It’s better to know(actually have to know) kT/C if you design ADCs.

First, let’s take a look at its inverse-derivation (for detailed explanation, please go for elsewhere, there are lots of…):

Then, comes the dessert of today – Equipartition Theorem ðŸ˜‰

**General Note 1:**

The equipartition theorem states that energy is shared equally amongst all energetically accessible degrees of freedom of a system. That is also to say, a system will generally try to maximise its entropy (i.e. how â€˜spread outâ€™ the energy is in the system) by distributing the available energy evenly amongst all the accessible modes of motion.

**General Note 2:**

The equipartition theoremÂ states that if a system is in thermal equilibrium with a reservoir of temperature T, then each quadratic degree of freedom of the system will possess an energy Â½kT. A â€˜quadratic degree of freedomâ€™ is one for which the energy depends on the square of some property.

**Specific Note here:**

The thermal fluctuations in the voltage on a capacitor is the ultimate origin of circuit noise. Since the energy stored on the capacitor is , the voltage on the capacitor constitutes a degree of freedom. Thus,

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All credits go for: 1) an old paper published in 1993 – “White Noise in MOS Transistors and Resistors”; 2) the online document which explains the equipartition theorem; 3) Dr. Timmy’s PhD thesis, where introduces the interesting year-1993-paper.

Btw, sorry for the equations, which are really ugly!

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Great useful post.