10.9  Voltage dividers and analog input

Step 1: Measure the resistors

Measure both resistors with the ohmmeter before they go into the board, and log the values. Put the measured values into 10.9.1 and predict the midpoint at 12.00 V. If the 33 kΩ resistor reads 32.8 kΩ and the 10 kΩ one 10.06 kΩ, the prediction is \(12.00 \cdot 10.06/42.86 = 2.82\) V, against 2.79 V from the nominal values.

Step 2: Build on the bench supply

Build the divider on the breadboard and feed it from the lab supply at 12.00 V with a current limit of 10 mA (the divider draws 0.3 mA). The midpoint is not connected to the ESP32 yet.

Measure the midpoint against the − rail and compare with the prediction from Step 1. The two should agree to within about 1 %, the accuracy of the ohmmeter and the voltmeter together.

On the car the divider hangs on the car rail after the fuse and the main switch (Figure 10.9.3), so it is dead when the car is switched off and does not drain the battery on the shelf. It draws \(12.6/43\,000\) A = 0.29 mA when the car is on, which a 2200 mAh pack would need over 7000 hours to empty. The 100 nF capacitor in the figure keeps the reading steady; Loading of a divider explains why it is there.

The ESP32 ADC

analogRead(pin) starts a conversion and returns a whole number. The ESP32’s ADC has 12 bits, so the result lies between 0 and \(2^{12} - 1 = 4095\). Internally the converter compares the input with a reference of about 1.1 V, and the median value is 1.1 V with a spread from 1.0 to 1.2 V between chips (ESP-IDF v4.4 ADC documentation). To measure larger voltages, an attenuator in front of the converter scales the input down. Arduino-ESP32 core 2.0.17 uses the strongest setting, ADC_11db, by default; we set it explicitly anyway with analogSetPinAttenuation(pin, ADC_11db) so the code does not depend on the default. (ESP-IDF 5 calls the same setting ADC_ATTEN_DB_12, which is closer to its true value.)

What range does 11 dB give? Espressif’s documents answer differently depending on how much error they accept. The ESP32 datasheet (v5.3, Table 4-4) gives an effective measurement range of 150 to 2450 mV, with a total error of ±60 mV after factory calibration, and adds that accuracy is worse for readings above about 3000 counts, i.e. about 2450 mV. The Arduino-ESP32 documentation lists 150 to 3100 mV as the measurable range for the same setting. Both are right: the converter responds up to about 3.1 V, but its characteristic bends near both ends. Below about 0.15 V the reading stays at or near 0, and toward the top it flattens and reaches 4095 before the input reaches 3.3 V. A pot turned to its stops therefore does not give a clean 0 and 4095, and code reading a throttle should treat the last few percent at each end as “fully on” and a small band around the middle as zero.

Between the ends the response is close to a straight line, but its slope and offset differ from chip to chip by up to about 6 % according to the datasheet. The function analogReadMilliVolts(pin) corrects for this using calibration values Espressif burned into each chip at the factory and returns millivolts at the pin. For the battery we go one step further and calibrate the complete chain, divider resistors included, against a bench supply, as described in Calibrating with two points.

The ESP32 has two converters. ADC1 serves GPIO 32 to 39. ADC2 serves several other pins (GPIO 0, 2, 4, 12 to 15 and 25 to 27) and is shared with the Wi-Fi radio: while Wi-Fi is running, a read from ADC2 fails. The car runs a Wi-Fi access point, so every analog input on the car uses ADC1. GPIO 34 to 39 are also input-only: they have no output driver and no internal pull-up or pull-down resistors. That makes them poor choices for anything else and good choices for analog inputs, which is why the pin map puts the battery on GPIO 34, the bench pot on GPIO 35 and the motor current sense on GPIO 36 and 39.

Loading of a divider

10.9.1 assumed that nothing draws current from the tap. Anything connected there, a multimeter, an ADC pin, a careless wire, appears as a load resistance \(R_L\) in parallel with \(R_2\), and the output drops to

\[ V_\text{out} = V_\text{in}\,\frac{R_2 \parallel R_L}{R_1 + R_2 \parallel R_L}, \qquad R_2 \parallel R_L = \frac{R_2 R_L}{R_2 + R_L}. \]

A convenient way to judge how much a load matters is to replace the divider by its Thévenin equivalent: an ideal source of the unloaded voltage \(V_\text{in} R_2/(R_1+R_2)\) behind the resistance \(R_\text{th} = R_1 \parallel R_2\). A load current \(I_L\) then lowers the output by \(I_L R_\text{th}\). The ESP32 input leaks at most 50 nA (datasheet Table 5-3), so its steady error is \(50\ \text{nA}\cdot R_\text{th}\), negligible for any \(R_\text{th}\) below a few hundred kΩ.

The ADC is not a steady load, though. At each conversion it connects a small sampling capacitor to the pin and charges it from the source. Through a large \(R_\text{th}\) that charge arrives slowly, the conversion sees a voltage that has not settled, and the reading comes out low and noisy. A high-resistance node also picks up interference more easily, and on the car there is a lot of it: the motor driver switches 10 A or more at 20 kHz a few centimetres away.

The remedy is a capacitor of about 100 nF from the tap to ground, close to the pin. It holds the voltage while the sampling capacitor draws its charge, and together with \(R_\text{th}\) it forms a low-pass filter that removes switching noise. Espressif specifies the ADC’s linearity with exactly such a 100 nF capacitor on the input (datasheet Table 4-3). The filter’s time constant \(\tau = R_\text{th} C\) must stay short compared with how fast the measured voltage changes, which for a battery is easy.

Choosing the resistances is therefore a compromise. Small resistors give a stiff, quiet node but draw current from the battery all the time and heat themselves. Large resistors save the battery but make the node soft. For a battery divider read a few times per second, a total of a few tens of kΩ with a 100 nF capacitor works well on the car, where the 3S pack holds thousands of mAh. A small battery that must last for weeks, like the remote’s, justifies a few hundred kΩ.

The potentiometer as an adjustable divider

A potentiometer is a resistive track with a terminal at each end and a sliding contact, the wiper, that touches the track somewhere between them. The resistance between the two end terminals is fixed, e.g. \(R = 10\) kΩ. The wiper splits the track into two parts that always add up to \(R\): a part \(R - R_b\) above the wiper and a part \(R_b\) below it. With the ends on 3.3 V and ground, the pot is a divider whose ratio we set by hand,

\[ V_\text{wiper} = 3.3\ \text{V}\cdot\frac{R_b}{R}. \]

On a linear pot \(R_b\) is proportional to the rotation angle, so the wiper voltage is too. The thumb joysticks in the kit contain two such pots, one per axis, held at the middle by a spring. At rest each axis sits near \(1.65\) V, and the reading moves up or down from there. The bench throttle in the labs is a pot on THROTTLE_POT_PIN (GPIO 35), wired as in Figure 10.9.5.

The value of the pot matters little for the ratio, but it sets the current and the source resistance seen by the pin. A 10 kΩ pot across 3.3 V draws \(3.3/10\,000 = 0.33\) mA. The resistance seen from the wiper is the two parts of the track in parallel, largest at mid-travel where it is \(R/4 = 2.5\) kΩ. A 100 Ω pot would draw 33 mA from the 3.3 V regulator for no benefit, and a 1 MΩ pot would make the pin sensitive to the loading and noise discussed below.

Before the pot touches the board, we check it as in the multimeter chapter: the resistance between the outer pins is about 10 kΩ, and with the outer pins on a 3.3 V bench supply the wiper voltage moves from near 0 to near 3.3 V. Only then does the wiper go to GPIO 35.

Filtering the readings

Successive readings of a steady voltage scatter by several counts, and on the car the motor adds its own disturbance. The datasheet’s advice is to sample several times and filter or average (notes to datasheet Table 4-3). batteryUpdate() is meant to be called in every pass of loop(), but it takes a reading only every BATT_SAMPLE_MS (10 ms), so the filter behaves the same however fast the loop runs. Each new reading \(x\) updates the filtered value \(y\) by

\[ y \leftarrow y + \frac{x - y}{N}, \]

an exponential moving average with \(N\) = BATT_FILTER_N = 32. Every reading moves the average a thirty-second of the way toward itself, which smooths the noise like an average of about 32 readings while needing one stored number. The time constant is about \(N \cdot 10\ \text{ms} = 0.32\) s, short enough to follow a battery and long enough to ignore a single bad sample. batteryBegin() seeds the filter with a plain average of 32 readings, so the first printed value is already correct.

The raw command in main.cpp uses a different method: it averages 256 fresh readings each time it prints. Averaging \(n\) independent readings reduces random noise by a factor \(\sqrt{n}\), here 16, and the result does not depend on what the filter saw before. That is what we want while turning the supply knob during calibration.

The low-voltage cutoff with hysteresis

A LiPo cell is damaged if it is discharged too far. We warn at 9.9 V (3.3 V per cell) and cut off at 9.6 V (3.2 V per cell). These are voltages under load. The pack has internal resistance, so the voltage at its terminals sags while the motor draws current and recovers when it stops. With, e.g., 30 mΩ for the pack and its wiring and 15 A drawn by the motor, the sag is \(0.030 \cdot 15 = 0.45\) V. A half-empty pack can therefore dip below 9.6 V on a hard acceleration and read 10.5 V a second later.

A single threshold would make the state flicker every time the voltage hovers around it, and the car would stutter. batteryUpdate() avoids this in two ways. A lower state is entered only when the filtered voltage has stayed below its threshold for BATT_TRIP_MS (1000 ms), so a short dip during acceleration does not trigger it. A higher state is entered only above BATT_RECOVER_V (10.2 V, 3.4 V per cell), well above the level that caused the change. The gap between the level that enters a state and the level that leaves it is called hysteresis. Between 9.6 and 10.2 V the state depends on where the voltage came from: a pack falling from full is still OK at 10.0 V, while a pack that has already given a warning stays in WARN at 10.0 V.

After a cutoff the motor stops, the sag disappears and the rest voltage may well climb above 10.2 V, which ends the cutoff and lets the car drive into the next one. Setting -D BATT_CUTOFF_LATCH=1 makes CUTOFF permanent until the ESP32 restarts, which is the safer choice for the car once it drives on its own battery. What to do in each state, limit the throttle in WARN and stop the motor in CUTOFF, belongs to the motor code; the monitor only reports.

Defaults and build flags

battery.h uses #ifndef to give each optional flag a default, so a project only sets what it needs. When any of the four calibration flags is missing, the header falls back to the ideal chain, 2597 counts at 9.0 V and 3636 counts at 12.6 V, and defines BATT_UNCALIBRATED. main.cpp prints this at start-up and on cal, so an uncalibrated monitor cannot go unnoticed. The calibration readings may have decimals (e.g. -D BATT_ADC_LO=2680.4), since the averaged raw output has one.