1 / 27 L.Angrisani University of Naples Federico II, ITALY [email protected] 14th International Magnetic Measurement Workshop 26-29 September 2005, Geneva,

L.AngrisaniUniversity of Naples Federico II, ITALY

[email protected]

14th International Magnetic Measurement Workshop26-29 September 2005, Geneva, Switzerland 1 / 27

Digital Signal Processing Approaches

for Varying Magnetic Field Measurements through a

Rotating Coils SystemL. Angrisani, R. Schiano Lo

Moriello L. Bottura, A. Masi


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Outline Introductive notes Limitations of the standard Fourier analysis in

the presence of analog bucking Proposals:

Quadrature demodulation-based method Extrapolation-based method

Performance comparison in simulated conditions

Results in actual experiments Conclusions


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Introduction

In superconducting magnets, dynamic effects like snapback can be estimated only through measurements in ramping conditions.

Due to current variation, field harmonics are both time and ramp rate dependent (e.g. decay phenomena).

The conventional Fourier analysis does not work properly. Being addressed to DC measurements, it assumes field harmonics to be constant during a coil turn.


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Introduction

Increasing coil speed is often impracticable, and it could prove unsuccessful in the presence of high ramp rates of the current.

To measure varying magnetic fields through a rotating coils system, two new digital signal-processing methods are proposed.

They both assume continuous coil rotation.


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The standard Fourier analysis

1,..,1,2

NiiNi

Magnetic flux as a function of the

angular position

Integration of Vcoil in the angular domain

Constant during a coil turn

Field Harmonics

][ DFT

15

11

)exp(Re)(n

nref

nn jnR

CK

1

12

nn

nref

n K

R

NC


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Problems with the standard Fourier analysis

A periodic, magnetic flux characterized by constant envelope…..

0 5 10 15 20 25 30

10-15

10-10

10-5

100

Field harmonic order

Fiel

d ha

rmon

ic a

mpl

itude

[T

]T0;T0

T10BT;10;T1

)cos()(

42

55

331

5

1

BB

BB

nBkΦn

n


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0 5 10 15 20 25 30

10-15

10-10

10-5

100


Fiel

d ha

rmon

ic a

mpl

itude

[T

]

The envelope of the main component (fundamental) increases linearly with the angular position, B3 and B5 being constant …..

T0;T0

T10T;10

T)1π20

()(

42

55

33

1

BB

BB

B

Imposed values

DFT results



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Hardware solution: analog bucking

0 5 10 15 20 25 30

10-20

10-15

10-10

10-5


Fiel

d ha

rmon

ic a

mpl

itude

[T

]


Measurement problems with B5

still survive.

Measurement problems with B3

still survive.

0;0

T10T;10

T10)1π20

()(

42

65

53

31

BB

BB

B


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Standard Fourier analysis characterization

LHC Cycle

Typical test current evolution


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Raw DataCn(t)

Flux samples()

Cn expectedinstantaneous

Cn estimatedHarmonic

coefficients

Instantaneous vs. each turn

Average

Bn(t) An(t)

Best EstimatorsBn_avg and An_avg

For each turn

Absolute and relative errors|Bn-Bn_avg|

|Bn-Bn_avg|/Bn_avg|An-An_avg|

|An-An_avg|/An_avg


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Difference greater than 1 unit for a 10A/s linear

ramp

4000 5000 6000 7000 8000 9000 10000 110000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

-3

R=10A/s

R=20A/s

R=30A/s

Absolute difference for B1

Current [A]

Ab

s(B

1-B

1 avg

)

[T

]

R=40A/s

R=50A/s

R=60A/s

R=70A/s

R=80A/s

R=90A/s

R=100A/s

Linear ramp: different ramp rates have been simulated


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0 10 20 30 400

1

2

3

4

5x 10-5

Number of coil turns

abs[

(B1-

B1a

vg)

[T

]

B1

0 10 20 30 400

0.2

0.4

0.6

0.8

1x 10 -4


abs[

(B3-

B3a

vg)

[T

]

B3

Parabolic ramp


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0 10 20 30 400

0.2

0.4

0.6

0.8

1

1.2x 10

-4


abs[

(B1-

B1a

vg)

[T

]

B1

0 10 20 30 400

0.5

1

1.5

2

2.5

3x 10 -4


abs[

(B3-

B3a

vg)

[T

]

B3

Exponential ramp


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Proposal 1: quadrature demodulation-based method

Ф(t)

sin(2πft)

FIR

FIR

tg-1

x2

Unwrap

Phase

Modulus

cos(2πft)

)1(Kn

( )2

( )2

0 0.05 0.1 0.15 0.2

-300

-250

-200

-150

-100

-50

0

50FIR filter frequency response

Frequency [Hz]

Am

plitu

de

[dB

]

1800 2000 2200 2400

-2

-1

0

1

2

3

Time [s]

[Wb]

Magnetic flux and its envelope

Magnetic fluxFlux envelope

Residual C1 evaluation: digital bucking

Quadrature demodulation scheme


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Sliding window FFT

Overlap ratio:

Flux samples()

I-Q Demodulation

Flux samples reconstruction

C2...C15=0

B1 estimation

STFT

STFT

+

-

Compensated Cn

Proposal 1: quadrature demodulation-based method

1N

N

1724 1734 1744 1754 1764 1774

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Mag

neti

c fl

ux [

Wb]

Standard analysis

N points window

one sample


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Proposal 1: typical results

Difference () between estimated and nominal

evolution of B1.

Comparison of the nominal evolution of B3 (green) to those

obtained through STFT (red) and demodulation+STFT (violet).


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Proposal 2 : extrapolation-based method

For each angular position, the flux samples (red dots) acquired in the last four coil turns are used to extrapolate the flux samples related to the next turn for different sampling times (black dots).


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Proposal 2 : extrapolation-based method

Time [s]

Mag

neti

c fl

ux

[Wb]

Time [s]Angle [rad]

Mag

netic

flu

x [

Wb]

Magnetic flux extrapolation

extrapolated flux samples

4 samples buffer

intersection plane


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Performance comparison: linear ramp

( )2

1

1 N

n nn

Absolute RMS error X XN

10

20

30

40

50

1,00E-05

1,00E-04

1,00E-03

1,00E-02

Absolute

RMS error

[T]

Ramp Rate [A/s]

B1

Standard analysis

STFT

Extrapolation


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Performance comparison: linear ramp

B5

B3


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Performance comparison: parabolic ramp

B1-B7

A1-A8


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Performance comparison: exponential ramp

B1-B7

A1-A8


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Proposal 2: experimental results

Some measurements have been carried out on a reference, resistive magnetic dipole.

Continuous coil rotation has been assured.

Coil speed has been equal to 1 turn/s.

Different ramp rates have been considered: 5A/s, 10A/s, 20A/s.


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Proposal 2: experimental results

As expected for the reference magnet measured, the first results seem to highlight that harmonic coefficients are independent of ramp rate.


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Comparison

Blue trace: proposal 1

Black trace: proposal 2

The evolution versus time of harmonic coefficients gained from proposal 2 result smoother than those from proposal 1.


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Conclusions Characterization of the standard Fourier analysis when

applied to compensated magnetic fluxes in non-stationary conditions.

A quadrature demodulation-based method for a reliable digital bucking has been proposed (proposal 1).

Harmonic coefficients are gained from the application of the standard Fourier analysis to digitally compensated flux samples.

An alternative method that extrapolates, for a given time instant, the flux value for each angular position in a coil turn has also been presented (proposal 2).


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Conclusions Many tests in simulated dynamic conditions have been carried

out to assess and compare the performance of the proposed methods.

Proposal 1 allows more accurate estimate of harmonic coefficients, but B1, than that granted by standard Fourier analysis.

Proposal 2 performs much better even in the presence of high ramp rates.

The reduced computational burden of proposal 2 make real-time tracking of harmonic coefficients feasible.

The first results obtained in actual experiments seem to confirm the reliability of proposal 2.

Documents

1 / 27 L.Angrisani University of Naples Federico II, ITALY [email protected] 14th International Magnetic Measurement Workshop 26-29 September 2005, Geneva,