KTC Noise Calculator

The KTC Noise Calculator computes thermal noise on capacitors from Boltzmann’s constant, absolute temperature and capacitance, aiding analogue circuit design.

KTC Noise Calculator Explained

KTC noise arises when a capacitor is charged through a resistor at a finite temperature and then disconnected. Due to thermal agitation of electric charge, the capacitor does not hold a perfectly stable voltage. Instead, it exhibits a random voltage fluctuation called thermal noise.

The classic result is that the mean‑square voltage noise on a sampled capacitor is given by ( overline{v_n^2} = frac{kT}{C} ). Here, (k) is Boltzmann’s constant, (T) is absolute temperature in kelvin, and (C) is capacitance in farads. The calculator uses this relationship to compute the noise voltage, typically reported as its root‑mean‑square (RMS) value.

Designers often compare KTC noise to the least significant bit (LSB) of an analog‑to‑digital converter (ADC). If the KTC noise is larger than one LSB, precision is fundamentally limited no matter how good the rest of the circuit is. The KTC Noise Calculator makes this comparison straightforward by tying physical constants, units, and derived results into one simple workflow.

The Mechanics Behind KTC Noise

Thermal noise originates from the random motion of charge carriers in resistive paths. When a capacitor is connected to a resistor at temperature (T), the resistor’s thermal noise appears across the capacitor. After the capacitor is disconnected, some of that randomness remains as noise voltage.

• A resistor at temperature (T) has a thermal noise spectral density of (4kTR) volts squared per hertz across it.
• When a capacitor is connected to this resistor, the resistor noise is low‑pass filtered by the RC network, limiting the noise bandwidth.
• Over enough time, the capacitor reaches thermal equilibrium with the resistor, accumulating a mean‑square voltage of (kT/C).
• When the sampling switch opens, the capacitor is isolated and “freezes” that random voltage, which becomes the sampled KTC noise.
• The RMS noise voltage is the square root of (kT/C), giving a practical single number to compare with signal levels.

This mechanism is fundamental and does not depend on the detailed implementation of the switch or resistor. Any sampled system that charges a capacitor from a resistive source at non‑zero temperature will face this limit. That is why KTC noise is often treated as an unavoidable baseline noise in precision sampling circuits.

KTC Noise Formulas & Derivations

The KTC noise expression comes from analyzing a resistor–capacitor network in thermal equilibrium using basic statistical physics. The key is the equipartition theorem, which states that each independent quadratic energy term has an average energy of ( frac{1}{2}kT ). For a capacitor with voltage noise (v_n), the stored energy is ( frac{1}{2} C v_n^2 ).

• Equipartition gives ( frac{1}{2} C overline{v_n^2} = frac{1}{2} kT ), so ( overline{v_n^2} = frac{kT}{C} ) (volts squared).
• The RMS noise voltage is ( v_{n,text{rms}} = sqrt{overline{v_n^2}} = sqrt{frac{kT}{C}} ), with units of volts.
• Boltzmann’s constant is (k approx 1.380,649 times 10^{-23},text{J/K}), where joule (J) is the unit of energy.
• If you want noise in terms of charge, the mean‑square noise charge on the capacitor is ( overline{q_n^2} = kTC ) in coulomb squared.
• To compare KTC noise with an ADC LSB, you can compute ( text{LSB} = frac{V_text{FS}}{2^N} ), then evaluate ( v_{n,text{rms}} / text{LSB} ).

The calculator uses these formulas to derive noise values from your selected temperature and capacitance. It can also express results in different units, such as microvolts RMS, so you can quickly compare them with data sheet specifications. While the derivation is simple, careful attention to units is essential to avoid errors in real designs.

What You Need to Use the KTC Noise Calculator

To compute KTC noise, you only need a few physical parameters and, optionally, some system‑level numbers. The core inputs define how much noise energy your sampled capacitor can store. Extra entries help relate that noise to resolution and performance targets.

• Capacitance (C) in farads (typically entered as picofarads or nanofarads).
• Temperature (T) in kelvin or degrees Celsius (the calculator converts Celsius to kelvin internally).
• Desired output unit for noise voltage (volts, millivolts, or microvolts RMS).
• Optional ADC full‑scale voltage (V_text{FS}) in volts, to compare noise to LSB size.
• Optional ADC resolution (N) in bits, to compute LSB size and noise‑to‑LSB ratio.

The calculator checks for edge cases, such as extremely small capacitances or unrealistic temperatures. Very small capacitors (for example, below 1 femtofarad) produce large noise values that may exceed circuit supply rails, which indicates an impractical design. Temperatures below absolute zero or negative capacitances are physically impossible and will be rejected with clear error messages.

How to Use the KTC Noise Calculator (Steps)

Here’s a concise overview before we dive into the key points:

1. Enter the sampling capacitance value and select the appropriate unit (F, nF, pF, or fF).
2. Enter the operating temperature, either in kelvin or degrees Celsius, and choose the correct unit.
3. (Optional) Enter your ADC full‑scale voltage and resolution in bits if you want noise‑to‑LSB comparisons.
4. Choose the preferred output unit for noise voltage, such as microvolts RMS.
5. Click the Calculator button to compute the KTC noise and any derived quantities, such as LSB size.
6. Review the reported RMS noise voltage, noise‑to‑LSB ratio, and any warnings about extreme or invalid inputs.

These points provide quick orientation—use them alongside the full explanations in this page.

Example Scenarios

Imagine a precision 16‑bit ADC using a 10 pF sampling capacitor at room temperature (25 °C, about 298 K) with a 2.5 V full‑scale range. Plugging these into the calculator, you obtain a KTC noise of roughly 64 µV RMS. The 16‑bit LSB is about 38 µV, so the KTC noise is about 1.7 LSBs. What this means

Consider a low‑power sensor front end that uses a large 1 nF hold capacitor at 85 °C (about 358 K) and a 1.8 V, 12‑bit ADC. Using the calculator, the KTC noise is around 0.7 µV RMS, whereas the LSB is about 440 µV. The KTC noise is far below the ADC quantization level, so capacitor size could possibly be reduced to save area or cost. What this means

Limits of the KTC Noise Approach

The KTC model captures only one source of noise: thermal sampling noise on the capacitor. Real circuits include many other contributors. These can raise the total noise floor well above the theoretical KTC limit, especially in integrated systems with complex switching behavior.

• Op‑amp input noise and bias currents add to the sampled noise and may dominate in high‑gain stages.
• Switch resistance and non‑idealities, such as charge injection, introduce extra noise and offset beyond pure KTC effects.
• Flicker noise (1/f noise) in MOS transistors becomes important at low frequencies and is not included in the KTC formula.
• Quantization noise from ADCs and digital processing can overshadow thermal noise in moderate‑resolution systems.
• Layout‑related coupling, such as substrate noise and supply ripple, adds deterministic and random disturbances that are not thermal.

Because of these factors, the KTC Noise Calculator should be treated as a baseline estimator, not a complete noise simulator. It is most useful during early design to set capacitor sizes and to understand fundamental limits. Detailed noise budgeting later should include additional models and, where possible, measurements from prototypes.

Units Reference

Correct units are critical when working with very small capacitances and low noise voltages. Mixing up prefixes or temperature scales can easily cause errors of several orders of magnitude. The table below summarizes the key quantities used in KTC noise calculations and their standard units.

When using the calculator, make sure each input is entered in the correct base unit or chosen from the proper prefix. For example, entering 10 when you mean 10 pF requires selecting “pF” rather than “F”, or the computed noise will be off by many orders of magnitude.

Tips If Results Look Off

If the KTC Noise Calculator returns values that seem far too high or too low, the cause is often a unit or temperature mistake. Another common issue is entering effective capacitance values that do not match the actual sampling capacitor in your circuit. A quick systematic check usually reveals the problem.

• Verify that capacitance is in farads, not just a bare number; use pF or nF prefixes correctly.
• Confirm that temperature is realistic and converted properly between °C and K.
• Recheck ADC full‑scale voltage and bit resolution if noise‑to‑LSB results appear inconsistent.
• Compare the calculated noise against a hand calculation of ( sqrt{kT/C} ) to validate the output.

After confirming these points, if results still do not match expectations, consider other noise sources in your design. The sampled KTC noise may be only a small part of the total noise, and circuit‑level simulations or lab measurements may be needed for deeper analysis.

FAQ about KTC Noise Calculator

Does KTC noise depend on the resistor value used to charge the capacitor?

No, the final KTC noise on a fully settled sampled capacitor depends only on temperature and capacitance, not on the resistor value. The resistor affects how quickly the capacitor reaches equilibrium, but not the equilibrium noise level itself.

Should I use kelvin or degrees Celsius for temperature?

The underlying formula uses absolute temperature in kelvin, but the calculator accepts both kelvin and degrees Celsius. If you enter Celsius, it automatically converts it to kelvin using T = °C + 273.15 before computing KTC noise.

Can the KTC Noise Calculator predict total system noise?

No, it only estimates the thermal sampling noise on the capacitor. To predict total system noise, you must add contributions from amplifiers, switches, quantization, reference noise, and other circuit elements, typically using root‑sum‑square methods.

When should I worry that KTC noise is too high?

Compare the RMS KTC noise with your signal amplitude and ADC LSB. If KTC noise is on the order of one LSB or larger, it will significantly degrade resolution, and you may need to increase capacitance, reduce temperature, or adjust system requirements.

KTC Noise Terms & Definitions

KTC Noise

KTC noise is the thermal noise voltage or charge stored on a capacitor after it is charged through a resistive path and disconnected, quantified by the relationship ( overline{v_n^2} = kT/C ).

Boltzmann’s Constant

Boltzmann’s constant, denoted (k), is a fundamental physical constant that links temperature to energy in statistical mechanics and has the value 1.380 649 × 10⁻²³ J/K.

RMS Noise Voltage

RMS noise voltage is the square root of the mean‑square noise voltage, providing a single effective value that represents the average energy of a random voltage fluctuation.

Sampling Capacitor

A sampling capacitor is the capacitor that holds a snapshot of an analog signal in a sample‑and‑hold or switched‑capacitor circuit, and it directly determines the magnitude of KTC noise.

Least Significant Bit (LSB)

The least significant bit is the smallest voltage step that an ADC can resolve, equal to the full‑scale input range divided by 2 to the power of the converter’s bit resolution.

Equipartition Theorem

The equipartition theorem is a principle in statistical physics stating that each independent quadratic term in the energy of a system contributes an average energy of one‑half kT at thermal equilibrium.

Thermal Noise

Thermal noise is random electrical noise generated by the thermal motion of charge carriers in conductors and resistive elements, proportional to both temperature and resistance.

Quantization Noise

Quantization noise is the error introduced when converting a continuous‑time or continuous‑amplitude signal into discrete digital levels, often modeled as a uniform random noise source.

References

Here’s a concise overview before we dive into the key points:

• K. Bult and G. Geelen, “The CMOS gain‑boosting technique,” IEEE Journal of Solid‑State Circuits
• Analog Devices MT‑048: Op Amp Noise – Calculation and Measurement
• B. Razavi, “Design Considerations for High‑Speed, Low‑Voltage ADCs,” IEEE Journal of Solid‑State Circuits
• Texas Instruments application report: Noise Sources in Delta‑Sigma ADCs
• P. R. Gray and D. J. Hamilton, “MOS Operational Amplifier Design – A Tutorial Overview”
• NIST Special Publication 330: The International System of Units (SI)

These points provide quick orientation—use them alongside the full explanations in this page.