Exercises: Op-Amps, ADC, and Sampling Theory

Chapter 3: Signal conditioning — getting clean numbers from messy analog signals

Self-assessment guide: Write your answer before expanding the details block. If you can answer 80% without peeking, you’ve mastered the analog signal chain.

Section A — Conceptual Questions

A1. Your colleague says “the STM32 has a 12-bit ADC, so we get 12 bits of accuracy.” You check the datasheet and see INL = ±3 LSB. How many bits of effective accuracy do you really have?

A2. A teammate reads the ICM-42688 IMU at 100Hz via SPI, and notices that accelerometer data sometimes has strange low-frequency oscillations that don’t correspond to robot movement. The IMU’s internal ODR is set to 100Hz with DLPF disabled. What’s likely causing this?

A3. Your colleague designs a non-inverting amplifier with gain = 100 using an LM358 op-amp (GBW = 1MHz). The sensor signal is 500Hz. They say “gain is 100, frequency is only 500Hz, well below 1MHz, so it should work.” What’s the actual bandwidth at gain = 100?

A4. You need to measure a 100kΩ thermistor with the STM32 ADC. A colleague suggests connecting the thermistor directly to the ADC pin (in a voltage divider with a 100kΩ reference resistor). The ADC is configured for maximum speed (shortest sampling time). Will this work?

A5. An op-amp is wired as a voltage follower but the output oscillates wildly at ~5MHz. The circuit looks correct (V+ from sensor, V- connected to Vout, supply pins decoupled). What’s likely causing the oscillation?

A6. Your ADC reference voltage comes from the STM32’s VDDA pin, which is connected to the 3.3V rail through a ferrite bead. The 3.3V rail comes from a switching regulator. Your colleague says “the 12-bit ADC gives us 0.806mV resolution, so we can detect signal changes smaller than 1mV.” What’s the flaw in this reasoning?

A7. A 1kHz signal is sampled at 1.5kHz (below the 2kHz Nyquist requirement). At what frequency does the aliased ghost appear? Can any digital filter applied after sampling remove it?

A8. An inverting amplifier has Rin = 10kΩ and Rf = 100kΩ. The input is a 0.2V signal from a temperature sensor. What is the output voltage? Is there a practical problem with using this circuit on a single-supply 3.3V system?

A9. Your STM32 12-bit ADC gives noisy readings (±8 LSB jitter) when measuring a stable DC voltage. You enable 16× hardware oversampling. What effective resolution do you now have? What’s the output rate if the base conversion rate was 1Msps?

A10. A comparator (op-amp without feedback) is used to detect when a battery voltage drops below 3.0V. V+ is connected to the battery through a voltage divider, V- is connected to a 1.5V reference. The system works but triggers repeatedly when the battery is near 3.0V. What’s happening and how do you fix it?

Section B — Spot the Bug

B1. Non-inverting amplifier gain calculation:

“I need gain = 5. Using the non-inverting formula: Gain = Rf/Rin = 5. So Rf = 50kΩ, Rin = 10kΩ.”

What’s wrong?

B2. Anti-alias filter design for a 500Hz ADC sample rate:

“Nyquist frequency = 500Hz. Anti-alias cutoff = 500Hz. Using R = 3.3kΩ, C = 100nF. f_c = 1/(2π × 3300 × 100e-9) = 482Hz. Close enough to 500Hz.”

Two errors. Find them.

B3. ADC voltage calculation:

STM32 12-bit ADC, Vref = 3.3V. ADC reads 3000. “Voltage = 3000 / 4096 = 0.732V”

Find the error.

B4. Oversampling calculation:

“I need 16-bit effective resolution from my 12-bit ADC. That’s 4 extra bits. Oversampling ratio = 4² = 16×. Sample at 16× the output rate.”

Find the error.

B5. Sensor signal conditioning chain:

“I have a 0–5V sensor feeding the STM32’s 3.3V ADC. I’ll add an RC anti-alias filter (R=3.3kΩ, C=100nF) directly on the sensor output, then it goes straight to the ADC pin.”

Two problems. Find them.

B6. A differential amplifier is used to measure current through a 0.1Ω shunt resistor carrying up to 10A:

“Voltage across shunt = 10A × 0.1Ω = 1V. Using matched resistors R1 = Rf = 10kΩ, gain = Rf/R1 = 1. Output = 1V. Feeding directly to 3.3V ADC.”

This gives wildly inaccurate readings. Why?

Section C — Fill in the Blank / From Memory

C1. Op-Amp Gain Formulas

C2. Op-Amp Golden Rules

When negative feedback is present:

• Rule 1: ______
• Rule 2: ______

When do these rules break down? (Name at least 3 cases): 1. ______ 2. ______ 3. ________

C3. ADC Key Values at 3.3V Vref

Formula: LSB = __ | SNR_max = ____

C4. Sampling Theory Formulas

1. Nyquist theorem: f_sample ≥ ______
2. Aliased frequency: f_alias = ______
3. Anti-alias filter cutoff target: f_c ≤ ______
4. Oversampling extra bits: extra = ______
5. Quantization noise RMS: σ = ______

C5. ADC Architecture Comparison

Section D — Lab / Calculation Tasks

D1. Non-Inverting Amplifier Design

A force sensor outputs 0–200mV. You need to amplify this to 0–3.0V for the STM32 12-bit ADC (Vref = 3.3V).

a) What gain do you need? b) Choose Rin and Rf (use standard E12 values). c) Verify the gain with your chosen values. d) What percentage of the ADC range does 3.0V use? e) What force sensor voltage change corresponds to 1 LSB?

D2. Anti-Alias Filter Design for IMU

Your SPI bridge reads the IMU at 100Hz. The IMU internal ODR is 1kHz. You want to add an external anti-alias filter on the analog accelerometer backup channel that samples at 500Hz.

a) What is the Nyquist frequency? b) Design a single-stage RC filter. Choose R and C values. c) At 1kHz (a common motor vibration frequency), what attenuation does this filter provide? d) Is single-stage enough, or do you need second-order? Justify with numbers.

D3. Oversampling vs. External ADC

You need 16-bit effective resolution at 100Hz output rate. Your STM32 has a 12-bit ADC capable of 2Msps.

a) How many extra bits do you need? What oversampling ratio? b) At what rate must the ADC sample? c) Is this feasible with the 2Msps ADC? d) An alternative: the ADS1115 is a 16-bit external ADC at 860sps over I2C. Compare the approaches.

D4. Aliasing in Practice

A robot wheel encoder produces a 450Hz signal at top speed. Your ADC samples at 1000Hz.

a) Does this violate Nyquist? By how much? b) Now the robot speeds up and the encoder reaches 600Hz (you didn’t expect this). Where does the aliased signal appear? c) The aliased signal looks like what real speed to the software? How would this affect the robot’s odometry?

D5. Complete Signal Chain Design

Design the analog conditioning chain for reading a 0–10V industrial pressure sensor with the STM32 12-bit ADC (Vref = 3.3V, 100Hz output needed, 1kHz sample rate).

Specify each stage with component values: a) Input protection b) Voltage scaling c) Buffering (if needed — justify) d) Anti-alias filter e) Calculate the sensor voltage resolution per LSB

Section E — Deeper Thinking

E1. In your STM32H7 ↔ Jetson Orin SPI bridge project, the ICM-42688 IMU has its own internal ADC (16-bit ΔΣ) sampling at 32kHz, an internal DLPF (anti-alias), and outputs digital data over SPI. You read it at 100Hz. Explain why this architecture (analog sensing → high-rate ADC → digital filter → decimation → SPI read) is fundamentally better than if you tried to read an analog accelerometer with the STM32’s 12-bit SAR ADC at 100Hz. Cover at least: aliasing, resolution, noise, and system complexity.

E2. The Nyquist theorem says you must sample at ≥ 2× the signal frequency. But in practice, audio CDs sample at 44.1kHz for a 20kHz audio bandwidth (2.205×, barely above 2×). Professional audio uses 96kHz or 192kHz (4.8× to 9.6× Nyquist). Why sample so much faster than Nyquist? Relate this to the practical limitations of anti-alias filters and reconstruction filters.

E3. You’re debugging a mysterious 2Hz oscillation in the robot’s position estimate. The IMU data looks clean (no 2Hz content when logged directly). The ADC on a separate analog sensor channel shows the 2Hz oscillation clearly. The analog sensor itself is DC-coupled with a bandwidth of 500Hz. The ADC is sampled at 100Hz. Your colleague says “the sensor must be picking up 2Hz vibration from somewhere.” Propose an alternative hypothesis involving aliasing. How would you test each hypothesis?

E4. Connect the op-amp voltage follower concept (gain = 1, impedance transformation) to a software analogy you already know. Think about a function that takes an input and returns the same value but changes some property of how downstream consumers interact with it. Then explain why this matters for the signal chain from voltage divider → buffer → RC filter → ADC in your robot.

def get_sensor_value():          # Voltage divider (high impedance source)
    return database.query(...)   # Slow, one reader, breaks under load

def buffered_get():              # Op-amp voltage follower
    value = get_sensor_value()   # Reads once (no loading)
    return value                 # Returns same value, but fast,
                                 # multiple readers OK