Johnson-Nyquist Thermal Noise By: Mohammad Ali Ahmadi Pajouh AUT 2007

Some Interesting Measurements

Noise? Noise is a complex composite of lots of things: 1. Thermal or Johnson Noise 2. Shot noise 3. Flicker or 1/f noise 4. Environmental noise

History! In 1927 J. B. Johnson observed random fluctuations in the voltages across electrical resistors. A year later H. Nyquist published a theoretical analysis of this noise which is thermal in origin. Hence this type of noise is variously called Johnson noise, Nyquist noise, or Thermal noise.

At any non-zero temperature we can think of the moving charges as a sort of Electron Gas trapped inside the resistor box. The electrons move about in a randomised way — similar to Brownian motion — bouncing and scattering off one another and the atoms.

At any particular instant there may be more electrons near one end of the box than the other.

If we note the meter reading at regular intervals (e.g. every second) for a long period we can plot a histogram of the results. we choose a ‘bin width’, dv, and divide up the range of possible voltages into small ‘bins’ of this size. We then count up how often the measured voltage was in each bin, divide those counts by the total number of measurements, and plot a histogram

We can now use this plot to indicate the likelihood or probability, that any future measurement of the voltage will give a result in any particular small range

NYQUIST EQUATION = 4kTR  f

For frequencies below a few gigahertz, Equation 1 gives the relationship between the -, the measured mean-squared voltage - (T), absolute temperature of the resistor - ( R), its resistance -  f, is the bandwidth in hertz over which the noise is measured. For a resistor of 1kΩ at room temperature and a 10 kHz bandwidth, the RMS noise voltage is 400 nV or 0.4 microvolts - k B is Boltzman’s constant (1.38 x Joules/Kelvin)

POWER SPECTRAL DENSITY OF THE RANDOM FLUCTUATIONS The frequency content of the random fluctuations are characterized by the power spectral density (PSD). PSD = power per unit frequency The noise spectrum of the random fluctuations is wide band or “white”noise. Therefore, the PSD of the random fluctuations is constant and the power is calculated by PSD (f) (f 1 - f 2 ), where  (f 1 - f 2 ) is the bandwidth of the measurement system. The mean-square of the random fluctuations can be reduced by reducing the bandwidth.

In communications, power is often measured in decibels relative to 1 milliwatt (dBm), assuming a 50 ohm resistance. With these conventions, thermal noise at room temperature can be estimated as:

Thermal Noise for Capacitances Johnson noise in an RC circuit can be expressed more simply by using the capacitance value, rather than the resistance and bandwidth values. The rms voltage noise on a capacitance C is independent of the resistor value, since bandwidth varies reciprocally with resistance in an RC circuit

Shot Noise Shot Noise occurs whenever any phenomenon can be considered as a series of independent events occuring at random. This occurs for carriers falling through a potential in one direction only. This is a non-equilibrium process and requires DC current flow. First seen in vacuum tubes. The spectrum is white and has a Gaussian Amplitude distribution.

Why? Arises because electrons are quanta of charge, and flow of charge across a junction is therefore a multiple quanta event, and has to be dealt with using statistics.

Flicker or 1/f Noise Inversely proportional to frequency Cause not well understood Recognized by frequency dependence Becomes significant in signals <100 Hz Worst for 0 Hz, or DC drift! Better in wire-wound or metallic film resistors, worse in standard carbon resistors. So can be improved by using different electronic components in your circuit

Where? 1. Current in Carbon Composition resistors 2. Current in Thin Metal Films (in the past, not today) 3. Current in Ionic Solutions 4. All Solid-State components; but especially Si MOSFET, GaAs 5. Body Sway 6. The Earth’s wobble on its axis 7. Magnitude of Ocean waves 8. Magnitude of Earthquakes 9. Magnitude of Thunder Storms 10. Magnitude of Tornados 11. Magnitude of Hurricanes 12. Classical and Jazz Music 13. Economic data

Environmental Noise Every wire in your instrument can act like an antenna to pick up electromagnetic energy and convert it into an electrical signal inside your instrument There are lots of things in the environment that are putting out Electromagnetic signals There are two region that are relatively noise free 3Hz to 60Hz and 1 kHz to 500 kHz Will often design instruments around these frequencies to take advantage to this low environmental noise

Hardware Solutions Grounding and Shielding Difference and Instrumentation Amplifiers Analog Filtering Modulation Signal Chopping: Chopping amplifiers

Noise from environment can often be dramatically reduced by shielding, grounding, and minimizing wire lengths. Shielding - surrounding a circuit or the wires in the circuit with a conducting material then attaching that to ground. Environmental EM radiation can’t penetrate the conductor where it gets absorbed Arranging optimum shielding is something of an art., something of hit and miss Very important in high resistance transducers (like glass electrodes or pH electrodes Here even tiny noise get dramatically amplified Wiring –avoid loops to reduce parasitic inductance.

To amplify signal but not noise, instruments typically use a difference amplifier for first stage of amplification If this isn’t enough a circuit called an instrumentation amplifier is used With these kinds of electronics can amplify signal 1000x and eject of noise

Analog Filtering Most common way of improving S/N is with a low-pass filter. Low pass means that it passes slow moving signals (low frequency) but it stops signals with high frequencies, the jiggling noise. Can be done with just a resistor and a capacitor. This method is most common used in older analog instruments. You have a switch on the consol where you change either the capacitor or the resistor, that changes the time response of the instrument. Does reduce high frequency noise, but can also damage signal, so you have to be careful how this is used.At other times can use a high pass filter to pass a high frequency signal but block a low frequency drift Also can use narrow band or band pass filters to let just certain frequencies through

1. J. Johnson, "Thermal Agitation of Electricity in Conductors", Phys. Rev. 32, 97 (1928), (the experiment). 2. H. Nyquist, "Thermal Agitation of Electric Charge in Conductors", Phys. Rev. 32, 110 (1928), (the theory).