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 5.3. Noise characteristics Reference: [4] The signal-to-noise ratio is the measure for the extent to which a signal can be distinguished from the background noise: SNR  SNSN References: [1] and [2] where S is the signal power, and N is the noise power. 5.3.1. Signal-to-noise ratio, SNR 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR

3. 3 SNR in  S in N in References: [1] and [2] 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR A. Signal-to-noise ratio at the input of the system, SNR in It is usually assumed that the signal power, S in, and the noise power, N in, are dissipated in the noiseless input impedance of the measurement system. Measurement objectMeasurement system vSvS Z S =R S + jX S Z in =R in + jX in Noiseless RLRL SNR in

4. 4 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR Example: Calculation of SNR in 1) S in , V S 2 Z in (Z S + Z in ) 2 2) N in , V n 2 Z in (Z S + Z in ) 2 3) SNR in  V S 2 V n 2 V S 2 4 k T R S N   Measurement objectMeasurement system vSvS Z S =R S + jX S Z in =R in + jX in Noiseless RLRL SNR in Note that SNR in is not a function of Z in.

5. 5 SNR o src SNR in SNR o src  SNR in References: [1] and [2] 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR Measurement objectMeasurement system vSvS Power gain, A p Noiseless SNR o src  S o N o src Z S =R S + jX S 1) The measurement system is noiseless.  S in A p N in A p  S in N in RLRL B. Signal-to-noise ratio at the output of the system, SNR o

6. 6 SNR o SNR o  SNR in References: [1] and [2] 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR Measurement objectMeasurement system vSvS Power gain, A p Noisy SNR o  S o N o Z S =R S + jX S 2) The measurement system is noisy. SNR in  S in A p (N in +N in msr ) A p  S in N in RLRL

7. 7 SNR o Noise factor is used to evaluate the signal-to-noise degradation caused by the measurement system (H. T. Friis, 1944). 5.3.2. Noise factor, F, and noise figure, NF 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF F  SNR in SNR o Measurement objectMeasurement system vSvS Power gain, A p Noisy Z S =R S + jX S SNR in RLRL

8. 8 SNR o SNR in The signal-to-noise degradation is due to the additional noise, N o msr, which the measurement system contributes to the load. 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Measurement objectMeasurement system vSvS Power gain, A p Noisy Z S =R S + jX S F  SNR in SNR o  SNR o src SNR o S o /N o src S o /N o    N o N o src    N o src + N o msr N o src   RLRL N o msr N o src    

9. 9 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF RSRS Measurement objectMeasurement system F  N o N o src V no 2 /R L 4 kTR S NB (G V A V ) 2 /R L  V no 2 4 kTR S NB (G V A V ) 2  vovo v in Example: Calculation of noise factor Voltage gain, A V RLRL e nS GVGV Here and below, we assume that the reactance in the source output impedance is compensated by the properly chosen input impedance of the measurement system (noise tuning).

10. 10 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF F  V no 2 4 kTR S B (G A V ) 2 The following three characteristics of noise factor can be seen by examining the obtained equation: 1. It is independent of the load resistance R L, 2. It does depend on the source resistance R S, 3.If the measurement system was completely noiseless, the noise factor would equal one. Reference: [2] Conclusions:

11. 11 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Noise factor expressed in decibels is called noise figure (NF) : NF  10 log F Due to the bandwidth term in the denominator there are two ways to specify the noise factor: (1) a spot noise, measured at specified frequency over a 1  Hz bandwidth, or (2) an integrated, or average noise measured over a specified bandwidth. C. Noise figure F  V no 2 4 kTR S NB (G A V ) 2 Reference: [2]

12. 12 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Reference: [2] We will consider the following methods for the measurement of noise factor: (1) the single-frequency method, and (2) the white noise method. E. Measurement of noise factor 1) Single-frequency method. According to this method, a sinusoidal test signal v S is increased until the output power doubles. Under this condition the following equation is satisfied: RSRS Measurement objectMeasurement system vSvS vovo v in Voltage gain, A v GvGv RLRL

13. 13 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Reference: [2] RSRS Measurement objectMeasurement system vSvS vovo v in 1) (V S G V A V ) 2 + V no 2  2 V no 2 V S  0 2) V no 2  (V S G V A V ) 2 V S  0 3) F  N o src V no 2 V S  0 (V S G V A V ) 2 4 kTR S NB (G V A V ) 2  V S 2 4 k T R S NB  Voltage gain, A V GVGV RLRL

14. 14 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Reference: [2] F  The disadvantage of the single-frequency method is that the noise bandwidth of the measurement system must be known. A better method of measuring noise factor is to use a white noise source. 2) White noise method. This method is similar to the previous one. The only difference is that the sinusoidal signal generator is now replaced with a white noise source: V S 2 4 k T R S NB

15. 15 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Measurement objectMeasurement system i n ( f ) vovo v in 1) (i n R S G R A V ) 2 NB + V no 2  2 V no 2 i n  0 2) V no 2  (i n R S G R A V ) 2 B i n  0 3) F  N o src V no 2 i n  0 (i n R S G R A V ) 2 NB 4 kTR S NB (G R A V ) 2  i n 2 R S 4 k T  RSRS Voltage gain, A v GRGR RLRL

16. 16  5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF F The noise factor is now a function of only the test noise signal, the value of the source resistance, and temperature. All of these quantities are easily measured. The noise bandwidth of the measurement system should not be known. i n 2 R S 4 k T The standard reference temperature is T 0 = 290 K for that k T 0 = 4.00  10  21. (H. T. Friis: NF, P a, and T 0.)

17. 17 Reference: [2] 5.3.3. V n  I n noise model The actual network can be modeled as a noise-free network with two noise generators, e n and i n, connected to its input (Rothe and Dahlke, 1956): RSRS Measurement objectMeasurement system vSvS vovo R in Noiseless AVAV RLRL In a general case, the e n and i n noise generators are correlated. enen inin 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. V n  I n noise model

18. 18 Reference: [2] The e n source represents the network noise that exists when R S equals zero, and the i n source represents the additional noise that occurs when R S does not equal zero, The use of these two noise generators plus a complex correlation coefficient completely characterizes the noise performance of a linear network. RSRS Measurement objectMeasurement system vSvS vovo R in Noiseless AVAV RLRL enen inin 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. V n  I n noise model

19. 19 Reference: www.analog.com Example: Input voltage and current noise spectra (ultralow-noise, high-speed, BiFET op-amp AD745) enen inin 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. V n  I n noise model

20. 20 The total equivalent noise voltage reflected to the source location can easily be found if we apply the following modifications to the input circuit: A. Total input noise as a function of the source impedance RSRS Measurement objectMeasurement system vSvS vovo R in Noiseless AVAV RLRL enen 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. V n  I n noise model inin

21. 21 e n at S =  4 kT R S + e n 2 + 2  e n i n + (i n R S ) 2 RSRS Measurement objectMeasurement system vSvS vovo enen inin R in Noiseless AVAV RSRS Measurement objectMeasurement system vSvS vovo i n R s Noiseless AVAV RLRL RLRL enen 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. V n  I n noise model R in

22. 22 RSRS Measurement objectMeasurement system vSvS vovo Voltage gain, A V We now can connect an equivalent noise generator in series with the input signal source to model the total input voltage of the whole system. We assume that the correlation coefficient in the previous equation  0. (For the case   0, it is often simpler to analyze the original circuit with its internal noise sources.) e n at S e n at S =  4 kT R S + e n 2 + (i n R S ) 2 RLRL Reference: [7] 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. V n  I n noise model Noiseless

23. 23 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. V n  I n noise model B. Measurement of e n and i n Measurement system Noiseless AVAV 1) R t = 0  4 kT R t + (i n R t ) 2   0 2) e n = v n o ( f ) / A V 1) (i n R t ) 2 >> 4 kT R t + e n 2 2) i n R t  v n o ( f ) / A V 3) i n  [v n o ( f ) / A V ] / R t Measurement system Noiseless AVAV RtRt v n o RLRL RLRL enen inin enen inin R in

24. 24 5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR 5.4. Noise matching: maximizing SNR The purpose of noise matching is to let the measurement system add as little noise as possible to the measurand. Influence Measurement System Measurement Object Matching +   x x

25. 25 5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR where, N at S and SNR at S are the noise power and the signal-to- noise ratio at the source location. We then will try and maximize the SNR o at the output of the measurement system by matching the source resistance. V S 2 N at S SNR o = SNR at S   V S 2 4kTR S + e n 2 + (i n R S ) 2    f ( R S ) Let us first find the noise factor F and the signal-to-noise ratio SNR o of the measurement system as a function of the source resistance: F = f ( R S ) and SNR o = f ( R S ). 4kTR S + e n 2 + (i n R S ) 2 4kTR S   N o N o src F   N at S N R    f ( R S ) S

26. 26 F 0.5, dB SNR 0.5, dB 5.4.1. Optimum source resistance 1 10 100 0.1 10 1 10 2 10 3 10 4 10 0 e n at S, nV/Hz 0.5 e n = 2 nV/Hz 0.5, i n = 20 pA /Hz 0.5 R S min F R S max SNR 5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 Measurement system noise e n = i n R n  R S opt = eninenin R S opt is called the optimum source resistance (also noise resistance). V S 2 4kTR S + e n 2 + (i n R S ) 2 SNR   4kTR S + e n 2 + (i n R S ) 2 4kTR S F   v S = e n ·1 Hz 0.5 i n R S enen  4kTR S Source noise

27. 27 SNR 0.5, dB F 0.5, dB R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 It is important to note that the source resistance that maximizes SNR is R S max SNR   0, whereas the source resistance that minimizes F is R S min F   R S opt. For a given R S, SNR cannot be increased by connecting a resistor to R S. V s 2 4kTR S + e n 2 + (i n R S ) 2 SNR   4kTR S + e n 2 + (i n R S ) 2 4kTR S F   5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance

28. 28 RSRS Measurement object vSvS SNR 0.5, dB F 0.5, dB R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 V S 2 4kTR S + e n 2 + (i n R S ) 2 SNR   4kTR S + e n 2 + (i n R S ) 2 4kTR S F   5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance Adding a series resistor, R, increases the total source resistance up to R S opt = R S + R and (!) decreases SNR. R S  R S opt

29. 29 V S 2 4kTR S + e n 2 + (i n R S ) 2 SNR   4kTR S + e n 2 + (i n R S ) 2 4kTR S F   SNR 0.5, dB RSRS Measurement object vSvS + R 5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance F 0.5, dB R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 4kTR S opt + e n 2 + (i n R S opt ) 2 4kTR S opt F    V S 2 4kTR S + e n 2 + (i n R S ) 2 SNR   V S 2 4kTR S  + e n 2 + (i n R S ) 2  SNR    R S  R   R S opt Adding a series resistor, R, increases the total source resistance up to R S opt = R S + R and (!) decreases SNR.

30. 30 Adding a parallel resistor, R, decreases by the same factor both the input signal and the source resistance seen by the measurement network, and therefore (!) decreases SNR. RSRS vSvS 5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance V S 2 4kTR S + e n 2 + (i n R S ) 2 SNR   4kTR S + e n 2 + (i n R S ) 2 4kTR S F   Measurement object SNR 0.5, dB F 0.5, dB SNR 0.5, dB R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 R S  R S opt

31. 31 SNR 0.5, dB F 0.5, dB SNR 0.5, dB R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 V S 2 4kTR S + e n 2 + (i n R S ) 2 SNR   4kTR S + e n 2 + (i n R S ) 2 4kTR S F   RSRS Measurement object vSvS R 5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance V S 2 4kTR S + e n 2 + (i n R S ) 2 SNR   Adding a parallel resistor, R, decreases by the same factor both the input signal and the source resistance seen by the measurement network, and therefore (!) decreases SNR. R S / [(R S  R)/R]   R S opt V S / [(R S  R)/R]  V S k  (R S  R)/R > 1 F 0.5, dB SNR 0.5, dB R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 k 2k 2k 2k 2 4kTR S opt + e n 2 + (i n R S opt ) 2 4kTR S opt F    V S 2 4kTR S + e n 2 + (i n R S ) 2 SNR   V S 2 4kTR S k + e n 2 k 2 + (i n R S ) 2 SNR   

32. 32 Conclusions. The noise factor can be very misleading: the minimization of F does not necessarily leads to the maximization of the SNR. This is referred to as the noise factor fallacy (erroneous belief). 5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance

33. 33 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR Methods for the increasing of SNR are based on the following relationship: SNR o = SNR in 1F1F The strategy is simple: to increase SNR o, keep SNR in constant while decreasing the noise figure:  SNR o = SNR in 1F 1F  The SNR at the output will increase because the relative noise power contributed by the measurement system will decrease. 5.4.2. Methods for the increasing of SNR

34. 34 A. Noise reduction with parallel input devices This method is commonly used in low-noise OpAmps: to increase the SNR, several active devices are connected in-parallel: 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR Reference: [7] i o sc v in k Measurement system R in roro roro g m v in enen inin enen inin

35. 35 Reference: [7] Home exercise: Prove that the following network is equivalent to the previous one. i o sc Equivalent measurement system v in R in k k g m v in 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR e n /k 0.5 i n k 0.5 rokrok

36. 36 i o sc Reference: [7] Measurement object vSvS k = e n / i n R S Equivalent measurement system v in R in k RSRS Thanks to parallel connection of input devices, it is possible to decrease the ratio, ( e n / i n ) p   ( e n / i n ) single / k, with no change in v S and R S, and hence in the SNR in. Note that SNR o cannot be improved if the R S is too large. SNR o = SNR o max and F = F min at R S = e n k i n  k g m v in 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR e n /k 0.5 i n k 0.5 rokrok

37. 37 Reference: [7] Home exercise: Prove that SNR o p = k SNR o single at F min 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR

38. 38 SNR in  (n V S ) 2 4 kT n 2 R S  const, SNR o  = SNR in. 1 F1 F RSRS Measurement object v in 1: n n vSn2 RSn vSn2 RS vovo Measurement system A V vSvS F  SNR in SNR o  RLRL B. Noise reduction with an input transformer 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR enen inin R in

39. 39 SNR o (1: n) = SNR o F F min Example: Noise reduction with an ideal input transformer 1 10 100 0.1 10 1 10 2 10 3 10 4 10 0 e n at S, nV/Hz 0.5 B = 1 Hz, e n = 2 nV/Hz 0.5, i n = 20 pA /Hz 0.5 enen i n R s R S n 2 v S n RSRS vSvS 1: n F 0.5, dB R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 SNR 0.5, dB SNR o (1: n) 0.5 1 F1 F   SNR o = SNR in Measurement system noise Source noise R S for minimum F SNR o (1: n) = n 2 SNR o F min n 2 = R S opt R S  4kTR s B 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR v S = e n ·1 Hz 0.5 F 0.5 F min 0.5 SNR o 0.5 SNR o (1: n) 0.5 SNR o F min 0.5

40. 40 To prove that let us consider the following. At noise matching, the amplifier sees R s =R n, thus the total noise at the amplifier output remains the same: N o (1:n) = N o F min. The transformer, however, increases the amplifier input voltage: v in (1:n) = n 2 v in F min. Therefore: 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR SNR o (1: n) = n 2 SNR o F min SNR o (1: n)    n 2 SNR o F min v in (1:n) 2 N o (1:n) n 2 v in F min 2 N o F min

41. 41 1 10 100 0.1 10 1 10 2 10 3 10 4 10 0 e n at S, nV/Hz 0.5 B = 1 Hz, e n = 2 nV/Hz 0.5, i n = 20 pA /Hz 0.5 enen i n R s R S n 2 v S n RSRS vSvS 1: n F 0.5, dB R S,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 SNR 0.5, dB Measurement system noise Source noise R S for minimum F  4kTR s B 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR v S = e n ·1 Hz 0.5 F 0.5 F min 0.5 SNR o 0.5 SNR o (1: n) 0.5 SNR o F min 0.5 Note that:  n 2 F F min SNR o (1: n) = SNR o F F min  F/F min n SNR o (1: n) 0.5

42. 42 v S n (R S + R 1 ) n 2 + R 2 RSRS vSvS 1: n R1R1 R2R2 Example: Noise reduction with a non-ideal input transformer 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR

43. 43 Reference: [4] Our aim in this section is to maximize the SNR of a three-stage amplifier. RSRS vSvS A V 1 A V 2 A V 3 vOvO e nS1 e nS2 e nS3 5.4.3. SNR of cascaded noisy amplifiers 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. SNR of cascaded noisy amplifiers

44. 44 Reference: [4] 2) V no 2 = [e nS1 2 A V1 2 A V2 2 A V3 2 + e nS2 2 A V2 2 A V3 2 + e nS3 2 A V3 2 ] NB 1) SNR o  SNR at S  V S 2 V no 2 /( A V1 2 A V2 2 A V3 2 ) 3) SNR o  V S 2 / NB e nS1 2 + e nS2 2 /A V1 2 + e nS3 2 /A V1 2 A V2 2 Conclusion: keep A V1 >> 1 to neglect the noise contribution of the second and third stages. 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. SNR of cascaded noisy amplifiers RSRS vSvS A V 1 A V 2 A V 3 vOvO e nS1 e nS2 e nS3

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