1. 1 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 msr is the signal power, and N msr 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

2. 2 Reference: [4] SNR msr . S msr N msr References: [1] and [2] It is usually assumed that the signal power, S msr, and the noise power, N msr, are dissipated in the noiseless input impedance of the measurement system. 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 msr Measurement objectMeasurement system Noiseless S msr RsRs S in R msr

3. 3 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR Example: Calculation of SNR at the input of a measurement system 1) S msr , V in rms 2 R in  R s + R msr  2 2) N msr , V n rms 2 R in  R s + R msr  2 3) SNR msr  V in rms 2 V n rms 2 V in rms 2 4 k T R   f n  Measurement objectMeasurement system Noiseless S msr RsRs S in R msr

4. 4 Reference: [4] References: [1] and [2] 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR B. Signal-to-noise ratio at the output of the system, SNR o RsRs Measurement objectMeasurement system RLRL S in Power gain, A P Noisy GPGP SNR o SNR o . S o N o

5. 5 Reference: [4] SNR o * . References: [1] and [2] 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR B. Signal-to-noise ratio at the output of the system, SNR o RsRs Measurement objectMeasurement system RLRL S in Power gain, A P Noiseless GPGP SNR o * S o N o *

6. 6 Noise factor is used to compare at the output the noise contribution of a system (amplifier) against the noise power delivered by the source (H. Friis, 1940s): 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 RsRs Measurement objectMeasurement system RLRL S in Power gain, A P Noisy GPGP SNR o

7. 7 Noise factor is used to compare at the output the noise contribution of a system (amplifier) against the noise power delivered by the source (H. Friis, 1940s): 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 RsRs Measurement objectMeasurement system RLRL S in SNR o * Power gain, A P Noiseless GPGP F  SNR o * SNR o SNR msr *  SNR o note that SNR o * = SNR msr * since the measurement system is noiseless.

8. 8 Noise factor is used to compare at the output the noise contribution of a system (amplifier) against the noise power delivered by the source (H. Friis, 1940s): 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 msr SNR o RsRs Measurement objectMeasurement system RLRL S in SNR o Power gain, A P Noisy GPGP SNR msr

9. 9 F , NoNo*NoNo* 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF where N o is the total noise power at the output, and N o * is the noise power at the output of the same but noiseless system (the output noise comes only from the source). A. Another definition for noise factor F  SNR msr SNR o  S msr /N msr * S o /N o  (S o /A P )/(N o * /A P ) S o /N o

10. 10 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF RsRs Measurement objectMeasurement system RLRL NoNo Power gain, A P Noisy GPGP Illustration: e ns

11. 11 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF RsRs Measurement objectMeasurement system RLRL No*No* Power gain, A P Noiseless GPGP Illustration: F , NoNo*NoNo* e ns

12. 12 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF RsRs Measurement objectMeasurement system RLRL e ns F  NoNo*NoNo* V no 2 /R L 4 kTR s B n (G V A V ) 2 /R L  V no 2 4 kTR s B n (G V A V ) 2  VoVo V msr Example: Calculation of noise factor Voltage gain, A V GVGV

13. 13 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 n (G A V ) 2 The following three characteristics of noise factor can be seen by examining the obtained equation: 1. It is independent of load resistance R L, 2. It does depend on source resistance R s, 3.If the measurement system were completely noiseless, the noise factor would equal one. References: [2] Conclusions:

14. 14 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) : References: [2] 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 B n (G A V ) 2

15. 15 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF References: [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 in (rms) is increased until the output power doubles. Under this condition the following equation is satisfied: RsRs Measurement objectMeasurement system RLRL V in VoVo V msr Voltage gain, A V GVGV

16. 16 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF References: [2] RsRs Measurement objectMeasurement system RLRL V in VoVo V msr 1) (V in G V A V ) 2 + V no 2  2 V no 2 V in  0 2) V no 2  (V in G V A V ) 2 V in  0 3) F  No*No* V no 2 V in  0 (V in G V A V ) 2 4 kTR s B n (G V A V ) 2  V in 2 4 kTR s B n  Voltage gain, A V GVGV

17. 17 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF References: [2] F V in 2 4 kTR s B n  The disadvantage of the single-frequency meted 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 current source:

18. 18 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF Measurement objectMeasurement system RLRL i in ( f ) VoVo V in 1) (i in R s G A V ) 2 B n + V no 2  2 V no 2 i t  0 2) V no 2  (i in R s G A V ) 2 B n i t  0 3) F  No*No* V no 2 i t  0 (i in R s G A V ) 2 B n 4 kTR s B n (G A V ) 2  i in 2 R s 4 kT  RsRs Voltage gain, A V GVGV

19. 19 i in 2 R s 4 kT  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. Neither the gain nor the noise bandwidth of the measurement system need be known.

20. 20 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF 1. Increasing the source resistance may decrease the noise factor, while increasing the total noise in the circuit. 2. If a purely reactive source is used, noise factor is meaningless, since the source noise is zero, making the noise factor infinite. 3.When the measurement system noise is only a small part of of the source thermal noise (as with some low-noise FETs), the noise factor requires taking the ratio of two almost equal numbers. this can produce inaccurate results. References: [2] The concept of noise factor has three major limitations: D. Limitations of the noise factor concept F  V no 2 4 kTR s B n (G A V ) 2

21. 21 Noise factors varies with the bias conditions, frequency, and temperature as well as source resistance, and all of these should be defined when specifying and comparing noise factors. Knowing the noise factor for one value of source does not allow the calculation of the noise factor at other values of resistance. This is because both the source noise and measurement system noise vary as the source resistance is changed. 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF References: [2] Noise factor is usually specified for matched devices and is a popular figure of merit in RF applications.

22. 22 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model Reference: [2] 5.3.3. Two source noise model A more recent (1956) approach and one that overcomes the limitations of noise factor, is to model the noise in terms of an equivalent noise voltage and current. The actual network can be modeled as a noise-free network with two noise generators, e n and i n, connected to its input: RsRs Measurement objectMeasurement system RLRL V in VoVo V msr enen inin R msr Noiseless AVAV

23. 23 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model 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 the network. At relatively low frequencies, the correlation between the e n and i n noise sources can be neglected. RsRs Measurement objectMeasurement system RLRL V in VoVo V msr enen inin R msr Noiseless AVAV

24. 24 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model Reference: www.analog.com Example: Input voltage and current noise spectra (ultralow noise, high speed, BiFET op-amp AD745) enen inin

25. 25 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model Assuming no correlation between the noise sources, the total equivalent input noise voltage of the whole system can be found by superposition: A. Total input noise as a function of the source impedance RsRs Measurement objectMeasurement system RLRL V in VoVo V msr enen inin R msr Noiseless AVAV

26. 26 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model V n in rms =  4kTR s B + V n rms 2 + (I n rms R s ) 2. RsRs Measurement objectMeasurement system RLRL V in VoVo V msr enen inin R msr Noiseless AVAV RsRs Measurement objectMeasurement system RLRL V in VoVo V msr i n R s Noiseless AVAV enen

27. 27 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model V n in rms =  4kTR s B + V n rms 2 + (I n rms R s ) 2. RsRs Measurement objectMeasurement system RLRL V in VoVo Voltage gain, A V V msr We now can connect an equivalent noise generator in series with input signal voltage source to model the total input voltage of the whole system. V n in

28. 28 V n in Measurement system noise 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model V n in rms =  4kTR s B + V n rms 2 + (I n rms R s ) 2. Example: Total equivalent input noise voltage as a function of R s 1 10 100 0.1 10 1 10 2 10 3 10 4 10 0 V n in rms, nV/Hz 0.5 B = 1 Hz, e n = 2 nV/Hz 0.5, i n = 20 pA /Hz 0.5 R s,  i n R s enen  4kTR s B Rs Rs Source noise

29. 29 5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model B. Measurement of e n and i n Measurement system RLRL V n o enen inin Noiseless AVAV e n = (V n o / B) / A V V n o rms >> (4 kT R t B + V n 2 ) 0.5 i n R s = (V n o / B) / A V i n = (V n o / B) / A V R s Measurement system RLRL V n o enen inin Noiseless AVAV RtRt

30. 30 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance 5.4. Low-noise design: noise matching Let us first find the signal-to-noise ratio SNR and the noise factor F of the measurement system as a function of the source resistance. We next will try and maximize the SNR at the output of the measurement system by matching the source resistance. 5.4.1. Maximization of SNR RsRs Measurement objectMeasurement system RLRL V in VoVo V msr enen inin Noiseless AVAV

31. 31 SNR 0.5, dB 1 10 100 0.1 10 1 10 2 10 3 10 4 10 0 v n in rms, nV/Hz 0.5 B = 1 Hz, e n = 2 nV/Hz 0.5, i n = 20 pA /Hz 0.5 i n R s V in = e n ·1 Hz 0.5 R n for minimum F R s for maximum SNR v n in rms = [4kTR s + e n rms 2 + (i n rms R s ) 2 ] 0.5 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance e n = i n R n F 0.5, dB R s,  10 1 10 2 10 3 10 4 10 0 -30 -20 -10 0 10 20 Measurement system noise  Rn =Rn = eninenin A. Noise resistance R n enen Source noise  4kTR s B R n is called noise resistance

32. 32 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance SNR msr = (n V in rms ) 2 4 kT n 2 R msr = const,   SNR o = SNR msr. 1 F1 F RsRs Measurement object V msr 1: n n 2 R s n V in RLRL VoVo inin Measurement system enen AVAV V in B. Transformer coupling F  SNR msr SNR o 

33. 33 Example: Transformer coupling 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance 1 10 100 0.1 10 1 10 2 10 3 10 4 10 0 v n in rms, 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 V in = e n ·1 Hz 0.5 R s n 2 V in n RsRs V in 1: n SNR 1: n = SNR F F min SNR 1: n = n 2 SNR min F 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 1: n 0.5 SNR min F 0.5   SNR o = SNR msr 1 F1 F Measurement system noise Source noise R n for minimum F n 2 = RnRsRnRs  4kTR s B

34. 34 C. Parallel connection of input stages 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance RsRs Measurement object Measurement system RLRL V in VoVo V msr enen inin Noiseless AVAV e n k / i n k = e n k 0.5 / i n k 0.5 R s = e n / i n k enen inin Noiseless AVAV k k = e n / i n R s 

35. 35 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance Example:

36. 36 D. SNR of cascaded noisy amplifiers 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance Reference: [4] Our aim in this Section is to maximize the SNR of a three-stage amplifier. For the sake of simplicity, let us assume that all the stages are identical in terms of noise, and their e n >> i n R s. RsRs V in A V 1 A V 2 A V 3 VoVo enen enen enen

37. 37 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance Reference: [4] 2) V no rms 2 = [(4kT R s +e n 2 ) A V1 2 A V2 2 A V3 2 + e n 2 A V2 2 A V3 2 + e n 2 A V3 2 ] B 1) SNR in  V in 2 V no 2 / A V1 2 A V2 2 A V3 2 3) SNR in  V in 2 / B (4kT R s +e n 2 ) + e n 2 /A V1 2 + e n 2 /A V1 2 A V2 2 Conclusion: keep A V1 > 5 to neglect the noise contribution of the second and third stages. RsRs V in A V 1 A V 2 A V 3 VoVo enen enen enen

38. 38 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Noise in diodes 5.4.2. Noise in diodes IDID i nd rdrd IDID e nd rdrd IDID 2) i n d 2 = 2 q I D = 2 k T / r d 1) r d  k Tq IDk Tq ID 3) e n d 2 = (2 k T / r d ) r d 2 = 2 k T r d, I D  k T q r d

39. 39 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. Noise in bipolar transistors 5.4.3. Noise in bipolar transistors i nb e nb i nc CC rr vv rbrb CC gmvgmv roro BC IBIB ICIC A. Small-signal equivalent circuit e nb 2 = 4 k T r b i nb 2 = 2 q I B i nc 2 = 2 q I C

40. 40 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. Noise in bipolar transistors i nb e nb i nc rr vv rbrb BC i n o B. Mid-frequency noise model gmvgmv RsRs RsRs

41. 41 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. Noise in bipolar transistors 3) e n = i n o /G v A g, G v = r  /(r b  + r  ), A g = g m Rs=Rs= 4) i n 2 = i n o /G i A g 2, G i = r  Rs=Rs= 1) i n o 2 = {[e nb r  /( r b + r  )] g m } 2 +[ i nb (r b II r  ) g m ] 2 + i nc 2 Rs=Rs= 2) i n o 2 = ( i nb r  g m ) 2 + i nc 2 R s  i nb e nb i nc rr vv rbrb BC gmvgmv RsRs i n o

42. 42 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. Noise in bipolar transistors 5) e n 2 = e nb 2  + (i nb r b ) 2 + [i nc ( r b + r  )/ g m r  ] 2 6) i n 2 = i nb 2  + [i nc /( g m + r  )] 2 i nb e nb i nc rr vv rbrb BC gmvgmv RsRs i n o

43. 43 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. Noise in bipolar transistors 5) e n 2 = e nb 2  + (i nb r b ) 2 + [i nc ( r b + r  )/ g m r  ] 2 6) i n 2 = i nb 2  + [i nc /( g m + r  )] 2 inin ICIC IBIB enen inin enen rr vv rbrb BC gmvgmv RsRs i n o

44. 44 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.4. Noise in FETs 5.4.4. Noise in FETs  due to the thermal noise of the base resistance,  the shot noise in both the collector and base currents,  and the flicker noise of the base current: The noise in bipolar transistors are: BJT e n 2 = 4kT r B + 2qI C r e 2 i n 2 = 2qI B + a A I B / f inin IDID IGIG enen

45. 45 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.5. Noise in differential and feedback amplifiers. 5.4.5. Noise in differential and feedback amplifiers

46. 46 5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.6. Noise measurements. 5.4.6. Noise measurements

47. 47 Next lecture Next lecture: D (diode) FID flicker noise RS thermal noise associated with RS SID shot noise TOT total noise J (JFET) FID flicker noise RD thermal noise associated with RD RG thermal noise associated with RG RS thermal noise associated with RS SID shot noise TOT total noise M (MOSFET) FID flicker noise RB thermal noise associated with RB RD thermal noise associated with RD RG thermal noise associated with RG RS thermal noise associated with RS SID shot noise TOT total noise

48. 48 Next lecture Next lecture: J (JFET) FID flicker noise RD thermal noise associated with RD RG thermal noise associated with RG RS thermal noise associated with RS SID shot noise TOT total noise

49. 49 Next lecture Next lecture: M (MOSFET) FID flicker noise RB thermal noise associated with RB RD thermal noise associated with RD RG thermal noise associated with RG RS thermal noise associated with RS SID shot noise TOT total noise

50. 50 Next lecture Next lecture: Q (BJT) FIB flicker noise RB thermal noise associated with RB RC thermal noise associated with RC RE thermal noise associated with RE SIB shot noise associated with base current SIC shot noise associated with collector current TOT total noise