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Electrical Circuits and Sensors · Module 10 of 10

Data Acquisition and Measurement Systems

The final link turns a continuous voltage into numbers a computer can store. Sample fast enough and resolve finely enough, and the digital record faithfully represents the physical world the sensor saw.

01

Readiness check

This closing module digitises the signal. Tick only what you can do closed-notes.

  • Recall that a sensor and conditioner produce an analogue voltage.
  • Evaluate a power of two, such as 210 = 1024.
  • Divide a voltage range into equal steps.
  • Recall a low-pass filter removes high frequencies.
  • Relate sample rate to the time between samples.
0 or 1 weak itemsContinue with this module.
2 weak itemsRevisit the conditioning filter in Module 9.
3 or more weak itemsConnect to coding for acquisition in Programming and Computation.
02

The core idea

Digitising a signal has two axes: time and amplitude. Sampling discretises time and must obey the Nyquist rate; quantising discretises amplitude into 2n levels, each one LSB apart, with a rounding error of half an LSB.

fs > 2 fmax (Nyquist)LSB = Vref / 2nmax quantisation error = LSB / 2

An analogue-to-digital converter (ADC) samples the conditioned voltage at regular intervals and assigns each sample to one of a finite set of levels. In time, the sampling theorem requires the sample rate fs to exceed twice the highest signal frequency; sampling too slowly folds high frequencies down into the signal band as aliases that cannot be removed afterward. In amplitude, an n-bit converter splits its reference span Vref into 2n equal steps; the size of one step is the least significant bit, LSB = Vref/2n, and the resolution of the system. Rounding each sample to the nearest level introduces a quantisation error of at most half an LSB. More bits mean finer steps and smaller error; a faster sample rate and an anti-alias filter protect the time axis.

The skill works when: you size the LSB from the bits and reference, and the sample rate from the highest frequency.
The skill breaks down when: the signal is sampled below the Nyquist rate, or the LSB is computed from 2n − 1 levels carelessly.
The concept. The ADC samples in time and rounds in amplitude to one of 2n levels. The step between levels is the LSB, and the rounding adds at most half an LSB of error.
03

The skills, taught in order

Five skills follow the signal through the acquisition chain, from sampling in time to resolving in amplitude.

10.1 The data acquisition chain

A data acquisition system runs from the conditioned sensor voltage through an anti-alias filter, a sample-and-hold, an ADC, and into memory. Each stage preserves or limits fidelity; the engineer matches the rate and resolution to the signal and its noise.

10.2 Sampling and the Nyquist rate

Sampling records the signal at discrete instants spaced by 1/fs. The sampling theorem requires fs > 2 fmax, twice the highest frequency present. Meeting it preserves the signal exactly in principle; missing it loses information irretrievably.

n bitsLevels 2nLSB at 5 VResolution
825619.5 mVcoarse
1010244.88 mVmoderate
1240961.22 mVfine
166553676.3 µVvery fine

Each added bit halves the LSB and the quantisation error. The reference voltage sets the absolute step size.

10.3 ADC resolution and the LSB

An n-bit ADC divides its reference span into 2n steps. One step, the least significant bit LSB = Vref/2n, is the smallest voltage change the converter can distinguish. It sets the resolution of the whole measurement, no matter how good the sensor.

10.4 Quantisation error

Rounding each sample to the nearest level leaves an error between −½ LSB and +½ LSB, so the maximum quantisation error is half an LSB. It behaves like added noise; more bits, or a reference matched to the signal range, reduce it.

10.5 Aliasing

If a frequency above half the sample rate is present, it appears in the data as a false lower frequency, an alias, indistinguishable from a real signal. An anti-alias low-pass filter before the ADC removes those frequencies so they cannot fold back.

Engineering connection: the anti-alias filter is the same first-order low-pass from Module 9, now placed to protect the sampler, and the digitised data feeds directly into Control Systems.

04

Worked example 1: resolution of a 10-bit ADC

A 10-bit ADC has a 5 V reference. Find the number of levels, the LSB voltage, and the maximum quantisation error.

Figure 1. The 5 V span is split into 1024 steps of 4.88 mV each. Any voltage is rounded to the nearest step, with an error no larger than half a step, 2.44 mV.
  1. ProblemFind the levels, LSB, and maximum quantisation error of the 10-bit ADC in Figure 1.
  2. Given / findn = 10, Vref = 5 V. Find 2n, LSB, and the maximum error.
  3. AssumptionsIdeal uniform quantiser over the full 0 to 5 V span.
  4. ModelNumber of levels is 2n; LSB = Vref/2n; maximum error is LSB/2.
  5. Equationslevels = 2nLSB = Vref/2nerrormax = LSB/2
  6. Solve210 = 1024 levels. LSB = 5/1024 = 4.88 mV. Maximum error = 4.88/2 = 2.44 mV.
  7. Check1024 × 4.88 mV = 5.00 V, recovering the full reference. The error of 2.44 mV is about 0.05 percent of full scale, the resolution limit of a 10-bit system.
  8. ConclusionThe 10-bit converter resolves to about 4.9 mV. A sensor signal finer than that is lost in quantisation, which is why bit depth matters.
Result. 1024 levels, LSB = 4.88 mV, maximum error = 2.44 mV.
05

Worked example 2: a 12-bit converter on a 3.3 V reference

A 12-bit ADC runs from a 3.3 V reference, common in microcontrollers. Find the LSB voltage and the maximum quantisation error, and compare its resolution to the 10-bit converter.

Figure 2. At the same reference voltage, going from 10 to 12 bits quadruples the level count and cuts the LSB to one quarter. Here, the smaller 3.3 V reference reduces the LSB further.
  1. ProblemFind the LSB and maximum error of the 12-bit ADC in Figure 2, and compare to 10-bit.
  2. Given / findn = 12, Vref = 3.3 V. Find LSB, the maximum error, and the comparison.
  3. AssumptionsIdeal uniform quantiser over the full 0 to 3.3 V span.
  4. ModelLSB = Vref/2n; maximum error = LSB/2.
  5. Equations212 = 4096LSB = Vref/2nerrormax = LSB/2
  6. Solve212 = 4096 levels. LSB = 3.3/4096 = 0.806 mV. Maximum error = 0.806/2 = 0.403 mV.
  7. CheckThe 12-bit converter has four times as many levels, and its 3.3 V span is smaller than the 10-bit converter's 5 V span. Both changes reduce the LSB, from 4.88 mV to 0.806 mV, so the voltage resolution improves by about six times.
  8. ConclusionBit depth dominates resolution. A 12-bit converter resolves under a millivolt even on a modest reference, fine enough for most conditioned sensor signals.
Result. 4096 levels, LSB = 0.806 mV, maximum error = 0.403 mV.
06

Misconceptions and diagnostics

MistakeSymptomDiagnostic questionCorrection
Sampling too slowlyFalse low-frequency tones appear"Is fs above twice fmax?"Meet the Nyquist rate and filter before sampling.
No anti-alias filterHigh-frequency noise folds into the band"What removes content above fs/2?"Place a low-pass filter ahead of the ADC.
Confusing bits with accuracyMore bits assumed to fix a noisy sensor"Is the sensor itself this fine?"Resolution is not accuracy; the sensor and noise still limit it.
Reference far above signalSignal uses few of the levels"Does the signal span the reference?"Match Vref or amplify the signal to use the full range.
07

Practice ladder

Level 1 · Direct skill

An 8-bit ADC has a 5 V reference. Find the LSB.

Show answer

LSB = Vref/2n = 5/256 = 19.5 mV.

Level 2 · Mixed concept

A signal contains frequencies up to 2 kHz. What is the minimum sample rate, and a sensible practical choice?

Show answer

Nyquist requires fs > 2 × 2 kHz = 4 kHz. In practice sample several times higher, say 10 kHz, and filter above 2 kHz to avoid aliasing.

Level 3 · Independent problem

A 12-bit ADC on a 10 V reference reads a conditioned signal. What is the smallest voltage change it can resolve, and the maximum quantisation error?

Show answer

LSB = 10/4096 = 2.44 mV; maximum error = LSB/2 = 1.22 mV.

Transfer task | Real engineering

You are logging a 50 Hz vibration with a conditioned signal of 0 to 3 V. Choose a sample rate and a bit depth, and justify both.

What good work looks like

Sample well above 100 Hz (say 1 kHz) with an anti-alias filter near 100 Hz, and use a 12-bit ADC with a reference near 3 V so the LSB is under a millivolt, comfortably finer than the signal detail.

08

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that my sample rate satisfies the Nyquist condition."
"Give me three ADC specs; I will find the LSB and error of each."
"Pick the ADC for me." Sizing the LSB and rate from the signal is the skill.
"Why is there aliasing?" Reasoning from fs and fmax is the point.

Portfolio task

Specify a complete acquisition for a real signal: a sample rate, an anti-alias corner, a bit depth, and a reference, then state the resulting resolution and quantisation error.

Must include: a sample rate meeting Nyquist, a bit depth, an LSB, and a maximum error.
09

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. State the Nyquist condition.

The sample rate must exceed twice the highest signal frequency, fs > 2 fmax.

2. What is the LSB?

Vref/2n, the smallest voltage step an n-bit ADC resolves.

3. How large is the quantisation error?

At most half an LSB.

4. What is aliasing, and how is it prevented?

High frequencies folding into the band; an anti-alias filter before the ADC removes them.

5. Does adding bits improve accuracy?

It improves resolution, but the sensor and noise still set the true accuracy.

TodayFinish this quiz and Levels 1 and 2 of the ladder.
+1 dayRe-derive the LSB and quantisation error from a blank page.
+3 daysSpecify an acquisition for a new signal.
+7 daysCombine sensor, conditioning, and acquisition into one measurement design.
+30 daysCarry these digitised signals into Control Systems, the recommended next course.
10

Textbook mapping

This module draws on Alexander and Sadiku, Fundamentals of Electric Circuits, 4th edition for the analogue front end, with sampling and conversion from standard measurement texts.

Topic in this moduleWhere to read more
Filtering and the analogue front endAlexander & Sadiku, Chapter 14
Sampling, ADC resolution, quantisationFigliola & Beasley, measurement systems
Aliasing and the Nyquist rateFigliola & Beasley, measurement systems

The analogue front end follows the 4th edition of Alexander and Sadiku; sampling and conversion draw on measurement texts such as Figliola and Beasley.