Editing Johnson–Nyquist noise (section)

=== Complex impedances ===
Nyquist's original paper also provided the generalized noise for components having partly [[electrical reactance|reactive]] response, e.g., sources that contain capacitors or inductors.<ref name=Nyquist/> Such a component can be described by a frequency-dependent complex [[electrical impedance]] <math>Z(f)</math>. The formula for the [[power spectral density]] of the series noise voltage is
:<math>
S_{v_n v_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Z(f)].
</math>
The function <math>\eta(f)</math> is approximately 1, except at very high frequencies or near absolute zero (see below).

The real part of impedance, <math>\operatorname{Re}[Z(f)]</math>, is in general frequency dependent and so the Johnson–Nyquist noise is not white noise. The RMS noise voltage over a span of frequencies <math>f_1</math> to <math>f_2</math> can be found by taking the square root of integration of the power spectral density:
:<math> V_\text{rms} = \sqrt{\int_{f_1}^{f_2} S_{v_n v_n}(f) df}</math>.

Alternatively, a parallel noise current can be used to describe Johnson noise, its [[power spectral density]] being
:<math>
S_{i_n i_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Y(f)].
</math>
where <math>Y(f) {=} \tfrac{1}{Z(f)}</math> is the [[electrical admittance]]; note that <math>\operatorname{Re}[Y(f)] {=} \tfrac{\operatorname{Re}[Z(f)]}{|Z(f)|^2} \, .</math>