Electronic Noise
Physics116C, 6/2/06
D. Pellett
Reference: Horowitz and Hill, The Art of Electronics, 2nd Ed., Ch. 7
Any original material copyright D. Pellett 2006
Electronic Noise
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Distinguish between Noise and Interference
Resistor at temperature T is a source of a random noise
voltage due to thermal fluctuations of charge carriers
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Called “Johnson” or “Thermal” noise
Spectral distribution is flat in frequency (“white noise”)
Other types of noise:
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Shot noise arises when charges cross a barrier
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Poisson statistics – also has flat frequency distribution
Flicker noise, approx. 1/f spectrum
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Has fractal properties suggesting chaotic origin
f α: α may not be exactly -1
Johnson Noise Derivation (1)
Thermal equilibrium
at temperature T
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Noise signal from left R travel right, absorbed in right R
Noise signal from right R travels left, absorbed in left R
Resistors come to thermal equilibrium, both R’s at same T,
power flowing left = power flowing right
(If not in equilibrium, hotter resistor sends more power until
equilibrium reached)
J.B. Johnson, Phys. Rev. 29, 367 (1927)
H. Nyquist, Phys. Rev. 32, 110 (1928)
Johnson Noise Derivation (2)
In the frequency interval ∆f,
let Pav = available power from each resistor.
Assume PavL = PavR = universal function of T:
Pav = <Vn2>av/R
The voltage source produces a wave propagating at velocity v on
the line.
Energy on line from left resistor:
E = ∫ Pav dt = Pav ∆t = Pav L/v
∆t
(where ∆t is the time to traverse the line, L = v∆t)
Total energy on line = 2Pav L/v
(from both R’s)
Johnson Noise Derivation (3)
Now suddenly short-circuit both ends of the line.
Energy is trapped in standing waves on the line.
Normal modes (the standing waves) within the
frequency range ∆f are excited.
Natural frequencies are f = nv/2L
Ends are now shorted
– get voltage nodes at each end
Density of states in frequency: dn = (2L/v) df
Number of states in the frequency interval ∆f :
Nf = (2L/v) ∆f
This is a quantum harmonic oscillator.
• Equally spaced levels, level spacing hf
• E = hf (n + ½) per photon in each level.
nλ/2 = L
f = v/λ = nv/2L
Johnson Noise Derivation (4)
Find average energy per mode*, assuming hf /kT
1
av. occupancy = 1/(exp(hf /kT) - 1) (Bose-Einstein Dist’n)
E = hf per photon in mode (ignoring zero point energy)
<E>av = average energy
<E>av = hf /(exp(hf /kT) - 1) ≈ kT
(if hf /kT
1).
(Get classical thermodynamic result in this approximation –
independent of f).
Total energy on line: Nf <E>av = (2L/v) ∆f <E>av = 2Pav L/v
or Pav ≈ kT∆f .
*See, for example, Feynman’s Lectures on Physics
Johnson Noise derivation (5)
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Relate to Vn:
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Since Vn2 just depends on ∆f and not directly on f, the spectrum
is flat in frequency – “white” noise (as long as hf /kT 1)
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For a 10 kΩ resistor at T = 300 K, ∆f = 10 kHz
<Vn>rms = (4 x 1.38x10-23 x 300 x 10000 x 10000)1/2 = 1.29 μV
Johnson Noise Vn Distribution
(B = bandwidth, ∆f )
Resistor Johnson Noise Data
Gaussian Vn dist’n
Waveform
(short and long
time scales)
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Flat frequency
spectrum
(fb = 16 kHz for
antialias filter)
Data collected using Johnson Noise Test Fixture from 116C Lab
Shot Noise
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Due to fluctuations in the number of independent
charge carriers crossing a boundary - random
process following Poisson distribution, quantized
by carrier charge: <In>rms = (2qIDCB)1/2
Gaussian In dist’n, uniform freq. dist’n (like Johnson
noise)
Example: current in a vacuum diode
(thermionic emission of carriers in hot cathode
at ground potential, collection on anode at
+HV)
Also photodiode or silicon strip leakage
current, BJT collector, base current, FET gate
leakage current, etc.
Not current crossing imaginary plane in
conductor or bulk semiconductor - carriers
are correlated (or shot noise effects averaged
over many collisions in short mean free paths,
reducing rms fluctuations - see paper by
Landauer)
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Flicker (1/f ) Noise
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Present in various electronic
components, e.g.,
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Carbon composite
resistors – have higher 1/f
noise than metal film or
wire-wound
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Amplifiers, particularly
evident at low frequencies
– see Op Amp spectra in
right-hand column
A.K.A. “Pink
noise” (emphasizes low
frequencies)
LT1793
1/f Noise Data
Gaussian Vn dist’n
Waveform –
note self-similar
time dist’ns
on different scales
(fractal-like)
Frequency
Spectrum
(log-log scales)
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Source: pseudorandom 1/f distribution from computer waveform
generator. (Note antialias filter cutoff in freq. spectrum near 20 kHz.)
BJT Noise Model
Base spreading resistance
≈ 40 Ω - 400 Ω
(decreases as current
increases)
JFET Noise Model
Non-Inverting Amplifier Noise Model
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LT1793 (first stage in Johnson Noise Test Fixture):
typical en = 6 nV Hz-1/2, in = 1 fA Hz-1/2 (f ≈ 1 kHz)
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AD797 (second stage amplifier):
typical en = 0.9 nV Hz-1/2, in = 2 pA Hz-1/2 (f ≈ 1 kHz)
LT1793 Noise Performance
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en and in are measured in noninverting configuration – evidently
can use “as is” in estimating noise
of your amplifier
AD797 Noise Performance
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This is the second stage
amplifier. In good design, its
noise contribution should be
negligible relative to the
amplified noise from the first
stage – the case here.
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One would use this amplifier as
the first stage if the signal
source impedance was < 1kΩ