hand we can ask design questions and answer them with numbers. Each question below changes one thing and keeps everything else as it is now.

7.1 How to read this section

• The "truth" is the twin's finite-heater model (Section 6.1): 17 point sources (25 for the long-needle case) in uniform, ideal soil.
• The "analysis" fits the ideal pulsed line source (Section 5.3) with \(q'\) = average power ÷ (number of resistors × 2.6 mm) and the nominal spacing, to one-second samples from heater-on to 1.5 times the peak time.
• Record length: fitting to 1.5 times the peak time needs about 145 s of record after heater-on in the dry soil at 8 mm, and about 220 s at 10 mm (Table B, last column). That is why the default cooling phase is 150 s (Section 7.8).
• No noise is added. The fitted values therefore show only the model error: what the ideal line misses about the real, finite heater. Real sensors add noise, spacing errors, needle heat capacity and contact effects (Section 6.3). The "ADC steps" column shows how big the signal is compared with the smallest step the electronics can resolve (about 0.0063 °C), which is a fair guide to how much noise will matter.
• Three measured Kansas soils cover the range of diffusivities:

The heater power is the same in every case (1.994 W average, 85% duty at 12 V), except the long-needle case, which needs different resistors (Section 7.5).

7.2 Two rules of thumb you can do on a calculator

From the short-pulse solution (Section 5.4), treating the pulse as if it were released all at once at its midpoint:

These are approximations. They work within a few percent while the pulse is short compared with \(r^2/(4\alpha)\), and fail for long pulses (Section 7.4). The comparisons below use the full pulsed line-source solution of Section 5.3, which is what the rules approximate. Check them on the present design (medium soil, \(r\) = 8 mm, \(t_0\) = 15 s, \(Q'\) = 677 J m⁻¹):

7.3 What if the needles were 6 or 10 mm apart?

By hand. Only \(r\) changes. \(r^2/(4\alpha)\) becomes 16.5 s at 6 mm and 45.7 s at 10 mm, so \(t_m \approx\) 24.0 s and 53.2 s (full solution: 26.0 s and 54.0 s). At 6 mm the time rule is 8% low, because the 15 s pulse is nearly as long as \(r^2/(4\alpha)\). The peak scales with \(1/r^2\): \(677/(2.718\,\pi \times 0.006^2 \times 2.345\times10^6) = 0.939\) °C at 6 mm and 0.338 °C at 10 mm (full solution: 0.909 and 0.336 °C).

What changes.

• Signal: 6 mm gives about 1.7 times today's peak, 10 mm about 0.64 times.
• Sensitivity to spacing error: since \(C \propto 1/r^2\), a 0.1 mm error costs \(1-(r/(r+0.1))^2\): 3.3% at 6 mm, 2.5% at 8 mm, 2.0% at 10 mm. Closer needles need tighter spacing tolerance.
• Model error: with the heater unchanged, a wider gap makes the heater "shorter" relative to the distance heat must travel, so the end effect grows: at 10 mm the fitted \(\lambda\) is +0.8% to +1.6% and \(C\) +0.4% to +0.6%; at 6 mm both stay within 0.1%.
• Not in this model: the needles themselves (2.77 mm outside diameter for 12 gauge) take up a larger share of a 6 mm gap, so the effects of needle radius and heat capacity that the ideal model ignores (Knight et al. 2012) become more important as the gap shrinks. Our 12-gauge needles are already thicker than the roughly 1.3 mm needles of typical research probes, so those effects are likely larger here than in most published sensors.

7.4 What if we heat for 8, 15 or 60 seconds?

The Kansas readings in this chapter were taken with a KD2 Pro SH-1, which heats for 60 s (Section 6.4). Today's dual-probe sensors typically heat for 8–15 s; ours defaults to 15 s, and 60 s remains available only as a setting.

By hand. The power per meter stays 45.11 W m⁻¹, so the energy is \(Q' = 45.11 \times t_0\): 361, 677 and 2706 J m⁻¹. For 8 s: \(t_m \approx 29.2 + 4 = 33.2\) s and \(\Delta T_m \approx 361/1282 = 0.282\) °C (full solution: 33.6 s and 0.281 °C). For 60 s the rules give 59.2 s and 2.11 °C, but the full solution gives 74.3 s and 1.867 °C: the rules of thumb fail. The time rule even puts the peak before the heater switches off at 60 s. A 60 s pulse lasts longer than the 29 s the heat needs to reach the thermistor, so it no longer behaves like a single burst. Use the full equations of Section 5.4.

What changes.

• Signal: grows with the energy, a little less than proportionally for long pulses: 8 s gives 44 ADC steps in the medium soil, 15 s gives 83, and 60 s (four times the energy of 15 s) gives 295, 3.6 times as many.
• Heating of the soil next to the needle: the soil touching the heater needle warms much more than the soil at the thermistors. Beside a resistor (between resistors it is about 10% less, and the model ignores the needle wall, which would even this out): in the medium soil 5.2 °C (8 s), 6.8 °C (15 s) and 10.5 °C (60 s); in the dry soil 14.4, 21.3 and 39.5 °C. Strong local heating can push water away from the heater, which changes the very property being measured; this model does not include it, so large local rises are a warning sign, not a computed error.
• Model error: small for 8 and 15 s. For 60 s it grows in high-diffusivity soil (+5.7% in \(\lambda\), −0.1% in \(C\) for the wet sandy soil), because by the time of the peak the heat has spread well past the ends of the heater.
• Time and battery: the energy drawn from the battery is proportional to \(t_0\), and a longer pulse delays the peak and the end of the measurement.

7.5 What if the needles were 3 inches long?

A 3-inch (76 mm) needle has room for a 65 mm heater: 25 resistors at 2.6 mm, heater midpoint about 40 mm below the board (the side thermistors would move there too). The catch is the 12 V supply: 25 × 3.3 Ω = 82.5 Ω would cut the power to about 1.4 W. Keeping about 2 W needs about the same total resistance as now, so 25 × 2.2 Ω = 55 Ω. By Ohm's law (Section 2.2) that gives 2.03 W average, spread over 65 mm: \(q' = 31.27\) W m⁻¹, 31% less than now.

• Signal: the peak falls from 0.522 to 0.362 °C in the medium soil (83 to 57 ADC steps). Heating 22 s instead of 15 s would restore it.
• Model error: essentially zero in all three soils. But with the effective length and a fit to 1.5 times the peak time, today's 44 mm heater is already within +0.5% in \(\lambda\) and +0.1% in \(C\) (worst of the three soils), so the gain is about half a percent.
• Bending: a needle held at one end bends in proportion to its free length cubed for a sideways force at the tip (to the fourth power for a force spread along it). The side needles, which set the spacing, now end about 32 mm below the board; moving the thermistors to the new heater midpoint would make them about 1.3–1.5 times longer, so they would bend roughly 2–4 times more for the same force during insertion. Because a 0.1 mm change in spacing already costs 2.5% in \(C\), this is likely to cost more accuracy than the longer heater gains.

7.6 What if we fit more, or less, of the curve?

This question matters more than the needle length. The peak's height and timing carry most of the information: the height mainly sets \(C\), the timing sets \(\alpha\). The tail is where the ideal line and the real sensor agree least. For the finite heater, the curve matches the ideal line through the peak and departs from it only late in the tail (Figure 11, right). The effects the model leaves out, such as the needles' own heat capacity, imperfect contact, heat left from a previous pulse and the soil's natural temperature drift, also show up mostly in the tail. So the further past the peak we fit, the more the fit is pulled by the part of the curve the ideal line gets wrong. This agrees with advice from G. J. Kluitenberg (personal communication, May 2026): fitting the tail, the part of least agreement, is a common mistake, and the fit should end just past the peak.

Without noise. Errors of the fitted \(\lambda\) / \(C\) for today's heater (8 mm, 15 s), by how far the fit runs:

With noise. The one argument for fitting further is that more points average out noise. To test it, we added independent random noise of 5 mK to each one-second reading (about one ADC step; the background is the mean of ten noisy readings), repeated each fit 200 times, and report the mean error ± the standard deviation of \(\lambda\):

• The error grows with the window, and the scatter shrinks only a little. Going from 2 to 3 times the peak time buys almost no precision but more than doubles the error in the medium and wet soils.
• The best balance is about 1.25–1.5 times the peak time. Ending right at the peak (1.1 times) lets the scatter grow in wet soil, because the peak time itself becomes uncertain.
• \(C\) hardly depends on the window (mean errors 0.0–0.5%, scatter ±0.1–0.4% in every case). The window matters for \(\lambda\) and \(\alpha\), which the tail drags upward.

The website and the planned firmware therefore fit from heater-on to 1.5 times the measured peak time. Readings recorded after that are kept, for plotting and quality checks, but not fitted. These tests assume independent noise and an ideal soil; the best window should be confirmed with bench data.

7.7 All cases in two tables

Table A. Signal. Peak rise at the thermistor and the maximum rise of the soil touching the heater needle (at its 1.385 mm outer radius, beside the middle resistor; point-source model, needle wall ignored).

Table B. What the analysis reports (fitted value, with the error relative to the true soil value in parentheses; fit from heater-on to 1.5 times the peak time).

7.8 Choosing settings: defaults and when to change them

Every setting below can be changed on the website and, in the planned firmware, on the sensor itself. The defaults are chosen to work reasonably in most soils with the present needles (8 mm spacing, 44 mm heater); change them when you know in advance what conditions you will measure.

When to change them (numbers from the same ideal model; confirm on the bench):

• Wet or sandy soils (high diffusivity). The peak comes early (about 20–25 s), so a shorter cooling phase (60–90 s) is enough. Wet soils give the smallest peaks, so a longer pulse helps the signal; in the wet sandy soil 20 s gives a 0.79 °C peak with +0.8% in \(\lambda\), and 30 s gives 1.12 °C with +1.7% (Section 7.4). Averaging repeated measurements is the other way to beat noise.
• Dry soils (low diffusivity). The peak comes late (90–100 s in oven-dry soil), so keep the 150 s cooling phase. The signal is large, but the soil next to the heater gets hot: about 21 °C at 15 s in the oven-dry example. Lowering the duty to 50% keeps a 0.54 °C peak and brings that down to about 12.5 °C; do not lengthen the pulse.
• How often to measure. Each pulse leaves heat behind that fades slowly, and the analysis subtracts a flat background. If the next measurement starts while the needles are still cooling, the falling background biases the result. Assuming identical pulses and no natural temperature drift, the added error in \(\lambda\) / \(C\) is:

  Interval	Dry	Medium	Wet sandy
  5 min	+19% / +17%	+4.0% / +3.6%	+1.8% / +1.4%
  15 min	+3.3% / +2.7%	+0.6% / +0.4%	+0.6% / +0.2%
  30 min	+0.9% / +0.7%	+0.3% / +0.1%	+0.5% / +0.2%
  60 min	+0.3% / +0.2%	+0.2% / +0.1%	+0.5% / +0.1%
  no previous pulse	+0.1% / ±0.0%	+0.2% / +0.1%	+0.5% / +0.1%

  Every 15 minutes is fine in moist soil; dry soil needs about 30 minutes, and hourly measurements are clean in every soil. Fitting a sloped background, rather than a flat one, would reduce this error and also handle the soil's natural daily temperature swing.
• Very dry soil or wider needles. If the website reports that the record ended before the peak, lengthen the cooling phase.

8. The whole chain in one table

For the Hodgeman soil (true values \(\alpha = 0.547\) mm² s⁻¹, \(\lambda = 1.283\), \(C = 2.345\)), as the website computes it: effective heated length, 14-bit readings once per second, fit to 1.5 times the peak time.

9. Glossary

10. References

1. Blackwell, J. H. (1954). A transient-flow method for determination of thermal constants of insulating materials in bulk. Part I—Theory. Journal of Applied Physics, 25, 137–144. doi:10.1063/1.1721592
2. Bristow, K. L., Bilskie, J. R., Kluitenberg, G. J., & Horton, R. (1995). Comparison of techniques for extracting soil thermal properties from dual-probe heat-pulse data. Soil Science, 160, 1–7. doi:10.1097/00010694-199507000-00001
3. Bristow, K. L., Kluitenberg, G. J., & Horton, R. (1994). Measurement of soil thermal properties with a dual-probe heat-pulse technique. Soil Science Society of America Journal, 58, 1288–1294. doi:10.2136/sssaj1994.03615995005800050002x
4. Campbell, G. S., Calissendorff, C., & Williams, J. H. (1991). Probe for measuring soil specific heat using a heat-pulse method. Soil Science Society of America Journal, 55, 291–293. doi:10.2136/sssaj1991.03615995005500010052x
5. Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of heat in solids (2nd ed.). Oxford University Press.
6. de Vries, D. A. (1963). Thermal properties of soils. In W. R. van Wijk (Ed.), Physics of plant environment (pp. 210–235). North-Holland.
7. Kluitenberg, G. J., Bristow, K. L., & Das, B. S. (1995). Error analysis of heat pulse method for measuring soil heat capacity, diffusivity, and conductivity. Soil Science Society of America Journal, 59, 719–726. doi:10.2136/sssaj1995.03615995005900030013x
8. Kluitenberg, G. J., Ham, J. M., & Bristow, K. L. (1993). Error analysis of the heat pulse method for measuring soil volumetric heat capacity. Soil Science Society of America Journal, 57, 1444–1451. doi:10.2136/sssaj1993.03615995005700060008x
9. Knight, J. H., & Kluitenberg, G. J. (2004). Simplified computational approach for dual-probe heat-pulse method. Soil Science Society of America Journal, 68, 447–449. doi:10.2136/sssaj2004.4470
10. Knight, J. H., Kluitenberg, G. J., Kamai, T., & Hopmans, J. W. (2012). Semianalytical solution for dual-probe heat-pulse applications that accounts for probe radius and heat capacity. Vadose Zone Journal, 11(2). doi:10.2136/vzj2011.0112
11. Parker, N., Kluitenberg, G. J., Redmond, C., & Patrignani, A. (2022). A database of soil physical properties for the Kansas Mesonet. Soil Science Society of America Journal, 86, 1495–1508. doi:10.1002/saj2.20465

Numbers in this chapter come from the HP-SDI12-NANO design files and the digital twin (docs/sensor-model.js, docs/engine.js) as of September 29, 2026. Self-heating depends on how the thermistor is potted and is an estimate, not a measurement.