35 ADC and DAC: Conversion Calculations

35.1 Start With the Decision

An ADC count is useful only when it maps back to the measured voltage. Work the scale, error, and range at the bench.

35.2 Route Overview

This is part 1 of 2. Continue with ADC and DAC: Interactive Design Calculators.

35.3 Part Objectives

• Test adc/dac worked examples with a concrete scenario and pass criteria.
• Validate adc/dac calculation links with a concrete scenario and pass criteria.

35.4 Start Simple

Make the Bench Value and the Screen Value Agree

Picture a light sensor producing 1.20 volts while a device shows a digital count and the dashboard shows a percentage. The test engineer must show how one known input becomes both outputs without hiding a reference value, scale factor, unit change, or rounding choice.

Write the known values first. Record input range, reference, number of steps, output unit, formula, and rounding rule. Calculate the expected count by hand, then run the same value through the device. Keep the code, board, settings, and meter reading with the result.

Try zero, mid-range, near-full-scale, just outside range, a changed reference, a negative input where allowed, and a value between two codes. Check clipping, rounding, output reconstruction, and the final display. Reverse the calculation to see whether the reported value can recover the original within the stated limit.

Do not use a neat digital result as proof that the physical input is safe or accurate. Keep urgent control on a path that handles out-of-range and failed-input states locally. A distant chart may show the result, but it cannot restore resolution lost at conversion.

This opening does not cover every converter design. Practitioner builds the input-to-output worksheet and bench check. Under the Hood examines step size, reference error, noise, sample timing, output levels, and uncertainty.

Use this value check:

• Write each known value.
• Keep each stated unit.
• Work the sum once.
• Test the mid point.
• Test both end points.
• Check the round trip.

Imagine a datasheet number, an ADC code, and a dashboard value that must all agree. Worked examples show the conversion path step by step so no scale factor, reference voltage, or rounding decision hides in the code. Start with one known input, calculate the expected digital value, and compare it with a measured result.

Phoebe’s Field Notes: Where the Converter Numbers Come From

The mathematical gist. A 10-bit, 3.3 V ADC has (q=V_{ref}/2^{10}=3.22) mV. At 1.65 V, the chapter’s endpoint rule gives (D=\lfloor(1.65/3.3)1023\rfloor=511), reconstructing 1.648 V within half an LSB. The same bit depth sets an ideal 62.0 dB SNR, while 8 kHz sampling places Nyquist at 4 kHz and folds a 20 kHz tone to that band edge.

Math Bridge · guided foundationsWhat does a 10-bit ADC actually report for 1.65 V?Let Eddie connect bin width, code rounding, reconstruction error, ideal SNR, Nyquist, and aliasing. Learn the maths

35.5 Learning Objectives

By the end of this section, you will be able to:

• Calculate ADC Output: Compute digital values from analog voltage inputs using the ADC conversion formula
• Select Reference Voltages: Evaluate and match Vref to sensor output ranges for maximum resolution
• Quantify Quantization Error: Derive measurement precision limits across different ADC resolutions
• Determine Nyquist Sampling Rates: Compute minimum and practical sampling frequencies for specific signal types
• Compute DAC Output Voltages: Derive specific analog voltages from digital input values
• Construct Gamma Correction Tables: Build non-linear mappings for perceptually accurate LED dimming

35.6 Prerequisites

Before diving into this chapter, you should be familiar with:

• ADC Fundamentals: Understanding of ADC operation and resolution
• Nyquist Sampling Theory: Understanding of sampling rate requirements
• Binary Fundamentals: Binary number systems and powers of 2

35.7 Why This Chapter Matters

Follow these connections in order: This is the chapter that turns the earlier ideas into engineering workflow. Students often think they understand a topic until they have to plug in numbers and justify a design choice. In real IoT work, this kind of calculation is what connects a sensor datasheet, a microcontroller spec, and an actual usable measurement or control signal. After this chapter, a beginner should be able to work through converter problems step by step instead of treating ADC and DAC settings as mysterious configuration values.

35.8 How To Use This Chapter

• Work the examples in order. They were chosen to move from direct formula use toward more realistic design decisions.
• Before reading each solution, try to identify the known values, the unknown value, and the governing formula yourself.
• If you get stuck, the goal is not speed. The goal is to learn a repeatable method: identify, calculate, verify, interpret.

35.9 ADC Conversion Formula

Start with the smallest complete loop: a voltage goes into the converter, a digital code comes out, and the code must still make physical sense when converted back.

The fundamental ADC conversion formula relates analog input voltage to digital output value:

$\text{Digital Output} = \left\lfloor \frac{V_{in}}{V_{ref}} \times (2^n - 1) \right\rfloor$

Where:

• $V_{in}$ = Analog input voltage (volts)
• $V_{ref}$ = Reference voltage (maximum measurable voltage)
• $n$ = ADC resolution (number of bits)
• $\lfloor \rfloor$ = Floor function (round down to nearest integer)

Inspect the labelled evidence in Figure 35.1 as the entry point to adc conversion formula. The visual is specifically about aDC conversion showing how continuous analog voltage is sampled and quantized into discrete digital steps.

First identify Temperature vs Resistance in Figure 35.1, then contrast it with Resistance (OHMs) and test the claim beside Relationship. These visible labels explain aDC conversion showing how continuous analog voltage is sampled and quantized into discrete digital steps. They keep adc conversion formula tied to evidence that can be inspected on the schematic, part, or trace.

35.10 10-bit Temperature ADC

Given:

• ADC Resolution: 10 bits (Arduino Uno)
• $V_{ref}$ = 3.3V
• $V_{in}$ = 1.65V (sensor output at 25C)

Calculate Digital Output:

$\text{Digital Output} = \left\lfloor \frac{1.65V}{3.3V} \times (2^{10} - 1) \right\rfloor$

$= \left\lfloor 0.5 \times 1023 \right\rfloor$

$= \left\lfloor 511.5 \right\rfloor$

$= 511$

Result: The ADC outputs 511 (binary: 0111111111)

Verification:

• 511.5 is exactly half of 1023, so the floor gives 511, confirming $V_{in}$ is half of $V_{ref}$
• This digital value would be read by analogRead(A0) as 511

35.11 LM35 Temperature Example

The LM35 is a popular precision temperature sensor that outputs 10mV per degree Celsius.

LM35 Specifications:

• Output: 10mV/C
• Temperature Range: 0-100C
• Voltage Range: 0-1.0V

Scenario: Measuring room temperature (25C) with 10-bit ADC

Step 1: Calculate Sensor Output Voltage

$V_{sensor} = \text{Temperature} \times 10\text{mV/C}$

$V_{sensor} = 25C \times 10\text{mV/C} = 250\text{mV} = 0.250V$

Step 2: ADC Configuration

For optimal resolution with 0-1V sensor range:

• Use $V_{ref}$ = 1.0V (external reference or ATmega internal 1.1V)
• Resolution: 10-bit → 1024 discrete values

Step 3: Calculate Digital Output

$\text{Digital Output} = \left\lfloor \frac{0.250V}{1.0V} \times 1023 \right\rfloor$

$= \left\lfloor 255.75 \right\rfloor = 255$

Step 4: Calculate Resolution (Temperature per ADC Count)

$\text{Resolution} = \frac{\text{Temperature Range}}{\text{ADC Counts}} = \frac{100C}{1024} = 0.0977C \text{ per count}$

This means each ADC step represents approximately 0.1C change.

Step 5: Convert ADC Reading Back to Temperature

$\text{Temperature} = \frac{\text{ADC Value}}{1023} \times 100C$

$\text{Temperature} = \frac{255}{1023} \times 100C = 24.93C \approx 25.0C$

Key Insight: By matching $V_{ref}$ to sensor range (1V instead of 5V), we maximize resolution and minimize wasted ADC range.

Comparison: 5V Reference vs 1V Reference

Using 1V reference gives 5x better temperature resolution by fully utilizing ADC range!

35.12 Quantization Error Example

Quantization error is the inherent inaccuracy introduced when converting continuous analog signals to discrete digital values.

Quantization Error Formula:

$\text{Max Error} = \pm \frac{1}{2} \text{LSB} = \pm \frac{V_{ref}}{2^{n+1}}$

Where LSB (Least Significant Bit) is the smallest voltage change the ADC can detect.

10-bit ADC with 3.3V Reference:

$\text{LSB Step Size} = \frac{3.3V}{2^{10}} = \frac{3.3V}{1024} = 3.22\text{mV}$

$\text{Max Quantization Error} = \pm \frac{3.22\text{mV}}{2} = \pm 1.61\text{mV}$

What This Means:

• Any voltage between 1.650V and 1.653V will round to the same digital value (512)
• Worst-case error: +/-1.61mV from actual voltage

Comparison Across ADC Resolutions:

Reducing Quantization Error:

1. Use Higher Resolution ADC

   • 12-bit vs 10-bit: 4x reduction in error (1.61mV → 0.403mV)
   • 16-bit vs 10-bit: 64x reduction in error

2. Oversampling and Averaging

   • Average N samples to gain extra resolution
   • Effective bits increase by log2(N)/2
   • Example: Average 16 samples with 10-bit ADC → 12-bit effective resolution

3. Match Vref to Signal Range

   • If measuring 0-1V signal, use 1V reference (not 3.3V or 5V)
   • Maximizes use of ADC’s available range

35.13 Nyquist Audio Sampling

The ADC examples handled voltage resolution. The next question is time resolution: how often must the system sample so the changing signal is still recoverable?

Given:

• Human voice frequency range: 300 Hz to 3.4 kHz
• Highest frequency: $f_{max}$ = 3.4 kHz

Calculate Minimum Sampling Rate:

$f_{sample} \geq 2 \times 3400\text{ Hz} = 6800\text{ Hz}$

Practical Implementation:

• Telephone systems: 8 kHz (provides margin above 6.8 kHz minimum)
• VoIP codecs: 8-16 kHz
• Audio recording: 44.1-48 kHz (for full 20 kHz human hearing range)

35.14 Vibration Accelerometer Example

Given:

• Machine vibration events: 0-100 Hz
• Highest frequency: $f_{max}$ = 100 Hz

Calculate Minimum Sampling Rate:

$f_{sample} \geq 2 \times 100\text{ Hz} = 200\text{ Hz}$

Practical Implementation:

• Industrial IoT: 500-1000 Hz (2.5-5x oversampling for safety margin)
• Allows detection of unexpected high-frequency faults
• Ensures anti-aliasing filter has gradual roll-off

Real-World Sampling Rates:

35.15 ESP32 Soil Moisture Example

35.16 Ultrasonic Distance Example

35.17 8-bit DAC Motor Control

Now reverse the conversion path. Instead of asking which code represents a measured voltage, ask which output voltage a digital code will command.

Given:

• DAC Resolution: 8 bits (0-255)
• $V_{ref}$ = 5V
• Motor controller accepts 0-5V (0V = stopped, 5V = full speed)

DAC Formula:

$V_{out} = V_{ref} \times \frac{\text{Digital Input}}{2^n - 1}$

Calculate Output Voltages:

Arduino Code Example (try on Wokwi):

// Control motor speed with DAC output
int speed_percent = 50; // 50% speed
int dac_value = map(speed_percent, 0, 100, 0, 255);

// On ESP32 with built-in DAC (8-bit, 0-3.3V output):
dacWrite(DAC1, dac_value); // Outputs ~1.65V (50% of 3.3V Vref)
// Note: For 0-5V motor control, add an op-amp gain stage after DAC

35.18 Gamma-Corrected LED Dimming

Human eyes perceive brightness logarithmically, not linearly. A “50% brightness” setting needs gamma correction.

Linear DAC (appears too bright):

• 50% digital value (128/255) → 2.5V → appears ~75% bright to human eye

Gamma-Corrected DAC (perceptually accurate):

$\text{Corrected Value} = \left(\frac{\text{Desired Brightness}}{100}\right)^{2.2} \times 255$

For 50% perceived brightness:

$\text{Corrected Value} = \left(\frac{50}{100}\right)^{2.2} \times 255 = (0.5)^{2.2} \times 255 = 0.2176 \times 255 \approx 56$

Comparison:

Arduino Gamma Correction Code:

// Gamma correction lookup table (saves computation)
const uint8_t gamma8[] = {
    0,  0,  0,  0,  0,  1,  1,  1,  1,  2,  2,  2,  3,  3,  4,  4,
    5,  5,  6,  7,  8,  8,  9, 10, 11, 12, 13, 15, 16, 17, 18, 20,
   21, 23, 24, 26, 28, 30, 32, 34, 36, 38, 40, 43, 45, 48, 50, 53,
   56, 59, 62, 65, 68, 71, 75, 78, 82, 85, 89, 93, 97,101,105,110,
  114,119,123,128,133,138,143,149,154,160,165,171,177,183,189,195,
  202,208,215,222,229,236,243,250,255 // ... (256 values total)
};

void setLEDBrightness(int percent) {
    int linear = map(percent, 0, 100, 0, 255);
    int corrected = gamma8[linear]; // Apply gamma correction
    analogWrite(LED_PIN, corrected); // PWM output
}

35.19 Knowledge Check Quiz

35.20 ADC/DAC Calculation Links

Key Insight: Every ADC calculation starts with matching Vref to the sensor’s output range to maximize resolution. The floor() function causes quantization error (±½ LSB), which sets the fundamental accuracy limit. For output, choose DAC (true analog) for audio/RF or PWM (cheaper, digital) for LED/motor control where fast switching is acceptable.

35.21 Continue to the Next Part

Carry this evidence into ADC and DAC: Interactive Design Calculators, which begins with Interactive Calculators.