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What is the Noise in Ideal Capacitors?

Published in Capacitor Theory 3 mins read

Ideal capacitors, being lossless components, do not generate thermal noise.

While ideal capacitors themselves are noise-free, it's crucial to understand the context in which capacitors are used. The term "kTC noise" often arises when discussing capacitors in circuits, particularly in relation to switched-capacitor circuits and reset noise. This isn't noise generated by the capacitor itself, but rather a limitation imposed by the unavoidable presence of resistance in real circuits, combined with the thermal energy present due to temperature.

Here's a breakdown:

  • Ideal vs. Real Capacitors: In theory, an ideal capacitor only stores energy and has no internal resistance. In reality, all capacitors have some equivalent series resistance (ESR). This ESR is what contributes to noise when the capacitor is part of a circuit.

  • kTC Noise: This noise voltage is given by:

    V_noise (rms) = sqrt(kT/C)

    Where:

    • k is Boltzmann's constant (approximately 1.38 x 10^-23 J/K)
    • T is the absolute temperature in Kelvin
    • C is the capacitance in Farads

  • Source of kTC Noise: Imagine a capacitor being connected to a resistor. Thermal energy in the resistor causes random fluctuations in voltage across the resistor. When the capacitor is connected, it charges to this fluctuating voltage. Even after the resistor is disconnected (e.g., in a switched-capacitor circuit), the capacitor holds this "noisy" voltage. Therefore, kTC noise is not inherently a property of the capacitor, but is instead determined by how a capacitor is charged with a resistor in a thermal environment.

  • Example: Consider a 1 pF capacitor at room temperature (approximately 300K). The kTC noise voltage would be:

    V_noise (rms) = sqrt((1.38 x 10^-23 J/K * 300 K) / (1 x 10^-12 F))
V_noise (rms) ≈ 64.3 μV

    This indicates that even a relatively small capacitance experiences a measurable amount of noise.

  • Practical Implications: kTC noise becomes important in sensitive analog circuits, data converters (especially sample-and-hold circuits), and other applications where low noise is critical. It sets a fundamental lower limit on the noise performance of many circuits.

In summary, while an ideal capacitor is theoretically noiseless, the combination of capacitance, resistance, and thermal energy in real-world circuits leads to phenomena like kTC noise, impacting the overall noise performance of the system. The noise originates from the resistor, not the capacitor itself, though the capacitor stores the noise voltage.

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