1 5. SOURCES OF ERRORS Fundamentals of low-noise design 5.5. Fundamentals of low-noise design

2 rdrd IDID e dsh rdrd i dsh IDID 5. SOURCES OF ERRORS Fundamentals of low-noise design Junction-diode noise model 4) i dsh 2 = 2 q I D = 2 k T / r d 3) r d  k Tq IDk Tq ID 5) e dsh 2 = (2 k T / r d ) r d 2 = 2 k T r d 2) i dsh 2  2 q ( I F  +  I S )  2 q ( I D  +  2I S )  2 q I D Junction-diode noise model 1) I D  I S e  I S  I F  I S V D /V T IDID At low frequencies and I D >> I S, i dn 2 = 2 q I D  K f I D f, K f = 2 q f f Note that dynamic resistances do not generate any thermal noise since them dissipate no power, v d i d = 0. i df i df r d

3 5. SOURCES OF ERRORS Fundamentals of low-noise design BJT noise model BJT noise model rbrb i csh B E C Noiseless v bt i bf i bsh i csh 2 = 2 q I C i bsh 2 = 2 q I B v bt 2 = 4 k T r b i bf 2  K f I B f NB: i cf  0 i ct  0

4 vsvs RSRS rr icic h fe i  roro rbrb BC i csh 5. SOURCES OF ERRORS Fundamentals of low-noise design BJT noise model A. Total input noise i bf i bsh ii v bt v n s (t)  v st (t)  v bt (t)   i bf (t)  i bsh (t)](R S  r b )  i csh (t) R S +r b +r  h fe 1) Total input noise vs. time, v n s (t). 2) Power spectral density of the total input noise, v n s 2 ( f ). v bt v n s ?

5 vsvs RSRS rr icic h fe i  roro C i csh 5. SOURCES OF ERRORS Fundamentals of low-noise design BJT noise model A. Total input noise i bf i bsh ii v n s (t)  v st (t)  v bt (t)   i bf (t)  i bsh (t)](R S  r b )  i csh (t) R S +r b +r  h fe 1) Total input noise vs. time, v n s (t). 2) Power spectral density of the total input noise, v n s 2 ( f ). v n s 2  4 k T (r b  R S )   i bf 2  i bsh 2 )(R S  r b ) 2  i csh 2 R S +r b +r  h fe 2 v n s ? rbrb v bt v n s B

6 v n s 2  4 k T (r b  R S ) +  2 q I C (R S  r b ) 2 h fe  2 q I C R S +r b +h fe V T / I C h fe 2 v n s 5. SOURCES OF ERRORS Fundamentals of low-noise design BJT noise model B. Optimum collector current I C opt  h fe V T    h fe  0.5  (R S  r b ) rr icic h fe i  roro rbrb BCii v bt v n s 2  4 k T (r b  R S )  i bsh 2 (R S  r b ) 2  i csh 2 R S +r b +r  h fe 2 RSRS i bf  0 Reference: [7]

7 5. SOURCES OF ERRORS Fundamentals of low-noise design BJT noise model C. e n  i n noise model e n 2  v n s 2  4 k T r b   i bf 2  i bsh 2 ) r b 2  i csh 2 r b +r  h fe 2 RS= 0RS= 0 i n 2   i bf 2  i bsh 2  v n s 2 R S 2 RS= RS=  i csh 2 h fe 2 vsvs RSRS rr icic h fe i  roro rbrb BCii enen inin v n s 2  4 k T (r b  R S )   i bf 2  i bsh 2 )(R S  r b ) 2  i csh 2 R S +r b +r  h fe 2

8 5. SOURCES OF ERRORS Fundamentals of low-noise design BJT noise model B E C e n 2  4 k T r b   i bf 2  i bsh 2 ) r b 2  i csh 2 r b +r  h fe 2 i n 2  i bf 2  i bsh 2  i csh 2 h fe 2 enen inin BJT e n  i n noise model f >> f f r b = 100  I C = 1 mA h fe = 100 e n  1.36 nV/Hz 0.5 i n  1.8 pA/Hz 0.5 e n / i n   756  R S = 756  i n R S = 1.4 nV/Hz 0.5

9 5. SOURCES OF ERRORS Fundamentals of low-noise design BJT noise model D. Optimum source resistance at I C opt vsvs RSRS rr icic h fe i  roro rbrb BCii R s opt  eninenin I C opt  r b  2   1 +  1+h fe enen inin I C opt

10 i gsh 2 = 2 q I G i df 2  K f I D f JFET noise model 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model i dt G S D Noiseless i gsh i df NB: i dsh   0 i dt 2 = 4 k T /(3/2 g m )

11 idid 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model g m v gs roro GD igig Equivalent small-signal model i gsh i dt i df v gs r gs

12 idid 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model 1/g m g m v gs roro GD igig i gsh i dt i df v gs r gs Equivalent small-signal model

13 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model ~1/g m g m v gs roro GD igig i gsh i dt i df v gs Equivalent small-signal model

14 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model idid vsvs RSRS ~1/g m g m v gs roro GD igig 1) Total input noise vs. time, v n s (t). v n s ? A. Total input noise i gsh i dt i df v gs

15 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model vsvs RSRS ~1/g m idid g m v gs roro GD 1) Total input noise vs. time, v n s (t). A. Total input noise i dt i df v n s ? i gsh i gsh R s v n s (t)  v st (t) + i gsh (t) R S   i df (t)  i dt (t)](1/g m ) v gs igig

16 v n s ? 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model vsvs RSRS ~1/g m idid g m v gs roro D i gsh R s i dt i df v n s G A. Total input noise 1) Total input noise vs. time, v n s (t). v n s (t)  v st (t) + i gsh (t) R S   i df (t)  i dt (t)](1/g m ) 2) Power spectral density of the total input noise, v n s 2 ( f ). v n s 2  4 k T R S + i gsh 2 R S 2   i df 2  i dt 2 )/g m 2 v gs igig

17 v n s ? 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model B. e n  i n noise model e n 2  v n s 2  i df 2  i dt 2 )/g m 2 R S = 0 i n 2   i gsh 2 v n s 2 R S 2 R S =  vsvs RSRS ~1/g m idid g m v gs roro D i gsh R s i dt i df v n s 2  4 k T R S + i gsh 2 R S 2   i df 2  i dt 2 )/g m 2 v n s enen inin G v gs igig

18 5. SOURCES OF ERRORS Fundamentals of low-noise design JFET noise model G S D enen inin e n 2  i df 2  i dt 2 )/g m 2 i n 2  i gsh 2 JFET e n  i n noise model f >> f f V p = 2 V I DSS = 10 mA I G = 10 pA e n  1.8 nV/Hz 0.5 i n  1.8 fA/Hz 0.5 e n /i n   1 M  R S = 1 M  i n R S = 1.8 nV/Hz 0.5 f >> f f r b = 100  I C = 1 mA h fe = 100 e n  1.36 nV/Hz 0.5 i n  1.8 pA/Hz 0.5 e n / i n   756  R S = 756  i n R S = 1.4 nV/Hz 0.5 BJT

19 i dt 2 = 4 k T /(3/2 g m ) i df 2  K f I D f MOSFET noise model 5. SOURCES OF ERRORS Fundamentals of low-noise design MOSFET noise model i dt G S D Noiseless i df NB: i gsh   0 i dsh   0

20 idid 5. SOURCES OF ERRORS Fundamentals of low-noise design MOSFET noise model 1/g m g m v gs roro D v n s (t)  v st (t)   i df (t)  i dt (t)](1/g m ) 1) Total input noise vs. time, v n s (t). 2) Power spectral density of the total input noise, v n s 2 ( f ). v n s 2  4 k T R S +  i df 2  i dt 2 )/g m 2 A. Total input noise i dt i df v n s ? vsvs RSRS G

21 v n s 5. SOURCES OF ERRORS Fundamentals of low-noise design MOSFET noise model B. e n  i n noise model e n 2  v n s 2  i df 2  i dt 2 )/g m 2 R S = 0 i n 2   0 v n s 2 R s 2 R S =  1/g m idid g m v gs roro D v n s 2  4 k T R S +  i df 2  i dt 2 )/g m 2 vsvs RSRS enen inin G

22 5. SOURCES OF ERRORS Fundamentals of low-noise design MOSFET noise model G S D enen e n 2  i df 2  i dt 2 )/g m 2 i n  0 MOSFET e n  i n noise model f >> f f V p = 2 V I DSS = 10 mA e n  1.8 nV/Hz 0.5 f >> f f V p = 2 V I DSS = 10 mA I G = 10 pA e n  1.8 nV/Hz 0.5 i n  1.8 fA/Hz 0.5 e n /i n   1 M  R S = 1 M  i n R S = 1.8 nV/Hz 0.5 JFET

Frequency response effect 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect rr icic h fe i  roro rbrb C i csh i bf i bsh ii v bt V CC i C C  C  V BB v s RSRS vsvs RSRS B The aim is to analyze the dependence of a transistor e n and i n on frequency and the operating point.

24 vsvs RSRS Ag Ag  1) Transconductance gain 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect rr icic h fe i  roro rbrb BC ii C  C  icvsicvs  is= 1is= 1 h fe [1/j  2  f  (C  +C  )]/[r  +1/j  2  f  (C  +C  )] R S + r b + r  II [1/j  2  f  (C  +C  )]   h fe /(R S +r b +r  ) 1+j  2  f      [(R S + r b ) II r  ](C  +C  ) isis A. Total input noise

25 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect rr icic h fe i  roro rbrb C i csh i bf i bsh ii v bt C  C  2) Power spectral density of the total input noise, v n s 2 ( f ). v n s 2  4 k T (R S +r b )   i bf 2  i bsh 2 ) (R S +r b ) 2  i csh 2 R S +r b +r  h fe 2 [1    2  f  ) 2 ] vsvs RSRS B v n s h fe /(R S +r b +r  ) 1+j  2  f  A g     [(R S + r b ) II r  ](C  +C  )

26 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect 3) e n and i n of the transistor. v n s 2  4 k T (R S +r b )   i bf 2  i bsh 2 ) (R S +r b ) 2  i csh 2 R S +r b +r  h fe 2 [1    2  f  ) 2 ] e n 2  v n s 2  4 k T r b   i bf 2  i bsh 2 ) r b 2  R S = 0 in2 in2  v n s 2 R S 2 R S =  i csh 2 r b +r  h fe 2 [1    2  f  en ) 2 ]  i bf 2  i bsh 2  [1    2  f  in ) 2 ] i csh 2 h fe 2  en  (r b II r  )(C  +C  )  in  r  (C  +C  )

27 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect rr icic h fe i  roro rbrb C ii C  C  vsvs RSRS B e n 2  4 k T r b   i bf 2  i bsh 2 ) r b 2  in2 in2  i csh 2 r b +r  h fe 2 [1    2  f  en ) 2 ] i bf 2  i bsh 2  [1    2  f  in ) 2 ] i csh 2 h fe 2 enen inin B. e n  i n noise model for high-frequencies

28 I C opt = 24 mA I C = 0.1 mA 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect e n ( f ) nV/Hz 0.5 A g A g max dB  f, Hz C. e n ( f ) for different I C r b   100  h fe   100 C   1 pF C  (1 mA)   100 pF r b   100  h fe   100 C   1 pF C  (1 mA)   100 pF e n 2  4 k T r b   i bf 2  i bsh 2 ) r b 2  i csh 2 r b +r  h fe 2 [1    2  f  en ) 2 ]

29 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect D. i n ( f ) for different I C I C opt = 24 mA I C = 0.1 mA i n ( f ) pA/Hz 0.5 A g A g max dB  f, Hz r b   100  h fe   100 C   1 pF C  (1 mA)   100 pF r b   100  h fe   100 C   1 pF C  (1 mA)   100 pF in2 in2  i bf 2  i bsh 2  [1    2  f  in ) 2 ] i csh 2 h fe 2

30 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect E. Noise simulation in PSPICE Frequency 1.0Hz10KHz100MHz1.0THz V(ONOISE)*1G/10 V(Out1)/V(V1:+)/10 V(INOISE)*1G

31 5. SOURCES OF ERRORS Fundamentals of low-noise design Comparison of the BJT, JFET and MOSFET Comparison of the BJT, JFET and MOSFET r b   40  h fe   500 r o   I C  1 mA r b   40  h fe   500 r o   I C  1 mA I DSS   2 mA V p   2 V r o   I D  1 mA v n s 2  4 k T R S + i gsh 2 R S 2   i df 2  i dt 2 )/g m 2 v n s 2  4 k T (r b  R S )   i bf 2  i bsh 2 )(R S  r b ) 2  i csh 2 R S +r b +r  h fe 2 v n s 2  4 k T R S +  i df 2  i dt 2 )/g m 2

32 5. SOURCES OF ERRORS Fundamentals of low-noise design Comparison of the BJT, JFET and MOSFET R S,  v n s nV/Hz 0.5 Power spectral density of the total input noise v n s as a function of R S I C opt The 1/f noise is neglected. The JFET gate current is neglected.

33 5. SOURCES OF ERRORS Fundamentals of low-noise design Frequency response effect Example: Comparison of an BJT and JFET in PSPICE R S = 10 k  R S = 100 

34 5. SOURCES OF ERRORS Fundamentals of low-noise design Comparison of the BJT, JFET and MOSFET Reference: [9] Conclusion: Guide for selection of the preamplifier k10 k100 k1 M10 M100 M1 G10 G100 G MOSFET Transformer coupling IC amplifiers BJT Source resistance, R S JFET

Noise analysis of a CE amplifier RSRS R C R E V CC V BB v s 5. SOURCES OF ERRORS Fundamentals of low-noise design Example circuit rr ioio h fe i  roro rbrb RERE RCRC B E C i csh vsvs RSRS i bf i bsh ii v et v bt v st v ct ro  ro  

36 Our final aim is to find and minimize the total input noise v n s. 5. SOURCES OF ERRORS Fundamentals of low-noise design Example circuit rr ioio h fe i  rbrb RERE RCRC B E C i csh vsvs RSRS i bf i bsh ii v et v bt v st v n s ? v ct Let us first find v n s by applying superposition.

37 A s  G s  G s  s fwd A OL 1  A OL   iovsiovs  1) Signal gain A s for v s, v st, v bt, and v et. 5. SOURCES OF ERRORS Fundamentals of low-noise design Example circuit rr ioio h fe i  rbrb RERE RCRC B E C vsvs RSRS ii A s  1RSrbrRE1RSrbrRE  h fe 1  h fe R E /(R E  R S  r b  r  )  00 v et v bt v st

38 A bf  G ibf  G bf  bf fwd A OL 1  A OL   i o i bf  2) Noise gain A bf for i bf and i bsh. 5. SOURCES OF ERRORS Fundamentals of low-noise design Example circuit rr ioio h fe i  rbrb RERE RCRC B E C vsvs RSRS i bf i bsh ii A bf  R S  r b  R E R S  r b  R E  r   h fe 1  h fe R E /(R E  R S  r b  r  )  00

39 A csh  G csh  G csh  csh fwd A OL 1  A OL   i o i csh  rr ioio h fe i  rbrb RERE RCRC B E C i csh vsvs RSRS ii 3) Noise gain A csh for i csh. 5. SOURCES OF ERRORS Fundamentals of low-noise design Example circuit A csh  R E R E  R S  r b  r    h fe 1  h fe R E /(R E  R S  r b  r  )  11

40 A ct  G ct  G ct  ct fwd A OL 1  A OL   i o i ct  4) Noise gain A ct for i csh. 5. SOURCES OF ERRORS Fundamentals of low-noise design Example circuit rr ioio h fe i  RERE RCRC B E C vsvs RSRS ii v ct /R C rbrb A csh  1RC1RC 

41 5) Total input noise vs. time, v n s. 5. SOURCES OF ERRORS Fundamentals of low-noise design Example circuit rr ioio h fe i  RERE RCRC B E C vsvs RSRS ii v n s v n s (t)  v st  v bt  v et (i bf  i bsh ) A bf A s   i csh A csh A s   v ct A ct A s   rbrb v n s 2 ( f )  4kT R SbE +(i bf 2  i bsh 2 ) R SbE 2 (R SbE  r  ) 2 h fe 2    i csh 2  4kT 1 R C A s 2   0 0 R SbE   R S  r b  R E

42 rr icic h fe i  RCRC BC rbrb vsvs RSRS (1+h fe ) R E E RERE E 6) e n  i n noise model. enen inin i n 2   i bf 2  i bsh 2  i csh 2 h fe 2 e n s 2 R S 2 R S =  e n 2  e n s 2  4 k T R bE   i bf 2  i bsh 2 ) R bE 2  i csh 2 (R bE +r  ) 2 h fe 2 R S = 0 ii R bE  r b  R E 5. SOURCES OF ERRORS Fundamentals of low-noise design Example circuit

43 R SbE 2 h fe I C opt  h fe V T    h fe  0.5  R SbE r b = 100 R S = 200 R E = 200 i bf 2 = 0 v bt 2 = 4 k T r b v et 2 = 4 k T R E i bsh 2 = 2 q I C /  i csh 2 = 2 q I C 7) Minimizing CE noise. v n s min 2   4 k T R SbE (1 + h fe ) 0.5 (1 + h fe ) 0.5  1 v n s 2  4 k T R SbE  2 q I C 2 q IC2 q IC R SbE +h fe V T /I C h fe e n s norm. dB h fe h fe =10 4 h fe =10 2 h fe =10 3 I C / I C opt e n s norm. dB SOURCES OF ERRORS Fundamentals of low-noise design Example circuit Reference: [7]

44 Next lecture Appendix: Noise analysis of the CE without applications of superposition Reference: [7]

45 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis Noise analysis of a CE amplifier rr icic h fe i  roro rbrb RERE RCRC B E C i csh v n s ? vsvs RSRS i bf i bsh ii v et v bst RSRS R C R E V CC V BB v s

46 rr ii icic h fe i  roro rbrb RERE RCRC B E C i csh v n s ? vsvs RSRS v et v bst 1) Disconnecting i bf and i bsh sources. i bf i bsh i bf i bsh 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

47 i bf i bsh i bf i bsh rr ii icic h fe i  roro RCRC C i csh RERE E rbrb B v n s vsvs RSRS ? v et v ne = v et  (i bf + i bsh ) R E 1) Disconnecting i bf and i bsh sources. v bst 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

48 rr icic h fe i  roro rbrb RERE RCRC B E C i csh v et v n s ? vsvs RSRS i bf i bsh v ne = v et  (i bf + i bsh ) R E 1) Disconnecting i bf and i bsh sources. ii ii v bst v bst  (i bf  i bsh ) (R s  r b ) 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

49 rr icic h fe i  roro rbrb RERE RCRC BC ro  ro   i csh v et E v n s ? vsvs RSRS v ne = v et  (i bf + i bsh ) R E 2) Disconnecting i bf and i bsh sources. v bst  (i bf  i bsh ) (R s  r b ) ii 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

50 rr icic h fe i  rbrb RCRC BC RERE i csh v et E v n s ? vsvs RSRS v ne = v et  (i bf + i bsh ) R E 2) Disconnecting i bf and i bsh sources. v bst  (i bf  i bsh ) (R s  r b ) ii 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

51 v et rr icic h fe i  rbrb RCRC BC RERE (1+h fe ) R E i csh E v n s ? vsvs RSRS v ne = v et  (i bf + i bsh ) R E 2) Disconnecting i bf and i bsh sources. v bst  (i bf  i bsh ) (R s  r b ) v ne = v et  (i bf + i bsh ) R E + i csh R E ii 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

52 rr icic h fe i  rbrb RCRC BC i csh v ne = v et  (i bf + i bsh ) R E + i csh R E E (1+h fe ) R E v n s ? v n s (t)  i c (t) R S +r b +r  +(1+h fe )R E h fe vsvs RSRS 3) Reflecting i bf and i bsh to v n s. v bst  (i bf  i bsh ) (R s  r b ) v n s (t)  v bst (t)  v et (t)   i bf (t)  i bsh (t)] R *  ? ii R *  R S  r b  R E 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

53 2) i c  h fe R S +r b +r  +(1+h fe )R E  i csh (t)  i csh (t) R E rr icic rbrb RCRC BC i csh (1+h fe ) R E i csc R E E v n s ? vsvs RSRS h fe i  1) v n s   i c (t), R S +r b +r  +(1+h fe )R E h fe 3) v n s   i csh (t) R E   i csh (t) R S +r b +r  +(1+h fe )R E h fe  i csh (t) R * + r  h fe 3) Reflecting i csh to v n s. ii R *  R S  r b  R E 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

54 rr icic h fe i  rbrb RCRC BC (1+h fe ) R E E v n s vsvs RSRS 4) Total input noise vs. time, v n s (t). ii v n s (t)  v bst (t)  v et (t)   i bf (t)  i bsh (t)] R *  i csh (t) R * + r  h fe R *  R S  r b  R E 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

55 rr icic h fe i  rbrb RCRC BC v n s vsvs RSRS v n s (t)  v bst (t)  v et (t)   i bf (t)  i bsh (t)] R *  i csh (t) R * + r  h fe v n s 2  4 k T R *   i bf 2  i bsh 2 ) R * 2  i csh 2 R * + r  h fe 2 (1+h fe ) R E E RERE E 5) Power spectral density of the total input noise, v n s 2. ii R *  R S  r b  R E 5. SOURCES OF ERRORS Fundamentals of low-noise design. Appendix: conventional noise analysis

56 Next lecture Next lecture: