Noise. Interference Noise

Transcription

1 Noise David Johns and Ken Martin University o Toronto (johns@eecg.toronto.edu) (martin@eecg.toronto.edu) University o Toronto 1 o 55 Intererence Noise Unwanted interaction between circuit and outside world May or may not be random Examples: power supply noise, capacitive coupling Improvement by... Reduced by careul wiring or layout These notes do not deal with intererence noise. University o Toronto o 55

2 Inherent Noise Random noise can be reduced but NEVER eliminated Examples: thermal, shot, and licker Improvement by... Not strongly aected by wiring or layout Reduced by proper circuit DESIGN. These notes discuss noise analysis and inherent noise sources. University o Toronto 3 o Time-Domain Analysis v n () t (volts) time (seconds) Assume all noise signals have zero mean University o Toronto 4 o 55

3 RMS Value T 1 rms ( ) T 0 -- v n()dt t 1 T suitable averaging time interval Indicates normalized noise power. I v n () t applied to 1Ω resistor, average power dissipated,, P diss (1) P diss rms ( ) ( ) V 1Ω nrms () University o Toronto 5 o 55 SNR 10log SNR I signal node has normalized signal power o, rms ( ) and noise power o, signal power noise power V xrms ( ) (3) V xrms ( ) rms ( ) SNR 10log log V xrms ( ) rms ( ) (4) When mean-squared values o noise and signal are same, SNR 0dB. University o Toronto 6 o 55

4 Units o dbm Oten useul to know signal s power in db on absolute scale. With dbm, all power levels reerenced 1mW. 1mW signal corresponds to 0 dbm 1 µw signal corresponds to -30dBm What i only voltage measured (not power)? I voltage measured reerence level to equiv power dissipated i voltage applied to 50 Ω resistor Also, can reerence it to 75 Ω resistor University o Toronto 7 o 55 dbm Example Find rms voltage o 0 dbm signal ( 50Ω reerence) What is level in dbm o a volt rms signal? 0 dbm signal (50 Ω reerence) implies V rms ( ) ( 50Ω) 1mW 0.36 Thus, a volt (rms) signal corresponds to.0 0 log dbm (5) (6) Would dissipate mw across a 50Ω resistor 80 mw corresponds to 10log( 80) 19 dbm University o Toronto 8 o 55

5 Noise Summation v n1 () t + v no () t v n () t i n1 () t i n () t ( rms) Voltage - v no () t v n1 () t + v n () t 1 ( rms) 1( rms) T -- v T [ n1 () t + v n () t ] dt 0 T ( rms) T 0 i no () t Current v n1 ()v t n ()dt t (7) (8) (9) University o Toronto 9 o 55 Correlation Last term relates correlation between two signals Deine correlation coeicient, C, T 1 -- T v n1 ()v t n ()dt t 0 C ( rms) ( rms) (10) ( rms) 1( rms) ( rms) + + C1( rms) ( rms) Correlation always satisies 1 C 1 C +1 ully-correlated in-phase (0 degrees) C -1 ully-correlated out-o-phase (180 degrees) C 0 uncorrelated (90 degrees) (11) University o Toronto 10 o 55

6 Uncorrelated Signals In case o two uncorrelated signals, meansquared value o sum given by ( rms) Two rms values add as though they were vectors at right angles When ully correlated ( rms) 1( rms) ( rms) sign is determined by whether signals are in or out o phase Here, rms values add linearly (aligned vectors) + ( ) 1( rms) ± ( rms) (1) (13) University o Toronto 11 o 55 Noise Summation Example 1( rms) 10µV, ( rms) 5µV, then ( rms) ( ) 15 which results in ( rms) 11.µV. (14) Note that eliminating ( rms) noise source same as reducing 1( rms) 8.7 µv (i.e. reducing by 13%)! Important Moral To reduce overall noise, concentrate on large noise signals. University o Toronto 1 o 55

7 Frequency-Domain Analysis With deterministic signals, requency-domain techniques are useul. Same true or dealing with random signals like noise. This section presents requency-domain techniques or dealing with noise (or random) signals. University o Toronto 13 o () Spectral Density ( µv) 100 µv Hz 3.16 Hz Spectral Density log( ) ( Hz) () ( Hz) Root-Spectral Density Periodic waveorms have their power at distinct requencies. Random signals have their power spread out over the requency spectrum. 100 log( ) University o Toronto 14 o 55

8 Spectral Density Spectral Density () Average normalized power over a 1 hertz bandwidth Units are volts-squared/hertz Root-Spectral Density () Square root o vertical axis (req axis unchanged) Units are volts/root-hertz (i.e. V Hz). Total Power rms ( ) ()d Above is a one-sided deinition (i.e. all power at positive requencies) 0 (15) University o Toronto 15 o 55 Root-Spectral Density Example () µv Hz Around 100 Hz, I measurement used 10 I measurement used RBW 100 () 10µV Hz RBW µv µv log( ) ( Hz) 30 Hz, measured rms 0.1 Hz, measured rms (16) (17) University o Toronto 16 o 55

9 White Noise () 10 µv Hz () w µv Hz log ( Hz) Noise signal is white i a constant spectral density () w (18) where w is a constant value University o Toronto 17 o 55 µv Hz () db/decade / Noise 10 ( ) () ( ) 1 noise corner / noise dominates 1000 white noise dominates () kv kv log ( is a constant) (19) In terms o root-spectral density () k v (0) Falls o at -10 db/decade due to Also called licker or pink noise. University o Toronto 18 o 55

10 Filtered Noise i() As () () Ajπ ( ) Vni() Output only a unction o magnitude o transerunction and not its phase Can always apply an allpass ilter without aecting noise perormance. Total output mean-squared value is ( rms) 0 () Ajπ ( ) i () Ajπ ( ) i()d (1) University o Toronto 19 o 55 i () nv Hz Noise Example 0 i () () log C 10 4 R 1kΩ 1 o πrc 0.159µF Ajπ ( ) ( db) As () o s π o 10 4 log () 0 nv Hz log University o Toronto 0 o 55

11 Noise Example From dc to 100 khz o input signal i( rms) d ( nv) ( 6.3 µv rms) 0 Note: or this simple case, i( rms) 0 nv Hz 100kHz ( 6.3 µv rms) For the iltered signal, (), () (3) () o (4) University o Toronto 1 o 55 Between dc and 100kHz ( rms) o Noise Example d 0 atan + o --- o ( nv) ( 0.79 µv rms) (5) Noise rms value o () is almost 1 10that o i () since high requency noise above 1kHz was iltered. Don t design or larger bandwidths than required otherwise noise perormance suers. University o Toronto o 55

12 Sum o Filtered Noise 1 () A 1 () s uncorrelated noise sources () 3 () A () s A 3 () s () I ilter inputs are uncorrelated, ilter outputs are also uncorrelated Can show () Ai ( jπ) i() i 13,, (6) University o Toronto 3 o 55 Noise Bandwidth Equal to the requency span o a brickwall ilter having the same output noise rms value when white noise is applied to each Example ( db) 0 0 Ajπ ( ) o o o 0 db/decade 10 o log ( db) 0 Noise bandwidth o a 1 st-order ilter is 0 A brick ( jπ) o log o o π x -- o π -- o University o Toronto 4 o 55

13 Noise Bandwidth Advantage total output noise is easily calculated or white noise input. I spectral density is w volts/root-hz and noise bandwidth is, then Example x ( rms) wx A white noise input o 100 nv/ Hz applied to a 1 st order ilter with 3 db requency o 1 MHz (7) ( ) rms ( rms) 15 µv π (8) University o Toronto 5 o 55 1/ Noise Tangent Principle Method to determine the requency region(s) that contributes the dominant noise Lower a 1/ noise line until it touches the spectral density curve The total noise can be approximated by the noise in the vicinity o the 1/ line Works because a curve proportional to 1 x results in equal power over each decade o requency University o Toronto 6 o 55

14 () 1/ Tangent Example 00 1/ curve Frequency (Hz) N 1 N N 3 N Consider root-spectral noise density shown above University o Toronto 7 o / Tangent Example 00 N d ln() 1 N ( nv) 0 d ( nv) N d 0 ( 10 3 ) ( nv) N d d d 00 π ( 00 )( 10 4 ) ( nv) (9) (30) (31) (3) University o Toronto 8 o 55

15 1/ Tangent Example The total output noise is estimated to be But µV rms ( rms) ( N 1 + N + N 3 +N 4 ) 1 N µV rms (33) (34) Need only have ound the noise in the vincinity where the 1/ tangent touches noise curve. Note: i noise curve is parallel to 1/ tangent or a wide range o requencies, then also sum that region. University o Toronto 9 o 55 Noise Models or Circuit Elements Three main sources o noise: Thermal Noise Due to thermal excitation o charge carriers. Appears as white spectral density Shot Noise Due to dc bias current being pulses o carriers Dependent o dc bias current and is white. Flicker Noise Due to traps in semiconductors Has a 1/ spectral density Signiicant in MOS transistors at low requencies. University o Toronto 30 o 55

16 Resistor Noise Thermal noise white spectral density R R (noiseless) V R() 4kTR R (noiseless) 4kT I R() R element models k is Boltzmann s constant JK 1 T is the temperature in degrees Kelvin Can also write V R () R nv Hz or 7 C 1k (35) University o Toronto 31 o 55 Diodes Shot noise white spectral density r d kt qi D (noiseless) V d() ktrd (orward-biased) element r d kt qi D (noiseless) models I d() qid q is one electronic charge C is the dc bias current through the diode I D University o Toronto 3 o 55

17 Bipolar Transistors Shot noise o collector and base currents Flicker noise due to base current Thermal noise due to base resistance (active-region) element V i () I i () (noiseless) 1 V i () 4kT rb g m V i () has base resistance thermal noise plus collector shot noise reerred back I i () has base shot noise, base licker noise plus collector shot noise reerred back University o Toronto 33 o 55 I C KI B I i () q IB β() model MOSFETS Flicker noise at gate Thermal noise in channel V g() I d() (active-region) element K V g() WLC ox I d() 4kT -- 3 gm model University o Toronto 34 o 55

18 MOSFET Flicker (1/) Noise K V g() WLC ox (36) K dependent on device characteristics, varies widely. W & L Transistor s width and length gate-capacitance/unit area C ox Flicker noise is inversely proportional to the transistor area, WL. 1/ noise is extremely important in MOSFET circuits as it can dominate at low-requencies Typically p-channel transistors have less noise since holes are less likely to be trapped. University o Toronto 35 o 55 MOSFET Thermal Noise Due to resistive nature o channel In triode region, noise would be I d() ( 4kT) rds where is the channel resistance r ds In active region, channel is not homogeneous and total noise is ound by integration or the case I d() 4kT -- 3 gm V DS V GS V T (37) University o Toronto 36 o 55

19 Low-Moderate Frequency MOSFET Model V i () (noiseless) (active-region) element V i () 4kT K g m WLC ox model Can lump thermal noise plus licker noise as an input voltage noise source at low to moderate requencies. At high requencies, gate current can be appreciable due to capacitive coupling. University o Toronto 37 o 55 Opamps I n-() (noiseless) () I n+ () (), I n- (), I n+ () values depend on opamp typically, all uncorrelated. element model Modelled as 3 uncorrelated input-reerred noise sources. Current sources oten ignored in MOSFET opamps University o Toronto 38 o 55

20 Why 3 Noise Sources? () () ignored Actual 0 R () I n- ()ignored Actual + ( I n- R) R () I n+ ()ignored Actual + ( I n+ R) University o Toronto 39 o 55 Capacitors Capacitors and inductors do not generate any noise but... they accumulate noise. Capacitor noise mean-squared value equals kt C when connected to an arbitrary resistor value. R C πrc Noise bandwidth equals R V R () 4kTR ( π ) o C () ( rms) π V R() -- o kt C ( rms) ( 4kTR) π πrc (38) University o Toronto 40 o 55

21 Capacitor Noise Example At 300 K, what capacitor size is needed to have 96dB dynamic range with 1 V rms signal levels. Noise allowed: Thereore C µv rms 10 1V rms ( ) 96 0 kt pF (rms) (39) (40) This min capacitor size determines max resistance size to achieve a given time-constant. University o Toronto 41 o 55 Sampled Signal Noise Consider basic sample-and-hold circuit φ clk v in C v out When φ clk goes low, noise as well as signal is held on C. an rms noise voltage o kt C. Does not depend on sampling rate and is independent rom sample to sample. Can use oversampling to reduce eective noise. Sample, say 1000 times, and average results. University o Toronto 4 o 55

22 C Opamp Example C V i R 1 circuit R R V o Use superposition noise sources uncorrelated Consider I n1, I n and I n- causing 1 () I n1 R 1 R I n- I n+ I n R equivalent noise circuit Vno () 1 R () ( I n1() + In() + In-() ) jπr C (41) University o Toronto 43 o 55 Opamp Example Consider I n+, and causing () R R 1 () ( I n+ ()R + () + Vn() ) jπc R (4) I R «R 1 then gain 1 or all req and ideal opamp would result in ininite noise practical opamp will lowpass ilter noise at opamp. I R» R 1, low req gain R R 1 and 3dB 1 ( πr C ) similar to noise at negative input however, gain alls to unity until opamp. (rms) t Total noise: + (43) t 1(rms) (rms) University o Toronto 44 o 55

23 Numerical Example Estimate total output noise rms value or a 10kHz lowpass ilter when C 160pF, R 100k, R 1 10k, and R 9.1k. Assume () 0nV Hz, I n ( ) 0.6 pa Hz opamp s 5MHz. t Assuming room temperature, I n pa Hz I n1 1.8 pa Hz 1. nv Hz (44) (45) (46) University o Toronto 45 o 55 Numerical Example The low req value o 1 () is ound by 0, in (41). 1 ( 0) ( I n1( 0) + In() 0 + In-( 0) )R Since (41) indicates noise is irst-order lowpass iltered, 1(rms) ( )( ) ( 100k) ( 147 nv Hz) ( 147 nv Hz) π ( 18.4 µv) π( 100kΩ) ( 160pF) (47) (48) University o Toronto 46 o 55

24 For ( 0) Numerical Example ( 0) ( I n+ ()R + () + Vn() ) ( 1 + R R 1 ) ( 4.1 nv Hz) 11 ( 65 nv Hz) Noise is lowpass iltered at o until 1 ( R R 1 ) o where the noise gain reaches unity until 5MHz Breaking noise into two portions, we have (rms) t (49) ( ) π πr C ( ) π + -- ( t ) 1 ( 74.6 µv) (50) University o Toronto 47 o 55 Numerical Example Total output noise is estimated to be (rms) 1(rms) (rms) + 77 µv rms (51) Note: major noise source is opamp s voltage noise. To reduce total output noise use a lower speed opamp choose an opamp with a lower voltage noise. Note: R contributes to output noise with its thermal noise AND ampliying opamp s positive noise current. I dc oset can be tolerated, it should be eliminated in a low-noise circuit. University o Toronto 48 o 55

25 CMOS Example Look at noise in input stage o -stage CMOS opamp V bias 5 V dd Q 5 1 V in+ Q 1 Q V in- 3 4 (output) Q 3 Q 4 V ss Equivalent voltage noise sources used since example addresses low-moderate requency. University o Toronto 49 o 55 CMOS Example g m1 R o (5) where is the output impedance seen at. R o g m3 R o 4 (53) g m g m3 (54) Found by noting that 5 modulates bias current and drain o Q tracks that o Q 1 due to symmetry this gain is small (compared to others) and is ignored. University o Toronto 50 o 55

26 CMOS Example () ( gm1 R o ) 1() ( gm3 R o ) + 3() (55) Find equiv input noise by dividing by g m1 R o g m3 eq () 1() + Vn3() g m1 Thermal Noise Portion For white noise portion, subsitute i() 4kT g mi (56) (57) eq () kt g m3 1 + kt g m1 g m1 g m1 (58) University o Toronto 51 o 55 CMOS Example Assuming g m3 g m1 is near unity, near equal contribution o noise rom the two pairs o transistors which is inversely proportional to. g m1 In other words, g m1 should be made as large as possible to minimize thermal noise contribution. 1/ Noise Portion We make the ollowing substitution into (56), i g mi µ i C W ox ---- L () Vn1() + Vn3 I Di i ( W L) 3 µ n () ( W L) 1 µ p (59) (60) University o Toronto 5 o 55

27 CMOS Example Now, letting each o the noise sources have a spectral density K i i() W i L i C ox (61) we have K 1 µ n i() K L C ox W 1 L 1 µ p W 1 L 3 (6) Recall irst term is due to p-channel input transistors, while second term is due to the n-channel loads University o Toronto 53 o 55 CMOS Example Some points or low 1/ noise For L 1 L 3, the noise o the n-channel loads dominate since µ n > µ p and typically n-channel transistors have larger 1/ noise than p-channels (i.e. K 3 > K 1 ). Taking L 3 longer greatly helps due to the inverse squared relationship this will limit the signal swings somewhat The input noise is independent o W 3 and thereore one can make it large to maximize signal swing at the output. University o Toronto 54 o 55

28 CMOS Example Some points or low 1/ noise Taking W 1 wider also helps to minimize 1/ noise (recall it helps white noise as well). Taking L 1 longer increases the noise due to the second term being dominant. University o Toronto 55 o 55