Thermal Noise of a Resistor


Thermal noise of a resistor

Figure 1. Thermal noise model of a resistor

The thermally agitated charge carriers introduce random fluctuations in the voltage across the conductor even the average current through it is zero. This noise was first reported by Johnson and explained the cause as the Brownian movement of charge carriers by Nyquist. Hence this noise is also called Johnson noise or Nyquist noise. The thermal noise of a resistor is modeled as a rms voltage source (\overbar{v_n}) in series with its resistance(R) as illustrated in Figure 1.

Noise power in a resistor at temperature ‘T’ is k T \Delta{f}. Maximum power is transferred to load, when load has same resistance as the noise source. The noise power delivered to load under this condition is called available noise power P_{\tiny AN}.

(1)   \begin{equation*} P_{\tiny AN} = (\frac{\overline{v_n}}{2})^{\tiny 2}.\frac{1}{R} = k T \Delta{f} \quad \Rightarrow \quad \boxed{\overline{v_n^2} = 4 k T R \Delta{f}} \end{equation*}

or voltage spectral density (S_v(f)) is

(2)   \begin{equation*} \boxed{S_{v}(f)=\frac{\overline{{v_n}^{\tiny 2}}}{\Delta{f}} =4 k T R} ---- \left({V^{\tiny 2}}/{Hz}\right) \end{equation*}

where,
k=1.38 \times 10^{\tiny -23} J/K is Boltzman constant
T is absolute temperature of resistor in Kelvin.
\Delta{f} is the measurement bandwidth

The spectral density of thermal noise is independent of frequency. Hence it is called white noise.

Rule of thumb:

At 300K, a 50\Omega resistor

  • has a spectral density of 8.28\times 10^{\tiny -19}V^2/Hz
  • generates 0.91 ~nV/\sqrt{Hz} noise voltage.
  • noise power k T \Delta f = -173.83 dBm over 1Hz BW.

Thermal Noise model of  a resistor

Norton equivalent representation of thermal noise is illustrated in Figure 2. and the noise current is given as
\overline{i_n^2} = \left({\overline{v_n}\over R}\right)^{2} = {4k T\over R}\Delta{f}

Figure 2. Norton equivalent model of thermal noise

Current spectral density is given by

(3)   \begin{equation*} \boxed{S_{i}(f)=\frac{\overline{{i_n}^{\tiny 2}}}{\Delta{f}} = {4 k T \over R}} ----\left({A^{\tiny 2}}/{Hz}\right) \end{equation*}

Intutively, it is the voltage one would see if measured across a resistor held at a temperature T using using a measurement system with a bandwidth of \Delta f. Therefore the noise increases with system bandwidth. A wideband system has higher noise floor than a narrow band system. Therefore sensitivity of Rx degrades as the bandwidth of the system increases. Eg. GSM Vs WLAN.

Depending on the circuit topology, one model may lead to simpler calculation than the other. Once the polarity is chosen it must be retained throughout the analysis of the circuit.

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