2. INTRODUCTION
Johnson–Nyquist noise (thermal
noise, Johnson noise, or Nyquist noise) is
the electronic noise generated by the thermal
agitation of the charge carriers (usually
the electrons) inside an electrical conductor at
equilibrium, which happens regardless of any
applied voltage. The generic, statistical physical
derivation of this noise is called the fluctuation-
dissipation theorem, where generalized
impedance or generalized susceptibility is used
to characterize the medium.
3. CONTENTS
Noise voltage and power
Noise current
Noise power in decibels
Noise at very high frequencies
4. Noise voltage and power
Thermal noise is distinct from shot noise, which
consists of additional current fluctuations that
occur when a voltage is applied and a
macroscopic current starts to flow. For the
general case, the above definition applies to
charge carriers in any type of
conducting medium (e.g. ions in an electrolyte),
not just resistors. It can be modelled by a
voltage source representing the noise of the non-
ideal resistor in series with an ideal noise free
resistor.
5. The one-sided power spectral density, or voltage
variance (mean square) per hertz of bandwidth, is
given by:
where kB is Boltzmann's
constant in joules per kelvin, T is the resistor's
absolute temperature in kelvin, and R is the resistor
value in ohms (Ω).
Use this equation for quick calculation, at room
temperature:
6. A resistor in a short circuit dissipates a noise power
of
The noise generated at the resistor can transfer to
the remaining circuit ; the maximum noise power
transfer happens with impedance matching when
the Thévenin equivalent resistance of the remaining
circuit is equal to the noise generating resistance. In
this case each one of the two participating resistors
dissipates noise in both itself and in the other
resistor. Since only half of the source voltage drops
across any one of these resistors, the resulting noise
power is given by
(where P is the thermal noise power in watts)
7. Noise current
The noise source can also be modeled by a current
source in parallel with the resistor by taking
the Norton equivalent that corresponds simply to
divide by R. This gives the root mean square value of
the current source as:
Thermal noise is intrinsic to all resistors and is not a
sign of poor design or manufacture, although resistors
may also have excess noise.
8. Noise power in decibels
Signal power is often measured
in dBm (decibels relative to 1 milliwatt). From
the equation above, noise power in a resistor
at room temperature, in dBm, is then:
where the factor of 1000 is present because the
power is given in milliwatts, rather than watts.
This equation can be simplified by separating
the constant parts from the bandwidth :
9. Noise at very high frequencies
The above equations are good approximations at frequencies
below about 80 gigahertz (EHF). In the most general case,
which includes up to optical frequencies, the power spectral
density of the voltage across the resistor R, in V2/Hz is given
by:
where f is the frequency, h Planck's constant, kB Boltzmann
constant and T the temperature in kelvin. If the frequency is
low enough, that means:
(this assumption is valid until few terahertz at room
temperature) then the exponential can be approximated by
the constant and linear terms of its Taylor series. The
relationship then becomes:
10. In general, both R and T depend on frequency.
In order to know the total noise it is enough to
integrate over all the bandwidth. Since the signal
is real, it is possible to integrate over only the
positive frequencies, then multiply by 2.
Assuming that R and T are constants over all the
bandwidth , then the root mean square (RMS)
value of the voltage across a resistor due to
thermal noise is given by
