1. 1 LECTURE 7. Contents 5.Sources of errors 5.1.Impedance matching 5.4.1.Non-energetic matching 5.4.2.Energetic matching 5.4.3.Non-reflective matching 5.4.4.To match or not to match? 5.2.Basic noise types 5.2.1.Thermal noise 5.2.2.Shot noise 5.2.3. 1/f noise 5.3.Noise characteristics 5.3.1. Signal-to-noise ratio, SNR 5.3.2. Noise factor, F, and noise figure, NF 5.3.3. Calculating SNR and input noise voltage from NF 5.3.4. V n  I n noise model 5.4.Noise matching 5.4.1.Optimum source resistance 5.4.2.Methods for the increasing of SNR 5.4.3. SNR of cascaded noisy amplifiers

2. 2 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. 5. SOURCES OF ERRORS. 5.2. Basic noise types 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 in Sections 5.2 and 5.3 the most fundamental types of noise and its characteristics. Influence Measurement System Measurement Object Matching +   x x

3. 3 Consider n carriers of charge moving with a velocity v through a conductor of length L. The induced current at the ends of the conductor is The fluctuations of this current Thus, two mechanisms contribute to the total noise:  the fluctuations in velocity, e.g. thermal noise,  the fluctuations in number, e.g. shot noise and flicker noise. t   5.2.Basic noise types 5. SOURCES OF ERRORS. 5.2. Basic noise types H. Spieler q n v L q n L i rms 2  v rms 2  n rms 2    2 q v L 2 L vivi q Q i 

4. 4 Reference: [1] 5. SOURCES OF ERRORS. 5.2. Basic 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. q n L i rms 2  v rms 2  n rms 2  2 q v L 2

5. 5 vn(t)vn(t) t f (vn)f (vn) vn(t)vn(t) 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise R V 66   VnVn Example: Resistor thermal noise T  0 2R()2R()  0 White (uncorrelated) noise en2en2 f 0 Normal distribution according to the central limit theorem

6. 6 C T e nC 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise To calculate the thermal-noise power density, e t 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: e t T

7. 7 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise 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: x z Illustration: The law of the equipartition of energy y

8. 8 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise Illustration: Resistor thermal noise pumps energy into the capacitor Each particle (mechanical equivalents of electron in the resistor) has three degrees of freedom CVC22CVC22 m i v i 2 2 C V C 2 2 = k T2k T2 In thermal equilibrium: The particle (a mechanical equivalent of the capacitor) has a single degree of freedom x y z

9. 9 5. SOURCES OF ERRORS. 5.2. Basic 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: H( f ) = e nC ( f ) e t ( f ) C V C 2 2 = k T2k T2 In thermal equilibrium: C T e nC R Ideal, noiseless resistor Noise source Real resistor e t T

10. 10  kT C 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise = =  nC 2 =  C  nC 2 2 kT 2 C C v nC (t) 2 2 0 According to the Einstein – Wiener–Khinchin theorem  e t 2 ( f )  H(j2  f)  2 d f   nC 2  R nC (  ) = 1  d f  0   e t 2 ( f ) 1+ (2  f RC) 2 et2( f )et2( f ) 4 RC e t 2 ( f ) = 4 k T R [V 2 /Hz]. PSD of resistor noise: C V C 2 2 =

11. 11 Amplitude spectral density of noise (ASD): e t =   4 k T R [V/  Hz]. Noise voltage, rms: V t =   4 k T R  [V]. 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise e t = 0.13   R [nV/  Hz]. At room temperature, ASD

12. 12 An equivalent noise bandwidth, ENB, is defined as the bandwidth of an equivalent-gain ideal rectangular filter that would pass as much power of white noise as the filter in question: D. Noise bandwidth vs. signal bandwidth 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise n w 2 NB  n w 2  H( j f )  2 d f.  0   H( j f )   H  max f 1 0.5 NB SB NB linear scale Equal areas LowpassBandpass 0 Signal bandwidth

13. 13 R C e no ( f ) SB   f c 1 2  RC 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise = e t 2 0.5  f c e t 2 NB = V no 2 =  e t 2   H( j f )  2 d f  0  Example: NB/SB for an RC filter = e t 2 1 1 + ( f / f c ) 2 d fd f  0  e t NB /SB  ? NB /SB  0.5 

14. 14 V t =   4 k T 1k  1Hz  4 nV V t =   4 k T 50  1Hz  0.9 nV V t =   4 k T 1M  1MHz  128  V Example: Thermal noise voltage, rms: 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise R V nR e t Noiseless filter

15. 15 Question: What about crest factor? 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise

16. 16 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. 5.2. Basic noise types. 5.2.1. Thermal noise E. Effect of dynamic networks on the noise pdf

17. 17 Input noise pdfInput noise autocorrelation Output noise pdfOutput noise autocorrelation 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise 1) First-order filtering of the Gaussian white noise. E. Effect of dynamic networks on the noise pdf

18. 18 Input noise pdfInput and output noise spectra Output noise pdfInput and output noise vs. time 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise 2) First-order filtering of the uniform white noise. E. Effect of dynamic networks on the noise pdf

19. 19 Input noise pdfInput noise autocorrelation Output noise pdfOutput noise autocorrelation 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise 2) First-order filtering of the uniform white noise. E. Effect of dynamic networks on the noise pdf

20. 20 We will show in this section that in thermal equilibrium any system that dissipates power generates thermal noise; and vice versa, any system that does not dissipate power does not generate thermal noise. For example, ideal capacitors, inductors, and dynamic resistances do not dissipate power and then do not generate any thermal noise. To prove the above, we will show that the following circuit can only be in thermal equilibrium if e nC = 0. F. Thermal noise in capacitors and inductors Reference: [2], pp. 230-231 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise RC e t e nC TT

21. 21 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 f  5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise e t e nC P RC  0  P CR  0 T = T

22. 22 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 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise G. Noise power at a capacitor V nC 2  = 4 k T R  NB  4 k T R 0.5  1 2  RC V nC 2  k TCk TC e t T SB

23. 23 To reduce the rms noise level across a capacitors, either capacitor value should be increased or temperature should be decreased. Reference: [5], p. 203 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise V nC 2  kT C R C V nC etet T

24. 24 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. H. Noise temperature, T n 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise R e n ( f ) R i n ( f ) R n a ( f ) e n 2 = 4 k T R [V 2 /Hz], i n 2 = 4 k T/ R [A 2 /Hz], n a   k T [W/Hz]. e n 2 4 R4 R e n ( f )

25. 25 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 ( f ) k Then, given a source's noise temperature T n, n a ( f )  k  T n ( f ). 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise n a ( f ) e n ( f ) Nonthermal sources of noise

26. 26 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. In contrast to thermal noise, shot noise cannot be reduced by lowering the temperature. Reference: Physics World, August 1996, page 22 5.2.2. Shot noise (רעש הברד) 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.2. Shot noise D I i www.discountcutlery.net q n L i rms 2  v rms 2  n rms 2    2 q v L 2

27. 27 dtdt D N. Gershenfeld, The physics of information technology. i(t)i(t) 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.2. Shot noise t i(t)i(t) i(t)  q  (t   t n ),  n = 1n = 1  1 T i sh ( f )  q  (t   t n ) exp[ j 2   f t] d t  q exp[ j 2   f t n ],  n = 1n = 1  ∫ - ∞ ∞  n = 1n = 1  A. Spectral density of shot noise dtdt q  i(t)i(t)

28. 28 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.2. Shot noise i sh ( f )  q exp[ j 2   f t n ],  n = 1n = 1   n = 1n = 1  i sh 2 ( f )  E [i( f ) i*( f )]  lim exp[ j 2   f t n ]  exp[  j 2   f t m ] TT q2Tq2T  m = 1m = 1   lim = q E [  i(t)  ]  q I, TT q2NTq2NT ASD: i sh ( f ) =  2 q . PSD: i sh 2 ( f ) = 2 q . N Pulses are random and do not come in groups  E [cross products]=0

29. 29 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.2. Shot noise B. Shot noise in resistors and semiconductor devices Through a p  n junction (or any other potential barrier), the electrons are transmitted randomly and independently of each other. Thus the transfer of electrons can be described by Poisson statistics. In this case, the shot noise has its maximum value at i sh 2 ( f ) = 2 q I. Shot noise is absent in a macroscopic, metallic resistor because the inelastic electron-phonon scattering smoothes out current fluctuations that result from the discreteness of the electrons, leaving only thermal noise. Shot noise does exist in mesoscopic (nm) resistors, although at lower levels than in a diode 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

30. 30 Example: Noise current, rms: R I sh =   2 q 1mA 1Hz  18 pA I sh =   2 q 1  A 1Hz  0.6 pA I sh =   2 q 1A 1Hz  566 pA 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise rdrd V R i sh (t) IDID it(t)it(t) (R II r d ) 2 (2qI D +4kT/R)NB Noiseless filter D Example: Shot noise measurements (find R for V out =V sh ±10% at I D =1 mA ).

31. 31 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.1. Thermal noise The aim is to measure the diode shot noise voltage (rms) with a 0.5% inaccuracy. 1.For I D  1 mA, calculate R for which the thermal noise contribution, V t, to the output rms voltage, V out, is 10% compared to the contribution of the diode shot noise, V sh. (Calculate V t and V sh as a function of r d and R.) Find the relative inaccuracy for the noise voltage (rms) and PSD. 2.Read “Noise analysis” in pspicead.pdf, pp. 333  339. 3. Simulate the circuit in SPICE and compare the calculated and simulated values of V t and V sh. The difference between the theoretical calculations and the simulation results should not be greater than 1%. To provide this, (i) set I D at 1 mA ±0.01%, (ii) find the exact value of r d, (iii) find the SPICE default temperature by analyzing the thermal noise of the resistor, (iv) do not consider contributions from other noise sources. 4. What one can do to maximize V sh ? Reach in your simulation V sh  15 nV/Hz 0.5, V t  1 nV/Hz 0.5. Homework: Simulate the previous example in SPICE

32. 32 The most general type of excess noise is 1/f or flicker noise. This noise has approximately 1/f power 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 device properties caused, for example, by electric current in resistors and semiconductor devices. 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] 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 devices generate additional or excess noise. 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise No unifying principle has been found for all the 1/f noise sources.

33. 33 References: [4] and [5] 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 bipolar, JFET, and MOSFET transistors have flicker noise, it is a significant noise source in MOS transistors, whereas it can often be ignored in bipolar transistors (and some modern JFETs). 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise q n L i rms 2  v rms 2  n rms 2 2 q v L 2 In electrical and electronic devices, flicker noise occurs only when electric current is flowing.

34. 34 An important parameter of 1/f noise is its corner frequency, f c, where its PSD equals the white noise level. A typical value of f f is 100 Hz to 1 kHz for BJT and JFET and 100 kHz for MOSFET. 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise f, decades PSD, i n 2 ( f ), dB f White noise Pink noise  10 dB/decade

35. 35 References: [4] and [5] Flicker noise is proportional to the dc (or average) current flowing through the device: 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise in2( f ) in2( f )  where K f is a constant that depends on the type of material, 1 < m < 3, and 1 < n < 3. We will assume: m  2 and n  1 for resistors, and m  1 and n  1 for transistors. K f I m f n

36. 36 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise The PSD 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: in2( f ) in2( f )  K f I 2 f ASD of flicker noise is directly proportional to the dc (or average) current flowing through the device: PSD R   J ~  P R R R 1PR1PR J J/3 PRPR 9PR9PR R R R R R R R R R R

37. 37 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise Example: Let us compare 1/f noise in 1 , 1 W and 1 , 9 W resistors for the same  dc current: i f 1W 2 ( f ) 1  9 W 1 A i f 9W 2 ( f )  ? 1  1 W 1 A

38. 38 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise 1/3 A 1 A i f 1W 2 / 9 1  1 W3  3 W 1  9 W i f   ·   / 3 i f   / 27 i f   / 9 f f f (9 W) White noise i f 2 ( f ), dB f f (1 W) i f 1W 2 ( f ) 1  9 W 1 A i f 9W 2 ( f )  ? 1  1 W 1 A 1W 9W in2( f ) in2( f )  K f I 2 f 

39. 39 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise 0100200300400500 -7.5 -5 -2.5 0 5 7.5 10 Sampling frequency, 1/  t (Hz) Time, (s) PSD white noise PSD flicker noise White noise Flicker noise 3300303000 1 100 0.01 0.0001 Example: Noise in a carbon resistor with and without a current. Question: Is flicker noise stationary? I  I 

40. 40 5. SOURCES OF ERRORS. 5.2. Basic noise types. 5.2.3. 1/f noise Homework: Repeat the simulations in the previous slide.

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