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Lecture 7: Noise Why Do We Care? Lecture 7: Noise Basics of noise analysis Thermomechanical noise Air damping Electrical noise Interference noise » Power supply noise (60-Hz hum) » Electromagnetic interference Electronics noise » Thermal noise » Shot noise » Flicker (1/f) noise Calculation of total circuit noise Reference: D.A. Johns and K. Martin, Analog Integrated Circuit Design, Chap. 4, John Wiley & Sons, Inc. ENE 5400 ΆýɋÕģŰ, Spring 2004 1 µóģµóģvʶ1Zýɋvʶ1Zýɋ Why Do We Care? Noise affects the minimum detectable signal of a sensor Noise reduction: The frequency-domain perspective: filtering » How much do you see within the measuring bandwidth The time-domain perspective: probability and averaging Bandwidth Clarification -3 dB frequency Resolution bandwidth ENE 5400 ΆýɋÕģŰ, Spring 2004 2 µóģµóģvʶ1Zýɋvʶ1Zýɋ 1 Time-Domain Analysis White-noise signal appears randomly in the time domain with an average value of zero Often use root-mean-square voltage (current), also the normalized noise power with respect to a 1-Ω resistor, as defined by: 1 T V = [ ∫ V 2 (t)dt]1/ 2 n(rms) T 0 n T = 1 2 1/ 2 In(rms) [ ∫ In (t)dt] T 0 Signal-to-noise ratio (SNR) = 10⋅⋅⋅ Log (signal power/noise power) Noise summation = + 0, for uncorrelated signals Vn (t) Vn1 (t) Vn2 (t) T T 2 = 1 + 2 = 2 + 2 + 2 Vn(rms) ∫ [Vn1(t) Vn2 (t)] dt Vn1(rms) Vn2(rms) ∫ Vn1Vn2dt T 0 T 0 ENE 5400 ΆýɋÕģŰ, Spring 2004 3 µóģµóģvʶ1Zýɋvʶ1Zýɋ Frequency-Domain Analysis A random signal/noise has its power spread out over the frequency spectrum Noise spectral density is the average normalized noise power over a 1-Hz bandwidth (unit = volts-squared/Hz or amps- square/Hz) It is common to use root spectral density (e.g., V/√Hz) V 2(f) n Vn(f) 100 10 µV2/Hz µV/√Hz 10 Spectral density 3.16 Root spectral density 1 1 f f ENE 5400 ΆýɋÕģŰ, Spring 2004 4 µóģµóģvʶ1Zýɋvʶ1Zýɋ 2 Filtered Noise Root-mean-squared voltage can also be obtained in the frequency domain using ∞ 2 = 2 Vn(rms) ∫ Vn ( f )df 0 Total mean-squared value from a filter output is: ∞ V 2 = ∫ A( j2πf ) 2V 2 ( f )df no(rms) 0 n You should avoid designing circuits with a larger bandwidth than your signal actually requires 2 2 π 2 2 Vn (f) Vno (f) = |A(j2 f)| Vn (f) A(s) Input noise output noise filter ENE 5400 ΆýɋÕģŰ, Spring 2004 5 µóģµóģvʶ1Zýɋvʶ1Zýɋ Example: Filtered Noise |A(j2πf)|, dB Vn(f) π A(s) = 1/(1 + s/2 fo) √ 0 nV/ Hz 20 3 fo = 10 f (Hz) f (Hz) 110 100 103 104 110 100 103 104 2 = Vno( rms ) ENE 5400 ΆýɋÕģŰ, Spring 2004 6 µóģµóģvʶ1Zýɋvʶ1Zýɋ 3 Noise Bandwidth |A (j2πf)| |A(j2πf)| π brick A(s) = 1/(1 + s/2 fo) 0 0 = f (Hz) f (Hz) fo fo fx π Noise bandwidth fx = fo/2 ENE 5400 ΆýɋÕģŰ, Spring 2004 7 µóģµóģvʶ1Zýɋvʶ1Zýɋ Minimum Detectable Signal Usually expressed in VOLTS Min. detectable signal = root spectral density × BW / sensitivity For example: g V/√Hz √Hz V/g ENE 5400 ΆýɋÕģŰ, Spring 2004 8 µóģµóģvʶ1Zýɋvʶ1Zýɋ 4 Thermomechanical (Brownian) Noise For mechanical sensors, thermomechanical noise due to air damping is the ultimate performance limit A dimensionless parameter, namely, the quality factor, is commonly used to describe the damping and also the Brownian noise ENE 5400 ΆýɋÕģŰ, Spring 2004 9 µóģµóģvʶ1Zýɋvʶ1Zýɋ First: Understand 2nd-order Lumped-Parameter System 2 2 ξω ω 2 Transfer function x(s)/F(s) = 1/(ms + bs + k) = 1/[m(s + 2 ns + n )] ξ ω Damping ratio = b / (2m n) » Underdamped (ξ < 1, overshoot), critically damped (ξ = 1), and overdamped (ξ > 1) ω 1/2 Natural frequency n = (k/m) ω ξ2 1/2ω Damped natural frequency d = (1 - ) n Force F(s) Displacement x(s) k b 1 x F ms2 + bs + k M ENE 5400 ΆýɋÕģŰ, Spring 2004 10 µóģµóģvʶ1Zýɋvʶ1Zýɋ 5 Cont’d: Transient-Response Characteristics ⇒ ω Reduce rise time tr have to increase n ⇒ ξ Reduce maximum overshoot Mp have to increase ∝ ξξωω Settling time ts 1/( n) Mp Displacement z(t) Displacement t r tr t ENE 5400 ΆýɋÕģŰ, Spring 2004 11 µóģµóģvʶ1Zýɋvʶ1Zýɋ Second: Definition of Quality Factor (Q) The most fundamental definition of Q is that it is proportional to the ratio of energy stored to the energy lost, per unit time: Q ≡ (energy stored) / (average energy dissipated per radian); dimensionless Q is related to the damping ratio: Q = 1 / (2ξ) Q can be measured using the half-power points on frequency spectrum: ω ω ω Q = n / ( h2 – h1) Note that the amplitude is amplified by Q times at resonance Q⋅A QA/√2 A amplitude ω ω ω ω ENE 5400 ΆýɋÕģŰ, Spring 2004 12 h1 h2 µóģµóģvʶ1Zýɋvʶ1Zýɋ 6 Derivation: Q = 1/2ξ Take the 2nd-order mechanical system for example: mx&& + bx& + kx = f (t) The rate of change of energy with time: (the minus sign indicate energy dissipation) dW = force×velocity = dt ⋅⋅⋅ ω Assume a simple harmonic motion of x(t) = X sin( dt), the dissipated energy per cycle is: 2π / ω ∆W = ∫ d Fx&dt = πbω X 2 t =0 d ω = ω − ξ 2 d n 1 ENE 5400 ΆýɋÕģŰ, Spring 2004 13 µóģµóģvʶ1Zýɋvʶ1Zýɋ Cont’d The stored total energy of the system: 1 1 1 W = kX 2 = mv2 = mX 2ω2 12 424 2434max 2 d for ξ<<1 The quality factor : Q = ENE 5400 ΆýɋÕģŰ, Spring 2004 14 µóģµóģvʶ1Zýɋvʶ1Zýɋ 7 Verify Q and Half-Power Bandwidth The amplitude at resonant is amplified by Q times of the amplitude (Xo) at d.c. ω Q ≈ n ω − ω h2 h1 ) Q o Q/√2 ∆ω Amplitude (X/X Amplitude ω ω ω h1 n h2 ω ENE 5400 ΆýɋÕģŰ, Spring 2004 15 µóģµóģvʶ1Zýɋvʶ1Zýɋ Derivations Standard 2nd-order system: ω 2 1 G(s) = n = s 2 + 2ξω s + ω 2 ω 2 ω n n 1− + j2ξ ω 2 ω n n ω ω ω ⋅⋅⋅ω At = n, the resonant amplitude is Q; at h = r n, the amplitude is Q/√2. Therefore: = Q = 1 G(s) = ω s j h 2 2 (()1− r 2 ) + (()2ξr )2 Also, we already know that: 1 Q = 2ξ ENE 5400 ΆýɋÕģŰ, Spring 2004 16 µóģµóģvʶ1Zýɋvʶ1Zýɋ 8 Cont’d Therefore we get: r 4 − r 2 (2 − 4ξ 2 ) + (1− 8ξ 2 ) = 0 2 = − ξ 2 − ξ + ξ 2 ≈ − ξ r1 1 2 2 1 1 2 2 = − ξ 2 + ξ + ξ 2 ≈ + ξ r2 1 2 2 1 1 2 ω 2 − ω 2 = 2ω 2 − 2ω 2 = ξω 2 h2 h1 r2 n r1 n 4 n Qω + ω = ω h2 h1 2 n ω 1 ∴ω − ω = 2ξω , n = = Q h2 h1 n ω − ω ξ h2 h1 2 ENE 5400 ΆýɋÕģŰ, Spring 2004 17 µóģµóģvʶ1Zýɋvʶ1Zýɋ Example: A Parallel RLC Tank Circuit + R C L V Iin out _ The impedance is zero at DC and at infinitely high frequency: Z = ω 0.5 At the resonant frequency n= 1/(LC) , the reactive terms are cancelled; Z = R Peak energy stored in the network at resonance is: = Etot ENE 5400 ΆýɋÕģŰ, Spring 2004 18 µóģµóģvʶ1Zýɋvʶ1Zýɋ 9 Cont’d The average power dissipated in the resistor at resonance: 1 P = I 2 R ave 2 pk The quality factor Q is: E R = ω tot = Q n Pave L /C Q = ω RC n Large R is good in parallel network How does the bandwidth (here it means ∆ω at the half-power ω ω ∆ω bandwidth) relate to Q; Let = n + , the -3 dB point, then 1 1 Z = ≈ = R / 2 −3dB 1 j 1 + [2∆ωω + ∆ω 2 ]LC + j2C∆ω ω n R nL R ENE 5400 ΆýɋÕģŰ, Spring 2004 19 µóģµóģvʶ1Zýɋvʶ1Zýɋ Cont’d Therefore we get ∆ω = 1 / (2RC) ω ω ⋅⋅⋅∆ω⋅∆ω The bandwidth h2 – h1 = 2 = 1 / RC We can verify that: ω 1/ LC R n = = = Q ω − ω h2 h1 1/ RC L / C ENE 5400 ΆýɋÕģŰ, Spring 2004 20 µóģµóģvʶ1Zýɋvʶ1Zýɋ 10 Air-Damping Analysis ENE 5400 ΆýɋÕģŰ, Spring 2004 21 µóģµóģvʶ1Zýɋvʶ1Zýɋ Couette Damping Motion-induced shear stress τ sets up a velocity gradient in the fluid between two plates Viscosity is the proportionality constant that relates τ = η⋅η⋅∂∂v/∂x; η µ ⋅⋅⋅ 2 ° ( air = 77 N s/m @ 25 C) Incompressible laminar flow; density change is negligible Linear velocity profile is valid for small gaps ∂v V τ = η x = η x Shear stress acting on the plate: B ∂y g ηA Total force acting on the plate: F = V B g {x { dx Bx dt Vx Movable plate y Linear velocity profile g x ENE 5400 ΆýɋÕģŰ, Spring 2004 22 µóģµóģvʶ1Zýɋvʶ1Zýɋ 11 Squeeze-Film Damping V g z FB Damping caused by squeezing air in and out of plates The distribution of gas pressure between plates is governed by the compressible gas-film Reynolds equation; assuming a small plate motion, the linearized Reynolds equation is: Pressure change P g 2 ∆P ∂ ∆P ∂ z Atmospheric pressure a ∇2( )− ( ) = ( ) η ∂ ∂ 12 eff Pa t Pa t g η Effective gas viscosity: η = η µ ⋅ 2 ° eff + 1.159 ( = 22.6 N s/m @1atm and 25 C) 1 9.638Kn λ Mean free path of air (70 nm @1atm and 25 °C) Knudsen number: K = n g ENE 5400 ΆýɋÕģŰ, Spring 2004 23 µóģµóģvʶ1Zýɋvʶ1Zýɋ Cont’d The equation can be solved by the eigenfunction method, assuming that ∆p(x,t) = ∆p(x)e-αt in an one-dimensional problem, followed by separation of variables 1 ()= ∆ ( ) total force F t Pa wl∫ P x,t dx 0 Squeeze-film damping exhibits damper (low-frequency) and spring-like (high-frequency) behaviors The solution can be simulated by an equivalent circuit model Lmn and Rmn depend on the plate size (w and l), gap, and effective viscosity (m,n are odd numbers) force π 4 (g − z) + = 2 Lmn (mn) L11 L13 L31 Lmn 64lwPa π 6 (g − z)3 velocity R = (mn)2 (m2l 2 + n2w 2 ) mn 768(lw )3η R11 R13 R31 Rmn eff _ Ref: J.
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