ponderomotive instabilities and microphonics

Thomas Jefferson National Accelerator FacilityOperated by the Southeastern Universities Research Association for the U.S. Department of Energy

PONDEROMOTIVE INSTABILITIES AND MICROPHONICS

A TUTORIAL

Jean Delayen

Thomas Jefferson National Accelerator Facility

SRF 2005 Cornell 10-15 July 2005

Operated by the Southeastern Universities Research Association for the U.S. Department of Energy


Some Definitions

• Ponderomotive effects: changes in frequency caused by the electromagnetic field (radiation pressure)—Static Lorentz detuning (cw operation)—Dynamic Lorentz detuning (pulsed operation)

• Microphonics: changes in frequency caused by connections to the external world—Vibrations—Pressure fluctuations

Note: The two are not completely independent.When phase and amplitude feedbacks are active, ponderomotive effects can change the response to external disturbances



Some History• M. M. Karliner, V. E. Shapiro, I. A. Shekhtman

Instability in the Walls of a Cavity due to Ponderomotive Forces of the Electromagnetic FieldSoviet Physics – Technical Physics, Vol. 11, No. 11, May 1967

• V. E. ShapiroPonderomotive Effects of Electromagnetic RadiationSoviet Physics JETP, Vol. 28, No. 2, February 1969

• M. M. Karliner, V. M. Petrov, I. A. ShekhtmanVibration of the Walls of a Cavity Resonator under Ponderomotive Forces in the Presence of FeedbackSoviet Physics – Technical Physics, Vol. 14, No. 8, February 1970

Stability conditions derived by comparing the rate of transfer of energy from the electromagnetic mode to the mechanical mode with the dissipation rate of the mechanical mode.Analysis valid when decay time of the electromagnetic mode much less than period of mechanical mode (τ Ωµ << 1), or when the rate of transfer of energy is very high



Some More History

• D. SchulzeMechanical Instabilities of a Superconducting Helical Structure due to Radiation PressureProc. 1970 Proton Linac Conference, Batavia

Ponderomotorishe Stabilität von Hochfrequenzresonatoren und ResonatorregelungssystemenDissertation, KFK-1493, December 1971Ponderomotive Stability of R.F. Resonators and Resonator Control Systems (ANL-TRANS-944, November 1972)

Analysis of stability of generator-driven superconducting resonator (arbitrary τ Ωµ ) using control systems methods (Laplace transforms, transfer functions) with and without phase and amplitude feedbackAnalysis of damping of mechanical vibrations by ponderomotive forces



Some More History

• J. R. DelayenPhase and Amplitude Stabilization of Superconducting ResonatorsDissertation, Caltech, 1978

Analysis of ponderomotive instabilities of resonators operated in self-excited loops, with and without feedback, using control systems methods.

Included I-Q detection

Used methods of stochastic analysis to quantify performance of feedback system in the presence of microphonics and ponderomotive effects

Later extended to include effects due to presence of beam loading and electronic damping.



Ponderomotive Effects

• Adiabatic theorem applied to harmonic oscillators (Boltzmann-Ehrenfest)

2

/

2 20 0

1If = 1, then is an adiabatic invariant to all orders

( ) (Slater)

Quantum mechanical picture: the number of photons is constant:

( ) ( )4 4

d

VV

d Udt

UU U o eU

U N

U d H r E r

ε

ωεω ω

ωω ω ω

ω

µ ε

∆ ∆∆ -Ê ˆ Ê ˆ fi =Ë ¯ Ë ¯

=

È= +ÎÚ

∼

2 20 0

(energy content)

( ) ( ) ( ) ( ) (work done by radiation pressure)4 4S

U dS n r r H r E rµ εξ∆

˘Í ˙

È ˘= - ◊ -Í ˙Î ˚Ú




2 20 0

2 20 0

( ) ( ) ( ) ( )4 4

( ) ( )4 4

Expand wall displacements and forces in normal modes of vibration ( ) of the resonator

( ) ( )

( ) (

S

V

vS

V

dS n r r H r E r

d H r E r

r

dS r r

r q r

µ

µ µν

µ µµ

µ εξωµ εω

φ

φ φ δ

ξ φ

∆È ˘◊ -Í ˙Î ˚= -

È ˘+Í ˙Î ˚

=

=

Ú

Ú

Ú

Â ) ( ) ( )

( ) ( ) ( ) ( )

S

S

q r r dS

F r F r F F r r dS

µ µ

µ µ µ µµ

ξ φ

φ φ

=

= =

Ú

Â Ú




2

2

2

Equation of motion of mechanical mode

(Euler-Lagrange)

1 (elastic potential energy) : elastic constant2

1= (kinetic energy) 2

d L L F L T Udt q q q

U c q c

qT c

µµ µ µ

µ µ µµ

µµ

µ µ

µ

Φ

ΩΩ

∂ ∂ ∂- + = = -∂ ∂ ∂

= Â

Â2

2

22

: frequency

= (power loss) : decay time

2

c q

q q q Fc

µ

µ µµ

µ µ µ

µµ µ µ µ µ

µ µ

ττ

τ

ΦΩ

ΩΩ+ + =

Â




22 2 2 20

0 0

The frequency shift caused by the mechanical mode is proportional to

2

Total frequency shift: ( ) ( )

Static frequency shift:

q

FU k V

c U

t t

µ µ

µµ µ µ µ µ µ µ

µ µ

µµ

µµ

ω µ

ωω ω ωτ

ω ω

ω ω

∆

∆ ∆ Ω ∆ Ω Ω

∆ ∆

∆ ∆

Ê ˆ+ + = - = -Á ˜Ë ¯

=

=

Â2

Static Lorentz coefficient:

V k

k k

µµ

µµ

= -

=

Â Â

Â



Ponderomotive Effects – Mechanical Modes

Fluctuations around steady state:0 (1 )

o

V V vµ µ µω ω δω

δD = D +

= +

Linearized equation of motion for mechanical mode:

2 2 22 2 ok V vµ µ µ µ µ µµ

δω δω δω δτ

+ +W = - W

The mechanical mode is driven by fluctuations in the electromagnetic mode amplitude.

Variations in the mechanical mode amplitude causes a variation of the electromagnetic mode frequency, which can cause a variation of its amplitude.

⇒ Closed feedback system between electromagnetic and mechanical modes, that can lead to instabilities.

( )2 2 22ok V n tµ µ µ µ µ µ

µ

ω ω ωτ

∆ ΩD + D +W = - +



Lorentz Transfer Function

TEM-class cavities ANL, single-spoke, 354 MHz, β=0.4

simple spectrum with few modes

( )

2 2 2

2 2

2 2

2 2

2( ) ( )2

o

o

k V v

k Vv

i

µ µ µ µ µ µµ

µ µµ

µµ

δω δω δω δτ

δω ω δ ωω ω

τ

+ +W = - W

- W=

W - +



Lorentz Transfer Function

SNS Med β Cryomodule 3, Cavity Position 1, Lorentz Transfer Function (5MV/m CW)

25

30

35

40

45

50

55

60

65

70

0 60 120 180 240 300 360 420

AM Drive Frequency (Hz)

Cav

ity D

etun

ing,

Res

pons

e R

atio

(dB

)

0

45

90

135

180

225

270

315

360

Rel

ativ

e Ph

ase

(deg

)

TM-class cavities (Jlab, 6-cell elliptical, 805 MHz, β=0.61)Rich frequency spectrum from low to high frequenciesLarge variations between cavities



GDR and SEL



Generator-Driven Resonator

• In a generator-driven resonator the coupling between the electromagnetic and mechanical modes can lead to two ponderomotive instabilities

• Monotonic instability : Jump phenomenon where the amplitudes of the electromagnetic and mechanical modes increase or decrease exponentially until limited by non-linear effects

• Oscillatory instability : The amplitudes of both modes oscillate and increase at an exponential rate until limited by non-linear effects



Generator-Driven Resonator

2 12oy k Vµ τ

- <

• Oscillatory:( )

( )

22 22

2 2

112

where : normalized detuning

o

g co

y k V

y

µµ

µ µ

ττ τ

τ ω ω

+ W<

W

= -

Approximate stability criteria in the absence of feedback:

The monotonic instability can occur on the low frequency side when the Lorentz detuning is of the order of an electromagnetic bandwidth.The oscillatory instability can occur on the high frequency side when the Lorentz detuning is of the order of a mechanical bandwidth.

Amplitude feedback can stabilize system with respect to ponderomotive instabilities

• Monotonic:



Self-Excited Loop

• Resonators operated in self-excited loops in the absence of feedback are free of ponderomotive instabilities. An SEL is equivalent to the ideal VCO.—Amplitude is stable —Frequency of the loop tracks the frequency of the cavity

• Phase stabilization can reintroduce instabilities, but they are easily controlled with small amount of amplitude feedback



Self-Excited Loop

where: tan no beam loading , 1, 1l ay k kϕ µµ

τθ ττ

Ω=

The stability boundary can be pushed arbitrarily far with amplitude feedback

General stability criteria without above restrictions (including beam loading) exist

Approximate stability criteria with phase feedback

• Monotonic:

• Oscillatory:

20

12aky k Vµ τ+

- <2

20 2 2

( 1)12 ( 1)

a

a

k ky k V

k kϕ

µµ µ ϕτ τ Ω

+<

+ +



Input-Output Variables

• Generator - driven cavity

• Cavity in a self-excited loop

Detuning (ω - ωc)

Field amplitude (Vo)Generator amplitude (Vg)

Cavity phase shift (θl)

Ponderomotive effects

Loop phase shift (θl)

Field amplitude (Vo)Limiter output (Vg)

Loop frequency (ω)

Pondermotive effects



Input-Output Variables Generator-Driven Resonator

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-6 -5 -4 -3 -2 -1 0 1 2

Detuning (Norm.)

Am

plitu

de (N

orm

.)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-6 -5 -4 -3 -2 -1 0 1 2

Detuning (Norm.)

Cai

ty P

hase

Shi

ft



Input-Output Variables Self-Excited Loop

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Loop Phase Shift

Am

plitu

de (N

orm

.)

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Loop Phase Shift

Det

unin

g (N

orm

.)



Ponderomotive Instabilities in GDR

0 / 2 72 MHz, 9 msec, 18 Hz/ 2 89 Hz, 85 sec, / 2 500 kHzkµ µ µ

ω π τ ωπ τ π

∆Ω

= = == = =



Ponderomotive Instabilities in SEL

2 2dY dX dX Y V Vdt dt dt

ϕ δω- = =



Ponderomotive Instabilities in SEL



Microphonics

• Microphonics: changes in frequency caused by connections to the external world—Vibrations—Pressure fluctuations

When phase and amplitude feedbacks are active, ponderomotive effects can change the response to external disturbances

( )2 2 22 2 ok V v n tµ µ µ µ µ µµ

δω δω δω δτ

Ω+ +W = - +



Microphonics

Two extreme classes of driving terms:

• Deterministic, monochromatic—Constant, well defined frequency—Constant amplitude

• Stochastic—Broadband (compared to bandwidth of mechanical mode)—Will be modeled by gaussian stationary white noise

process



Microphonics (probability density)

SNS M02, CAVITY 3 BACKGROUND MICROPHONICS HISTOGRAM

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-20 -15 -10 -5 0 5 10 15 20

Cavity Detuning (Hz)

Prob

abili

ty o

f Occ

urre

nce

Std Dev = 2.2 Hz

SNS M03, CAVITY 1 BACKGROUND MICROPHONICS HISTOGRAM

0

0.01

0.02

0.03

0.04

0.05

0.06

-20 -15 -10 -5 0 5 10 15 20Cavity Detuning (Hz)

Prob

abili

ty o

f Occ

urre

nce

Std Dev = 5.6 Hz

Single gaussian

Noise driven

Bimodal

Single-frequency driven

Multi-gaussian

Non-stationary noise

805 MHz TM 805 MHz TM 172 MHz TEM



Microphonics (frequency spectrum)

TM-class cavities (JLab, 6-cell elliptical, 805 MHz, β=0.61)Rich frequency spectrum from

low to high frequenciesLarge variations between cavities

SNS M02, Cavity 3, Bkgnd Microphonics Spectrum, 1W

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0 60 120 180 240 300 360

Frequency (Hz)

RM

S A

mpl

itude

(Hz

rms)

TEM-class cavities (ANL, single-spoke, 354 MHz, β=0.4)Dominated by low frequency (<10 Hz) from pressure fluctuations

Few high frequency mechanical modes that contributelittle to microphonics level.



Probability Density (histogram)

0

0.5

1

1.5

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

δω/δωmax

p(δω

)

0

0.1

0.2

0.3

0.4

0.5

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

δω/δωσ

p(δω

)

Harmonic oscillator (Ωµ,τµ) driven by:

Single frequency, constant amplitude White noise, gaussian

2 2max

1( )p δωπ δω δω

=-

212

1( ) exp2

pω

δωσω

δωσ π

È ˘Ê ˆÍ ˙Á ˜Í ˙Á ˜Ë ¯Í ˙Î ˚

-=



Autocorrelation Function

-1

-0.5

0

0.5

1

0 2 4 6 8 10

τ

r(τ)

-1

-0.5

0

0.5

1

0 2 4 6 8 10

τ

r(τ)

( )( ) cos( )(0) d

RrR

δωδω

δω

ττ ω τ= = /( )( ) cos( )(0)

Rr eR

µτ τδωδω µ

δω

ττ τΩ -= =

0

1( ) ( ) ( ) lim ( ) ( )T

x TR x t x t x t x t dt

Tτ τ τ

Æ•= + = +Ú

Single frequency, constant amplitude White noise, gaussianHarmonic oscillator (Ωµ,τµ) driven by:



Stationary Stochastic Processes

x(t): stationary random variable

Autocorrelation function:

Spectral Density Sx(ω): Amount of power between ω and dω

Sx(ω) and Rx(τ) are related through the Fourier Transform (Wiener-Khintchine)

Mean square value:

0

1( ) ( ) ( ) lim ( ) ( )T

x TR x t x t x t x t dt

Tτ τ τ

Æ•= + = +Ú

1( ) ( ) ( ) ( )2

i ix x x xS R e d R S e dωτ ωτω τ τ τ ω ω

π• •-

-• -•= =Ú Ú

2 (0) ( )x xx R S dω ω•

-•= = Ú



Stationary Stochastic Processes

For a stationary random process driving a linear system

( ) ( ) ( ) ( )

( ) ( ) [ ][ ] [ ]

( )

2 2

2

22

0 0

: auto correlation function of ( ) ( )

( ) ( ) : spectral density of ( ) ( )

( ) ( ) ( )

( )

y y x x

y x

y x

y x

x

y R S d x R S d

R R y t x t

S S y t x t

S S T i

y S dT i

ω ω ω ω

τ τ

ω ω

ω ω ω

ω ωω

+• +•

-• -•

+•

-•

= = = =

È ˘Î ˚

=

=

Ú Ú

Ú

( )x t ( )y t( )T iω



Performance of Control SystemResidual phase and amplitude errors caused by microphonicsCan also be done for beam current amplitude and phase

fluctuations

2

Assume a single mechanical oscillator of frequency and decay time

excited by white noise of spectral density Aµ µτΩ



Performance of Control System

( )

( ) ( )( )

( ) ( )( )

2

22 2 2 222 2 2

22

22 2 22

22

22 2 22 2

2

2

2

2

2

2

ex

a

a ex

ex

d

G i di

G i

G i G i d di

G i

G i G i d di

A A A

v A

A

µµ

µ µµ

µµ

µµµ

ϕµ

µ ϕµµ

µ

ωω ω

ω ωτ

ω

ω ω ω ωω ωτ

ω

ω ω ω ωω ω

τ

πτδω

δ δωπτ

δϕ δωπτ

Ω

Ω

Ω

Ω

Ω

Ω

+• +•

-• -•

+• +•

-• -•

+• +•

-• -•

- + +

- + +

- + +

< >= = =

< >= = < >

< >= = < >

Ú Ú

Ú Ú

Ú Ú



The Real WorldProbability Density

0.00001

0.0001

0.001

0.01

0.1

-15 -10 -5 0 5 10 15

Hz

Hz-1

Probability Density

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-15 -10 -5 0 5 10 15

Hz

Hz-1

Microphonics

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 0.02 0.04 0.06 0.08 0.1

Time (s)

Hz

Normalized Autocorrelation Function

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

Sec



The Real WorldProbability Density

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-10 -5 0 5 10

Hz

Hz-1

Microphonics

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.02 0.04 0.06 0.08 0.1

Time (sec)

Hz

Probability Density

0.00001

0.0001

0.001

0.01

0.1

-10 -5 0 5 10

Hz

Hz-1


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

Time (sec)



The Real WorldProbability density

0.00001

0.0001

0.001

0.01

0.1

-20 -15 -10 -5 0 5 10 15 20

Hz

Hz-1

Microphonics

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.02 0.04 0.06 0.08 0.1

Time (sec)

Hz


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Time (sec)

Probability density

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-20 -15 -10 -5 0 5 10 15 20

Hz

Hz-1



Piezo control of microphonics

MSU, 6-cell elliptical 805 MHz, β=0.49

Adaptive feedforward compensation



Piezo Control of Microphonics

FNAL, 3-cell 3.9 GHz



Piezo Control of Microphonics

LANL 402.5 MHz scruncher cavity, QL= 109

Phase-locked with piezo only



Final Words

• Microphonics and ponderomotive instabilities issues in high-Q SRF cavities were “hot topics” in the early days (~70s), especially in low-β applications

• They were solved and are well understood• They are being rediscovered in medium- to high-β

applications• Today’s challenges:

—Large scale (cavities and accelerators): need for optimization—Finite beam loading

• Small but non-negligible current (e.g. RIA)• Low current resulting from the not quite perfect cancellation of 2 large

currents (ERLs)

ponderomotive instabilities and microphonics

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