1 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] 5.2.Noise types 5. SOURCES OF ERRORS Noise types Thermal noise

2 Reference: [1] 5. SOURCES OF ERRORS Noise types Thermal noise 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.

3 vn(t)vn(t) t f(vn)f(vn) vn(t)vn(t) 5. SOURCES OF ERRORS Noise types Thermal noise R V 66   V n rms Example: Resistor thermal noise Normal distribution according to the central limit theorem T  0 2R()2R()  0 White (uncorrelated) noise en2en2 f 0

4 C e nC 5. SOURCES OF ERRORS Noise types Thermal noise To calculate the thermal noise power density, e n 2 ( f ), of a resistor, which is in thermal equilibrium with its surrounding, we temporarily connect a capacitor to the resistor. R Ideal, noiseless resistor Noise source Real resistor A. Noise description based on the principles of thermodynamics and statistical mechanics (Nyquist, 1828) From the point of view of thermodynamics, the resistor and the capacitor interchange energy: enen

5 5. SOURCES OF ERRORS Noise types Thermal noise Illustration: The law of equipartition of energy m v 22m v 22 Each particle has three degrees of freedom m i v i 2 2 m i v i 2 2 = m v 22m v 22 = 3= 3 k T2k T2 In thermal equilibrium:

6 5. SOURCES OF ERRORS Noise types Thermal noise C V 22C V 22 = k T2k T2 In thermal equilibrium: Illustration: Resistor thermal noise pumps energy into the capacitor Each particle has three degrees of freedom CV 22CV 22 m i v i 2 2

7 5. SOURCES OF ERRORS Noise types 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: R Ideal, noiseless resistor Noise source Real resistor C e nC H( f ) = e nC ( f ) e nR ( f ) e nR C V 22C V 22 = k T2k T2 In thermal equilibrium:

8  kT C 5. SOURCES OF ERRORS Noise types Thermal noise = =  nC 2 =  C  nC 2 2 kT 2 C C v nC (t) 2 2  According to the Wiener–Khinchin theorem (1934), Einstein (1914),  e nR 2 ( f )  H(j2  f)  2 e j 2  f  d f   nC 2  R nC (  ) = 1  d f     e nR 2 ( f ) 1+ (2  f RC) 2 e nR 2 ( f ) 4 RC e nR 2 ( f ) = 4 k T R [V 2 /Hz]. Power spectral density of resistor noise:

9 SHFEHFIRR 10 GHz 100 GHz 1 THz 10 THz 100 THz 1 GHz  e nR P ( f )  2  e nR ( f )  2 f 5. SOURCES OF ERRORS Noise types Thermal noise e nR P 2 ( f ) = 4 R [V 2 /Hz]. B. Noise description based on Planck’s law for blackbody radiation (Nyquist, 1828) h f e h f /k T  1 A comparison between the two Nyquist equations: R = 50 , C = 0.04 f F R = 50 , C = 0.04 f F

10 5. SOURCES OF ERRORS Noise types Thermal noise The Nyquist equation was extended to a general class of dissipative systems other than merely electrical systems. e qn 2 ( f ) = 4 R + [V 2 /Hz]. h f e h f /k T  1 Zero-point energy  f(t) h f2h f2 C. Noise description based on quantum mechanics (Callen and Welton, 1951)  e qn ( f )  2  e nR ( f )  2 SHFEHFIRR 10 GHz 100 GHz 1 THz 10 THz 100 THz 1 GHz f Quantum noise

11 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 T strd = 290 K (standard temperature: k T strd = 4.00  10  21 ) 10 Log dB. 2 e h f /k T  1 f, Hz Ratio, dB SOURCES OF ERRORS Noise types Thermal noise

12 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 B of an ideal filter is its actual bandwidth. For practical filters, B is greater than their 3-dB bandwidth. For example, an RC filter has B = 0.5   f c, 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. D. Equivalent noise bandwidth, B Reference: [4] 5. SOURCES OF ERRORS Noise types Thermal noise

13 R C e n o ( f ) f c = =  f 3dB 1 2  RC 5. SOURCES OF ERRORS Noise types Thermal noise e n in = e n in  f c V n o rms 2 =  e n o 2 ( f ) d f  0   = e n in 2  H( f )  2 d f  0  Example: Equivalent noise bandwidth of an RC filter = e n in ( f / f c ) 2 d fd f  0  V n o rms 2 = e n in 2 B

14 fcfc Equal areas f /f c B = 0.5   f c  1.57 f c R C e n o 0 5. SOURCES OF ERRORS Noise types Thermal noise e n in  e n o  2  e n in  f /f c fcfc B 0.5 Equal areas  e n o  2  e n in  2 f c = =  f 3dB 1 2  RC Example: Equivalent noise bandwidth of an RC filter

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

16 V n rms =   4 k T 1k  1Hz  4 nV V n rms =   4 k T 50  1Hz  0.9 nV Amplitude spectral density of noise: e n =   4 k T R [V/  Hz]. Noise voltage: V n rms =   4 k T R   f n [V]. V n rms =   4 k T 1M  1MHz  128  V Examples: 5. SOURCES OF ERRORS Noise types Thermal noise

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

18 Input noise pdfInput noise autocorrelation Output noise pdfOutput noise autocorrelation 5. SOURCES OF ERRORS Noise types Thermal noise 1) First-order filtering of the Gaussian white noise.

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

20 Input noise pdfInput noise autocorrelation Output noise pdfOutput noise autocorrelation 5. SOURCES OF ERRORS Noise types Thermal noise 2) First-order filtering of the uniform white noise.

21 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. e n 2 = 4 k T R [V 2 /Hz], i n 2 = 4 k T/ R [I 2 /Hz], n a  = k T [W/Hz]. e n 2 4 R4 R F. Noise temperature, T n 5. SOURCES OF ERRORS Noise types Thermal noise

22 Any thermal noise source has available power spectral density n a ( f )  k T, where T is defined as the noise temperature, T = T n. 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 T n : T n ( f ) . n a 2 ( f ) k Then, given a source's noise temperature T n, n a 2 ( f )  k  T n ( f ). 5. SOURCES OF ERRORS Noise types Thermal noise

23 Example: Noise temperatures of nonthermal noise sources Cosmic noise: T n = 1 … K. Environmental noise: T n (1 MHz) = 3  10 8 K. T V n ( f ) 2 = 320  2 (l/ ) k T = 4 k T R  l  5. SOURCES OF ERRORS Noise types Thermal noise

24 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 e nC = 0. G. Thermal noise in capacitors and inductors RC Reference: [2], pp e nR e nC 5. SOURCES OF ERRORS Noise types Thermal noise

25 Reference: [2], p. 230 In thermal equilibrium, the average power that the resistor delivers to the capacitor, P RC, must equal the average power that the capacitor delivers to the resistor, P CR. Otherwise, the temperature of one component increases and the temperature of the other component decreases. P RC is zero, since the capacitor cannot dissipate power. Hence, P CR should also be zero: P CR  [e nC ( f ) H CR ( f ) ] 2 /R  where H CR ( f )  R /(1/j2  f+R). Since H CR ( f ) , e nC ( f ) . RC e nR e nC f  5. SOURCES OF ERRORS Noise types Thermal noise

26 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 : Reference: [5], p. 202 R C V nC e nR 5. SOURCES OF ERRORS Noise types Thermal noise H. Noise power at a capacitor V nC rms 2  = e nR 2  H( f )  2 d f  0   4 k T R  B  4 k T R 0.5  1 2  RC V nC rms 2  kT C

27 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 SOURCES OF ERRORS Noise types Thermal noise V nC rms 2  kT C R C V nC e nR

28 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 V n  rms in the following circuit against kT/C. How do you account for the difference? (The operational amplifier is assumed ideal and noiseless.) 5. SOURCES OF ERRORS Noise types Thermal noise Home exercise: 1nF 1k 1pF V n  rms V n o rms vsvs

29 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. Reference: Physics World, August 1996, page Shot noise 5. SOURCES OF ERRORS Noise types Shot noise R I i

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

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

32 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 = q n. 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 + d t) as P 1 (d t) = n d t. Next, we derive the probability that no electrons pass the cross- section in the time interval (0, t + d t): P 0 (t + d t ) = P 0 (t) P 0 (d t) = P 0 (t) (1  n d t). A. Statistical description of shot noise 5. SOURCES OF ERRORS Noise types Shot noise

33 This yields with the obvious initiate state P 1 (0) = 0. This yields with the obvious initiate state P 0 (0) = SOURCES OF ERRORS Noise types Shot noise =  n P 0 d P0d td P0d t The probability that exactly one electrons pass the cross- section in the time interval (0, t + d t) P 1 (t + d t ) = P 1 (t) P 0 (d t) + P 0 (t) P 1 (d t) = P 1 (t) (1  n d t) + P 0 (t) n d t. =  n P 1 + n P 0 d P1d td P1d t

34 In the same way, one can obtain the probability of passing the cross section  electrons, exactly: 5. SOURCES OF ERRORS Noise types Shot noise =  n P N + n P N  1 d PNd td PNd t P N (0) = 0. which corresponds to the Poisson probability distribution. P N (t) = e  n t, (n t) N N !(n t) N N ! By substitution, one can verify that

35 N = SOURCES OF ERRORS Noise types Shot noise Illustration: Poisson probability distribution P N (t) = e  1 t (1 t) N N ! t N = 20 N = 30 0

36 The average number of electrons passing the cross-section during a time interval  can be found as follows 5. SOURCES OF ERRORS Noise types Shot noise N  = e  n  = n  e  n  = n , (n) N N !(n) N N ! and the squared average number can be found as follows:  n = 0n = 0  (n   ) N  1 (N  1) !  n = 1n = 1  N  2 = N  2 e  n  = [N (N  1) + N ] e  n  (n) N N !(n) N N !  n = 0n = 0   n = 0n = 0  (n) N N !(n) N N ! = n  n     e  n  = n  n    .  n = 2n = 2  (n   ) N  2 (N  2) !

37 We now can find the average current of the electrons, I, and its variance, i rms 2 : 5. SOURCES OF ERRORS Noise types Shot noise I = i  = (q /    N   = q n, i rms 2 = (q /      N 2  = (q /     . The variance of the number of electrons passing the cross- section during a time interval  can be found as follows  N 2 = N  2  N  ) 2 = n .

38 Hence, the spectral density of the shot noise 5. SOURCES OF ERRORS Noise types Shot noise I n 2 = 2 q  f s. i n ( f ) =  2 q . B. Spectral density of shot noise Assuming  = 1/ 2 f s, we finally obtain the Schottky equation for shot noise rms current

39 5. SOURCES OF ERRORS Noise types 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 2 q I. 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

40 The most general type of excess noise is 1/f or 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. Reference: [3] /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 excess noise. 5. SOURCES OF ERRORS Noise types /f noise No unifying principle has been found for all the 1/f noise sources.

41 References: [4] and [5] 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. 5. SOURCES OF ERRORS Noise types /f noise

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

43 References: [4] and [5] Flicker noise is directly proportional to the density of dc (or average) current flowing through the device: 5. SOURCES OF ERRORS Noise types /f noise i n 2 ( f )  , a A J 2 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.

44 5. SOURCES OF ERRORS Noise types /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: f, decades fcfc White noise 1 A i n 1W ( f ) 2  a A J 2 f 1  1 W i n 1W 2 ( f ), dB

45 5. SOURCES OF ERRORS Noise types /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  1 W 1 A i n 1W 2 ( f )  a A J 2 f 1  9 W 1/3 A 1 A i n 9W 2 ( f )  a A J 2 9 f f, decades i n 1W 2 ( f ), dB fcfc White noise

46 5. SOURCES OF ERRORS Noise types /f noise f, decades fcfc 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: White noise 1 A 1  9 W 1/3 A 1 A i n 9W 2 ( f )  a A J 2 9 f i n 1W 2 ( f )  a A J 2 f 1  1 W i n 1W 2 ( f ), dB

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

48 5. SOURCES OF ERRORS Noise types /f noise Example: Simulation of 1/f noise

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