Professional Article. What is noise?

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Camilla Manning
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1 Professional Article What is noise?

Fachbericht Spektrumanalyse What is noise? Noise, explained simply and easy to understand. Everybody knows the noise of a waterfall. A multitude of water drops collide and drop deep.

Noise of all kinds is ubiquitous, also in electronics. If one picks up the sound of a waterfall with a microphone and displays it on a scope a picture like number 2 may result.

2 Fachbericht Spektrumanalyse What is noise? Noise, explained simply and easy to understand. Everybody knows the noise of a waterfall. A multitude of water drops collide and drop deep. The superposition of all the sound waves created yields the sound characteristic of a waterfall. This noise contains all frequencies from the infrasound to the ultrasonic ranges. Noise of all kinds is ubiquitous, also in electronics. If one picks up the sound of a waterfall with a microphone and displays it on a scope a picture like number 2 may result. At any time the signal v(t) can not be predicted into the future. The signal is completely stochastic. However, certain parameters of a noise signal may well be defined such as the statistical distribution of the frequencies and their amplitudes contained in the noise, also possible regularities of the signal like an amplitude modulation as they may be caused by external influences. The Brown movement shows that statistics also reign in the world of microscopic particles. Small particles in air or water are perturbed statistically by other particles and thus change their position in space. If one would measure their coordinates over time a similar signal would result as shown in picture 2. Picture 2 Statistical signal vs. time. Noise in electronics Noise is ubiquitous in some form or other also in all electronics. Noise sets the limit for signal amplification. The stereo set may emit audible noise in pianissimo sequences. Noise is the reason for the extremely slow signal transmission from the Voyager. Causes of noise In the following different causes resp. sources of noise are discussed. The treatment is intended to remain practical. Thermal noise (also called Johnson or Nyquist noise) Picture 1 noisy waterfall: unmanageable multitude of events and frequencies This noise is caused by the Brown movement of charges in all ohmic resistances and is already present without any voltage or current. An open circuit noise voltage according to equation (1) can be measured 2

3 Formulae across a resistor R, see the table above. Vn is the rms value of the noise voltage, the instantaneous values are like in picture 2. Short circuiting the resistor will cause a noise current according to equation (2). If one connects an ideal noise-free resistor R2 (can be realized by cooling a resistor down to zero degrees Kelvin) in parallel to a resistor R1 a noise power according to equation (3) will be transferred from R1 to R2. R2 = R1 is called power matching. ktb is the maximum available noise power from a resistor R. T is usually assumed to be 290 K, the bandwdth 1 Hz, this leads to a power density according to equation (4) of W/Hz. Shot noise Shot noise can only be generated if there is a current flow. However, a current flow does not necessarily cause shot noise. The cause of shot noise is the fact that the smallest charge possible is the elementary charge e. The movement of the electrons which constitute the current flow in a wire follows a certain pattern of mutual dependence, this may be compared to a column of marching soldiers. The number of electrons which pass a given cross section area per unit of time is constant, the current is continuous, there is no shot noise. If a wirewound resistor is inserted only the thermal noise voltage can be measured across its terminals (eq. (1)). This changes if the charge carriers have to overcome a potential barrier. This requires kinetic energy which is distributed statistically. This may be analogous to a number of walkers strolling along the same street but with different speeds. The result is a certain spread of the carrier density around its average value. Equation (5) shows the value of the noise current caused by a dc current I provided there is a potential barrier to overcome. 3

Fachbericht Spektrumanalyse Typical examples of shot noise sources are e.g.: Diode and transistor leakage currents, gate leakage currents of FETs.

4 Fachbericht Spektrumanalyse Typical examples of shot noise sources are e.g.: Diode and transistor leakage currents, gate leakage currents of FETs. Photo and dark currents of photo diodes and photo cells. Anode and screen grid currents in electron tubes. Normal currents in pn junctions, e.g. of semiconductor diodes, may be considered in practice as shot noise free when measured directly or as a voltage drop across a resistor. The shot noise is not temperature dependent (5). Often the current (e.g. bias, leakage current) causing the shot noise is temperature-dependent, so cooling the component will decrease the noise because this current will decrease. Other kinds of noise Carbon resistors show a marked noise when current is flowing, metal film resistors much less. This noise increases with increasing current, hence it is mostly called current noise. This is caused by the composition of the resistive layer, consisting of minute particles. In order for the current to flow potential barriers between those particles have to be overcome. This is quite similar to shot noise. The amount of noise is heavily dependent on the quality of the material and the manufacturing process, consequently, wirewound resistors are free from this kind of noise. In case of doubt the manufacturer s data sheet should be consulted. In critical applications metal film or wirewound resistors are called for. Also, the size of the component is vital, the smaller the worse. This is especially important with SMD parts. Socalled flicker noise occurs mostly at low frequencies (Hz to KHz) and is also strongly dependent upon the quality of the components. In electron tubes it is caused by the continuously changing emission areas of the cathode. With semiconductors there are diverse causes which are partly unknown. Hence flicker noise can not be calculated. The only source of information are again the data sheets. However, due to its nature flicker noise does not remain constant over life, also it is good practice to make measurements of one s own because the spread of "typical" values is enormous. There are further noise types (popcorn noise in integrated analog circuits, high frequency noise at very high frequencies, Barkhausen noise in coils and transformers), however, in practice, those are of minor importance. Last to be mentioned is the avalanche noise. It appears in zener diodes in the avalanche region (above appr. 6.2 V), also in pn junctions operated beyond their breakdown voltage, in gas discharge tubes, avalanche photo diodes. It is quite strong compared to the current causing it and it is temperature dependent. Also it varies enormously from component to component. For the practical application of zener diodes it follows that they should never be directly connected to the inputs of low noise amplifiers, always a low pass RC filter should be inserted. Amplitude characteristics Theory and practice show that for most kinds of noise ( e.g. thermal and shot noise) the amplitudes are distributed following a Gauss curve (picture 3). Picture 3: Amplitude distribution of noise following a Gauss curve. For the instantaneous values only probabilities can be given, those are higher for small than for large amplitudes. Theoretically, arbitrarily high positive or negative amplitudes v(t) are possible, however, this is highly unlikely. Mostly v(t) will remain within the area C, seldomly in area B, rarely in area A. The integral of v(t) taken over some longer time will always remain zero. Zero is the average value of the noise. The standard deviation δ is equal to the rms value of the voltage. The probability that v(t) will remain 4

5 at any given time within +- δ is 68.3 %. For an interval of +- 2 δ the probability rises to 95.5 %, for +- 3 δ it is 99.9 %. Measurement of noise intensity The intensity of noise is usually expressed in rms values, see eqs. (1) to (5). If no true rms measurement instrument with sufficient bandwidth is available a normal ac voltmeter may be used. It must be noted, however, that such instruments measure the linear average of the rectified signal and that they are calibrated to show the rms values of sine waves. Any other waveform will render erroneous results. The quotient of the rms and rectified values is 1.11 for sine waves, but 1.25 for noise with a Gauss distribution. Hence, in order to obtain the correct reading for Gauss noise the value indicated by the ac voltmeter must be multiplied by a factor of A more convenient but not as accurate noise measurement uses a scope. v(t) is displayed at medium intensity. The width of the noise band shown is measured as accurately as possible, this is the peak-to-peak value. As there is no true peak value with Gauss noise this is somewhat arbitrary. The rms value of the noise is given by: V rms, noise 0.2 V pp This is valid for analog scopes (6) The accuracy thus achieved is 10 to 20 %, quite acceptable for noise measurements. Mathematically eq. (6) refers to an interval of δ which means a probability of 99 %. BFor DSOs eq. (7) is adequate: V rms, noise V pp (7) The reason for the difference is that such voltage excursions which are infrequent will also be stored and used for the calculation. With analog scopes they remain invisible due to the limited persistence of the phosphor. The smaller factor corrects for this difference. Frequency characteristics The most relevant kinds of noise, thermal and shot noise, have a constant power density with respect to frequency, i.e., each interval along the frequency axis contains the same noise power. This is called white noise, see picture 4. Considering the noise voltage or current rather than the noise power and further considering that P (Tilde) V hoch 2 resp. P (Tilde) I hoch 2, it follows that in a n times larger interval there will be n times the power, but only n times the noise voltage or current. P n (f) Picture 4 Power density of white noise f s 3THz The fall-off of the power density at extremely high (THz) frequencies is explained by quantum theory, this is of no consequency for practical electronics. From eqs. (1) to (5) the important rule follows that for uncorrelated noise sources the noise powers will add up, but the noise voltages or currents will add only "geometrically", i.e. according to eq. (8). Noise bandwidth In practice the bandwidth of noise is important, see pictures 5a and b. Picture 5a shows the amplitude vs. frequency characteristic of a lowpass. The bandwidth is given by a 3 db drop in amplitude. If white noise is applied to a lowpass also frequency components above the bandwidth (shown in the shaded area beyond fo) will pass. Components below fo will already be attenuated. This need not be considered with an ideal lowpass (picture 5b). For the real world lowpass the "noise bandwidth" f o is the bandwidth an ideal lowpass would have in order to deliver the same noise power from white noise as the practical lowpass with bandwidth fo. Noise and signal bandwidths are thus not identical. For a first order lowpass eq. (9) holds: B noise = (π/2) B signal For a simple bandpass consisting of a 1st order lowpass (bandwidth fo) and a 1st order highpass (bandwidth fl) eq. (10) holds. For higher order filters with steep slopes noise and signal bandwidths differ less and less. The calculations presented here must f δ 5

6 Fachbericht Spektrumanalyse be considered in measurements and calculations, otherwise errors of up to 57 % are possible. If noise generators are used errors can not be made because any errors will cancel in the result. Picture 5 Noise and signal bandwidths of a real and an ideal lowpass Pink noise Apart from white noise there is pink noise as shown in picture 6, low frequencies are pronounced. Pink noise is important in acoustics because here a kind of noise is preferred where the power is not constant per absolute frequency interval, but per relative interval (octave). In order to realize this goal Picture 6: Power density for pink noise (both scales are logarithmically) the power density must decrease with increasing frequency as demonstrated in picture 6. Pink noise is derived from white noise by frequency filtering. It is also called "1/f noise". The flicker noise mentioned earlier has about the same characteristics, but may differ in slope. The question whether at 0 Hz the amplitude must reach infinity can be answered with no. For special applications other forms of noise are derived from white noise such as f 2 or triangular noise which is used for FM measurements. Another popular type of noise is one with a frequency response adapted to that of the human ear. Noise measurements at high frequencies One method of measuring noise at high frequencies consists of a bandpass preceding a rms responding voltmeter; the bandpass suppresses any noise outside the frequency range of interest. When measuring e.g. the output noise of an amplifier or the like the value obtained is divided by the amplification factor in order to obtain an equivalent noise at the input. It is thus assumed that all noise sources are concentrated at the input and constitute a noise generator there while the amplifier is considered noise-free. This noise referred to the input corresponds to a signal of equal amplitude such that the signal-to-noise ratio becomes 1:1 or o db. This value is also called the maximum usable sensitivity. It is evident that several parameters must be known in order to achieve correct results: the sensitivity of the rms voltmeter, the noise bandwidth of the bandpass (also perhaps its attenuation) and the amplification factor of the unit under test. As quite a few possible errors can add simpler methods are more popular. The basic principle is like this: Measurements using a noise generator A calibrated noise signal from a noise generator is applied to the unit under test. At first the generator remains turned off. Using the above mentioned set-up (bandpass and rms voltmeter) the indication on the voltmeter (noise power at the output) is recorded, the absolute value is of no consequence. 6

7 Now the noise generator is turned on, its output is adjusted so that the voltmeter reading doubles indicating a doubling of noise power (or the noise voltage increases by Wurzel 2). As the two noise sources involved (generator and unit under test) generate uncorrelated noise the noise powers add linearly. The equivalent noise power referred to the input is thus equal to the output noise power of the generator. The advantages of this method are obvious: in place of a rms voltmeter a customary hf voltmeter can be used, the amplification factors of the unit under test and the bandpass need not be known. All that is needed is a calibrated noise generator delivering white noise in the frequency range of interest. Of course, it is preferable to have a generator with as wide a frequency range as possible to provide for future measurements. Customary noise parameters k. T 0 - unit and noise figure. Equations (3) and (4) show that the noise power output of the source impedance R1 of a generator to a noise-free receiver is k x To x B. Picture 7 shows a generator G (signal or noise generator), Z is the characteristic impedance of the cable, R2 the input impedance of the receiver which mostly will not exist as a fixed resistor. In order to avoid reflections caused by mismatching it is necessary that R1 = Z = R2. Z is mostly 50 ohms, 75 ohms are used in video circuits, radio and tv receivers use 60 or 240 ohms. As described earlier at first the generator remains off. (G in picture 7 assumed to be a short). The receiver will receive a noise power of value k x To x B from R1. Then the generator will be turned on and its output increased until the output of the receiver doubles (Eq. (11)). The factor n or the noise figure F hence is the number of kxtoxb units required for doubling of the receiver output. (+ 3 db noise power). This figure would be 1 for a noise-free receiver. Picture 7 Connection of a receiver to a generator Noise figure Eq. (12) defines the noise figure in logarithmic units: F = 10 log. F. (12) The noise-free receiver thus has a logarithmic noise figure of 0 db. Noise temperature Instead of turning on the noise generator it would also be possible to heat R1. Eq. (3) shows that the noise power delivered by R1 to R2 will rise proportional to the Kelvin temperature. The temperature rise necessary to reach doubling of the receiver output minus 290 K is the noise temperature of the receiver input. Eq. (13) expresses: T noise = (F 1) T 0 (13) The ideal noise-free receiver has a noise temperature of 0 K. This is a purely mathematical figure which has nothing to with the real temperature of the receiver. The noise temperature may indeed be smaller than the real temperature, also much higher in the case of suboptimum receivers. Audio noise measurements Because of the requirement of matching in high frequency circuits the characteristic impedances are the same throughout, this is why it is more convenient to use noise powers for calculations. In audio circuits the impedances vary widely, power matching is uncommon. Hence it is more convenient to use voltages or currents. The maximum usable sensitivity of amplifiers etc. is thus expressed in noise voltages or currents (V/ Hz, A/ Hz) referred to the inputs. This is demonstrated with these two examples: Input noise voltage The input impedance of the electrometer amplifier shown in picture 8 is extremely high, theoretically infinite and in any case much greater than the source impedance. Therefore it does not make sense to define an input power. In order to determine the maximum usable sensitivity the equivalent input noise voltage must be found. The measurement set-up is similar to that used in high frequency circuits. A bandpass is connected to the output, its output connected to an ac voltmeter. 7

8 Fachbericht Spektrumanalyse Picture 8: Voltage amplifier using an operational amplifier A noise generator is hooked up to the input. Its output impedance, mostly 50 ohms, may be considered as a short with respect to the amplifier input impedance. The generator noise given in kxto units would be impractical here, V/ Hz units are more appropriate. On the other hand RH must not become too large as its stray capacity ( pf) end-to-end and to ground will bypass high frequencies either to the input or to ground thus causing errors. Thus a compromise must be found. Too large here also applies to the physical size as the capacities depend on this, too. At first the noise at the output is measured while the generator remains off. Then the generator will be turned on and its output increased until the voltmeter indication increased by 2. The noise voltage density to which the generator is set is now equal to the input noise voltage density. Input noise current Picture 9 shows a circuit for the amplification of current signals such as produced by photo diodes. As the input signal here is a current it makes sense to express the noise in A/ Hz. The input voltage in this kind of circuit will be practically zero. (Current sink.) The measurement is similar to the foregoing except for the addition of RH which converts the generator output voltage into a current. Picture 9 Current-to-voltage converter circuit using an operational amplifier The following rules will help to prevent measurement errors: R H should be connected with short leads directly from the coaxial cable to the amplifier input. R H RH should be large so that its noise contribution is neglegible (Eq. (2)). This requirement is adequately fulfilled if RH 10 x R. 8

Thermal Johnson Noise Generated by a Resistor

Thermal Johnson Noise Generated by a Resistor Complete Pre- Lab before starting this experiment HISTORY In 196, experimental physicist John Johnson working in the physics division at Bell Labs was researching