Electronic Noise
Noise is everywhere  the audible noise we hear, the electronic noise we
measure and, usually, try to avoid. Noise reminds us that we live in a real
universe. On this page I'm going to look at some of the sources of unwanted noise that
are found in electronic circuits.
Note: I say unwanted noise as dedicated noise sources will not be covered on this page.
Thermal Noise
Thermal noise, or Johnson Noise, is noise energy generated in conducting materials
due to thermal effects. The simplified relation between noise power, \(P_n\) Watts,
temperature, \(T\) Kelvin, and bandwidth, \(B\) Hz, is
$$
P_n = 4 k T B
$$
where \(k\) is Boltmann's constant, \(1.38 \times 10^{23}\).
Note there is no mention of resistance. Since
$$
P = \frac{V^2}{R}
$$
it follows that the thermal noise voltage generated by a resistor \(R\)
for a given bandwidth \(B\) Hz at temperature \(T\) Kelvin is then
$$
V_n = \sqrt{4kTBR}
$$
An excellent article on resistor noise is
"Resistor Noise—reviewing basics, plus a Fun Quiz" recently published in EDN. One interesting thing to note is shown in the graph in figure 1: due to temperature, \(T\), being in degress Kelvin, changes in ambient temperature from 55°C to +125°C have very little effect on a resistor's thermal noise voltage.
The Effect Of Parasitic Capacitance
Due to the physical construction of resistors they all have some parasitic capacitance between the terminals, moreso when the resistor is soldered down onto a PCB, especially when you factor in trace capacitance.
Because this capacitor is in parallel with the resistor terminals it forms a singlepole lowpass filter, the classic RC filter. This has the effect of articially limiting the bandwidth of the resistor's noise, setting
an upper bound on the maximum noise that a resistor can inject into a circuit.
In the literature (e.g., Lundberg2002) this effect of parallel capacitance is called the kT/C noise. For a given \(R\) and \(C\) we know that
the noise bandwidth is
$$
\Delta f = \frac{1}{2\pi} \int_0^{\infty}\frac{1}{1+(\omega R C)^2} d\omega = \frac{1}{2\pi}\frac{\pi}{2RC} = \frac{1}{4RC}
$$
This gives us the bandwidth within which lies the total noise generated by this resistorcapacitor combination. Put this into the noise equation and we get a surprising result:
$$
V_n = \sqrt{4kT\frac{1}{4RC}R} = \sqrt{\frac{kT}{C}}
$$
Put simply, the upper bound on a resistor's thermal noise voltage is not determined by the resistor value, but by the parasitic capacitance of that resistor. And that is determined by the physical size of that resistor and
the associated PCB trace layout.
But what does that mean? Well, according to Vishay we could be looking at around 0.04pF parasitic shunt capacitance for small 0603 surface mount resistors.
Putting that into the above equation and assuming the device is at room temperature (298 Kelvin) we find the maximum noise we get from any resistor in that size is 320μV. Over the audio bandwidth
that is the same noise as an ideal capacitorless 250MΩ resistor.
This result has two (at least) conclusions for designing audio circuits. Firstly, system bandwidth is still the dominant noisecontrol mechanism that designers must use to manage noise.
And secondly, where resistors are being used in lowfrequency circuits (e.g., CV manipulation or bias compensation resistors for Bipolar opamps)
putting a small capacitor in parallel with the resistor will reduce the noise voltage considerably.
For example, consider a 10k bias compensation resistor connected to an opamp. By itself that would inject 1.8μV of noise into the circuit.
As there is no signal across this resistor we can connect a 100nF capacitor across it
(a common value used for supply rail decoupling) which would drop the noise contribution down to 0.2μV. That's a 19dB reduction!
E24 Rresistor Values, Noise Density and Noise Voltage
The table below lists the voltage noise densities and voltage noises for two decades
in the E24 resistor series. These figures have been calculated at T=296K (about 23°C),
and the noise voltages for the a bandwidth of 20Hz20kHz (B = 19980Hz).
For other resistance decades you can use this table and scale the noise voltages
accordingly. Since noise voltage is proportional to the square root of resistance
you need to do a doubledecade jump. For example, for a 100Ω resistor we
look at the row for 10kΩ (100 x 10^{2}) and divide the voltages
by 10 to give a noise voltage
density of 1.279nV/√Hz, or 180.7nV over the audio bandwidth. For signal
levels in dBu add or subtract 10dB per resistor decade. So our 100Ω
resistor being two decades below 10k has 112.6dBu  20db = 132.6dBu noise.
Resistance  Noise Density  Noise Voltage (20Hz to 20kHz)  dBu (0dBu = 775mV) 
1k0  4.04 nV√Hz  0.57 μV  122.6 
1k1  4.24 nV√Hz  0.60 μV  122.2 
1k2  4.43 nV√Hz  0.63 μV  121.8 
1k3  4.61 nV√Hz  0.65 μV  121.5 
1k5  4.95 nV√Hz  0.70 μV  120.9 
1k6  5.11 nV√Hz  0.72 μV  120.6 
1k8  5.42 nV√Hz  0.77 μV  120.1 
2k0  5.72 nV√Hz  0.81 μV  119.6 
2k2  6.00 nV√Hz  0.85 μV  119.2 
2k4  6.26 nV√Hz  0.89 μV  118.8 
2k7  6.64 nV√Hz  0.94 μV  118.3 
3k0  7.00 nV√Hz  0.99 μV  117.9 
3k3  7.34 nV√Hz  1.04 μV  117.4 
3k6  7.67 nV√Hz  1.08 μV  117.1 
3k9  7.98 nV√Hz  1.13 μV  116.7 
4k3  8.38 nV√Hz  1.19 μV  116.3 
4k7  8.77 nV√Hz  1.24 μV  115.9 
5k1  9.13 nV√Hz  1.29 μV  115.6 
5k6  9.57 nV√Hz  1.35 μV  115.2 
6k2  10.07 nV√Hz  1.42 μV  114.7 
6k8  10.54 nV√Hz  1.49 μV  114.3 
7k5  11.07 nV√Hz  1.57 μV  113.9 
8k2  11.58 nV√Hz  1.64 μV  113.5 
9k1  12.20 nV√Hz  1.72 μV  113.0 
10k  12.79 nV√Hz  1.81 μV  112.6 
11k  13.41 nV√Hz  1.90 μV  112.2 
12k  14.01 nV√Hz  1.98 μV  111.8 
13k  14.58 nV√Hz  2.06 μV  111.5 
15k  15.66 nV√Hz  2.21 μV  110.9 
16k  16.17 nV√Hz  2.29 μV  110.6 
18k  17.15 nV√Hz  2.42 μV  110.1 
20k  18.08 nV√Hz  2.56 μV  109.6 
22k  18.96 nV√Hz  2.68 μV  109.2 
24k  19.81 nV√Hz  2.80 μV  108.8 
27k  21.01 nV√Hz  2.97 μV  108.2 
30k  22.15 nV√Hz  3.13 μV  107.9 
33k  23.23 nV√Hz  3.28 μV  107.4 
36k  24.26 nV√Hz  3.43 μV  107.1 
39k  25.25 nV√Hz  3.57 μV  106.7 
43k  26.51 nV√Hz  3.75 μV  106.3 
47k  27.72 nV√Hz  3.92 μV  105.9 
51k  28.87 nV√Hz  4.08 μV  105.6 
56k  30.26 nV√Hz  4.28 μV  105.2 
62k  31.84 nV√Hz  4.50 μV  104.7 
68k  33.34 nV√Hz  4.71 μV  104.3 
75k  35.01 nV√Hz  4.95 μV  103.9 
82k  36.61 nV√Hz  5.18 μV  103.5 
91k  38.57 nV√Hz  5.45 μV  103.0 
OpAmp Noise Voltage, Noise Current, and Noise Resistance
When considering an opamp for lownoise design we find two useful numbers
in the datasheet: equivalent input noise voltage, and equivalent input noise current.
The equivalent input noise voltage represents the internal noise voltages inherent
inside the opamp, and is modelled as a single voltage source in series with the
noninverting input.
The equivalent input noise current represents the current noise inside the
opamp, and is modelled as a current source placed across the input terminals.
The optimal external source resistance (i.e., the resistance
seen by the opamp terminals) is the ratio of noise voltage to noise current:
$$
r_n = \frac{v_n}{i_n}
$$
What this tells us is that lower source resistor values will not help much since the noise
performance will be dominated by the noise voltage, while a higher source resistance
will generate increasing levels of noise voltage due to the noise current flowing
in them. We also want to keep the external resistors low to minimize thermal noise.
Some examples (figures are typical, at 1kHz unless stated) sorted in order
of noise voltage:
Opamp  e_{n} (nV√Hz)  i_{n} (pA√Hz)  r_{n} (Ω) 
AD797A  0.9  2.0  450 
LM4562  2.7  1.6  1k7 
NE5534A  3.5  0.4  8k8 
LM833  4.5  0.5  9k0 
NE5532A  5  0.7  7k1 
OP275  6  1.5  4k0 
OPA2134  8  0.003  2M7 
TL072  18  0.01  1M8 
A Word on Noise Gain
As mentioned above, the internal noise sources within an opamp are referred
to the noninverting (+) input. While this sounds innocuous enough, it has an
important ramification: the gain applied to the opamp noise, the Noise Gain,
is that of a noninverting amplifier, which is 1 greater than that of an inverting
amplifier! At low gains this can make a huge difference: for example,
a unitygain inverting buffer has a Noise Gain of 2.
Minimising Thermal Noise
Thermal noise is characterised by three parameters: bandwidth, temperature, and resistance.
It has no dependence on the resistor type, so a carbon composition resistor will have the
same thermal noise as a superexpensive metal foil resistor (excess noise
(see below) will be different though).
Bandwidth is a system parameter so will be determined by the design parameters.
Temperature is a little more controllable, although unless you're prepared to
emerse your circuit in liquid nitrogen there's not much you can really do.
So the main parameter for controlling thermal noise is resistance. And from
the table above it is clear that lower resistance is better with less thermal
noise.
Shot Noise
Shot noise is noise caused by the discrete nature of electrical current flow. As such it
has a very simple model relating the shot noise current to the actual current flowing in a
conductor:
$$
i_{sn} = \sqrt{2qIB}
$$
where q is the charge on an electron (1.6x10^{19} eV), I is the current flow,
and B is the measurement bandwidth as before.
Of particular note is that the shot noise only depends on the current flow and the bandwidth, it
is independent of the temperature and of the resistance and type of the conductor.
Excess Noise
Work in progress!
Due to structure of resistive materials. Carbon comp the worst, then CF, then MF, then metal foil (very expensive!)
Larger physical size has lower excess noise. 0.5W better than 0.25W better than 0.125W.
Also lower resistance has lower excess noise. Check manufacturer datasheets.
