Tips & Tricks for Experimentalists 1

Noise and Servo Loops

An introduction to the description and control of dynamic systems

Dr. Uwe SterrPhysikalisch-Technische Bundesanstalt (PTB)AG 4.31: Unit of LengthBundesallee 10038116 BraunschweigPaschenbau Room 118aTel: 0531 592 4310uwe.sterr@ptb.de

MenueNoise• description• noise types• dynamic systems

Break• hands-on experience with spectrum analyzers

Feedback Control• feedback• stability• examples

• hands-on experience with spectrum analyzers

References, Books

Noise

• E. Rubiola, V. Giordano, K. Volyanskiy, and L. Larger, Phase and frequency noise metrology, arXiv:0812.0180, (2008)

• F. Riehle, Frequency Standards: Basics and Applications, Wiley-VCH 2004• Agilent/HP Technical Notes :

http://www.hpmemory.org/news/an150http://www.home.agilent.comhttp://www.home.agilent.com

Feedback Control• J. Bechhoefer, Feedback for Physicists: A tutorial essay on control,

Rev. Mod. Phys. 77, 783-836 (2005) • U. Tietze, Ch. Schenk, Halbleiter-Schaltungstechnik / Electronic Circuits• LTSpice, Simulation Software, http://www.linear.com/designtools/software/

Description of Noise

)(tV

t

System

Distinguish:• Noise: Random Perturbations

not easily avoidable• Interference, Pickup, Oscillations:

Avoidable –first try to suppressuse different methods

)(tV

Description of Noise

)(tVRandom variable

)(VPDistribution functionvery often Gaussian

Mean Square deviation

)(tV

)(tV

t

∫−

−=2/

2/

22 ))()((1

T

T

rms dttVtVT

V

Mean Square deviation

22 )()( tVtV −=

t

V

P(V)

<V>RMS: Root Mean Square Value

∫−

−2/

2/

2))()((1

T

T

rms dttVtVT

V

Description in Time Domain

Fluctuating Value V(t), e.g. Frequency, Temperature, Position ...

Vuncorrellated random values„white noise“

t

„random walk“

Description in the Time Domain

Autocorrelation

∫−

τ+⋅=τ2/

2/

0 )()(1

),(T

T

dttVtVT

tC

e.g. uncorrellated „white noise“ )()( τδτ rmsVC =

Spectral Analysis

Intuition of a “Spectrum” S(f): Spectral dispersion Measure average power behind filter

http://www.pinkfloyd.com/music/albums.php

∫=2

1

)(f

f

dffSP

Spectral Analysis

Idea: Spectral Filtering Measure average power after each filter

http://de.wikipedia.org/wiki/Zungenfrequenzmesser

Spectral analysis and Fourier transform

∫−

−=2/

2/

2)(1

)(~ T

T

iftiT dtetV

TfV πFourier transformation over

time interval T

∫−

=2/

2/

22 )(1 T

T

dttVT

Vtotal power: ∫∞

∞−

= dffVT

V T |)(~

|1 22Parseval’s theorem

Interpretation: dffVT |)(~

| 2 describes power in frequency range f ... f+df

V(t) is a real valued signal, thus: )(~

)(~ * fVfV −=

usually the single sided Spectral Power Density is used:T

fVfS T

V

2|)(~

|2)( =

i.e. normalized ∫∞

=0

2 )( dffSV Vrms

Units: Power related quantities

Power in Watt: • optical by direct power measurement• electrical from voltage current: P = U2/R = I2R

needs to specify resistance – often R = 50 Ω

Power related units – squared quantities (temporal average: rms)voltage: Su in V2/Hzu

current: Si in A2/Hzfrequency: Sf in Hz2/Hzphase: Sφ in rad2/Hz

Logarithmic units: decibel: 10·log10(P/Pref)dBm : Pref = 1 mW (RF, typically at 50 Ω)dBmV : “Pref“ = (1 mV)2

HzHz/,HzV/,HzA/=VS

Inside a real RF Spectrum Analyzer

Typically the power in the resolution bandwidth is displayed: “dBm”

To get spectral power density divide by bandwidth: “dBm/Hz”

Old Device – Full Analog Settings

Wiener-Khintchin Theorem

∫∞ τπ=τ

0

2)()( dfefSC ifV∫

∞ τπ− ττ=0

2)()( deCfS ifV

Spectral Power Density and Autocorrelation are Fourier-Transform Pairs

C( )τExample:

Exponential Correlation

2

0 2/1

1)(

iffSV π+τ

=

|/| 0)( ττ−=τ eC

20

22 /14

1)(

τ+π=

ffSV

Lorentzian

C( )τ

τ0

S(f)

f0

Example:

Linear Systems - Filter

Physical Restrictions on the System :linearitycausalitytime invariance ∫

∞

−= )()()( τττ datVtV

SystemVin

Vout

time invariancefinite energyreal-valued

Time Domain – Frequency Domainconvolution -> multiplication

with with real function a(t) analytic function

∫ −=0

)()()( τττ datVtV inout

∫∞

∞−

= dtetafA iftπ2)()(

Phase and amplitude response of are not independent: Kramers-Kronig relation

)( fA

)( fA

Linear Systems – Low Pass Filter

Bode Diagram:

cffiRCfifA

+=

+=

1

1

21

1)(

π

RCfc π2

1=cutoff frequency:

F. Riehle, Frequency Standards

cff

cff

Kramers-Kronig relation:gain 1/f leads to -90° phase shift

Shot Noise

Independent events at random times ti , e.g. electrons from thermal emissionphoto effect,photons from a laser

∑=

−=TN

i

ittgtV0

)()(tti

V(t)

=i 0

Spectral Power Density2

)(~2)( fg

T

NfSV =

current shot noise eIfSi 2)( =

photon shot noise ν= hPfSP 2)(

ln the limit of δ-pulses: constant SV(f) - “white noise”

tti

Poisson statistic: Average N, fluctuations N

Thermal Noise

Resistor Noise:

white noise of:

)( kThf <<

R

In a resistor, the electrons are not independent.Thus there is no shot noise, butwhite noise from thermal fluctuations:

white noise of:

voltage U

current I

noise power

kTRfSU 4)( =

R

kTfSI

4)( =

dBm/Hz174)( −== kTfSP

Resistor Noise vs. Resistance

10

100

1000

10000

77 K (nV

/Hz1/

2 )

300 K

kTRfSU 4)( =

Low voltage noise circuits need low value resistors!50 Ω @ 300 K: 0.91 nV/Hz1/2

100 101 102 103 104 105 106 107

0.01

0.1

1

77 K

SU

1/2 (

nV/H

z

R (Ω)

Thermal Noise vs. Shot Noise

R

I

U

0.1

1

10

100

R = 1 MΩ

shot noise

Si (

pA/H

z1/2 )

R = 1 kΩ

When a resistor is used to measure a current with shot noise, thermal and shot noise add:

total noise: 224)( ReIkTRfSU ⋅+=

nUIRe

kT ==2

mV502 =

e

kT

both contributions are equal:

at room temperature (T=300 K):

10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

0.01

0.1

I (A)

Amplifier Noise

In an amplifier, its voltage and current noise add, depending on the source resistance:

Example: bipolar ultra low noise OpAmp LM1028LM1028

Watch out for 1/f noise !often it dominates below 1 kHz

Amplifier Noise

Choice of best amplifier depends on source resistance

Amplifier Noise

Noise Figure F:outout

inin

NS

NSF

/

/=

input output

Applicable to RF and optical amplifiers.:

• input noise: thermal noise of system input impedance (e.g. 50 Ω)• output noise: amplified input signal & noise plus internal noise

in RF domain: noise figure < 1 dB for optimized, narrow band amplifiers

Quantum Noise

Noise Figure F:

outout

inin

NS

NSF

/

/=

In optical amplifiers e.g. Erbium Doped Amplifier – Amplification by stimulated emission

Input shot noise – Signal + Quantum Fluctuations

For phase insensitive amplification the uncertainty relations need to be fulfilled

Necessary addition of noise (spontaneous emission)

Minimum noise figure (at high gain) Factor 2, i.e. 3 dB

Brillouin amplifiers have much higher noise factor because of thermally excited modes.

H. A. Haus and J. A. MullenQuantum Noise in Linear Amplifiers,Phys. Rev. 128, 2407-2413 (1962)

EDFA Quantum Noise

noise figure and (b) amplifier gain as a function of the length for several pumping levels.

K. KikuchiGeneralised formula for optical-amplifier noise and its application to erbium-doped fibre amplifiers,Electron. Lett. 26, 1851-1853 (1990)

Feedback Control

Feedback control:

• Measure deviations from set-point

• Act back on system

• repeat ...• repeat ...

steam engine controller

system A(f)

controller G(f) -

-disturbance

correction

output Y(f)

error signal reference

Stability Conditions

system A(f)

controller G(f) -

-disturbance

correction

output Y(f)

error signal reference

D(f)

Example:

Temperature controlsystem ~ low pass

Different Units: • output/error: Temperature• correction: Heater Voltage

Transfer functions:

System: Kelvin/Volt

Controller: V/Kelvin

Loop (output – output): K/K

Error Supression

system A(f)

controller G(f) -

-disturbance

correction

output Y(f)

error signal reference

D(f)

open loop Gain: A(f)·G(f)

without servo / open loop: output error Y0(f) = A(f )·D(f)

closed loop: suppression of disturbances: Y(f) = A(f )·(D(f) – G(f)·Y(f))

Y(f) = A(f )·D(f)/(1+G(f))

suppression Y(f) / Y0(f) = 1/(1+A(f) ·G(f))

Optimum: Make controller Gain as large as possible!

Loop Gain – Stability Conditions

system A(f)

controller G(f) -

-

disturbance

correction

output Y(f)

error signal reference

Optimum: Make controller Gain as large as possible!

Limit: because of phase shifts, the system eventually will oscillate

Stability conditions: • Loop gain has to circle point -1 in the complex plane (Nyquist criterion)• Phase at unity gain frequency < 180°

K. J. Aström and R. M. MurrayFeedback Systems, An Introduction for Scientists and Engineers,Princeton Univeristy Press, (2011) online at http://www.cds.caltech.edu/~murray/amwiki/index.php/Main_Page

Stability Conditions – P controller

P-Controller

system + controllerloop gain

system

phase margin

system + controllersystem and

Stability Conditions – P controller

Proportional-controller:

• rather robust• fist approach

• remaining DC-error ~ 1/loop gain

Phase margins:90°

60°

45°

Systerm response

~ 1/loop gain 45°

From first test of servo with pure P-controller:increase gain until oscillations starts • critical gain• critical frequencyhelps to estimate parameters for optimized loop filter

PI-Controller

system + controllersystem

Stability Conditions – PI controller

phase margin

system + controllerloop gain

system + controller

system

PI controller

optimize integral part:

proportional controller

optimizedPI-controller

Error Signal

PI-controller

integral part will remove remaining error for constant conditons

top: integral part too slowslow approach towards zero error

bottom:integral part too fastringing

PID-Controller

system + controllersystem

Stability Conditions – PID controller

phase margin

system + controller

system

system

PID controller

output:

• top: PI controller• bottom: PID controller

differential part:

• can compensate low- pass behaviour of the system

• allow larger bandwidth• improves phase margin

• noise issues• gain has to be limited

PID controller – more flexible

Servo design for an ultrastable laserlo

op g

ain

(dB

)

40 dB/decade

dB/decade

“multiple integrators ”

• high gain at low frequencies, where the perturbations are largest

• leads to phase shift ~ 270° at lower frequencies

• no problem for stability, as long as phase margin at unity gain (~ 3 MHz) OK

H. Stoehr, PhD Thesis (2004) frequency (Hz)

loop

gai

n (d

B)

dB/decade

dB/decade

dB/decade

dB/decade

OK

• poor transient behaviour – to lock, first use fewer and gain-limited integrators

Laser frequency stabilization

“Servo Bump ”

noise increasesaround unit-gain frequency

noise will further increase

out-of loop error spectrum

H. Stoehr, F. Mensing, J. Helmcke and U. Sterr, Diode Laser with 1 Hz Linewidth,Opt. Lett. 31, 736-738 (2006)

noise will further increase and finally system oscillates there with increasing gain

Servo loop for an ultrastable laser

“Simulation Tool ”

sophisticated tools for frequency, noise and time-domain analysis are freely available,

e.g. LTSpice, PSpice

Temperature controller

Temperature controller

Temperature Sensor interfaceprovides 100 mV/°CAD590

sensor1 µA/K

Temperature controller

Temperature set pointprovides 100 mV/°C

set point knob

Temperature controllerPI controllertwo OP amps

Temperature controllerPower amplifier for thermo-electric elements

from http://xkcd.com/730/