1. ET8.017Delft University of TechnologyEl. Instr.Electronic Instrumentation Lecturer: Kofi MakinwaK.A.A.Makinwa@TUDelft.NL015-27 86466Room. HB13.270 EWI buildingAlso involved:Saleh Heidary(S.H.Shalmany@TUDelft.NL)ET8.017 OutlineEl. Instr.Delft University of TechnologyMonday Sept. 24:Deterministic and Random ErrorsCommon-mode Rejection RatioPower supply Rejection RatioGain error 1

2. ET8.017 Deterministic ErrorsEl. Instr.Delft University of Technology Two kinds of errors (uncertainty):a. Deterministic errors e.g. source loading, offset, gain errorb. Random (stochastic) errors e.g. thermal noise Assuming that the output of a system z = f(a,b,c, ...) Then the total error due to deteministic parameter uncertainty is: f (a, b, c,...)f (a, b, c,...)f (a, b, c,...) z = a +b +c + ... a bcwhere z denotes the uncertainty in z due to parameter a, which isspecified with uncertainty a etc..The derivatives above are called sensitivities. ET8.017Additive ErrorsEl. Instr.Delft University of Technology Example: Combining additive errors: For z = a + b or z = a - b with a the uncertainty in a and b the uncertainty in b, the overall uncertainty follows as: ( a b) ( a b) z =a + b = a + bab By extension, the effect of deterministic errors (e.g. offset, gain error etc.) in a linear time-invariant (LTI) system can be determined by superposition.2 3. Noise, interference &ET8.017Delft University of Technology distortionEl. Instr. Interference is caused by external signals e.g. 50 Hz hum Noise is caused by stochastic processes in the measurement system itself e.g. thermal noise. Noise and interference are usually additive error signals they can both be modeled by extra error sources to the system Note that interference and noise should not be confused with signal distortion, which is caused non-linearities in either the amplitude or the frequency domain. ET8.017Delft University of TechnologyNoise + 50 Hz interference El. Instr.3 4. ET8.017Delft University of TechnologyThermal noiseEl. Instr.ALL resistors exhibit thermal noiseThermal noise has a uniform (white) frequency spectrumnoise power density: Pn= 4kT (W/Hz)where k is Boltzmanns constant and T is absolute temperatureAt room temperature (293 K): Pn= 4kT = 1.62 x 10-20 (W/Hz)For a resistor: P = Vrms2/R = Irms2R=> noise voltage with spectral density: vn = (4kTR) (V/Hz1/2)=> noise current with spectral density: in = (4kT/R) (A/Hz1/2) ET8.017Delft University of Technology Thermal noise (1) El. Instr.(Thermal) noise is stochastic, i.e. its amplitude at a given time tcannot be predictedThermal noise has a Gaussian amplitude distribution v(t) t V2 2 1 2V p (V )dV = e n dV Vn 24 5. ET8.017Delft University of Technology Thermal noise (2) El. Instr.In the presence of thermal noise, the detection limit is usuallyspecified as the standard deviation (or sigma) VnA good estimate of its peak value (e.g. as seen on a wide-bandscope) is 5-sigma v(t)t V2 2 1 2V p (V )dV = e n dVVn 2 ET8.017Delft University of TechnologyShot noise El. Instr. Shot noise is the result of the fact that an electrical current is the result of the motion of discrete charge carriers. This can be compared with hail on a roof: even if the average current (mass flow) is constant, the amount of charge carrriers (ice lumps), measured in different intervals with the same length t, will, in general, be different. The relative fluctuation becomes greater as t becomes smaller and smaller. Like thermal noise, shot noise has a uniform frequency spectrum and so is also a form of white noise. If a current I consists of charge carriers with charge q that move independently of each other, then the spectral noise current is given by in = (2qI) (A/Hz1/2). This equation applies to p-n junctions (diodes, transistors, etc.) but not to metallic conductors where there is more long-range correlation between the movement of the charge carriers. In such conductors, the shot noise will thus be smaller!5 6. ET8.017Delft University of TechnologyCalculating shot noise El. Instr.in is proportional to thein = 2qI(A/Hz1/2)square-root of current!I n (rms) = in B = 2qI DC B(A)with bandwidth B = f 2 f1in f1f2frequencyExample (for B = 10 Hz): I = 1 A noise current In(rms) = 57 nA (0.000006%) I = 1 pA noise current In(rms) = 56 fA (5.6%) Shot noise is important at low current levels ET8.017Delft University of Technology1/f (flicker) noiseEl. Instr. Both thermal and shot noise are the result of fundamental physical processes and so form a fundamental detection limit. Apart from these types of noise, real electronic components also suffer from several sources of excess noise. Resistors exhibit current-dependent 1/f noise: Vn(rms) = cI where the constant c is determined by resistor quality (price). Transistors (especially MOSFETs) exhibit area-dependent 1/f noise: where Vn(rms) 1/sqrt(Area) 1/f noise has a 1/f POWER spectrum, i.e. the spectral noise power per decade (or per octave) is constant. 1/f noise is sometimes referred to as pink or flicker noise6 7. ET8.017Delft University of Technology1/f noise example El. Instr.3 instancesof 1/f ruis Peak amplitude increases for as measurement time increases?! x~t More and more important for long (low-frequency) measurements!ET8.017 Real noise spectrumEl. Instr.Delft University of Technology Spectrum of a critically damped micro-mechanical structureTwo spectral regimes: a. 1/f noise andb. frequency independent white noise. 7 8. ET8.017Delft University of Technology On-line noise demoEl. Instr. http://socrates.berkeley.edu/~phylabs/bsc/Supplementary/NoiseGenerator.html ET8.017 Non-electrical noiseEl. Instr.Delft University of Technology Noise is also present in non-electrical domains Thermal noise some kind of energy dissipation In the mechanical domain this is due to damping In very small structures (MEMS), Brownian noise is also important In the thermal domain this is due to thermal resistance Shot noise small amount of energy-carrying particles In the optical domain, photon shot noise is significant8 9. ET8.017Error propagation El. Instr.Delft University of Technology Errors (uncertainties) due to: a. Noise, b. Electro-Magnetic Interference (EMI) are referred to as Stochastic (random) sources of error (uncertainty). The effect of stochastic error(s) on overall system specifications can be evaluated using Gauss law of error propagation:z = f ( a , b , c ,...) f ( a , b, c,...) f ( a , b, c,...) f ( a , b, c,...) 2 2 2 z2 = a + 2 b2+ c + ...2 a a ,b , c ,... b a ,b , c ,...c a ,b , c ,...where z2 denotes the variance (uncertainty) in parameter z withaverage value z due to parameter a, which is specified withuncertainty a etc..Error propagation and ET8.017Delft University of Technologyequivalent input errorEl. Instr. Combining additive errors: For z= a+ b or z= a-b the overall uncertainty follows as: z =ab (a b) (a b) 22 z2 = a + 2 b 2=a +b2 2a a ,b , c ,... b a ,b , c ,... Hence, the variances of stochastic errors in a linear system can be linearly added. In case of distributed noise voltages or currents:9 10. Error propagation andET8.017Delft University of Technologyequivalent input error El. Instr. Combining additive errors: For z= a+ b or z= a-b, the overall uncertainty follows as: z =ab (a b) (a b) 2 2 z2 = a + 2 b2=a +b2 2 a a ,b , c ,... b a ,b , c ,... Hence, the variances of stochastic errors in a linear system can be linearly added.un ,eq = un1 + un 22 2 In case of distributed noise voltages or currents:in ,eq = in1 + in 22 2Provided these are independent !Error propagation andET8.017Delft University of Technologyequivalent input error El. Instr. Noise paradox: What is the equivalent noise voltage in a circuit with two resistors connected to a noise current source ? Adding the thermal noise powers of R1 and R2 to the noise power due to in flowing through R1 and In flowing through R2: ( a ) un21,eq = un2,R1 + un2,R 2 +in2 R12 +in2 R22 =un ,R1 + un ,R 2 +in ( R12 + R2 ) 222210 11. Error propagation and ET8.017Delft University of Technologyequivalent input errorEl. Instr. Noise paradox: What is the equivalent noise voltage in a circuit with two resistors connected to a noise current source ? Adding the thermal noise powers of R1 and R2 to the noise power due to in flowing through R1 and In flowing through R2: ( a ) un21,eq = un2,R1 + un2,R 2 +in2 R12 +in2 R22 = Xun ,R1 + un ,R 2 +in ( R12 + R2 ) 2222 Adding the thermal noise powers of R1 and R2 to the noise power due to in flowing through R1 plus R2:( b ) un22,eq = un2,R1 + un2,R 2 +in2 ( R1 + R2 )2 VET8.017Equivalent input sourcesEl. Instr.Delft University of Technology General approach for finding equivalent input noise sources: 11 12. ET8.017Equivalent input sourcesEl. Instr.Delft University of Technology General approach for finding equivalent input noise sources:Note: equivalent noise isindependent of sourceimpedance!Extreme case 1:open input. u2 a = (in1 Ri A1 ) 2 + (in 2 Ro ) 2 + un 2 22 u2 b = (in ,eq Ri A1 ) 22 u2 a = u2 b2 2 Hence:un 2 + in 2 Ro2 22 in ,eq = in1 +222 2A1 RiET8.017Equivalent input sourcesEl. Instr.Delft University of Technology General approach for finding equivalent input noise sources:Note: equivalent noise isindependent of sourceimpedance!Extreme case 2:short-circuited input.u2 a = (un1 A1 ) 2 + (in 2 Ro ) 2 + un 2 2 2 u2 b = (un ,eq A1 )2 2 u2 a = u2 b 22 Hence:un 2 + in 2 Ro2 22un ,eq = un1 + 22 2 A1 12 13. ET8.017 Equivalent input sourcesEl. Instr.Delft University of Technology General approach for finding equivalent input noise sources: un 2 + in 2 Ro222un ,eq = un1 + 222A1 un 2 + in 2 Ro222in ,eq = in1 + 22 2 2 A1 Ri Recommendations for low-noise performance: Focus design effort on first stage: low noise and high gain, A1. ET8.017 Noise bandwidth El. Instr.Delft University of TechnologyNoise in components is usually specified in terms of the noisespectral power, sn. In case of a resistor: snu ,Z = 4k BT Z = 4k BTR [ V 2/Hz]4k BT 4k BT sni ,Z ==[ A 2/Hz] ZRNoise bandwidth is the bandwidth of a theoretical low-pass filter withan abrupt cut-off frequency at B. A practical low-pass filter will have amore gradual transition from pass-band to stop-band, H(). vnAt 20C and into a 10kHz bandwidth, a 10k resistor generates 1.3 Vrms. 0 B frequency 13 14. ET8.017Noise bandwidth El. Instr.Delft University of Technology is defined as the bandwidth of a brick wall filter with an abrupt cut-off frequency at B. A practical system is specified by a low-pass filter with a more gradual transition from pass-band to stop-band, H(). The Noise bandwidth B of a practical low-pass filter can be obtained from its transfer function, H() as follows:1 Pn = H o B = H ( ) d B = H ( ) d 2 2 2 20 Ho 0Noise bandwidth ET8.017Delft University of Technology of a first-order systemEl. Instr. 1 Pn = H o B = H ( ) d B = H ( ) d 2 2 22 0 Ho 0 d 1 For Ho=1: B = = arctg ( ) == c [rad / s ] , 1+ ( ) 22 002 For noise calculations, a correction factor /2 is required 14 15. ET8.017Noise in passive componentsEl. Instr.Delft University of TechnologyNoise in impedance, Z, is primarily due to the dissipating (resistive)part, R:snu ,Z = 4k BT Z = 4k BTR [ V 2/Hz]4k BT 4k BT sni ,Z == [ A 2/Hz] ZRReactive components are not free of noise due to parasitics: d24k T 2 k TunC = in Z ( ) d = in R p2 2222 =in R p 2 2= B Rp = B1 + ( R p C ) 4 R p C 2 R p 2002R p C C ET8.017Noise in passive componentsEl. Instr.Delft University of TechnologyReactive components are not free of noise due to parasitics:In a capacitor: d24k T 2 k TunC = in Z ( ) d = in R p2 2222 =in R p 2 2= B Rp = B1 + ( R p C ) 4 R p C 2 R p 2002R p C CSimilarly,in an inductor: 2d 4k BTRs 1d k BTinL = 2un d = un 2 = 2 = Z ( ) Rs2 + ( L ) 2 2 2 200 Rs0 L L 1+ Rs 15 16. Ex. Calculation equivalentET8.017Delft University of Technologyinput noise sourcesEl. Instr.Trans-impedance amplifier: Uo/Ii= (-)Rf Ideal current source, thus in,eq only (open-input) 2 2un1 2 unR uo1 = un1 +in1 R 2 + unR = uo 2 = in ,eq R 22 22f 2 22f Hence,2 in ,eq = +in1 + 2R2f Rf For: 2 4.10 9 1.65.10 20 sni,1= 1 pA/sqrtHz, 2in ,eq = 10 -24 + 4 += 2.8.10 24 [A2 /Hz] 10 10 4 Rf=10 k and 4kBT= 1.65 10-20 J2in ,eq = 1.67 pA/ Hz @300 K ET8.017Signal-to-noise ratio (SNR)El. Instr.Delft University of TechnologySNR is defined as the ratio between signal power and noise powerexpressed in dB: VS (rms) 2 SP V (rms) = 10 log S = 10 log S = 20 log V (rms) N PN VN (rms) N Notes:Since the form of the signal and that of the noise may differsignificantly, SNR should not be calculated from signal amplitudesbut from rms values.A given SNR is always associated with a certain bandwidth.For maximum SNR, the bandwidth of a measurement system shouldbe about the same as the expected signal bandwidth16 17. ET8.017 Noise matching El. Instr.Delft University of TechnologyBest noise performance implies that readout adds no noise to thenoise power already included in the source.The noise added by the readout is specified in the Noise factor, F:SNRinputF== SNRoutputPs ( input ) Pn ( source ) Pn ( source ) + Pn ( readout ) P ( readout ) == 1+ n Ps ( output ) Pn ( source ) Pn ( source ) H ( ) Pn ( source + readout ) 2How can we optimize the NF of a practical system?ET8.017 Noise matching El. Instr.Delft University of Technology Noise matching in a practical system:Pn ( readout )22 2 SNRinputu +i RNF = = 1+= 1+ n n g SNRoutput Pn ( source )4k BTRg Minimum at:NF 4k BTRg 2in Rg - 4k BT un +in Rg=2 2 2 2 () = 0 R2 = 2un2Rg ( 4kBTRg ) 2g,optin2 2un Resulting in a minimum noise factor: NFmin = 1+ 4k BTRg Minimum NF indicates best redistribution of noise over un,eq and in,eq at the minimum noise power, Pn(readout), derived before. 17 18. ET8.017Noise matchingEl. Instr.Delft University of Technology Noise matching using a series resistance? P n ( source + readout ) = 4k BTRg + 4k BTRs +un +in ( Rg + Rs ) 2 224k BT ( Rg + Rs ) +un +in ( Rg + Rs )Rs un + in ( Rg + Rs )2 2 2 2 2 2NF == 1++4k BTRgRg4k BTRg- Noise matching at higher noise level.- Attenuation of Ug Noise matching using series resistance is counter productiveET8.017 Analog to Digital Conversion El. Instr.Delft University of Technology v(t) t Signal is quantized in both amplitude and in time 18 19. ET8.017Quantization Noise El. Instr.Delft University of Technology Range: 0 - VrefProbability distr. function p(V) Number of bits: n1/q Number of levels: 2n V step-size: q = Vref.2-n-q q error: -q < V < q For a busy signal (one that jumpsbetween many quantization levels),this error can be modeled asrandom noise with a uniform pdf The associated standard deviation q 2 1 1e= 2 V dV = 12 q 2e is then given by: q q ET8.017Delft University of Technology Sources of interference El. Instr. Interference Interference sourcesensitive system Coupling mechanismelectrostatic discharge conductionintegrated circuittransformer (50 Hz)magnetic fieldaudio systemSwitched-mode PSU electric fieldhigh-impedance sensorHF digital circuitelectromagnetic field FM radio19 20. ET8.017Delft University of Technology Interference via conduction El. Instr.Coupling via a common impedanceE.g. via a common supply line orVsa ground connectionVCircuit 1 : HF digital circuit i1 is a rapidly varying current (high di/dt ) i1V = R i1 + L di1/dt time VR L i1 VsCircuit 1Circuit 2 Common impedances:ET8.017Delft University of Technology solutions El. Instr. Quick fix: use a bypass capacitor near a HF circuit to locally deliver its rapidly changing supply currents (this can be seen as a form of filtering) Good design: avoid shared impedances di/dtC Local charge storage Seperate supply lines20 21. ET8.017External sources of error El. Instr.Delft University of Technology Ground loops Additive error due to IaRa in series with ug.ET8.017External sources of error El. Instr.Delft University of Technology Ground loopsstar connection tolimit error due to IaRa. 21 22. ET8.017Delft University of TechnologyElectric & magnetic coupling El. Instr.Electric (capacitive) coupling is due to changing voltages, i.e. dV/dtMagnetic (inductive) coupling is due to changing currents, i.e. dI/dt dI/dt magnetic field dV/dt electric field 1source ofinterference1sensitive 2 2 circuitShared magnetic fluxCapacitance b/w two conductorsbetween two loops In the same circuit there can be multiple coupling mechanisms Depends on source and load-impedances, size and geometry ET8.017Delft University of Technology Electromagnetic interferenceEl. Instr. When an antenna is driven by a source, it creates an electromagnetic field L an electric fieldstoringsbron a magnetic field In a given situation, the dominant field depends on the shape and size of the antenna, the frequency and the distance to the antenna Field strengths vary with: 1/R2 and 1/R3 voor electric and magnetic fields, resp. 1/R for EM-waves (R = distance to antenna) R < : E and M fields can dominate (near field) R > : EM waves can dominate (far field) The following applies to EM waves: L = /2: good antenna (lots of EM interference)R L < /20: bad antenna (less problems)22 23. Interference reduction: ET8.017Delft University of Technology Filtering El. Instr.Filtering can help suppress interference that is coupled into acircuit via its supply linesFiltering can also be used on signal lines if the interference doesnot occur at the same frequencies as the wanted signalinterference Not goodfilter GoodSupply cable circuitwith interference filtercircuit Supply cable With interference ET8.017Delft University of Technology Shielding: Coax cable El. Instr.If the outer conductor of a coaxcable (on 1 side) is grounded,electric field cannot reach the innerconductor. The outer conductoralso forms the return path and thesignals reference.If the outer is grounded at bothsides (often the case) a groundloop will be created. This canlead to interference because ofthe voltage differences between ground loopthe two grounds and/or magneticcoupling via the loop.23 24. ET8.017Delft University of Technology Twisted pairEl. Instr. signal currentinduced current M Magnetic field induces current Attn [ dB ]in coilsuntwisted pair (ref) 0 coils are twistedtwisted pair 43 currents in neighboringuntwisted pair + alu foil shield 3 loops have opposite signsuntwisted pair + steel shield32 currents cancel each other net induced current is zero50 Hz Magnetic shielding ET8.017Delft University of TechnologyTwisted pair (2) El. Instr.E Rs+V RLAn electrical field Couples capacitively to both conductors Generates equal currents in both signal and return lines Signal line: current flows through Rs//RL generates +V Return line: current flows to ground (low R)Result: electric fields are not shielded!Solutions: Add capacitive shielding (e.g. a conducting outer layer) Balance the circuit24 25. ET8.017Delft University of TechnologyTwisted pairs: BalancingEl. Instr.(+V) - (+V) = 0 Icm+V+-+V Balancing: Use differential inputs and outputs Ensure that capacitively coupled currents in both lines are the same (i.e. that they are common-mode signals) and that they see the same impedance The current-induced voltages will then cancel each other out Adding an outer shield will then result in excellent suppression of both magnetic and electric interferenceET8.017Delft University of Technology Shielding: EM wavesEl. Instr. EM waves will not penetrate a closed conducting enclosure if it is atleast 10 skin depths thick The enclosure does not need to be grounded EM waves can leak into the enclosure via: Cables that enter the enclosure1 = Use filters and/or shield the cables ... f Openings in the enclosure such as: = skin depth plastic knobs = conductivity displays = permeability Ventilation holes f = frequency Poorly designed connectors 1 = 8.7 dB attn. box1cables box2 (= 2.7 x)Completely closed system 25 26. ET8.017Delft University of TechnologyTypes of cableEl. Instr. untwisted pair coaxial twin-axial twisted-pair twisted-pair shieldedMost types of cable can be obtained with an extra outer shieldfor even better suppression of EM interferenceET8.017 ShieldingEl. Instr.Delft University of TechnologyShielding to reduce capacitive couplingNon-shieldedinterconnect:Ziui , g = ug Zg + ZiZ g // Z iui ,n = un1 Z g // Z i +j CcApplying an electrically Cc ui ,n / un conductive shield: Z i // Z a ui , g =ugZ g + Z i // Z a Z i // Z aZi BW ( ui , g / ug ) 26 27. ET8.017 Active guarding (1)El. Instr.Delft University of Technology Active guarding reduces capacitive coupling, and maintains BWApplying an electricallyconductive shield:Active guarding:No current flowingthrough Za. Hence:Za=Ui/Ia >> ZaET8.017 Active guarding (2)El. Instr.Delft University of Technology Active guarding reduces capacitive coupling, and maintains BWApplying an electricallyconductive shield:Active guarding:Ia= ui-(1-)ui]/Za.Hence: Za= Ui/Ia =uiZa/ui= Za/Stability: positive !27 28. ET8.017Delft University of Technology Suppressing electric fieldsEl. Instr. Knowing how electric (capacitive) coupling arises, it is easy to avoid this type of interference: Reduce the signal impedancesThe coupling cap and the signal impedance form a voltagedivider Drop the frequency and amplitude of the interfering signalUsually not possible Decrease the coupling capacitanceIncrease the distance between interference and circuitGround floating conductors between interference and circuit Break any return path for interferenceThese usually form ground loops Capacitive shielding:Shielded cables (e.g. coax)Conducting box around sensitive circuitsGround the shielding!ET8.017Delft University of Technology Assignment 3aEl. Instr.Rs Iout gm Givirtualground Vs of nextstage A voltage-to-current converter with trans-conductance Gm= 1 A/V is used to readout a sensor with source resistance Rs= 4 k. The voltage-to-current converter consists of a transconductor with gm= 10 mA/V followed by a current amplifier with Gi= 100 The transconductors equivalent noise voltage and current are: un = 1 nV/Hz and in = 0.5 pA/Hz. Its input resistance is 100k. The current amplifiers equivalent noise voltage and current are: un = 100 nV/Hz and in = 10 pA/Hz. Its input resistance is 10. Calculate the converters equivalent noise voltage and current 28 29. ET8.017Delft University of Technology Assignment 3bEl. Instr.Rsgm Givirtual ground Vsof next stage Calculate the detection limit due to noise of the readout, if the bandwidth of interest lies between 20Hz and 20kHz. The source also produces both 1/f and thermal noise. Assuming that the latter is due to the 4k resistor and that the 1/f corner frequency is 1kHz, calculate the new detection limitET8.017Delft University of Technology Assignment 3bEl. Instr.220V/50HzCk1 Ck2Ck3Rsgm Givirtual ground Vsof next stage The circuit is mounted close to a power cable and so suffers from capacitively-coupled 50Hz interference. Calculate the input-referred voltage if Ck1 = Ck2 = Ck3 =1pF What can be done to reduce the power-line coupling? 29