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Electrical Noise, Thermal or White Noise

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What Is Noise?

Noise has been defined in several manners, but the most known definition is that noise is any undesirable signal, disturbance or interference that affects a desired signal, and generally of random nature.

However the term random is not always true, as for a defective electrical motor which can produce a spark each time it rotates and at a periodical pattern, or produced by the spark plugs of a combustion motor, in each case it could interfere with the desired signal but it well could not be considered random, this tutorial will deal  only with the electrical noise type, not the acoustic one.

 

Types of Noise

Noise has been classified by the nature of its own origin or it source, the most encountered type of random noise are thermal (also called white noise), flicker, shot and Johnson noise.

Thermal or White Noise

The movement of conductive electrons in a resistor due to temperature produces a random noise, in accordance with  H.Nyquist the electrical noise generated in a resistor R  in a frequency bandwidth is:

 

 

     R

 

        (Vn)2  = 4kTRB

 

Where:

Vn = rms value of the thermal noise voltage

R = The resistive component of the impedance given in ohms

T= Absolute temperature given n Kelvin degrees

k = The Boltzmann’s constant  = (1.38 x 10-23 J/oK)

 

Also the noise can defined in terms of current by the circuit and equation below:

 

 

          (In)2 = 4kTGB                            G = 1/R

 

 

where:

In = rms value of the current

k = The Boltzmann’s constant  = (1.38 x 10-23 J/oK)

T= Absolute temperature given n Kelvin degrees

G = Is the conductance of the circuit  

The transmission of noise is proportional to the bandwidth of the circuit therefore is highly recommended that bandwidth of the circuit never exceed the required bandwidth in order to meet the expected signal to noise ratio SNR.

 

Example:

Calculate the RMS noise voltage produced by a resistor of 270 kilohm in a bandwidth of 106  Hz at a temperature of 25o C.

Temperature T,  in oK = 273 + 25 = 298 oK

4kT =1.644 x 10-20

(Vn)2  = 4kTRB

4kTRB = 1.644 x 10-20  x 270000 x  106

Vn = √(4kTRB)    = 66.62 microvolts

 

For more complex circuits containing several resistors a Thevenin equivalent circuit can be calculated to reduce the multiple resistor to a total resistor equivalence RT.

Nyquist established that the rms noise produced by a circuit with a complex impedance, this is resistive and reactive components is:

(Vn)2 = kT B  R(f) df

Where R(f) is the real or resistive part of the impedance at the frequency f integrated over the desired bandwidth range.

Note that for extremely sensitive circuits the ohmic resistance of the reactive elements also must taken into consideration, specially at high frequencies.

 

Thermal noise has been also called white noise because is has shown to have a uniform frequency spectrum in the order of 1013  hertz