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 ΆŠýɋÕģŰ, Spring 2004 1 µóģµóģvʶ1ZýŠɋvʶ1ZýŠɋ

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

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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 ΆŠýɋÕģŰ, Spring 2004 3 µóģµóģvʶ1ZýŠɋvʶ1ZýŠɋ

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 ΆŠýɋÕģŰ, Spring 2004 4 µóģµóģvʶ1ZýŠɋvʶ1ZýŠɋ

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

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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 )

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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

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Minimum Detectable Signal

Usually expressed in VOLTS
Min. detectable signal = root spectral density × BW / sensitivity

For example: g V/√Hz √Hz V/g

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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

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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

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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

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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 ΆŠýɋÕģŰ, Spring 2004 12 h1 h2 µóģµóģvʶ1ZýŠɋvʶ1ZýŠɋ

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

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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 =

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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 ΆŠýɋÕģŰ, Spring 2004 15 µóģµóģvʶ1ZýŠɋvʶ1ZýŠɋ

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

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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

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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

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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

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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

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10 Air-Damping Analysis

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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

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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

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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. J. Blech, “On isothermal squeeze films,” J. Lubrication Tech., vol. 105, pp. 615-620, 1983. ENE 5400 ΆŠýɋÕģŰ, Spring 2004 24 µóģµóģvʶ1ZýŠɋvʶ1ZýŠɋ

12 Derivation of Brownian Noise

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Brownian Motion

The zigzag motion performed by suspended particles in a liquid or a gas; named after the English botanist Robert Brown
Fluctuations from thermodynamic equilibrium
Derived by Einstein and Smoluchowski (using different methods from kinetic theory) in 1906
Experimentally verified by Perrin, Svedberg, Seddig, and others
A simpler derivation can follow Langevin’s method

Zigzag particle motion

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13 Thermomechanical Noise

A good reference can be found in T.B. Gabrielson, “Mechanical- thermal noise in micromachined acoustic and vibration sensors,” IEEE Trans. on Electron Devices, ED-40(5), p. 903-909, 1993
Brownian noise force:

∆ 2 = Fn

× -23 kb: Boltzmann’s constant (1.38 10 J/K) T : temperature in Kelvin b : damping coefficient (unit = N/(m/s)) ∆f: measuring bandwidth (unit = Hz)

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Here is the Derivation

A particle of mass M with arbitrary acceleration: && = − & + () mr br {Ftotal 1 Friction force Other influences combined Multiply by r:

Equation (2) becomes: 1 d 2r 2 1 dr2 m + b = r ⋅ F + mv2 2 dt2 2 dt total

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14 Cont’d

Using the Equipartition theorem: 1 2 1 2 1 mv x = mv y = k T 2 2 2 b = × −23 kb 1.3803 10 J / K(Boltzmann cons tant) 1 2 1 2 1 2 mv = mv x + mv y = k T 2 2 2 b

Therefore:

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Cont’d

By integration from t = 0 to t = ∆t to obtain the traveled distance:

4kbT 4k T ∆r 2 = r 2 − r 2 = b ∆t = b {o b ∆f =0

4kbT ∴∆r = b ∆f

To get the Brownian force, we first get the noise velocity:

∆r 2 4k T / b ∆v = = b ∆t ∆t ∆ = ⋅⋅⋅ ∆ = ∆ then Fn b v 4kbTb f

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15 Noise Models for Circuit Elements

Thermal noise is due to thermal excitation of charge carriers in a conductor; occurs in all resistors above absolute zero temperature
Was first observed by J.B. Johnson (Johnson noise) and analyzed by H. Nyquist in 1928
a white noise
Shot noise occurs in pn junctions because the dc bias current is not continuous and smooth due to pulsed current caused by individual flow of carriers
Was first studied by W. Schottky in 1918
Depend on bias current; also a white noise
Flicker noise (1/f noise) arises due to release of trapped charges in semiconductors
A significant noise source in MOS transistors

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Equivalent Resistor Model with Added Noise

Spectral density functions:

2 = VR ( f ) 2 = I R ( f )

R (noiseless) R (noisy) = R (noiseless) 2 = IR (f) V 2(f) * R

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16 Forward-Biased Diodes

Spectral density functions: 2 = Id ( f ) 2 = Vd ( f )
Small-signal resistance of the diode rd does not contribute any thermal noise: k T = VD /(kbT / q) ⇒ = b ID Ise rd qId

rd (noiseless) diode (noisy) = r (noiseless) 2 = d Id (f) V 2(f) * d

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MOS Transistors in the Active Region

Spectral density function is the sum of the flicker (1/f) noise and thermal noise
1/f noise voltage source and thermal noise voltage source 2 = V f ( f ) 2 = Vt ( f )
The equivalent voltage source of 1/f and thermal noise sources 2 = 2 ()+ 2 () Vi ( f ) V f f Vt f

= *2 Vi (f) noisy ENE 5400 ΆŠýɋÕģŰ, Spring 2004 34 µóģµóģvʶ1ZýŠɋvʶ1ZýŠɋ

17 Operational Amplifiers (Op-amps)

Noise is modeled using three uncorrelated input-referred noise 2 2 2 sources: Vn (f), In+ (f), and In- (f)
Op-amps with MOSFET input transistors can ignore current noises

2 In- (f) _ Noiseless op-amp * + 2 Vn (f) 2 In+ (f)

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Capacitors and Inductors

Capacitors and inductors do not generate any noises; however they accumulate noise generated by other noise sources
The rms noise voltage Vno across a capacitor is equal to 1/2 (kbT/C) , regardless of the resistance seen across it (why?)

2 = Vno()rms

Vno

R (noiseless) C 2 VR (f)=4kbTR *

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18 Cont’d

1/2 The rms noise current Ino across inductor is equal to (kbT/L) , regardless of the resistance seen across it

Ino 2 IR R (noiseless) L

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Sampled Signal Noise

The sample-and-hold circuit for A/D or D/A conversion, when turned off, the noise as well as the desired signal is held on the capacitor C
Over-sampling of the signal helps to reduce the noise level
Signal values add linearly, whereas their noise values add as the 2 2 2 0.5 root of the sum of squares (i.e., neq = (n1 + n2 + n3 + …) )

φ clk

Vin Vout

C Ron

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19 Op-amp Example

Cf Cf

In1 Inf Rf R1 In- V _ R i 1 R V _ f + o Vno(f) R + 2 * * R V V 2 n2 n In+

Low-pass filter Equivalent noise model 2 R V 2 ( f ) = [I 2 ( f ) + I 2 ( f ) + I 2 ( f )] f no1 n1 nf n− + π 1 j2 fRf C f 2 R / R V 2 ( f ) = [I 2 ( f )R2 +V 2 ( f ) +V 2 ( f )]1+ f 1 no2 n+ 2 n2 n + π 1 j2 fC f R f 2 = 2 + 2 Vno ( f ) Vno1( f ) Vno2 ( f ) ENE 5400 ΆŠýɋÕģŰ, Spring 2004 39 µóģµóģvʶ1ZýŠɋvʶ1ZýŠɋ

Example: CMOS Differential Input Pair

Goal: to obtain an equivalent input-referred voltage noise by summing up all the contributed noises at the output, then dividing by the gain of the circuit V DD 16 k T 16 g 1 Vn5 2 = b + m3 2 Vneq ( f ) ( ) ( )kbT ( ) ( ) Q 3 g 3 g g * 5 m1 m1 m3

Vin+ V Noise output due to various sources: Q Q in- * 1 2 * V V Vno,1 Vno,2 n1 n2 = = g R V m1 o no Vn1 Vn2

Q3 Q4 V V * * no,3 = no,4 = Vn3 Vn4 gm3Ro Vn3 Vn4 VSS

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20 Cont’d

Due to symmetry, the drain of Q2 will track that of Q1:

Vno,5 gm5 = (can be neglected) Vn5 2gm3 Output noise value: 2 = 2 2 + 2 2 Vno( f ) 2(gm1Ro ) Vn1( f ) 2(gm3Ro ) Vn3( f )

Output value can be related back to an equivalent input noise value

by dividing it by the gain, gm1Ro 2 = 2 + 2 2 Vneq ( f ) 2Vn1( f ) 2Vn3 ( f )(gm3 / gm1)

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Cont’d

Thermal noise (1/f noise neglected) of each transistor can be substituted: 2 = 2 1 Vni ( f ) 4kbT( )( ) 3 gmi W g = 2µ C ( ) I mi i ox L i Di

Can increase gm1 (more current) to minimize thermal noise contribution. Remember the price you pay for noise reduction is POWER!!

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21