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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/
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