Michael David Bryant 1 11/1/07 Signal Conditioning Issues: Filtering Ideal frequency domain filter Ideal time domain filter Practical Filter Types Butterworth Bessel Chebyschev Cauer Elliptic Noise & Avoidance Signal to noise ratio S/N Mechanical Noise sources Electrical Noise sources Thermal noise Shot Noise Flicker Noise Noise Figure Grounding & Shielding Ground loops Example: Improper Grounding Example: Grounding of Off-the-shelf components Example: Unavoidable ground loops Power Supply Feedback Loops Stray Capacitances Prescription: Stray Capacitances Impedance Matching
Michael David Bryant 2 11/1/07 2 Filtering Goal: remove unwanted signal components increase signal to noise ratio Ideal frequency domain filter passband: constant gain; stopband: zero gain phase angle unaffected H ( f ) stopband passband stopband - f h - f l f l f h f (Hz) low-pass: fl = 0 f fc = fh high-pass: fl = fc f < fh = band-pass shown: fl f fh
Michael David Bryant 3 11/1/07 3 H ( f ) passband stopband passband f L f l f h f (Hz) band reject: stopband between passbands 0 f fl fl f fh Intended for steady state operation transient response altered overshoot(s) possible
Michael David Bryant 4 11/1/07 Ideal time domain filter f(t) time domain filter f(t -!) f(t) f(t) f(t -!)! t (sec) Time domain: steady state response delayed by τ Frequency domain: gain unchanged added phase lag φ = τ ω proportional to frequency ω = 2π f Signal not distorted, only delayed in time
Michael David Bryant 5 11/1/07 Practical Filter Types Simultaneously approximate both ideal filters gain: alter phase compromise: proportional to ω time delay Butterworth (maximally flat in passband) gain flat gain over passband milder attenuation in stopband K/ω n c H(ω) = 1 + (ω/ωc) 2n phase angle approximately linear in passband multiple arctan terms φ 45 n (ω/ωc ) little phase distortion, only time delay step response: delayed with small overshoot ωc : -3 db cutoff frequency; n: # poles (dynamic order of filter) K: gain factor poles equi-spaced on circle in s plane H db H db Φ H Φ H 10 10 10 10 ω/ω c Figure : n = 5 pole Butterworth filter with K/ω n c = 1.
Michael David Bryant 6 11/1/07 Bessel (approximates constant time delay in passband) transfer function based on Bessel polynomials G(s) = bo K bo + b1 s +...+ bn-1 s n-1 + s n bk = (2k - 1)! k! (n - k)! (2 τ o ) k-n ; k = 0, 1,..., n n: filter order, τo : time delay of filter nearly linear phase in passband step response: delayed with overshoot nil stopband rolloffs comparable to Butterworth poles on circle in s plane, spaced differently from Butterworth H db H Φ db H Φ H 10 10 10 10 ω Figure : n = 5 pole Bessel filter.
Michael David Bryant 7 11/1/07 Chebyschev (equi-ripple) gain very fast initial roll-off, just outside passband compromise: slight ripple (distortion) in passband H(ω) = K 1 + ε 2 C 2 n (ω) K: gain factor; Cn(ω ) : nth order Chebyschev polynomial K ε: passband gain "ripple" bounded by K and 1 + ε 2, ripple = 10 log10 1 + ε 2 measured in db cuttof frequency ωc = cosh( 1 n cosh-1 1 ε ) phase angle & overshoot behavior crudely similar to Butterworth poles on ellipse in s plane Im s H db H db Re s 10 10 10 10! Figure : n = 5 pole Chebyschev filter with ε = 0.5, K = 1. Maximum passband ripple 0.97 db and ωc 1.035.
Michael David Bryant 8 11/1/07 Cauer Elliptic (anti-aliasing) filters equi-ripple in passband with equi-ripple stopband floor steepest drop off just outside passband stopband attenuation & floor specified to digitizing level of A/D converter attenuation floor specified (db = 20 log10 ) poles on ellipse in s plane, zeroes on Imaginary axis outside ellipse Im s H db H db Re s 10 10 10 10! Figure : n = 5 pole elliptic filter: poles of prior figure with zeros ± 1.1 j and ± 1.5 j. The dashed line represents the stopband attenuation floor
Michael David Bryant 9 11/1/07 Noise & Avoidance Noise: unwanted (AC) signal riding on desired signal Noise = (actual signal) (desired signal) Noisy system model: communication channel adds noise n(t) n(t) x(t) Transmitter + y(t) Receiver x'(t) (S/N) i (S/N) o x(t): input signal, band limited to W n(t): noise signal, added by channel y(t) = x(t) + n(t): noisy signal x' (t): recovered signal
Michael David Bryant 10 11/1/07 Signal to Noise Ratio (SNR or S/N) SNR: average signal power to average noise power SNR = [x'(t)]2 S = [n(t)] 2 No B f(t) = 1 T T f(t) dt average over T; [f(t)] 2 denotes average power t=0 White noise: Random signal Equal power, all frequency bands Term from white light N o db white noise: constant power over entire spectrum f (Hz) Goal: keep SNR large as possible maximize numerator increase signal power S minimize denominator decrease bandwidth B (No fixed) General Prescription: Minimize Bandwidth B
Michael David Bryant 11 11/1/07 Electrical Noise sources external: reduce with proper grounding & shielding lightning sun radio transmissions power source contamination & feedback cosmic radiation other system signals internal electrical noise: cannot eliminate, only minimize thermal shot flicker General Prescription: Practice proper grounding & shielding techniques shield especially sensitive equipment use smallest practical resistors & DC bias currents modulate critical signals to higher frequency bandwidths
Michael David Bryant 12 11/1/07 Thermal noise Temperature: measure of kinetic energy of particle collection Charged particles in motion electric current Random, thermally excited electron currents sum statistically produce noise voltage Vn across resistances power density spectrum: v 2 n = 4 k T R (V 2 /Hz) Boltzmann's constant k = 1.33 x 10-23 J/ K T: temperature ( K) R: resistance (Ω) fh V 2 n = fl v 2 n df = 4 k T R B Bandwidth: B = (fh - fl ) Bias & gain resistors in active electronics (e.g. Op-Amps) add this noise. Prescription: Reduce R and/or B, and T if possible.
Michael David Bryant 13 11/1/07 Real Resistor with Thermal noise R R + Ideal Resistor V n = {4kTRB} 1/2 - Real Resistor Real Resistor = Ideal Resistor + Thermal noise source For noise calculations, replace Ideal with Real V n : rms value of white noise
Michael David Bryant 14 11/1/07 Shot Noise DC current = uniform avalanche of charge carriers electron arrivals random & discrete, sum is DC current arrivals many small pulses, synthesizes white noise I 2 n = 2 e I DC B electronic charge: e = 1.6 x 10-19 coulomb DC Bias currents in active electronics add this noise. Prescription: Reduce DC bias currents I DC and bandwidth B Unique problem: I DC blocked by coupling capacitors, but high frequency (AC) noise In passes through capacitors
Michael David Bryant 15 11/1/07 15 Flicker Noise Present in active devices (Op-Amps) due to random thermal currents noise density vn (or in) increases as 1/f below cutoff fc 10 to 100 Hz special problem for LOW frequency active devices transducers amplifiers v n 1/f f c f (Hz) noise power: V 2 n fh = fl v 2 n df Prescription: Modulate signal to higher frequency range (> fc )
Michael David Bryant 16 11/1/07 16 Mechanical Noise sources External: reduce with thermal/mechanical isolation & control shock & vibration thermal expansions other system signals Internal: minimize with good lubrication, cooling, design, maintenance, etc. gear chatter & backlash: spur gears bearing noise: wear foreign particles in contact elements belt slip excessive friction with slip/stick resonance ANY DISSIPATIVE ELEMENT Nonlinearities: harmonic generation gear backlash harmonics in stator field General Prescription: Replace mechanical components with electrical/optical Precision components (bearings) Thermal & environmental controls Preload components (gears & bearings) to minimize backlash Careful vibration isolation
Michael David Bryant 17 11/1/07 17 Mechanical Noise Model Analogy: Dissipative elements (resistances) = Resistor Like circuit, dissipative elements = resistance + noise voltage source. Bond graph: Replace resistance R with grouping R, 1, and S e R: R R: R S e : V n {4kTRB} 1/2 1 A Ideal dissipative element Real dissipative element A
Michael David Bryant 18 11/1/07 Justification Dissipative elements Resistors, dampers, friction, etc. ALL Irreversible processes: convert usable energy to heat Entropy increases (irreversible entropy produced) Disorder increases Random (white) noise calibrates amount of disorder
Michael David Bryant 19 11/1/07 Noise Figure Total predicted noise sums vectorially V 2 total = V 2 1 + V 2 2 +... + V 2 m + (I 1 Req ) 2 +... Noise Figure NF = 10 log10 all noise (mean square) source noise typically: NF between 0.1 & 3 db goal: minimize NF
Michael David Bryant 20 11/1/07 Root Mean Square (RMS) AC amplitude I AC of I(t) = I AC sin 2πt T Definition: Equivalent average AC/DC heating of resistor, over period τ + T I 2 DC R = 1 T { I AC sin 2πt T }2 R dt = I 2 τ + T AC R T { sin 2πt T }2 dt = I 2 AC R T t = τ t = τ T 2 RMS value = equivalent DC: I RMS = I DC equivalent = I AC 2
Michael David Bryant 21 11/1/07 Grounding & Shielding: Ground loops one and only one node = ground other "ground" nodes connected via conductors (nonzero impedances) currents voltage drop other "ground" nodes ground + - pre-amp sensor i s i a power stage R sa actuator proper (solid lines): each element in circuit has separate lead to ground improper (dashed lines): grounding leads haphazardly patched together input signal to preamp Vi = Vsensor - is Rlead - ia Rsa 1442443 144424443 small for small is = 0 with proper grounding
Michael David Bryant 22 11/1/07 Example: Improper Grounding Rsa one foot length of wire, Rlead 10 foot length of wire common copper wire gauges: 10-1 ρcu 10 1 Ω per 1000 ft signal source Vsensor m V to V Rlead m Ω, is m A is Rlead µ V, SMALL! actuator signal ia 1 A, Rsa 10-1 m Ω ia Rsa 10-1 m V, SIGNIFICANT voltage drop! Reduction technique: separate ground leads, NO sharing
Michael David Bryant 23 11/1/07 Example: Grounding of Off-the-shelf components e 1 in e 1 out e 2 in e 2 out e 1 out e 2 in g 1 V 1 g V 2 g g 2 lead R 1 i 1 i 2 lead R 2 system ground or earth ground Off-the-shelf components 1 and 2: output of 1 = input to 2 Note 1& 2 have common ground. ground loop can result: ik over ground lead R lead k V k g = ik R lead k at g k Input presented to 2 by 1 (e 2 in with respect to g 2 ): e 2 in = e 1 out + [ V 1 g - V 2 g 14243 "noise" ] ; Desire: e 2 in = e 1 out
Michael David Bryant 24 11/1/07 Example: Unavoidable ground loops Minimize effects with Common Mode Rejection Ratio of differential amplifier: E 1 out e 1 out = { with respect to g 1 e 1 out + [ V 1 g - V 2 g ] with respect to g 2 for rejection R3 = R1, R4 = R2 (match carefully)
Michael David Bryant 25 11/1/07 Input to Op-Amp with respect to g 2 e 2 in = - R 2 R1 { e 1 out + [ V 1 g - V 2 g ] } R 3 = R 1, R 4 = R 2 + R4 R3 + R4 { 1 + - R 2 R1 } [ V 1 g - V 2 g ] - R 2 R1 CMRR large e 1 out High Common Mode Rejection Ratio and resistor matching gives e 2 in - R 2 R1 e 1 out (proportional)
Michael David Bryant 26 11/1/07 Power Supply Feedback Loops Unwanted feedback paths through power source AC signals from amp stage 2 feeds back through common power source V± AC feedback alters/contaminates signals in amp stage 1 V ± positive voltage regulator R V ± 1 2 Prescription: separate power supplies use voltage regulators & decoupling networks
Michael David Bryant 27 11/1/07 Stray Capacitances white (conducting ground) green (nonconducting ground) black (115 VAC) C s "plate" C s + C s "plate" R in - stray capacitances Cs couple electric fields to circuit nodes spurious voltages induced wires act as capacitance plates, air acts as dielectric medium
Michael David Bryant 28 11/1/07 Prescription: Stray Capacitances C s Grounded shield protects (surrounds) sensitive circuit parts 1. Limit induced voltages from Cs by shielding shield grounded short circuit electric field (induced voltages) to ground input/output leads insulated from ground shield CAUTION: improper attachment of shield to multiple grounds ground loops 2. Distance between source & sensitive components reduces E 1/r
Michael David Bryant 29 11/1/07 C s C s NO YES C s NO YES single attachment point to avoid ground loops
Michael David Bryant 30 11/1/07 AC Power @ Steady State & Impedance + V(t) - i(t) Z 2 terminal device AC excitation & response: v(t) =V o cos (ωt + η) i(t) = I o cos (ωt + ν) frequency: ω = 2π/T, amplitudes: V o, I o, phases: η, ν Complex representation: Voltage: v(t) = ½ (V e jωt + V* e -jωt ) = V o cos (ωt + η) Current: i(t) = ½ (I e jωt + I* e -jωt ) = I o cos (ωt + ν) Complex amplitudes: V= V o e jη, I= I o e jν Complex conjugate: * note: Re v(t) = Vcos (ωt + η), Im v(t) = Vsin (ωt + η) Instantaneous Power: P = v i = ¼ (V e jωt + V* e -jωt ) (I e jωt + I* e -jωt ) = ½ Re [VI* ] + ½ Re [VI e j2ωt ] = ½ Re [VI* ] + ½ V o I o cos (2ωt +η+ν) Average Power: P avg = 1 T " +T # Pdt = 1 T " " +T # v(t)i(t)dt = 1 2 Re[V * I] " P avg = ½ Re [VI* ]
Michael David Bryant 31 11/1/07 Impedance: ratio of effort to flow Z = V/I Also expressed as complex number Z = R + j X = Z e j Z R = Re(Z): resistance X = Im(Z): reactance With V = I Z, I= I o e jν P avg = ½ Re [VI* ] = ½ Re [Z II* ] P avg = ½ I o 2 Re [Z ] = ½ I o 2 R P avg = ½ I o 2 R Average power to impedance Z to R = Re(Z): resistance
Michael David Bryant 32 11/1/07 Impedance Matching Goal: maximize average power transfer, from source (network) to load V Th + - Z Th I L + Z L - V L Source impedance: ZTh = RTh + j XTh Load impedance: ZL = RL + j XL Source Load Impedance: Z = R + j X = Z e j Z R = Re(Z): resistance (real) part of impedance X = Im(Z): reactance (imaginary) part of impedance Load Current: I L = V TH Z TH + Z L Voltage: V L = I L Z L = V TH Z TH + Z L Z L Average Power to load: P avg = ½ Re [VI* ] = ½ I o 2 Re [Z ] = ½ I o 2 R I= I o e jν I o = I PL = VL IL = [IL ] 2 RL = VTh 2 (RTh + RL) 2 + (XTh + XL) 2 RL f(t) = 1 T T t=0 f(t) dt average over T; [f(t)] 2 average power
Michael David Bryant 33 11/1/07 Choose XL and RL to maximize power transfer from source to load Maximize PL with respect to XL and RL Result XL = - XTh, RL = RTh ZL = Z * Th maximum power transferred Pmax = 1 4 VTh 2 RTh