When studying practical examples of filter circuits, the 1st-order RC low-pass filter is the foundational building block for signal conditioning. For a 2 kHz cutoff frequency driving a 10 kSPS ADC, the exact component values you need are R = 10 kΩ and C = 8.2 nF (C0G/NP0 dielectric). This yields a theoretical cutoff of 1,941 Hz, providing adequate attenuation at the 5 kHz Nyquist limit while avoiding the common trap of ADC charge kickback. Below is the complete exam-style walkthrough, algebraic proof, and bench verification process.
The Problem Statement: ADC Anti-Aliasing Filter
Requirements:
- Target cutoff frequency ($f_c$): 2,000 Hz (to provide a guard band below the 5 kHz Nyquist frequency).
- Calculate the exact resistor and capacitor values.
- Select standard E12/E24 component values and recalculate the actual cutoff frequency.
Method Selection and The Hidden Trap
Which theorem applies and why? We use AC steady-state analysis via the s-domain voltage divider theorem. Because we are analyzing sinusoidal attenuation to find the -3 dB roll-off point, treating the capacitor as a complex impedance ($Z_c = 1 / j\omega C$) allows us to derive the standard cutoff frequency formula: $f_c = 1 / (2\pi RC)$.
The Trap in this Problem: Most textbook examples of filter design ignore the load. If you blindly solve the algebra by picking a massive resistor (e.g., $R = 1\text{ M}\Omega$) and a tiny capacitor (e.g., $C = 82\text{ pF}$) to achieve the same 2 kHz cutoff, you will fail on the bench. Why? ADC charge kickback.
When a SAR ADC samples, an internal switch closes, connecting the input pin to an internal sampling capacitor (typically 10–30 pF). This internal capacitor must charge to the input voltage within a fraction of the acquisition time. If your external filter capacitor ($C$) is too small, it will share its charge with the ADC's internal capacitor, causing a massive voltage droop. The external resistor ($R$) must then replenish this charge. If $R$ is too large (like 1 MΩ), the RC time constant of the replenishment loop is too slow, resulting in severe conversion errors that manifest as non-linearity or missing codes. Furthermore, a 1 MΩ resistor will form a voltage divider with the ADC's input impedance, introducing a DC offset error due to the 1 nA leakage current ($V_{error} = I \times R = 1\text{ nA} \times 1\text{ M}\Omega = 1\text{ mV}$, which is nearly 1 LSB for a 12-bit ADC on a 3.3V rail).
Step-by-Step Algebraic Solution
To avoid the trap, we must constrain our resistor value. A 10 kΩ resistor is high enough not to load the 50 Ω sensor, but low enough to quickly recharge the ADC sampling capacitor and minimize leakage-induced DC offset.
- Start with the cutoff frequency formula:
f_c = 1 / (2π × R × C) - Rearrange to solve for C:
C = 1 / (2π × R × f_c) - Substitute the knowns ($R = 10,000\ \Omega$, $f_c = 2,000\text{ Hz}$):
C = 1 / (2π × 10,000 × 2,000) - Calculate the denominator:
2 × 3.14159265 × 10,000 × 2,000 = 125,663,706 - Divide to find C in Farads:
C = 1 / 125,663,706 = 7.9577 × 10^-9 F - Convert to nanofarads (nF):
C = 7.957 nF - Select the nearest standard E12 capacitor value:
The closest standard value is 8.2 nF. - Recalculate the actual cutoff frequency with the chosen part:
f_c_actual = 1 / (2π × 10,000 × 8.2 × 10^-9)
f_c_actual = 1 / 0.00051522
f_c_actual = 1,941 Hz
At the 5 kHz Nyquist frequency, this 1,941 Hz filter will provide an attenuation of -8.8 dB (a voltage ratio of 0.36), which is sufficient for a 12-bit ADC anti-aliasing guard band.
Sanity Check and Independent Verification
Order of Magnitude & Units Check:
Ohms ($\Omega$) are Volts/Amps, which is Volts/(Coulombs/second). Farads (F) are Coulombs/Volts. Multiplying them: $(V / (C/s)) \times (C/V) = \text{seconds}$. The inverse of the RC product is $1/\text{seconds}$, which is Hertz (Hz). The units balance perfectly.
For the order of magnitude: $R \approx 10^4$, $C \approx 10^{-8}$. The product is $10^{-4}$. The inverse is $10^4$. Dividing by $2\pi$ (roughly 6) yields a value in the low thousands ($1.6 \times 10^3$ to $2 \times 10^3$). Our exact answer of 1,941 Hz sits perfectly in this expected range.
How to verify the answer independently on the bench:
Do not trust the math until you see it on a scope. Build the circuit on a breadboard. Set your function generator to output a 2.0 Vpp sine wave. Connect Oscilloscope Channel 1 to the filter input and Channel 2 to the filter output. Sweep the frequency from 100 Hz up to 10 kHz. At exactly 1,941 Hz, the output amplitude on Channel 2 must read 1.414 Vpp (which is $2.0\text{ Vpp} \times 0.707$, representing the -3 dB point). If the amplitude reads higher, your capacitor is likely a degraded X7R part exhibiting DC bias capacitance drop, or your resistor is out of tolerance.
Component Decision Tree and Final Pick
Choosing the right physical component is just as critical as the math. Use this decision matrix to lock in your bill of materials (BOM).
| Condition / Constraint | Required Action | Concrete Pick |
|---|---|---|
| Source impedance is 50 Ω | Keep R < 100× source to avoid loading the sensor signal. | R = 10 kΩ |
| ADC internal sampling cap = 20 pF | External C must be > 10× sampling cap to absorb charge kickback without droop. | C = 8.2 nF (8200 pF) |
| High precision / wide temp range | Avoid X7R/Y5V dielectrics; they lose up to 50% capacitance under DC bias or temperature shifts. | C0G / NP0 Ceramic |
| Resistor noise / drift matters | Avoid carbon composition; use low-noise film. | 1% Metal Film (e.g., Vishay MRS25) |
| Final Default Recommendation (Use for 95% of 10-100 kSPS ADC setups) | 10 kΩ 1% Metal Film, 8.2 nF C0G 50V | |
Frequently Asked Questions
Why not use an active op-amp filter instead of a passive RC?
For a simple 1st-order roll-off, an active filter adds unnecessary cost, board space, and op-amp noise. Passive RC is preferred unless you need a sharp 2nd-order (or higher) Butterworth/Chebyshev roll-off or require gain.
Can I use a multilayer ceramic capacitor (MLCC) with X7R dielectric if I can't find C0G?
You can, but you must derate it. An X7R capacitor rated at 8.2 nF might only provide 6 nF of actual capacitance at room temperature with a 3.3V DC bias applied. If you must use X7R, select a value two steps higher (e.g., 15 nF) and verify the cutoff frequency on the bench. For rigorous examples of filter design in precision metrology, always default to C0G/NP0.
For deeper reading on ADC driving circuits and filter topologies, refer to the All About Circuits AC Textbook chapter on low-pass filters and Texas Instruments Precision Labs ADC training series.
