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# 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 single-pole low-pass 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 resistor-capacitor 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 capacitor-less 250MΩ resistor.

This result has two (at least) conclusions for designing audio circuits. Firstly, system bandwidth is still the dominant noise-control mechanism that designers must use to manage noise. And secondly, where resistors are being used in low-frequency circuits (e.g., CV manipulation or bias compensation resistors for Bipolar op-amps) 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 20Hz-20kHz (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 double-decade jump. For example, for a 100Ω resistor we look at the row for 10kΩ (100 x 102) 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.

ResistanceNoise DensityNoise 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

### Op-Amp Noise Voltage, Noise Current, and Noise Resistance

When considering an op-amp for low-noise 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 op-amp, and is modelled as a single voltage source in series with the non-inverting input.

The equivalent input noise current represents the current noise inside the op-amp, and is modelled as a current source placed across the input terminals.

The optimal external source resistance (i.e., the resistance seen by the op-amp 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:

Op-ampen (nV√Hz)in (pA√Hz)rn (Ω)
LM4562 2.71.6 1k7
NE5534A 3.50.4 8k8
LM833 4.50.5 9k0
NE5532A 5 0.7 7k1
OP275 6 1.5 4k0
OPA2134 8 0.0032M7
TL072 18 0.01 1M8

#### A Word on Noise Gain

As mentioned above, the internal noise sources within an op-amp are referred to the non-inverting (+) input. While this sounds innocuous enough, it has an important ramification: the gain applied to the op-amp noise, the Noise Gain, is that of a non-inverting amplifier, which is 1 greater than that of an inverting amplifier! At low gains this can make a huge difference: for example, a unity-gain 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 super-expensive 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.