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5. SOURCES OF ERRORS. 5.2. Noise types . 5.2.1. Thermal noise. 5.2. Noise types. In order to reduce errors, the measurement object and the measurement system should be matched not only in terms of output and input impedances, but also in terms of noise.

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  1. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise 5.2. Noise types In order to reduce errors, the measurement object and the measurement system should be matched not only in terms of output and input impedances, but also in terms of noise. The purpose of noise matching is to let the measurement system add as little noise as possible to the measurand. We will treat the subject of noise matching in Section 5.4. Before that, we have to describe the most fundamental types of noise and its characteristics (Sections 5.2 and 5.3). Reference: [1]

  2. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise 5.2.1. Thermal noise Thermal noise is observed in any system having thermal losses and is caused by thermal agitation of charge carriers. Thermal noise is also called Johnson-Nyquist noise. (Johnson, Nyquist: 1928, Schottky: 1918). An example of thermal noise can be thermal noise in resistors. Reference: [1]

  3. vn(t) 6s 2s Vn rms f(vn) 2R(t) en2 White (uncorrelated) noise t f 0 0 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Example: Resistor thermal noise vn(t) T 0 R V t Normal distribution according to the central limit theorem

  4. C enC 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise A. Noise description based on the principles of thermodynamics and statistical mechanics (Nyquist, 1828) To calculate the thermal noise power density, en2( f ), of a resistor, which is in thermal equilibrium with its surrounding, we temporarily connect a capacitor to the resistor. Real resistor R Ideal, noiseless resistor en Noise source From the point of view of thermodynamics, the resistor and the capacitor interchange energy:

  5. mv2 2 mi vi 2 2 mv2 2 kT 2 In thermal equilibrium: = =3 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Illustration: The law of equipartition of energy Each particle has three degrees of freedom mivi 2 2

  6. CV2 2 CV2 2 kT 2 In thermal equilibrium: = 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Illustration: Resistor thermal noise pumps energy into the capacitor Each particle has three degrees of freedom mivi 2 2

  7. enR CV2 2 kT 2 In thermal equilibrium: = 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Since the obtained dynamic first-order circuit has a single degree of freedom, its average energy is kT/2. This energy will be stored in the capacitor: enC ( f ) enR ( f ) H( f ) = Real resistor R C Ideal, noiseless resistor enC Noise source

  8. According to the Wiener–Khinchin theorem (1934), Einstein (1914), CsnC2 kT kT CvnC(t)2 = =  snC2 = . 2 2 C 2  enR 2(f)H(j2pf)2ej2pf tdf  snC2 = RnC (0) = 0  kT enR2(f) 1 = . = enR2(f) df =  C 4RC 1+ (2pfRC)2 0 Power spectral density of resistor noise: enR2(f)= 4kTR [V2/Hz]. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

  9. enR P(f)2 enR(f)2 1 R = 50 W, C= 0.04 fF 0.8 0.6 0.4 0.2 f 1 GHz 10 GHz 100 GHz 1 THz 10 THz 100 THz SHF EHF IR R 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise B. Noise description based on Planck’s law for blackbody radiation (Nyquist, 1828) hf ehf /kT-1 enR P2(f)= 4R [V2/Hz]. A comparison between the two Nyquist equations:

  10. Zero-point energyf(t) eqn (f)2 enR (f)2 8 6 Quantum noise 4 2 0 f SHF EHF IR R 1 GHz 10 GHz 100 GHz 1 THz 10 THz 100 THz 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise C. Noise description based on quantum mechanics (Callen and Welton, 1951) The Nyquist equation was extended to a general class of dissipative systems other than merely electrical systems. hf ehf /kT-1 hf 2 eqn2(f)= 4R + [V2/Hz].

  11. Ratio, dB 120 100 80 60 40 20 f, Hz 0 102 104 106 108 1010 1012 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise The ratio of the temperature dependent and temperature independent parts of the Callen-Welton equation shows that at 0 K there still exists some noise compared to the Nyquist noise level at Tstrd= 290 K (standard temperature: kTstrd = 4.0010-21) 2 ehf /kT-1 10Log dB.

  12. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise D. Equivalent noise bandwidth, B An equivalent noise bandwidth,B,is defined as the bandwidth of an equivalent-gain ideal rectangular filter that would pass as much noise power as the filter in question. By this definition, the Bof an ideal filter is its actual bandwidth. For practical filters, Bis greater than their 3-dB bandwidth. For example, an RC filter has B= 0.5pfc, which is about 50% greater than its 3-dB bandwidth. As the filter becomes more selective (sharper cutoff characteristic), its equivalent noise bandwidth, B, approaches the 3-dB bandwidth. Reference: [4]

  13. =en in2H(f)2 df  0 1 1 + (f / fc )2 = en in2 df = en in20.5p fc  Vn o rms2= en in2B  0 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Example: Equivalent noise bandwidth of an RC filter R  en o(f) Vn o rms2=en o2(f)df  0 C en in 1 2pRC fc = = D f3dB

  14. en o2 en o2 B=0.5pfc = 1.57fc en in2 en in2 Equal areas Equal areas 0.5 fc B 1 0.5 0.1 fc f /fc 0.01 0.1 1 10 100 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Example: Equivalent noise bandwidth of an RC filter R en o 1 C en in f /fc 1 2pRC fc = = D f3dB 1 0 2 4 6 8 10

  15. Butterworth filters: 1 1+ ( f /fc )2n H( f )2 = 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Example: Equivalent noise bandwidth of higher-order filters First-order RC low-pass filterB= 1.57fc. Two first-order independent stages B= 1.22fc. • second order B= 1.11fc. • third order B= 1.05fc. • fourth order B= 1.025fc.

  16. Noise voltage: At room temperature: Vn rms = 4kTRDfn[V]. en = 0.13R[nV/Hz]. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Amplitude spectral density of noise: en = 4kTR[V/Hz].

  17. Examples: Vn rms = 4kT1MW1MHz = 128 mV Vn rms = 4kT1kW1Hz = 4 nV Vn rms = 4kT50W1Hz = 0.9 nV 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

  18. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise 1) First-order filtering of the Gaussian white noise. E. Normalization of the noise pdf by dynamic networks Input noise pdf Input and output noise spectra Output noise pdf Input and output noise vs. time

  19. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise 1) First-order filtering of the Gaussian white noise. Input noise pdf Input noise autocorrelation Output noise pdf Output noise autocorrelation

  20. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise 2) First-order filtering of the uniform white noise. Input noise pdf Input and output noise spectra Output noise pdf Input and output noise vs. time

  21. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise 2) First-order filtering of the uniform white noise. Input noise pdf Input noise autocorrelation Output noise pdf Output noise autocorrelation

  22. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise F. Noise temperature, Tn Different units can be chosen to describe the spectral density of noise: mean square voltage (for the equivalent Thévenin noise source), mean square current (for the equivalent Norton noise source), and available power. en2 = 4kTR[V2/Hz], in2 = 4kT/R[I2/Hz], en2 na = kT[W/Hz]. 4R

  23. It is a common practice to characterize other, nonthermal sources of noise, having available power that is unrelated to a physical temperature, in terms of an equivalent noise temperature Tn: na2( f ) Tn ( f )  . k Then, given a source's noise temperature Tn, na2( f )  kTn ( f ) . 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Any thermal noise source has available power spectral density na( f )  kT ,where T is defined as the noise temperature, T = Tn.

  24. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Example: Noise temperatures of nonthermal noise sources Cosmic noise: Tn= 1 … 10 000 K. Environmental noise: Tn(1 MHz) = 3108 K. T Vn2(f)= 320p2(l/l)kT = 4 kT RS l << l

  25. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise G. Thermal noise in capacitors and inductors Ideal capacitors and inductors do not dissipate power and then do not generate thermal noise. For example, the following circuit can only be in thermal equilibrium if enC = 0. R C enR enC Reference: [2], pp. 230-231

  26. f > 0 f > 0 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise R C enR enC In thermal equilibrium, the average power that the resistor delivers to the capacitor, PRC, must equal the average power that the capacitor delivers to the resistor, PCR. Otherwise, the temperature of one component increases and the temperature of the other component decreases. PRC is zero, since the capacitor cannot dissipate power. Hence, PCR should also be zero: PCR= [enC(f)HCR(f)]2/R = 0, where HCR(f)= R/(1/j2pf+R). Since HCR(f) 0, enC (f)= 0. Reference: [2], p. 230

  27. =enR2H(f)2 df  0 VnC rms2 R C VnC enR 1 2pRC = 4kTR 0.5p kT C VnC rms2= 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise H. Noise power at a capacitor Ideal capacitors and inductors do not generate any thermal noise. However, they do accumulate noise generated by other sources. For example, the noise power at a capacitor that is connected to an arbitrary resistor value equals kT/C: = 4kTRB Reference: [5], p. 202

  28. R C VnC enR kT C VnC rms2= 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise The rms voltage across the capacitor does not depend on the value of the resistor because small resistances have less noise spectral density but result in a wide bandwidth, compared to large resistances, which have reduced bandwidth but larger noise spectral density. To lower the rms noise level across a capacitors, either capacitor value should be increased or temperature should be decreased. Reference: [5], p. 203

  29. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise Home exercise: Some feedback circuits can make the noise across a capacitor smaller than kT/C, but this also lowers signal levels. Compare for example the noise value Vn erms in the following circuit against kT/C. How do you account for the difference? (The operational amplifier is assumed ideal and noiseless.) 1nF 1k Vn erms Vn o rms 1k 1pF vs

  30. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise 5.2.2. Shot noise R Shot noise (Schottky, 1918) results from the fact that the current is not a continuous flow but the sum of discrete pulses, each corresponding to the transfer of an electron through the conductor. Its spectral density is proportional to the average current and is characterized by a white noise spectrum up to a certain frequency, which is related to the time taken for an electron to travel through the conductor. In contrast to thermal noise, shot noise cannot be reduced by lowering the temperature. I ii www.discountcutlery.net Reference: Physics World, August 1996, page 22

  31. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise Illustration: Shot noise in a conductor R i I t Reference: [1]

  32. I 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise Illustration: Shot noise in a conductor R i I t Reference: [1]

  33. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise A. Statistical description of shot noise We start from defining n as the average number of electrons passing a cross-section of a conductor during one second, hence, the average electron current I = qn. We assume then that the probability of passing through the cross-section two or more electrons simultaneously is negligibly small. This allows us to define the probability that an electron passes the cross-section in the time interval dt = (t, t+dt) as P1(dt) = ndt. Next, we derive the probability that no electrons pass the cross-section in the time interval (0, t+dt): P0(t+dt) = P0(t) P0(dt) = P0(t) (1-ndt).

  34. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise This yields with the obvious initiate state P0(0) = 1. dP0 dt = -nP0 The probability that exactlyone electrons pass the cross-section in the time interval (0, t+dt) P1(t+dt) = P1(t) P0(dt) + P0(t) P1(dt) = P1(t) (1-ndt) + P0(t) ndt . This yields with the obvious initiate state P1(0) = 0. dP1 dt = -nP1 +nP0

  35. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise In the same way, one can obtain the probability of passing the cross section N electrons, exactly: dPN dt = -nPN+nPN -1 . PN(0) = 0 By substitution, one can verify that (nt)N N! PN (t) = e- n t, which corresponds to the Poisson probability distribution.

  36. (1t)N N! PN (t) = e-1 t N=10 N=20 N=30 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise Illustration: Poisson probability distribution 0.12 0.1 0.08 0.06 0.04 0.02 t 0 10 20 30 40 50

  37. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise The average number of electrons passing the cross-section during a time interval t can be found as follows   (nt)N N! (nt)N-1 (N -1)! S S Nt= e- nt = nt e- nt = nt , n=1 n=0 and the squared average number can be found as follows:   (nt)N N! (nt)N N! S S Nt2= Nt2 e- nt= [N (N -1) + N ] e- nt n=0 n=0  (nt)N-2 (N -2)! S = nt +(nt)2 e- nt = nt +(nt)2. n=2

  38. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise The variance of the number of electrons passing the cross-section during a time interval t can be found as follows sN2= Nt2 - ( Nt )2 = nt. We now can find the average current of the electrons, I, and its variance, irms2: I = it= (q/t) Nt= qn, irms2=(q/t)2 sN2= (q/t)I .

  39. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise B. Spectral density of shot noise Assuming t=1/2 fs,we finally obtain the Schottky equation for shot noise rms current In2=2 qI fs . Hence, the spectral density of the shot noise in(f)=  2 qI .

  40. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise C. Shot noise in resistors and semiconductor devices In devices such as tunnel junctions the electrons are transmitted randomly and independently of each other. Thus the transfer of electrons can be described by Poisson statistics. For these devices the shot noise has its maximum value at 2qI. Shot noise is absent in a macroscopic, metallic resistor because the ubiquitous inelastic electron-phonon scattering smoothes out current fluctuations that result from the discreteness of the electrons, leaving only thermal noise. Shot noise may exist in mesoscopic (nm) resistors, although at lower levels than in a tunnel junction. For these devices the length of the conductor is short enough for the electron to become correlated, a result of the Pauli exclusion principle. This means that the electrons are no longer transmitted randomly, but according to sub-Poissonian statistics. Reference: Physics World, August 1996, page 22

  41. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise 5.2.3. 1/f noise Thermal noise and shot noise are irreducible (ever present) forms of noise. They define the minimum noise level or the ‘noise floor’. Many devises generate additional or excessnoise. The most general type of excess noise is 1/for flicker noise. This noise has approximately 1/f spectrum (equal power per decade of frequency) and is sometimes also called pink noise. 1/f noise is usually related to the fluctuations of the devise properties caused, for example, by electric current in resistors and semiconductor devises. Curiously enough, 1/f noise is present in nature in unexpected places, e.g., the speed of ocean currents, the flow of traffic on an expressway, the loudness of a piece of classical music versus time, and the flow of sand in an hourglass. No unifying principle has been found for all the 1/f noise sources. Reference: [3]

  42. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise In electrical and electronic devices, flicker noise occurs only when electric current is flowing. In semiconductors, flicker noise usually arises due to traps, where the carriers that would normally constitute dc current flow are held for some time and then released. Although both bipolar and MOSFET transistors have flicker noise, it is a significant noise source in MOS transistors, whereas it can often be ignored in bipolar transistors. References: [4] and [5]

  43. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise An important parameter of 1/f noise is its corner frequency, fc, where the power spectral density equals the white noise level. A typical value of fc is 100 Hz to 1 kHz. in 2(f), dB Pink noise White noise f,decades fc

  44. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise Flicker noise is directly proportional to the density of dc (or average) current flowing through the device: a AJ2 f in2(f)= , where a is a constant that depends on the type of material, and A is the cross sectional area of the devise. This means that it is worthwhile to increase the cross section of a devise in order to decrease its 1/f noise level. References: [4] and [5]

  45. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise For example, the spectral power density of 1/f noise in resistors is in inverse proportion to their power dissipating rating. This is so, because the resistor current density decreases with square root of its power dissipating rating: 1 W, 1 W 1 A a AJ2 f in 1W(f)2= in 1W2(f), dB White noise f,decades fc

  46. a AJ2 9f in 9W2(f)= 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise For example, the spectral power density of 1/f noise in resistors is in inverse proportion to their power dissipating rating. This is so, because the resistor current density decreases with square root of its power dissipating rating: 1 W, 1 W 1 A a AJ2 f in 1W2(f)= 1/3 A in 1W2(f), dB 1/3 A 1 A 1/3 A 1 W, 9 W White noise f,decades fc

  47. a AJ2 9f in 9W2(f)= 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise For example, the spectral power density of 1/f noise in resistors is in inverse proportion to their power dissipating rating. This is so, because the resistor current density decreases with square root of its power dissipating rating: 1 W, 1 W 1 A a AJ2 f in 1W2(f)= 1/3 A in 1W2(f), dB 1/3 A 1 A 1/3 A 1 W, 9 W White noise f,decades fc

  48. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise Example: Simulation of 1/f noise Input Gaussian white noise Input noise PSD Output 1/f noise Output noise PSD

  49. 5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise Example: Simulation of 1/f noise

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