Johnson–Nyquist noise

Thermal noise, thermal noise, resistor noise, Nyquist noise, Johnson noise or Johnson - Nyquist noise called, is a largely white noise, which arises from the thermal motion of charge carriers in electric circuits. The frequency spectrum of the noise resistance was investigated experimentally by John Bertrand Johnson and justified theoretically simultaneously by Harry Nyquist Theodor.

• 6.1 Black waveguide and Black Body Radiation
• 6.2 Capacitive load
• 6.3 Dissipation and storage

• 7.1 resistor with parallel capacitance
• 7.2 Quantum theory of limited AKF resistance noise
• 7.3 White noise 7.3.1 Stationary sequence of impulse functions
• 7.3.2 Exponentialimpulse

Appearance

Thermal noise manifests itself with unloaded ohmic resistors as thermal resistance noise, often called simply resistance noise. The thermal motion of the electrons generates the line noise power and the noise voltage at the terminals of the two-terminal network. The present case of a short circuit or open circuit values ​​can generally be defined as noise spectral power density. To be proportional to absolute temperature. When unloaded component, the noise power is independent of the electrically conductive medium, however, can the carrying of DC device current noise added, which may lie with the carbon film resistor far above the thermal noise. When current-carrying semiconductor is created excess noise by modulating the load current - voltage impression when - due to thermally induced fluctuation of the number of carriers in the conduction band and valence band, and thus the conductivity.

Johnson experimented in the years 1927/28, at a temperature between the boiling point of nitrogen and water resistance with very different material. Were used under different carbon layer, copper and platinum resistors as well as a variety of electrolyte -filled capillaries.

Johnson announced Schottky 've realized in 1918 from theoretical considerations that thermal noise of conduction electrons must be discovered with tube amps, but with a resonant circuit at the amplifier input the searched effect will be masked by the shot noise. Nyquist quoted Schottkys work because lessons from excitation to derive the electro-dynamic noise power from thermodynamics and statistical mechanics.

Physical justification

The conduction electrons electrically conductive materials ( metals, semiconductors ) take part in the largely uncontrolled, thermally excited motion of the components of the atomic level and move at random and undirected. They contribute at room temperature to a small extent to the specific heat, and their random motion is at the terminals of the dipole in question here finite electrical noise performance. The conduction electrons generate large rate statistically independent of voltage and current pulses of finite, of short duration, resulting in the superposition of broad frequency distribution, which is often perceived as a noise source in electrical white spectrum. The noise power spectrum extends from zero frequency up to a cutoff frequency whose value is determined by the thermally still noticeably excitable quantum of electromagnetic harmonic components. The first calculation of the noise spectrum of Nyquist makes the equipartition law of thermodynamics use. - A finite DC component is not observed; they could not be regarded as an incidental component cf. thermoelectricity. To a broken symmetry would be necessary for no reason be seen as noise in the resistance thermodynamic equilibrium is provided.

The resistance noise is characterized here by the white borders in wide frequency range. Another issue is the description by the amplitude distribution of the instantaneous values ​​of voltage or current. Experience has shown that there is a normal distribution ( Gaussian distribution ) with zero mean, the dispersion parameter is given by the noise power. In particular, you can choose an arbitrarily large amplitude can be expected with exponentially decreasing probability.

The conditional stochastic amplitude statistics that noise voltages must be measured under real quadratic rectification. Johnson used for this purpose ( according to electronic amplification ) has a thermal converter, which causes the generation of heat by the supplied noise power, a temperature increase. This is measured with a thermocouple, the linear time -averaged thermal noise voltage is the average of the voltage proportional to the square. This measurement procedure is somewhat generalized by the definition of the autocorrelation function formulated mathematically. The conversion factor for the Thermoumformers is measured with a well defined by a DC voltage output.

Noise variables

Observed at a resistor in the course of time variations of the open-circuit voltage analogous to the random variations in the Brownian motion. The average of these voltages equal zero. As the noise size of the root mean square value of the voltage is measured after electronic amplification, which can be converted into the rms value. The average voltage is proportional to the square of the absolute temperature, the size of the electrical resistance and the bandwidth of the measuring arrangement.

The influence of the bandwidth is not easily identifiable with a broadband structure, the amplitude statistics can be quite good judge here. Its variance is given by. The amplitude statistics can be well determined narrowband. Narrowband the influence is a clear proportional to recognize at -centered range of the attack and decay through which the components of the noise spectrum to be modulated.

• Resistance noise is an expression of the thermal coupling of electrodynamic fluctuations. You can be illustrated by observations on the range of services on the routes chosen by Schottky and Nyquist.

The Nyquist formula, the following relation for the noise voltage at idle ago:

With the effective idle noise voltage

Consequently

Here are the Boltzmann constant, the absolute temperature and the resistance of the rushing dipole. is the authorized bandwidth. The total noise is obtained by integrating over all frequencies.

Dually, the time-averaged square noise current in case of short circuit calculated to

With the effective short-circuit noise current

For the generality of the formula of Nyquist and their importance to profound questions of physics Ginsburg provides comprehensive information.

Equivalent circuit and current account

The equivalent circuit diagram of a rushing resistance as a concentrated component is the series connection of the noise-free imaginary resistance R as a source resistor to be performing noise voltage source which emits the open circuit voltage squared. For viewing with a noise current source, a current source generator from the short-circuit current square the ideal internal resistance R is connected in parallel.

In short the rushing ohmic resistance dissipates even the generated power

Because the full source voltage drops across it.

In each of the two power adjustment rushing ohmic resistances in each other and themselves dissipated even the power

Because half the source voltage drops across them. This is the maximum deliverable power from a source and is called the available power. This term makes a circuit of contingencies and of independent and suitable for a general discussion by the thermally activated, but electro- dynamically mediated energy exchange between the two rushing, coupled to a heat bath of temperature T is symmetrical resistors.

These four dissipated noise powers add up to back the short-circuit power, which is consequently also total generated in this arrangement. The two interconnected for power adjustment resistors work, taken as a unit of resistance, short circuit and their power dissipated is only on the size and thus also as for each resistor.

• The power dissipated in a purely resistive circuit with power adjustment determined independently of the size and alone thermodynamically by the available power
• With this formulation in square sizes as the current account already recognized by Schottky claim is met manifest, if it were, the above coupling of thermal fluctuations on electrodynamic called appearance. Fluctuation energy of the order of the mean thermodynamic quantum exchanges each electromagnetic mode with the heat bath.

The formulation as the current account is not necessary to use the magnitude of electrical resistance and clear because of this generality, the proposed use of the lemma thermal noise. Performance is due to the necessary quadratic rectification anyway the actual measured size.

Quantum theoretical extension

The integration of the above equations over the entire frequency range leads to the ultraviolet catastrophe. A strictly white spectrum also requires the unrealistic participation arbitrarily short duration pulses for the excitation of the harmonic components. Therefore, the quantum-mechanical expansion is necessary for high frequencies. Nyquist made ​​this already. The later realized quantum mechanical zero-point energy is mentioned as a possible non-thermal noise source occasionally.

Nyquist formula

For sufficiently high frequencies or correspondingly low temperatures must also already specified by the Nyquist formula )

Be used. The quantum theoretical cut-off frequency was used recently been defined by

At room temperature (300 K) = 6.25 × 1012 it amounts Hz

• Above, the thermal resistance is not spectrally white noise ), but decreases with increasing frequency according to the Boltzmann factor exponentially.

• For low frequencies or sufficiently high temperature, the quantum theory, advanced formula is expected to be over in the low-frequency value.

Zero-point energy

A contribution of zero-point energy to the thermal noise is sometimes put forward for discussion. The zero-point energy is required by the Heisenberg Uncertainty Principle and is the harmonic oscillator as fully corrected quantum mechanical formula

Often suggested. With this formula, the ultraviolet catastrophe would be introduced strengthened again. The zero-point energy is thermal processes such as thermal noise for the exchange of energy to a load resistor is not available. The latter, the quantum mechanical approach quite directly expressing formulation obviously requires that the high frequencies with sufficient or sufficiently low temperatures is only the zero-point vibrational attributable to and exchanged with power matching between source and load resistance available power spectral density.

• This required changes in state of half quantum.

For the maser has been shown that the zero-point energy is not enhanced.

Power spectrum

The power spectrum emphasizes the fact individually each electromagnetic frequency component independently of the other frequency vibrations having to concede its own thermal degree of freedom, equipartition theorem. Nyquist shows this for the electromagnetic case, an imaginary tour of a circuit ( nichtdissipativen ) reactance between the power contained in matching resistors. Had the harmonic oscillations of different frequency is not equal to strongly coupled to the heat bath, it could increase the temperature of the warmer on average in contradiction to the second law of thermodynamics, the colder resistance.

• Each electromagnetic spectral component is independent on the rushing dipole in detailed balance with the heat bath and has due to its electromagnetic nature two thermal degrees of freedom.
• The required quantum theoretical complement shows that these independent frequency components which require a minimum energy of a photon, which is evident at high quantum by their thermal excitation is hampered by "freezing" due to low temperature.

The power spectrum for the available power of any ohmic resistance is defined by )

With the low frequency value

Note: The power spectral density of the dimension of energy.

For power adjustment

The total power available is

The limited by the quantum theory of effective bandwidth, assuming a constant throughout white adopted spectral power

The total power available at room temperature (300 K), P = 4.26 · 10-8 watts.

Black waveguide and Black Body Radiation

Two ohmic Two poles of the same frequency-independent resistance in the heat bath of the absolute temperature are connected by a lossless line of characteristic impedance, see real wave impedance. Because of this adjustment according to the characteristic impedance are on the line only progressive waves of both propagation directions. Influences of standing waves due to reflection do not exist, and consequently is not available frequency selectivity. In this wiring already exists power adjustment.

• The ideal line - of any length and a defined characteristic impedance - is interconnected so that the coupling of thermal fluctuations is based on electrodynamic by the thought of spatially extended electromagnetic waves.

The electromagnetic waves emitted by the line noise resistors and completely absorbed in the other.

• The simultaneously rushing and dissipating resistors convey the setting and maintenance of the thermodynamic equilibrium between the energy content of the electromagnetic waves and the heat bath, see the fluctuation-dissipation theorem.

The transmitted power on the other, resistance does not interfere with the thermodynamic equilibrium, in the middle is not directed energy transport takes.

• This with respect to the propagation of electromagnetic processes of the black along the waveguide, as the array ) will here called one-dimensional array is a three-dimensional electrical equivalent of the black body radiation. )

• The low-frequency noise spectrum is obtained by Nyquist considerations to the above arrangement, he applied by the equipartition theorem to the spectral components of the electromagnetic waves, represented by the capacitive and inductive coding of cable with energy storage or per line length. As a line he imagined an ideal coaxial cable from the wave resistance.

• At high frequencies, he looked at quantum and corrected the formula of the white spectrum corresponding to the results of Planck's formula.

In the low frequency region the excitation of electromagnetic waves is not reduced quantum theory. The white spectrum states that by means of the line through each spectral component frequency of the fluctuation energy available from a transfer to another resistance. It corresponds to two degrees of freedom, which is in accordance with the electromagnetic nature of the transmission mechanism. Electric and magnetic field at each control one degree of freedom and, therefore, according to the equipartition theorem, the average per fluctuation energy.

The Niederfrequenznäherung in shape is a factor of the number of excited photons. Almost 1010 quantum condensed 1 kHz at room temperature in the electromagnetic wave of frequency f =, of the potentially quantum- like character of the shaft has no obvious to bear. - The spectral component f of an electromagnetic wave can contain any number of quantum hf, cf photons and bosons.

The Hochfrequenznäherung with leads to the Boltzmann factor corresponding to the reduced availability of correspondingly large amounts of energy in the heat bath. The quantum can be thermodynamically stimulate efficient and high-yield only up to the order, larger quanta are at comparatively small thermally available energy frozen in the sense of freezing, for example, the rotational degrees of freedom of the specific heat at low temperatures.

At T = 0.05 K is GHz and with the quantum theoretical frequency limit would be just plain noticeable, only in about half the time the electromagnetic mode would be occupied by a photon. For frequencies up to 1 GHz, however, the ideal blackbody waveguide can hardly be realized with sufficient accuracy using common electrical means.

A comparison: the top, the total power P = 4.26 was calculated · 10-8 watts for room temperature. At T = 300 K is also already emitted by the blackbody according to the Stefan- Boltzmann law of an area 10-10 m2 approximately the same performance 4.6 × 10-8 watt in the half-space.

• ) This "black body " so allows the investigation of the radiation itself, unaffected by material properties of the radiating body, an almost ideal case, the experimental verification of complete abstraction, a theoretical concept. ' Emphasis in this quote are the author Walther Gerlach ( 1936) made. He goes on to describe the path to the study of Planck's formula: the development of the relationship between radiant energy and wavelength have not led to so many facts which were to order only, but directly to the physical law.
• ) A clear difference for cavity radiation will particularly pointed out that facilitates the Nyquist corresponding to the formula. The finite spectral power of the resistance noise ranges as white spectrum to arbitrarily small frequencies disappears the black body radiation on the other hand proportional to 0, as a result of radiation in a finite solid angle, the frequency in calculating the density of states is received (the number of the oscillators in the frequency interval ). The black fiber the emission of the available power is led to matched load resistor, however, one-dimensional, characterized the number of oscillators per frequency interval 1 s Nyquist. The states closely spaced f with energy hf can ever be occupied with many photons according to the mean occupation density

Capacitive load

The rushing resistance working to the ideal capacitor of capacitance.

The idle voltage range of the thermal noise is reduced at the capacitive load to the square value of the voltage divider ratio.

Each ohmic resistance as a component is a small stray capacitance in parallel, the range of its terminal voltage is in practice)

In thermal equilibrium, according to the formula for the energy to a capacitor in a capacitor voltage, the average power

Is stored, recently replaced by the low- frequency value. The capacitor is continuously supplied and removed in about the duration, the correlation time, such as the energy.

The effective bandwidth of the RC element is defined by

• The capacitor is coupled via the resistance to the heating bath and stores the average power
• The capacitor has a thermodynamic degree of freedom, as befits an energy store. Both statements apply to the inductance.

The stored energy to the complementary energy of the thermally generated by R in the effective frequency interval total energy dissipated in R itself.

This balance is known by charging a capacitor with a constant voltage and can be derived from the principle of minimum entropy production. Of course, the power generated outside of the effective bandwidth is dissipated in R itself; because with increasing the resistance works increasingly in short circuit.

The RC time constant and thus the effective frequency band just fall out so that the thermal degree of freedom of the capacitor is sufficient.

Corollary 1: Every real capacitor is the equivalent circuit diagram of an ideal capacitor connected in parallel, finite insulation resistance, whereby he learns the coupling to a heat bath. Therefore, the real capacitor stores the supplied, dependent only on the temperature is average energy according to the effective noise voltage on the capacitor to which the absolute value electron charges are stored in the agent. To a capacitor of 1 pF at room temperature, the effective noise voltage of 64 microvolts required by 402 elementary charges, which are transported in the middle of the random voltage fluctuations. Is reminded of the fact 0 and 0

Corollary 2: The basic proportionality of the noise power to the absolute temperature is immediately apparent when the square noise voltage is measured with high impedance across a capacitor. A wire resistance is expediently a rousing resistance because it allows very large temperature changes; according to the formula influenced his inevitable temperature dependence is not the result of measurement in this circuit. This arrangement is suitable for an impressive demonstration experiment. must always be so large that the intrinsic noise of the amplifier does not interfere.

• The result shows a particularly impressive that the component resistor serves only as an intermediary between the thermal storage heat bath and the electrical memory. In a magnetic memory the same applies.

Dissipation and storage

In fact, the voltage spectrum as a quantum theoretical formula would have to be integrated, but it reaches up to the electrical cut-off frequency band of a real capacitor limits the effective range at 300 K, far below the quantum theoretical cut-off frequency

This fact is exploited in the following to calculate the in rushing resistance dissipated even under capacitive load power. In contrast to the above, here is the tension square to look across the resistor itself, which is measured with the absolute square of the complex voltage divider factor. The power dissipated in

By adding to the electrical division factor in the Inter grandees 1 and is subtracted and -1 is included in this division factor, is available with the quantum theoretical cutoff frequency first

The integral of the first term, the short-circuit power in the R itself, has been evaluated above, is integral over the second - calculated approximation, by simplifying the factor is set equal to 1 because the frequency band further reaches out substantially to generally - usually in an excellent as the electro- technical reasons to. The result obtained is directly expressed with the bandwidth or the effective bandwidths

The second term is small compared to the first, which is the average power dissipated in R overall performance in short-circuit. This is reduced by the capacitive load to the output by the capacitor voltage reduces the voltage drop across R, and the current in the circuit. Capacitor voltage and current are out of phase, indicative of the energy storage and transport of reactive power in time ½ RC.

Autocorrelation function

The shock processes and the emission and absorption processes in the resistor material extending uniformly distributed in time in the middle, as long as the resistance does not age. As far as the resistance noise is stationary. The award of a timestamp as t = 0 has no meaning for the general characterization of the noise. This eliminates the distinction of odd and even component of the electromotive force, so that the tangent of the phase angle than the usual measure of the ratio is not an important indicator for the stationary noise itself Consequently, should the mathematically invariant description instead of the Fourier transform of u ( t), the amplitude spectrum, square sizes are chosen as above, the power spectrum. They already contain sufficient information on the temporal structure.

As information about the amplitudes facilitates the usual comparison to a DC voltage of the same heat generation. Except the timing structure, the above-mentioned amplitude distribution to be evaluated. The two distributions are independent, however, a restriction of the frequency band influencing the dispersion of the amplitude distribution. For white spectrum belongs not necessarily a normal distribution of instantaneous values ​​, as it exists in the resistance noise. )

To characterize the stationary noise over time remains not only the mean voltage squared.

• Invariant there is rather a temporal relationship to be described, the inner of which is measured by the auto-correlation function:

The abbreviation ACF is introduced for the autocorrelation function. The ACF is independent of the time direction: u ( t ) and u ( t ) have the same autocorrelation function. The definition formula can immediately recognize that the award at any time as a new reference time by no influence.

The AKF has at its maximum

Is the power dissipated by the terminal voltage in the resistor R performance.

The AKF is always an even function of. This means that there is no causal sequence is indicated by the time t. Nevertheless, and not independent, it can not change arbitrarily fast. The power spectrum for example, sets its upper frequency limit the effective fastest possible mode change.

The AKF is won the equivalent of the frequency spectrum for the point-in- ( or local ) level of description ( time domain ). The latter describes the internal context for the description level with harmonic oscillations ( frequency range).

• Depending on the intention or the metrological requirements, one or the other of the equivalent representation is chosen.

Actually a mathematical transformation due the equivalent representation of the process by the stationary ACF or by the frequency spectrum. The proof was in Vienna and Chintchin by noting that the Fourier transform yields the desired result:

Is defined on the grounds of the symmetry of the transformation formula for negative frequencies. Therefore, it should be noted defined above in accordance with the measuring process only.

Resistance noise spectra are real as autospectra, even functions of frequency. The position of the sign in the exponent is far convention, it will be chosen as indicated in terms of cross-correlation functions, in which the causal linkage is an object of analysis.

In the transform pair right in the integrand are replaced by the complex exponential and the integration limits 0 and because even functions are transformed. This is the classical Wiener- Chintchin formulation, often even by the metrology stand closer is replaced.

Resistor shunt capacitance

The ACF to the spectrum of the terminal voltage of the resistor lying parallel stray capacitance

The power dissipated in parallel lying capacitor of capacitance in rushing resistance itself is

The normalized ACF is determined solely by the statistical relationship

The mean correlation length is defined by

• This wiring of the rushing resistance forces the noise an average correlation time on, it is equal to its time constants, see above.
• Large correlation periods are represented with exponentially less werdendem weight.

Excursus on the metrological significance of the correlation time. Measuring the noisy rash of a measuring instrument, requires many independent readings for a sufficient statistic for the calculation of the mean and its error with the desired accuracy. ( Gaussian noise is to be an advantage. )

• The minimum required measurement time is calculated from the number required for the aspired accuracy of individual measurements multiplied by a small multiple of the correlation time of the fault.

Quantum theory, limited ACF of the resistance noise

The ACF for the quantum theory limited range of available power is calculated hereunder.

Note 1: defined to go into this formula.

Note 2: In the above, the correlation function of the terminal voltage having been treated, is now the power dimension.

It follows, first the above- calculated total power available

The normalized ACF of the quantum mechanically limited noise spectrum again describes the internal temporal structure alone

Shows that

• The quantum theoretically limited noise has a correlation time of about
• The large correlation times are weighted proportionally.

So that by way of example shows that a weak decrease in the spectrum has a sharp correlation function, and vice versa. The capacitance is proportional to limited spectrum associated with an exponential decay of the statistical weight increasing correlation times. - The quantum theory limited spectrum falls off exponentially with increasing frequency in practice, its correlation function finally approximated only in accordance with

White noise

On the issue of the wide range in inner relation of short duration and vice versa, the extreme case is cited. The white spectrum correspond to any short-duration operations. A pulse that passes already in the making, can serve and is mathematically well defined with the Dirac distribution. From this brief any object, only those values ​​can be specified for finite. Nevertheless, it is this delta function due to the mean value property

To represent physical situations. )

Necessarily leads to correlation functions: Because no square distribution can be formed, must be used to calculate the performance of the ACF, see convolution integral, be used:

The voltage pulse at the time

Generates the voltage surge

The unit 1 Vs and has the AKF arbitrarily short correlation time

As well as the white frequency spectrum

Conversely, the arbitrarily narrow frequency band at

On the AKF any far-reaching periodic correlation

With 0 the correlation length is arbitrarily large. When the DC voltage is

Register here we can speak of infinitely large correlation length at also strictly localized spectrum.

Stationary series of shock functions

Above-defined voltage pulses to be generated independently from each other are equally probable at any time, with the mean number density per time interval, they form a stationary sequence. The surges p are equal often provided with positive or negative sign, so that the linear average, the DC component disappears. The pulses are statistically independent. Such a design could be considered a first approach for a description of the thermal noise. However, the instantaneous values ​​satisfy obviously not a normal distribution ( bell curve ).

The statistical independence allows the easy specification of the AKF this episode with the help of the theorem of Campbell:

The AKF ( dimension capacity of the SI unit 1 W after division by a resistor R ) does not change its course, the correlation time is vanishingly small. The frequency spectrum ( energy dimension of the unit 1 Ws after division by the resistor R, as a power for each frequency bandwidth ) also does not change except for the factor

Exponentialimpulse

• Under the assumptions of Campbell's theorem to add the square sizes power and energy without the center inner temporal relationship of the pulse sequence - measured by the AKF - to change, statistical overlap of pulses of finite duration ( incoherent superposition ) is permitted, although the resulting amplitude spectrum is changed.

To illustrate be in the pulse sequence described above - under appropriate conditions - the impact functions Exponentialimpulse

Replaced. The ACF and the frequency spectrum, a Lorentz profile of the modified voltage include:

The terms in square brackets see note)

Is the power dissipated across the resistor R. By the product of the degree of overlap can be adjusted.

ACF and spectrum have the same function of and f as in the noise of the resistor with a parallel capacitor, see above, although the individual pulses are certainly much different. According discharges with time constants a capacitor through a resistor.

• While the RC - filtered noise resistance of the invariant inner connection is imposed in accordance with, he is here determined by the single-process ahead.
• From the spectrum before or can not be on deterministic single processes or random draw conclusions.

) Comment: A finite DC component is created when all Exponentialimpulse be included with fixed (positive) sign in the result. Equal proportions of the individual pulses as here are coherent and therefore must be added as amplitudes. Consequently, which is added ( after division by R) to the DC power ACF and the corresponding term of the spectrum.