measurements of field quality using harmonic coils · a “dipole coil” is therefore sensitive to...

US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.

Measurements of Field Quality Using Harmonic CoilsAnimesh K. Jain

Brookhaven National Laboratory, Upton, New York 11973-5000

[US Paticle Accelerator School on Superconducting Accelerator Magnets, January22-26, 2001, Houston, Texas, USA.]

1. Radial and Tangential Coils

2. A Dipole Coil

3. General Case of a 2m-pole Coil

4. Flux Through a Coil of an Arbitrary Shape

5. Rotating Coil of an Arbitrary Shape5.1 Tangential Coil as a special case5.2 Radial Coil as a special case

6. An Array of Rotating Coils

7. Imperfect Motion of Rotating Coils7.1 Transverse Vibrations7.2 Torsional Vibrations

8. Examples of Practical Coils – HERA and RHIC

9. Analog and Digital Bucking

10. Magnet Dependent Bucking Algorithm

11. Coil Construction Errors11.1 Finite Size of Coil Windings11.2 Random Variation of Radius11.3 Random Variation of Angular Position11.4 Random Variation of Opening Angle (tangential coil)11.5 Unequal Radii of the two grooves (tangential coil)11.6 Offset in Rotation Axis


12. Systematic Errors in Coil Parameters12.1 Error in Radius12.2 Error in Angular Position12.3 Error in Opening Angle

13. Effect of a Finite Averaging Time

14. Errors in Placing the Coil in the Magnet14.1 Coil Axis different from the Magnet Axis14.2 Sag of the Measuring Coil14.3 Coil Axis Tilted relative to the Magnet Axis

15. Calibration of a Five-Winding Tangential Coil15.1 Radii and the Opening Angle15.2 Angular Positions15.3 “Tilt” of Tangential Winding (unequal radii of the two grooves)


A Radial Coil (or Bθ Coil)

Number of Turns = NLength of the coil = L

A radial coil has a flat loop of wire whose plane coincides with the radialplane of the rotating cylinder. The two sides of the loop are located at radii R1

and R2, as shown above. The flux through the coil at an angular orientation θis:

∫Φ( ) ( , ) ( ) cos( )θ θ θ αθ= =

−

⌠

⌡

−

=

∞

∑NL B r dr NL C nr

Rn n dr

R

R

ref

n

nn

R

R

1

2

1

21

1

Φ( ) ( )cos( )θ θ α=

−

−=

∞

∑NLR

nR

RR

RC n n nref

ref

n

ref

n

nn

2 1

1

If the coil rotates with an angular velocity ω and θ = δ is the angular positionat time t = 0, then θ = ωt + δ. The flux as a function of time is then:

Φ( ) ( )cos( )tNLR

nR

RR

RC n n t n nref

ref

n

ref

n

nn

=

−

+ −=

∞

∑ 2 1

1

ω δ α

R2

θ

ω

X

Y

R1

Bθ


A Radial Coil (or Bθ Coil)

Number of Turns = NLength of the coil = LAngular position at time t=0: δ

Φ( ) ( )cos( )θ θ α=

−

−=

∞

∑NLR

nR

RR

RC n n nref

ref

n

ref

n

nn

2 1

1

Φ( ) ( )cos( )tNLR

nR

RR

RC n n t n nref

ref

n

ref

n

nn

=

−

+ −=

∞

∑ 2 1

1

ω δ α

The voltage signal induced in the radial coil is:

V tt

NLRR

RR

RC n n t n nref

ref

n

ref

n

nn

( ) ( )sin( )= −

=

−

+ −=

∞

∑∂Φ∂

ω ω δ α2 1

1

The amplitude of the voltage signal is proportional to the angular velocity.For analysis based on voltage signals, it is essential to control the angularvelocity and make corrections for any speed fluctuations. The integratedvoltage signal gives the flux, which is independent of angular velocity.

The above expressions assume that the two sides of the coil loop are locatedon the same side of the origin, as shown in the figure. If the two sides arelocated on opposite sides of the origin, as is true for many practical coils, thenone should replace R1 by –R1 in the above equations.

R2

θ

ω

X

Y

R1

Bθ


A Tangential Coil (or Br Coil)

Number of Turns = NLength of the coil = LOpening Angle = ∆

A tangential coil has a loop of wire whose plane is at right angles to the radiusvector through the center of the loop. The two sides of the loop are bothlocated at a radius of Rc, as shown above. The flux through the coil at anangular orientation θ is:

∫Φ∆

∆

∆

∆

( ) ( , ) ( ) sin( )/

/

/

/

θ θ θ θ α θθ

θ

θ

θ

= =

−

⌠

⌡−

+−

=

∞

−

+

∑NL B R R d NL C nR

Rn n R dr c c

c

ref

n

nn

c2

21

12

2

Φ∆

( ) sin ( )sin( )θ θ α=

−=

∞

∑2

21

NLR

nR

Rn

C n n nref c

ref

n

nn

If the coil rotates with an angular velocity ω and θ = δ is the angular positionat time t = 0, then θ = ωt + δ. The flux as a function of time is then:

Φ∆

( ) sin ( )sin( )tNLR

nR

Rn

C n n t n nref c

ref

n

nn

=

+ −=

∞

∑2

21ω δ α

Rc

∆

θ

ω

X

Y

Br


A Tangential Coil (or Br Coil)

Number of Turns = NLength of the coil = LOpening Angle = ∆Angular position attime t=0: δ

Φ∆

( ) sin ( )sin( )θ θ α=

−=

∞

∑2

21

NLR

nR

Rn

C n n nref c

ref

n

nn

Φ∆

( ) sin ( )sin( )tNLR

nR

Rn

C n n t n nref c

ref

n

nn

=

+ −=

∞

∑2

21ω δ α

The voltage signal induced in the tangential coil is:

V tt

NLRR

Rn

C n n t n nrefc

ref

n

nn

( ) sin ( ) cos( )= −

= −

+ −=

∞

∑∂Φ∂

ω ω δ α221

∆

The amplitude of the voltage signal is proportional to the angular velocity.For analysis based on voltage signals, it is essential to control the angularvelocity and make corrections for any speed fluctuations. The integratedvoltage signal gives the flux, which is independent of angular velocity.

The radius, Rc, of the coil should be maximized to get good signal strength forhigher harmonics. The opening angle, ∆, should be large enough to giveenough signal and small enough so that sin(n∆/2) does not vanish for higherharmonics of interest (∆ << 2π/nmax). Typically, ∆ ~ 15 degrees.

Rc

∆

θ

ω

X

Y

Br


Sensitivity of a Tangential Coil to Various Harmonics (Rc = Rref)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0 2 4 6 8 10 12 14 16 18 20 22 24

Harmonic Number (n )

Sin

(n∆

/2)

∆ = 10 deg.

∆ = 15 deg.

∆ = 20 deg.

Typically, one is interested in precise measurement of about 15 harmonics. Itis clear from the above plot that a tangential coil with an opening angle of 20degrees rapidly loses sensitivity for higher harmonics, although it is moresensitive to lower harmonics as compared to a coil with 15 degrees openingangle. On the other hand, the sensitivity of a coil with an opening angle ofonly 10 degrees peaks at 36-pole term, which is of little relevance foraccelerator physics. Such a narrow angle, therefore, sacrifices sensitivity forlower harmonics of interest with no useful outcome. It is clear from this plotthat optimum value of the opening angle is ∆ ~ 15 degrees.


A “Dipole Coil” : Radial or Tangential ?

A “Dipole Coil” is a coil with aspecific geometry which has the“Dipole Symmetry”, namely anantisymmetry under rotation by πradians. The flux through this coilcan be calculated by treating it asa radial coil with R1 = –Rc andR2 = +Rc, oriented at an angle θ,as shown. The flux through thecoil can also be calculated bytreating it as a tangential coilwith an opening angle of πradians, oriented at an angle ofθ’= θ +π/2. Both approachesgive the same result.

[ ]

Φradialref c

ref

nc

ref

n

nn

ref

n

c

ref

nn

n

NLR

nR

RR

RC n n n

NLR

nR

RC n n n

( ) ( )cos( )

( ) ( )cos( )

θ θ α

θ α

=

−

−

−

=

− − −

=

∞

=

∞

∑

∑

1

11 1

Similarly,

Φ Φtang. tang.( ) ( ) sin ( )sin( )

sin ( )cos( )

′ = + =

+ −

=

−

=

∞

=

∞

∑

∑

θ θπ π

θπ

α

πθ α

2

2

2 2

2

2

1

1

2

NLR

nR

Rn

C n nn

n

NLR

nR

Rn

C n n n

ref c

ref

n

nn

ref

n

c

ref

n

n

The terms with n = even vanish in both the expressions. The flux through adipole coil is therefore given by:

ΦDipoleref

nn odd

c

ref

n

nNLR

nR

RC n n n( ) ( )cos( )θ θ α=

−

==

∞

∑2

1

A “Dipole Coil” is therefore sensitive to only the odd harmonics, i.e., dipole,sextupole, decapole, etc. Such a coil is almost universally used in both radialand tangential coil systems for “bucking” the main dipole field.

Rc

X

Y

∆ = π

θ'θ

R2=Rc

R1 = –Rc


A Multipole Coil of Order m (A 2m-Pole Coil)

Rc

X

Y

θπ/m

θ'=θ+π/(2m)

A multipole coil of order m, or a 2m-pole coil is a coil with special geometrythat has m loops connected in series, as shown in the figure. For any angularposition characterized by the angle θ, the loops span the angular region of θto θ+(π/m), (θ+2π/m) to (θ+3π/m), (θ+4π/m) to (θ+5π/m), and so on. Theflux through such a coil as a function of θ can be easily calculated by treatingit as an array of m identical tangential coils with opening angle of ∆ = π/m and having angular positions of θ’ = θ+(π/2m), θ’+2π/m , θ’+4π/m, and soon.

The flux through the first segment of the coil is:

Φ Φ11

1

2

2

2

2

( ) ( ) sin ( )sin( )

Im sin ( ) ( )

θ θπ

θ α

π θ α

= ′ =

′ −

=

=

∞

=

∞′−

∑

∑

tang.NLR

nR

Rnm

C n n n

NLR

nR

Rnm

C n e

ref c

ref

n

nn

ref

n

c

ref

ni n n n



Rc

X

Y

θπ/m

θ'=θ+π/(2m)

The flux through the first segment of the coil is:

Φ Φ11

2

2( ) ( ) Im sin ( ) ( )θ θ

π θ α= ′ =

=

∞′−∑tang.

NLR

nR

Rnm

C n eref

n

c

ref

ni n n n

The contribution due to harmonics which are EVEN MULTIPLES OF m VANISHES

due to the sin(nπ/2m) factor. Let us now consider the total flux through thearray of loops:

[ ]Φ( ) Im (

Imexp( )

exp( / );

/ /θ

ππ

θ π π

θ

= + + +

=−

−

=

=

=

∞′

=

∞′

∑

∑

X

X

nn

in i n m i n m

nn

in

e e e m

ei n

i n mnm

1

2 4

1

1

1 21 2

0

K terms)

unless integer

where, X nref c

ref

ninNLR

nR

Rnm

C n e n=

−2

2sin ( )

π α = 0 for (n/m)=even integer.

Therefore, all terms in the summation vanish, except for those values of nwhich are ODD MULTIPLES OF m.



Rc

X

Y

θπ/m

θ'=θ+π/(2m)

Φ( ) Imexp( )

exp( / );θ

ππ

θ=−

−

=

==

∞′∑ X n

n

inei n

i n mnm1

1 21 2

0 unless integer

where, X nref c

ref

ninNLR

nR

Rnm

C n e n=

−2

2sin ( )

π α = 0 for (n/m)=even integer.

Therefore, all terms in the summation vanish, except for those values of nwhich are ODD MULTIPLES OF m. The total flux for the 2m-pole coil can bewritten as:

Φ( ) ( )cos( )

( )

θ θ α=

−

== +

∞

∑2

2 1

mNLR

nR

RC n n nref

n mn k m

c

ref

n

n

where k is any integer, including zero. If the coil rotates with an angularvelocity ω and θ = δ is the initial angular position of the coil, then θ = ωt +δ. The flux and the voltage at any time t are:

Φ( ) ( ) cos( )

( )

tmNLR

nR

RC n n t n nref

n mn k m

c

ref

n

n=

+ −

== +

∞

∑2

2 1

ω δ α

V tt

tm NLR

RR

C n n t n nrefn m

n k m

c

ref

n

n( )( )

( )sin( )

( )

= − =

+ −

== +

∞

∑∂Φ∂

ω ω δ α2

2 1

The results for a dipole coil are obtained by putting m = 1. Quadrupole coils(m = 2) are commonly used for bucking in tangential coil systems. Sextupoleand other higher order coils may be used for special applications.


Flux Through a Coil of Arbitrary Shape

We consider a coilmade up of a loop ofwire running parallel tothe Z-axis as shown inthe figure. The shape ofthe coil is defined by apath from the point z1

to z2 in the complexplane. The length of theloop is L along thenegative Z-axis, asshown.

d r = an element alongthe path from z1 to z2.

d s = $ | |n rd L =element of area definedby the line element d r

The flux through the area element d s is given by B. d s. The area element isgiven by the vector:

d d L d L dx dy L dy dx Ls n r z r z x y x y= = × = × + = − +$ | | ( $ ) $ ( $ $ ) ( $ $ )

∴ = ⋅ = + − + = −d d B B dy dx L B dx B dy Lx y y xΦ B s x y x y( $ $ ).( $ $ ) ( )

Let us evaluate the integral of the complex field B(z) from z1 to z2:

B z zz

z

z

z

1

2

1

2

( ) ( )( ) ( ) ( )d B iB dx idy B dx B dy i B dx B dyy x y x x y∫ ∫ ∫ ∫= + + = − + +

This leads us to the general result for a loop with N turns:

Φ =

=

−

−

∫ ∑=

∞NL d

NLR

nC n in

R Rref

nn

ref

n

ref

n

Re ( ) Re ( )exp( )B z zz z

z

z

1

2

1

2 1α

Lz1

z2

Bn̂

d r

L

X

Z

Yd s


A Rotating Coil of Arbitrary Shape

Let us consider a rotating coilof arbitrary shape formed by aloop of wire passing throughtwo points in the X-Y plane.In general, both the radial andthe azimuthal coordinates ofthese two points will bedifferent. The radial coil is aspecial case where theazimuthal coordinates of boththe points are either the same,or differ by π. Similarly, thetangential coil is a specialcase where the radialcoordinates of the two pointsare the same. Any angularposition of the coil ischaracterized by an angle θ measured from an “initial position”. If z1 and z2denote the locations of the two points in the complex plane at any instant,then the flux through the coil of length L and with N turns is:

Φ( ) Re ( )exp( )θ α=

−

−

=

∞

∑NLR

nC n in

R Rref

nn

ref

n

ref

n

1

2 1z z

From the figure: z z z z1 1 0 2 2 0= =, ,exp( ); exp( )i iθ θ . Substituting in theabove expression for the flux, we get:

Φ( ) Re exp( ) ( )exp( )θ θ α= −

=

∞

∑K nn

nin C n in1

where Kn is the “SENSITIVITY FACTOR” for the order n defined by:

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,

X

Y

θ z1,0

z2,0z1

z2


Tangential Coil as a Special Case of a General Coil

X

Y

θ z1,0

z2,0z1

z2

Rc

∆

θ

ω

X

Y

Br

Φ( ) Re exp( ) ( )exp( )θ θ α= −

=

∞

∑Knn

nin C n in1

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,

For a tangential coil, when θ is measured from the X-axis,

z z1 0 2 02 2, ,exp( / ); exp( / )= = −R i R ic c∆ ∆

∴ = −

Knref c

ref

n

iNLR

nR

Rntang. sin

2

2∆

The Sensitivity Factor for a tangential coil is purely imaginary. The flux at theangular position θ is given by:

Φ∆

tang ( ) sin ( )sin( )θ θ α=

−=

∞

∑2

21

NLR

nR

Rn

C n n nref c

ref

n

nn

which is the same expression as derived by directly integrating the radialcomponent, Br(Rc,θ), over the angular extent of the coil.


Radial Coil as a Special Case of a General Coil

X

Y

θ z1,0

z2,0z1

z2

R2

θ

ω

X

Y

R1

Bθ

Φ( ) Re exp( ) ( )exp( )θ θ α= −

=

∞

∑Knn

nin C n in1

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,

For a radial coil, when θ is measured from the X-axis, z z1 0 1 2 0 2, ,;= ± =R R .It should be noted that z1,0 = +R1 when R1 and R2 are on the same side of thecenter and z1,0 = –R1 when R1 and R2 are on the opposite sides of the center.

∴ =

−

±

Knref

ref

n

ref

nNLR

nR

RR

Rradial 2 1

The Sensitivity Factor for a radial coil is purely real. The flux at the angularposition θ is given by:

Φ radial ( ) ( ) cos( )θ θ α=

−

±

−=

∞

∑NLR

nR

RR

RC n n nref

ref

n

ref

n

nn

2 1

1

which is the same expression as derived by directly integrating the azimuthalcomponent, Bθ(r,θ), over the radial extent of the coil.


An Array of Rotating Coils

Let us consider an array of M coils mounted on the same rotating system. Letthe sensitivity factor of the j-th coil for the n-th harmonic be denoted by

Knj j M( ) , , , , .= 1 2 3 L Let all these coils be connected either in series, or in

opposition, to generate a combined signal. The total flux through this array ofcoils is the algebraic sum of the fluxes through individual coils:

Φ Φ( ) ( )θ ε θ==

∑ jj

M

j1

where εj = +1 if the j-th coil is connected in series, and εj = –1 if the j-th coilis connected in opposition. From the general formula for the flux through anindividual coil, we obtain:

Φ( ) Re ( )exp( )exp( )

Re ( )exp( )exp( ) ;

Re ( )exp( )exp( )

Re ( )exp( )exp( )

( )

( )

( )

θ ε α θ

ε α θ ε

ε α θ

α θ

= −

= −

−

= −

= =

∞

=

∞

=

=

∞

∞

∑ ∑

∑∑

∑∑

∑

jj

M

nj

nn

j nj

nnj

M

j

j nj

j

M

nn

n nn

C n in in

C n in in

C n in in

C n in in

1 1

11

11

1

K

K

K

K

since is real.

==

=

where Kn is the overall sensitivity of the array. From the above equation, it isclear that the sensitivity of an array of M coils connected either in series or inopposition is given by an algebraic sum of the sensitivities of the individualcoils:

K Kn j nj

j

M=

=∑ε ( )

1

If the individual coils are properly designed and εj are appropriately chosen,the overall sensitivity of an array of coils to a particular harmonic (or severalharmonics) can be made zero. This is the principle used in “bucking”.


Transverse Vibrations of the Rotation Axis

Let us consider a rotating coil ofa general shape whose rotationaxis has a displacement as thecoil rotates. This displacement,D(θ), of the rotation axis in thecomplex z-plane may be a func-tion of the azimuthal angle, θ.The positions of the two sides ofthe coil loop at any angularposition, θ, are given by:

z z D z z D1 1 0 2 2 0= + = +, ,exp( ) ( ); exp( ) ( )i iθ θ θ θ

The expression for flux at any angular position, θ, is:

Φ( ) Re ( )exp( )θ α=

−

−

=

∞

∑NLR

nC n in

R Rref

nn

ref

n

ref

n

1

2 1z z

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, , SENSITIVITY FACTOR FOR n-th HARMONIC

Case I: A Pure Dipole Field:

For a pure dipole field (n = 1), the expression for flux involves only thequantity z2 – z1 = [z2,0 – z1,0].exp(iθ), which is independent of thedisplacement, D(θ). Thus, the flux linked through a coil in a pure dipole fieldis unaffected by transverse displacements of the rotation axis. This result isnot too surprising because displacements in a pure dipole field do not produceany feed down harmonics.

z1,0

z2,0

z1 = z1,0 eiθ + D(θ)

z2 = z2,0 eiθ + D(θ)

D(θ)

θ

X

Y


z1,0

z2,0

z1 = z1,0 eiθ + D(θ)

z2 = z2,0 eiθ + D(θ)

D(θ)

θ

X

Y

Transverse Vibrations of the Rotation Axis

z z D

z z D1 1 0

2 2 0

= +

= +,

,

exp( ) ( )

exp( ) ( )

i

i

θ θ

θ θ

Φ( ) Re ( ) exp( )θ α=

−

−

=

∞

∑NLR

nC n in

R Rref

nn

ref

n

ref

n

1

2 1z z

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,

Case II: A Pure 2n-Pole Field:

An approximate expression for the flux in a pure 2n-pole field can beevaluated by using a binomial expansion and neglecting terms of the secondand higher order in [D(θ)/Rref], assuming that |D(θ)| << Rref. We get:

[ ]Φn nin in

ni n

ref

ine C n e en

RC n en n( ) Re ( ) Re

( ) ( )( )( )θ

θθ α θ α≈ +−

+−−

− −K KD

11 1

L

The flux picked up by a rotating coil in a pure 2n-pole field is, in general,affected by transverse displacements of the rotation axis. To a firstapproximation, the effect on the flux is proportional to the amplitude of thedisplacement and the sensitivity of the coil to the 2(n–1)-pole terms. It shouldbe noted that the highest power of D(θ) in the expression for the flux from a2n-pole field is (n – 1). Thus, only the first term in the above expressionsurvives for a pure dipole field, whereas the first two terms represent thecomplete expression for flux in a pure quadrupole field. For fields of highermultipolarities, other higher order terms are also present, but can be neglectedin practice if the condition |D(θ)| << Rref is satisfied.

If the coil is replaced by an array of coils whose combined sensitivity to the2(n – 1)-pole term is zero, then the effect of small transverse vibrations ispractically eliminated. This is the basis for bucking the (n – 1)th harmonic.


z1,0

z2,0

z1 = z1,0 eiθ + D(θ)

z2 = z2,0 eiθ + D(θ)

D(θ)

θ

X

Y

Periodic Transverse Motion of the Rotation Axis

z z D

z z D1 1 0

2 2 0

= +

= +,

,

exp( ) ( )

exp( ) ( )

i

i

θ θ

θ θ

Φ( ) Re ( ) exp( )θ α=

−

−

=

∞

∑NLR

nC n in

R Rref

nn

ref

n

ref

n

1

2 1z z

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,

If the displacement amplitude, D(θ), is a periodic function of θ :

D D( ) exp( )θ θ==−∞

∞

∑ pp

ip

In a pure 2n-pole field:

[ ]Φn nin in

ni n

ref

ine C n e en

RC n en n( ) Re ( ) Re

( ) ( )( )( )θ

θθ α θ α≈ +−

+−−

− −K KD

11 1

L

D D D D( ) ( ) ( ) ( )θ θ θ θe e ei np

p n

i p nn p

i p n

p n

−

=− +

∞+ −

− + −− − +

=

∞= + +∑ ∑1

2

11

1

Re ( ) Re ( )( ) ( )D K D K*-

*−

=

∞− − +

−−

−=

∞− +∑ ∑=p

p n

i p nn

inp

p n

i p nn

ine C n e e C n en n11

11

θ α θ α

[ ]Φn nin in

np n

i p n p

ref

in

nn

ref

inn

p n

i p n p

ref

in

e C n e en

RC n e

nR

C n e en

RC n e

n n

n n

( ) Re ( ) Re( )

( )

Re( )

( ) Re( )

( )

( )

* ( )*

θ θ α θ α

α θ α

≈ +−

+−

+−

−−

=− +

∞+ − −

−− + −

−=

∞− + −

∑

∑

K KD

KD

KD

12

1

11

11

1

1 1

The amount of “Spurious” 2m-pole harmonics in the measured flux is:

′ ≈ −

+

− ′ − − − − − − − −C m e nR

C n eR

C n eim n

m

m n

ref

in n

m

m n

ref

inm n n( ) ( ) ( ) ( )( )*

( )*

α α α1 1 1 1 1KK

D KK

D

For a sin(pθ) displacement, possible values of m = (p+n–1), (p–n+1), (n–p–1)


Torsional Vibrations of the Rotation Axis

z1,0

z2,0

z2 = z2,0 exp[iθ + iT(θ)]

T(θ)

θ


X

Y

Let us consider a type of rotational imperfection where the position of therotating coil at angular postion θ is not at θ, but at an angle of θ + T(θ), asshown in the figure. Such an imperfection may result either from an actualtorsional vibration of the rotating coil, or it could be due to errors in thetriggering of the data acquision. In general, the angular shift is a function ofthe angle. The position of the coil is characterized by:

z z z z1 1 0 2 2 0= + = +, ,exp[ ( )]; exp[ ( )]i iT i iTθ θ θ θ

The flux through the coil is given by:

Φ( ) Re ( ) exp( )θ α=

−

−

=

∞

∑NLR

nC n in

R Rref

nn

ref

n

ref

n

1

2 1z z

The sensitivity of the “perfect coil” to the 2n-pole field is defined as:

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,


z1,0

z2,0


T(θ)

θ


X

Y

Torsional Vibrations of the Rotation Axis

Φ( ) Re ( )exp( )θ α=

−

−

=

∞

∑NLR

nC n in

R Rref

nn

ref

n

ref

n

1

2 1z z

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,

z z

z z1 1 0

2 2 0

= +

= +,

,

exp[ ( )]

exp[ ( )]

i iT

i iT

θ θ

θ θ

Φ( ) Re exp( ).exp{ ( )} ( ) exp( )θ θ θ α= −

=

∞

∑ Kn nn

in inT C n in1

In practice, the angular error, T(θ), is expected to be very small. Therefore,

exp{ ( )} ( )inT inTθ θ≈ +1

Φ( ) Re . ( ) Re ( ) . ( )θ θθ α θ α≈

+

−

=

∞−

=

∞

∑ ∑K Knin in

nn

in in

ne C n e inT e C n en n

1 1

To a good approximation, the amplitude of distortion in a given harmoniccomponent of the flux seen by the coil is proportional to the amplitude of thedistortion as well as the sensitivity of the coil to the 2n-pole terms.

If the magnet has only one dominant harmonic, then the effect of torsionalvibrations can be minimized by making the sensitivity of the coil (or an arrayof coils) zero for that particular hamonic. This is the basis for bucking outthe dominant harmonic term from the pick up signal. It should be notedthat if the magnet has large allowed or unallowed multipoles, the effect oftorsional vibrations is not completely cancelled.


z1,0

z2,0

z2 = z2,0 exp[iθ + iT(θ)]T(

θ)

θ


X

Y

Torsional Vibrations of the Rotation Axis: Periodic Displacements

Φ( ) Re . ( )

Re ( ) . ( )

θ

θ

θ α

θ α

≈

+

−

=

∞

−

=

∞

∑

∑

K

K

nin in

n

nin in

n

e C n e

inT e C n e

n

n

1

1

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,

If the displacement amplitude, T(θ), is a periodic function of θ :

T ippp

( ) exp( )θ θ==−∞

∞

∑T

T e e einp

p n

i p nn p

i p n

p n

( )( )

( ) ( )θ θ θ θ= + +=− −

∞+

− −− −

= +

∞

∑ ∑T T T1 1

[ ] [ ]Re ( ) Re ( )( ) ( )in e C n e in e C n ep ni p n in

p ni p n inn nT K T K* *

−− − −

−−= −θ α θ α

In a pure 2n-pole field:

[ ]

[ ]

Φn nin in

p np n

i p n in

n nin

p np n

i p n in

e C n e in e C n e

in C n e in e C n e

n n

n n

( ) Re ( ) Re ( )

Re ( ) Re ( )

( )

( )

* * ( )

θ θ α θ α

α θ α

≈ +

+ + −

−

=− −

∞+ −

−−

−= +

∞−

∑

∑

K T K

T K T K

1

1

The amount of “Spurious” 2m-pole harmonics in the measured flux is:

′ ≈

−

− ′−

−− −C m e in C n e C n eim n

mm n

in n

mm n

inm n n( ) ( ) ( )*

*α α αKK

TKK

T

If T(θ) has a simple angular dependence of the form Tcos(pθ) or Tsin(pθ)then it will produce spurious harmonics corresponding to 2(n+p)-pole and2|(n–p)|-pole field.


Example of a Practical Radial Coil with Bucking

HERA Dipole Coil

Coil A: Main Coil

Coil B: Dipole Bucking Coil

Coil C: For compensating anyangular misalignmentof Coils A and B.

Condition for Bucking theDipole field component:

N r r N r rA B( ) ( )2 1 3 4− = +

HERA Quadrupole Coil

The compensated signal is:

V V V VA B cbucked = − −

Coil D: For compensating any angularmisalignment of Coils.

Condition for Bucking the Dipole fieldcomponent:N r r N r r N r rA B C( ) ( ) ( )2 1 3 4 5 6 0+ − + − + =

Condition for Bucking the Quadrupolefield component:

N r r N r r N r rA B C( ) ( ) ( )22

12

42

32

62

52 0− − − − − =

Other designs also exist with a similar philosophy.


Example of a Practical Tangential Coil: the RHIC Coils

Tangential Winding: 15 degrees opening angle, 30 TurnsDipole Buck Windings: 3 Turns each, at ±49.260 deg. wrt tangentialQuad Buck Windings: 3 Turns each, at ±24.840 deg. wrt tangential

Analysis is based on the Voltage signals, rather than the integrated voltagesignals (the flux). The bucked signal is defined as:

V V f V f V f V f Vbucked tangential DB1 DB2 QB1 QB2= + + + +1 2 4 5

With the above design, f1 = f2 = f4 = f5 = –1 to buck out the dipole and thequadrupole components in the bucked signal.

All RHIC measuring coils are built with the same basic design. The radii ofthe windings are scaled to suit the aperture of the magnet to be measured.


Analog Bucking

In Analog Bucking, the various coil signals are added before doing anyanalysis. The FFT is also carried out on one (or more) signal directly to getthe main harmonic component. The summing circuit must be tuned toprecisely cancel (a bucking factor of at least several hundred) the dipole andthe quadrupole components.

Data AcquisitionV(t), ∫V.dt

FFT

SU

M Data AcquisitionV(t), ∫V.dt FFT

"Main" Coil

Other Coils

Calculatethe MainHarmonic

CalculateOtherHarmonics

Digital Bucking

In digital bucking, all the coil signals are acquired without any summing. Thesumming coefficient for each signal is determined from a FFT analysis. Thesecoefficients are then calculated based on which two harmonics are to beeliminated in the bucked signal. The bucked signal is then digitallyconstructed and Fourier analyzed to get the harmonics.

V

VV

VV

FFT

Coi

l Sig

nals Calculate

BuckingFactors

(MagnetSpecific)

CalculateBuckedSignal

FFT

CalculateHarmonicAmplitudesandPhases


Bucking Algorithm for the RHIC Coil inVarious Magnets

Five Windings:

DB1: Dipole Coil

DB2: Dipole Coil

T: Tangential

QB1: Quad Coil

QB2: Quad Coil

DB1,DB2: Sensitive to Dipole, Sextupole, Decapole, etc. terms.QB1,QB2: Sensitive to Quadrupole, Dodecapole, etc. terms.T: Sensitive to all harmonics of interest.

Goal: To buck the most dominant, and the next lower order harmonic forany magnet. (Not achieved for Octupole and Decapole magnets)

MagnetType

UseDB1,DB2to buck

UseQB1,QB2to buck

Calculation of Harmonics

Dipole Dipole Quadrupole(optional)

♦ Dipole from DB1,DB2♦ Quad from QB1,QB2 (if used)♦ Rest from Bucked signal.

Quadrupole Dipole Quadrupole♦ Dipole from DB1,DB2♦ Quad from QB1,QB2♦ Rest from Bucked signal.

Sextupole Sextupole Quadrupole♦ Sextupole from Tangential♦ Quad from QB1,QB2♦ Rest from Bucked signal.

Decapole Decapole Quadrupole(Optional)

♦ Decapole from Tangential♦ Quad from QB1,QB2♦ Rest from Bucked signal.

Dodecapole Decapole Dodecapole♦ Dodecapole from Tangential♦ Decapole from DB1,DB2♦ Rest from Bucked signal.


Calculation of the Bucked Signal

DB1: N1, R1, δ1

DB2: N2, R2, δ2

T: N3, R3, δ3, ∆QB1: N4, R4, δ4

QB2: N5, R5, δ5

V V f V

f V f V

f V

bucked tangential DB1

DB2 QB1

QB2

= +

+ +

+

1

2 4

5

The values of the coefficients f1, f2, f4, f5 are calculated from the Fourieranalysis of the individual signals in such a way as to completely cancel two ofthe harmonics chosen according to the type of the magnet being measured.

If n1 is the harmonic to be cancelled with the DB1 and DB2 windings, thenthe design values of f1 and f2 are given by:

{ }{ }{ }{ }

fNN

RR

nn

n n

fNN

RR

nn

n n

n

n

13

1

3

1

1 3 2

1 2 1

1 1

23

2

3

2

1 1 3

1 2 1

1 1

1

1

2 2

2 2

=

−−

=

−−

sin ( )sin ( )

sin sin

sin ( )sin ( )

sin sin

δ δδ δ

π

δ δδ δ

π

∆

∆

Similarly, if n2 is the harmonic to be cancelled using the QB1 and QB2:

{ }{ }{ }{ }

fNN

RR

nn

n n

fNN

RR

nn

n n

n

n

43

4

3

4

2 3 5

2 5 4

2 2

53

5

3

5

2 4 3

2 5 4

2 2

2 2 4

2 2 4

2

2

=

−−

=

−−

sin ( )sin ( )

sin sin

sin ( )sin ( )

sin sin

δ δδ δ

π

δ δδ δ

π

∆

∆

n1 f1 f2 n2 f4 f5

1 –1.00 –1.00 2 –1.00 –1.003 –2.26 –2.26 6 –2.06 –2.065 +7.57 +7.57 — — —


Effect of the Finite Size of the Coil Windings

In practice, the coil windingsare not point-like. Toaccommodate the necessarynumber of turns, the windingmust have a finite crosssection. This could introduceerrors in the measurement ofthe amplitude of the harmonics.Although typical winding crosssections are rectangular, it isconvenient to approximate itwith a sector of an annulus, asshown in figure. The windingis assumed to have an angularwidth of (2α) and thickness(2δ). The mean position of thewinding is denoted by z0 = R.exp(iφ). The sensitivity factor of the winding tothe 2n-pole term involves the quantity zn. The value of zn averaged over thecross section of the winding is:

∫ ∫ [ ][ ]( )

exp( )

( )( )

( ) ( )

( )

exp( )sin( )

( )( / )

sin( )

.

( ) ( )

z

z

navg

n

R

Rn n in in

n

n n

n

n n

r dr in d R R e e

in n

R inn

n

R R

n R

nn

R R

= =+ − − −

+

=

+

− −

+

=

+

− −

−

+

−

++ + + −

+ +

+ +

δ

δ

φ α

φ αφ α φ αφ φ

α δ

δ δ

αδ

φα

α

δ δ

δ

αα

δ δ

2 2 4 1

1 1

2 1

1 1

1 1

1 1

0

1 1

2 1( )( / )n R+ δ

If the winding is assumed to be point like and located at the geometric center,the above formula gives the error in estimating the amplitude of the 2n-poleterm. Expanding in a power series, it can be shown that the leading correctionterms are of the second order in α and (δ/R).

X

Y

2δ

φ

α

z0

z0 = R exp(iφ)


Effect of a 1 mm × 1 mm Winding Cross Section

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16

Harmonic Number (n)

Err

or

in A

mpl

itude

(%

)

25mm Coil Radius10mm Coil Radius

For a 25 mm average radius of the measuring coil, the errors introduced witha 1 mm wide and 1 mm thick winding are negligible for all the harmonics ofinterest. For a smaller radius coil, the errors are more pronounced, asexpected. However, even for a 10 mm radius coil, the errors for theharmonics of interest are still less than one percent. The finite size may,however, be a serious limitation in measuring the transfer function of highermultipole magnets (such as sextupoles, octupoles, etc. correctors) with a smalldiameter measuring coil.


Random Variation of Winding Radius Along the Length

L

R(z)

z

R

Let us consider one segment of the coil loop in either a radial or a tangentialcoil. The radius is assumed to vary randomly along the length L of the coilwith a mean value of Rc and a standard deviation of σR. The effectivesensitivity factor of the coil for the n-th order harmonic is proportional to Rn.

[ ]R z R zL

R z dz R z dzL

R z R dzc

L

c

L

R c

L

( ) ( ); ( ) ; ( ) ; ( )= + = = = −∫ ∫ ∫ε ε σ1

01

0 0

2 2

0

[ ]∫1

11

2

11

2

02

2

0

2

LR z dz

RL

nR

zn n

Rz dz

Rn n

R

nL

cn

c c

L

cn R

c

( ) ( )( )

( )

( )

= + +−

+

⌠

⌡

≈ +−

ε ε

σ

L

The sensitivity factor for the n-th harmonic is given by:

K Kn nideal R

c

n nR

≈ +−

11

2

2( ) σ

In order to keep the error in the amplitude of the n = 15 term less than 1%,we should have σ R cR≤ −10 2 . A somewhat tighter tolerance may be requiredif such a coil is to be used to determine the transfer function in a magnet ofhigher multipolarity, such as in a dodecapole corrector magnet.


Random Variation of Angular Position Along the Length (Twist)

L

δ(z)z

δ

Let us consider a tangential coil in which the radii and opening angle areuniform along the length. However, the angular position, δ, is assumed tovary randomly along the length L of the coil with a mean value of δc and astandard deviation of σδ.

[ ]δ δ ε δ δ ε σ δ δδ( ) ( ); ( ) ; ( ) ; ( )z zL

z dz z dzL

z dzc

L

c

L

c

L

= + = = = −∫ ∫ ∫1

01

0 0

2 2

0

The flux seen by the coil for the 2n-pole field is:

∫Φn c n

Lt

LC n n t n n z n dz( ) ( )sin( ( ) )∝ + + −

1

0ω δ ε α

Expanding sin[nε(z)] and cos[nε(z)] in power series and retaining only theterms up to the second order, it is easy to show that:

∫Φn c n

L

c n

L

c n

tL

C n n t n nn

z dznL

C n n t n n z dz

C n n t n nn

( ) ( )sin( ) ( ) ( )cos( ) ( )

( )sin( )

∝ + − −

⌠

⌡ + + −

= + − −

11

2

12

22

00

22

ω δ α ε ω δ α ε

ω δ α σδ


K Kn nideal n

≈ −

12

22σδ


Random Variation in Opening Angle Along the Length (Tangential Coil)

L

∆(z)

z

∆

Let us consider a tangential coil in which the radii and the mean angularposition are uniform along the length. However, the opening angle, ∆, isassumed to vary randomly along the length L of the coil with a mean value of∆c and a standard deviation of σ∆.

[ ]∆ ∆ ∆ ∆ ∆ ∆∆( ) ( ); ( ) ; ( ) ; ( )z zL

z dz z dzL

z dzc

L

c

L

c

L

= + = = = −∫ ∫ ∫ε ε σ1

01

0 0

2 2

0

The flux seen by the coil for the 2n-pole field is:

Φ∆

n

L

Ln z

dz( ) sin( )

θ ∝

⌠⌡

12

0

Expanding sin[nε(z)/2] and cos[nε(z)/2] in power series and retaining onlyup to the second order terms, it is easy to show that:

∫Φ∆ ∆

Φ∆

∆

nc

L

cL

nc

Ln n

z dznL

nz dz

n n

( ) sin ( ) cos ( )

( ) sin

θ ε ε

θ σ

∝

−

⌠

⌡ +

∝

−

12

18 2 2

21

8

22

00

22


K Kn nideal n

≈ −

18

22σ∆


Imperfect Tangential Coil : Unequal Radii of the Two Grooves

Let us consider aslightly imperfecttangential coil wherethe two sides of the coilloop are not at the sameradius. Such animperfection can bereal, resulting fromunequal depths ofgrooves in the coilform. Even with aperfectly built coil,such an imperfectionwill be apparent if therotation axis does notexactly coincide withthe geometric center ofthe windings. It isassumed here that the wires are located at radii of Rc – ε and Rc + ε.

Φ( ) Re exp( ) ( )exp( )θ θ α= −

=

∞

∑Knn

nin C n in1

;

Kz z

nref

ref

n

ref

nNLR

n R R=

−

2 0 1 0, ,

z z1 0 2 02 2, ,( ) exp( / ); ( ) exp( / )= − = + −R i R ic cε ε∆ ∆

z z2 0 1 0 2 2

22 2

, , ( ) exp( / ) ( ) exp( / )

sin cos

n nc

nc

n

cn

c

R in R in

R in n

Rn

− = + − − −

≈ −

+

ε ε

ε

∆ ∆

∆ ∆

The first term can be identified to be related to the sensitivity of the perfectcoil. The second term implies that both amplitude and phase errors areintroduced in the sensitivity factor. Also, for coils such as the dipole coil with∆ = π, the flux for a perfect coil is zero for even harmonics. This is no longerthe case with an imperfect coil. However, the allowed terms for a dipole coilare not affected since cos(nπ/2) = 0 for odd values of n.

Rc

∆

θ

ω

X

Y

R c – ε

Rc + ε


Imperfect Tangential Coil : Unequal Radii of the Two Grooves

Rc

∆

θ

ω

X

Y

R c – ε

Rc + ε

z z2 0 1 02 2 2

2 2, ,/ /( ) ( ) sin cosn n

cn in

cn in

cn

cR e R e R i

n nR

n− = + − − ≈ −

+

−ε εε∆ ∆ ∆ ∆

Assuming that sin(n∆/2) is not zero, as is the case for the harmonics ofinterest in a practical tangential coil (∆≈15°):

z z2 0 1 0 22

12

22, , sin cot sin exp( )n n

cn

ccn

n niRn

inR

niR

nA in− ≈ −

+

= −

∆ ∆ ∆ελ

where An is an amplitude correction term and λn is a phase error given by:

AnR

n nR

nn

c c= +

≈ +

12

12 2

22

2 22ε ε

cot cot∆ ∆

λε ε

nc cn

nR

nR

n=

≈

−12 2

1tan cot cot∆ ∆

The amplitude error is of the second order in (ε/Rc) and can generally beneglected. For typical values of (ε/Rc) ~ 10–3, the phase error could be severalmilli-radians for the lowest order harmonics. The phase error reduces with theorder of the harmonic as roughly (1/n). The expression for flux is given by:

Φ∆

( ) sin ( )sin{ ( )}θ θ α λ≈

− −=

∞

∑2

21

NLR

nR

Rn

A C n n nref c

ref

n

nn n n


Offset in the Rotation Axis

K z z

K z z z z

nn n

nn n

∝ −

′ ∝ + − +

2 0 1 0

2 0 0 1 0 0

, ,

, ,( ) ( )∆ ∆

∆ ∆ ∆KK

z z z z

z zn

n

n n

n n=+ − +

−−

( ) ( ), ,

, ,

2 0 0 1 0 0

2 0 1 01

The sensitivity factor for the dipole term is not affected. For other harmonics,

( )( )

∆ ∆∆

∆K

Kzn

n k

n

c

k n k

nn

k n k R

=

−

=

−−

∑tangential

!!( )!

sin

sin

( )

1

10 2

2

( )∆∆

KK

zn

n k

nk

n k n k

n nn

k n kR R

R R

=

−

−−=

− − −

∑radial

!!( )!1

1

02 1

2 1

Usually, a first order approximation is adequate:

( ) [ ][ ]

( )

∆∆

∆ ∆

∆∆

∆

∆K

Kz

z z

z zK

Kz

KK

z

n

n

n n

n nn

n c

n

n

n

n

n n

n n

n nR

nR R

R R

≈−

−

≈

≈

−−

− − −

− −

02 0

11 0

1

2 0 1 0

0

12

2

02

11

1

2 1

, ,

, ,

( )

;sin

sintangential

radial

These results may also be used to estimate the effect of a “bow” or a bend inthe measuring coil. Different sections of such a coil will rotate about an axiswhich is offset from the geometric center by different amounts. An upperbound on the resulting effect can be obtained by equating ∆z0 to the total bendin the measuring coil.

X

Y

z1,0

z2,0

∆z0

RotationAxis


Offset in the Rotation Axis

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

0 2 4 6 8 10 12 14 16Harmonic Number (n)

Err

or

in A

mp

litu

de

(%) f

or

0.1

mm

Axi

s O

ffse

t

Tangential Coil (25mm, 15 deg.)Radial Coil (25mm, 8mm)

Percent error in the amplitudes of various harmonics with a 0.1 mm offset inthe rotation axis. The tangential coil is assumed to have a radius of 25 mmand opening angle of 15 degrees, while the radial coil is assumed to have theradii R2 = 25 mm and R1 = 8 mm (~ R1/3).


Systematic Errors in Coil Parameters (Calibration Errors)

The coil parameters of primary interest are the radius (R), the angular positionat the start of the data acquision (δ), and in the case of a tangential coil, theopening angle (∆). A systematic error in the knowledge of these parameterswill result in systematic errors in the calculation of the field parameters,namely the amplitudes C(n) and the phase angles αn.

Systematic Error in the Radius:

KK

Knn n

nR n

RR

∝ ∴ =

;∆ ∆

where Kn is the sensitivity factor for the n-th harmonic and ∆R is thesystematic error in the radius R. For a (∆R/R) ~ 10–3, the systematic error inthe amplitude of the 20-pole term will be ~ 1%.

Systematic Error in the angular Position:

A systematic error, εδ , in the initial angular position, δ, leads to the sameerror in the determination of all the phase angles. This would give rise toskew terms in a purely normal magnet, and vice versa.

[ ]α α ε α α ε εδ δ δn n n nC n in C n in n i n→ − − → − +; ( ) exp( ) ( )exp( ) cos( ) sin( )

The multipoles in a magnet are generally expressed in a reference framewhere the main field component has a zero phase angle. In this case, therewill be no systematic error in the multipoles, since the phase angles relative tothe main field still remain the same. However, when accurate determinationof the field direction of the main component is required, such a systematicerror is unacceptable. Efforts must be made to periodically check thecalibration, and/or correct for the errors by making measurements from thelead and non-lead ends of the magnet. For a 2m-pole magnet, the measuredphase angles from the lead and the non-lead ends are:

[ ]( ) ( )[ ]

( ) ( )[ ]

α α ε απ

α ε

ε α α

α α α

δ δ

δπ

π

lead non lead

lead non lead

lead non-lead

= − = + −

− −

∴ = + − − −

= − + + −

−

−

mm

m

mm

mm

m

m; ( )

{ ( ) }

{ ( ) }

1 12

1 1

1 1

12 2

12 2


Systematic Error in the Opening Angle of a Tangential Coil

The sensitivity factor, Kn, of a tangential coil depends on the opening angle,∆, as:

Knn

∝

sin∆2

For a systematic error ε∆ in the opening angle,

∆ ∆∆

KK

n

n

n n=

2 2

cot ε

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 2 4 6 8 10 12 14 16

Harmonic Number (n)

Err

or

in A

mpl

itude

(%

)

Effect of a 1 mrad systematic error in the determination of the opening angleof a tangential coil (∆ = 15 degrees). The error is significant for the lowerorder harmonics. In a typical 5 winding tangential coil system, the dipole andthe quadrupole terms are obtained from the dipole (∆ = 180 degrees) and thequadrupole windings (∆ = 90 degrees) which are practically insensitive to thiserror.


Effect of a Finite Averaging Time

In the acquisition of voltage data from the RHIC tangential coils, the signalsare averaged over one power line cycle to get rid of any AC noise on thesignals. At a typical angular speed of one revolution every 3.5 seconds and apower line frequency of 60Hz, the coil rotates about 1.7 degrees during onepower line cycle. This motion during data integration can cause errors.

If ∆t is the averaging time, the n-th harmonic component in the measuredvoltage signal is:

∫

[ ]

V tt

n t n n dt

tn t n n n t n t n n

n tn t

n t nt

n

n nt

t t

n n

n

( ) cos( )

sin( ) sin( )

sin( / )/

cos

∝ + −

= + − + − + −

= + +

−

+1

1

22 2

∆

∆

∆ω δ α

ω δ α ω∆ ω δ α

ω∆ω∆

ω δω∆

α

Amplitude Correction Factor = sin( / )

/.

n tn t

nt

Tω∆

ω∆2

21 164 2

2

≈ −

∆

Effective Angle Calibration = ′ = + = +

δ δ ω∆ δ π( / )tt

T2

∆

where T is the period of rotation. For ∆t = 1/60 sec. and T = 3.5 sec,(∆t/T) ~ 4.8×10–3.

The amplitude error is 0.004% for the dipole term and is 0.84% for the30-pole term. This effect is negligible. However, the angle calibration isaffected by roughly 0.86 degrees (15 mrad). Fortunately, the error is harmonicindependent, and can be absorbed in the calibration of the coil, as long as theangular velocity is kept the same. Considerable error will result, for example,if the coil were to rotate in the opposite direction (ω → –ω).

In practice, the coil rotation period may not be the same during calibrationand measurements. It is necessary, therefore, to specify the rotation speed ofthe measuring coil along with the angular parameters. Corrections must beapplied to the calibration values based on the actual rotation speed during themeasurements according to the above equations.


Rotation Axis Different from the Magnetic Axis

If the rotation axis of the coil isnot coincident with the magneticaxis of the magnet, the measuredharmonic coefficients areaffected by feed down.

O: Center of Measuring CoilO’:Magnetic Center

z0 = x0 + iy0 = Location ofmagnetic center in themeasuring coil frame.

If C(n) and αn are the measuredparameters in the measuring coilframe, then the parameters in the magnet’s frame are given by:

′ − ′ = −−

− −

=

∞ −

∑C n in C k ikk

n k n Rn kk n ref

k n

( )exp( ) ( )exp( )( )!

( )!( )!α α

11

0z

Magnetic center is defined as the location where an appropriate harmonic iszero. For example, for a 2m-pole magnet other than a dipole, the magneticcenter is defined as the location where the 2(m–1)-pole term is zero.

′ − = − + −

+ +

+ +

+

=

− − ′ − − −

− − +

− − +

− −

+

+

C m e C m e m C m e

C m e

C m e

i m i m imR

m m i mR

m m i mR

m m mref

mref

mref

( ) ( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

( ) ( )

1 1 1

1

2

0

1 1

12

12

16

23

1 1 0

1 0

22 0

α α α

α

α

z

z

z L

For most magnets, terms other than C(m) are small. For small offsets,

{ }zz0

0 111

1 11

R mC m i m

C m imm R

ref

m

mref

≈ −

−− − −

−≠ <<−

( )( )exp ( )

( )exp( ); ;

αα

X

Y

X '

Y '

O

O'

r 0

x0

y0ξ

z0 = x0+ iy0

= r0. exp (i ξ)

Meas. Coil Frame

Magnet Frame


Rotation Axis Different from the Magnetic Axis

X

Y

X '

Y '

O

O'

r 0

x0

y0ξ

z0 = x0+ iy0

= r0. exp (i ξ)

Meas. Coil Frame

Magnet Frame

′ − = − + −

+ +

+ +

+

− − ′ − − −

− − +

− − +

− −

+

+

C m e C m e m C m e

C m e

C m e

i m i m imR

m m i mR

m m i mR

m m mref

mref

mref

( ) ( ) ( ) ( )

( )

( )

( ) ( )

( ) ( )

( ) ( )

1 1 1

1

2

1 1

12

12

16

23

1 1 0

1 0

22 0

α α α

α

α

z

z

z L

For dipole magnets, no “natural” definition of a center can be used. Variousstrategies are used to define the center of a dipole magnet. For example, onecould argue that the very high order unallowed terms are not sensitive tosmall construction errors, and hence must be zero. If so, one could pick m inthe above expression to be a sufficiently high order allowed term andcalculate the center by requiring C’(m–1) to be zero. Of course, this worksonly if C(m) itself has sufficient strength. Because of the large values of m,the measured coefficients are of comparable strengths for both the allowedand the unallowed harmonics, even with relatively small offsets. As a result, itis often necessary to use higher order terms to calculate the offset. In manycases, ambiguous results may be obtained because of the non-linear nature ofthe equations. To resolve this, it is best to find a offset that willsimultaneously minimize several unallowed harmonics, rather than just one.

Other strategies for dipoles include the hysteretic centering, or minimizationof current dependence of quadrupole terms (cold data) or an “ugly quad”method, used with much success for both warm and cold data at RHIC.


The Quadrupole Configured Dipole (“Ugly Quad”) Method

UPPERCOIL

LOWERCOIL

I

I

DIPOLE MODE

UPPERCOIL

LOWERCOIL

I

I

QCD MODE

The quadrupole configured dipole method relies on powering the two coilhalves of a dipole magnet with opposite currents to produce a strong skewquadrupole field, instead of a dipole field, as shown above. This requires acenter tap connection on the magnet. The allowed harmonics are now theskew quadrupole, skew octupole, skew dodecapole, and so on. Several ofthese allowed harmonics are quite strong, and feed down from any one ofthem could be used to calculate the center.

Since two separate power supplies are required in this mode, it is important tobalance the current in the two halves with great accuracy, otherwise aspurious dipole field will also be generated that would affect centeringcalculations. The sensitivity to any current mismatch can be greatlyminimized by using feed down from the skew octupole term, rather than thedominant skew quadrupole term.

This method has been used at RHIC with great success. The results fromQCD method have very little noise (typically only a few microns) and agreevery well with the centers calculated by making high order unallowed termszero.


Sag of the Measuring Coil Due to its Own Weight

h

–L/2 +L/2

Zr0(Z)

For a long and thin measuring coil, the weight of the coil itself may beenough to cause a sagitta in the coil. In this case, each subsection of the coilrotates about its local geometric center. However, the location of this centervaries along the length of the magnet, as shown by the dashed line in theabove figure. This is different from a “bow” or bend in the coil. Varioussubsections of the coil see harmonics that are in a frame which is slightlydisplaced from the adjacent subsections. If r0(Z) is the vertically downwardoffset at axial position Z, the measured coefficients are given by:

{ }′ − ′ = −

− − −

− −

⌠

⌡

=

∞

−

−

∑C n in C k ikk i k n

n k n Lr ZR

dZn kk n ref

L

L k n

( )exp( ) ( )exp( )( )!exp ( )

( )!( )!( )

/

/

α απ1

112 0

2

2

For a parabolic profile given by r Z hZ

L0

2

212

( )( / )

= −

, it can be shown that:

{ }′ − ′ = −

− − −

− −−

− +

=

∞ −

∑C n in C k ikk i k n

n k nk n

k nh

Rn kk n ref

k n

( )exp( ) ( )exp( )( )!exp ( )

( )!( )![ ( )]!!

[ ( ) ]!!α α

π1

12

2 12

where (2k)!! = 2.4.6.8. ... 2k, (2k+1)!! = 1.3.5. ... (2k+1) and 0!! = 1.

For small values of h, the effect of sag is the same as a uniform displacementof the coil by an amount (2h/3). If the measured data are corrected for theoffset of the measuring coil axis, most of the errors due to sag are alsosubtracted out, except for terms of second and higher orders in (h/Rref) whichcan be neglected for reasonable values of h.


Measuring Coil Axis Tilted wrt the Magnet Axis

Meas. Coil Axis

Lmagnet

ξr0X

YMagnet Axis

r0

Y X

Average displacement of the measuring coil = 0Displacements at the two ends are given by (r0, ξ) and (r0, ξ+π)

If the field quality of the magnet is uniform along the length, then the oddorders of feed down from one half of the magnet will be cancelled by thecorresponding feed down from the other half of the magnet. The even ordersof feed down from the two halves will add to each other. The measuredcoefficients in this case are given by:

′ − ′ =−

− +−

− −

=− =

∞ −

∑C n inC k ik

k nk

n k nr i

Rnk

k nk n

ref

k n

( )exp( )( )exp( )

( )( )!

( )!( )!exp( )

( )

αα ξ

11

10

even

It should be noted that the summation includes only those values of k forwhich (k–n) is even. The lowest order correction term is of second order in(r0/Rref), and can be neglected in most cases for dipole and quadrupolemagnets, since there can not be any second order feed down from the mainfield component. However, for sextupoles and magnets of highermultipolarity, there can be a second order feed down from the main harmonic,leading to large errors even with relatively small tilt. For example, the dipolefield component will be incorrectly measured in a sextupole magnet, thequadrupole component in an octupole magnet, and so on.

Often, magnets have rather large harmonics in the lead end region which areabsent in the non-lead end region. In this case, even the first order terms fromthe two halves will not cancel each other, causing large errors. Examples ofsuch errors are in the measurement of integral decapole terms in a quadrupolemagnet having large dodecapole terms in the lead end region.


Effect of Tilt on Measurements of Sextupole Magnets

Granite Table Measurements: Measuring Coil Axis Parallel to Magnet AxisVertical Dewar Measurements: Measuring Coil Axis may not be Parallel.

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

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

Normal Octupole (Granite Table) in units

Nor

mal

Oct

upol

e (D

ewar

) in

uni

ts

-0.149 (Sigma: 0.5173)

Octupole terms are not affected and show good correlation

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30 35 40 45

Normal Dipole (Granite Table) in units

Nor

mal

Dip

ole

(Dew

ar)

in u

nits

-5.345 (Sigma: 9.0409)

Dipole terms are affected and show poor correlation


Calibration of a Five-Winding Tangential Coil

Rc

∆

θ

ω

X

Y

R c – ε

Rc + ε

1. Dipole Buck Winding (DB1): R1, δ1

2. Dipole Buck Winding (DB2): R2, δ2

3. Tangential Winding (T): R3, δ3, ∆, ε4. Quadrupole Buck Winding (QB1): R4, δ4

5. Quadrupole Buck Winding (QB2): R5, δ5

Total Number of Parameters required = 12.

If a pure Dipole AND a pure Quadrupole magnet are available, with welldefined phase angles, then the angle parameters for the four buck windingscan be obtained without any difficulty. The measured angle of the tangentialin the two fields may not be the same due to the tilt, ε. However, with wellcalibrated dipole and quadrupole fields, one could estimate the tilt using theexpressions for the effect of a tilt.

• What if such calibrated magnets are not available ?

• How to get the various radii ?

• How to get the opening angle, ∆ ?

It is possible to get all the angles relative to each other, all the radii relative toeach other, the absolute value of the opening angle as well as the tangentialtilt — all without using any knowledge of the field strength or direction !


Calibration of a Five-Winding Tangential Coil: Radii, ∆

Rc

∆

θ

ω

X

Y

R c – ε

Rc + ε

Effect of a “Tilt” in the Tangential winding:

Amplitude Corr. Factor = AnR

nn

c= +

12

22ε

cot∆

: NEGLECT (2ND ORDER)

Phase Correction λε

ncn

nR

n=

−12

1tan cot∆

; δ δ λ3 30( )n n= +

Vj(n) = Amplitude of n-th harmonic in the j-th winding.

In a Dipole field, assuming a measuring coil longer than the magnet:

( )V N R V N R V N R1 1 1 2 2 2 3 3 31 1 1 2( ) ; ( ) ; ( ) sin /∝ ∝ ∝ ∆

∴

=

=

RR

VV

NN

RR

VV

NN

2

1

2

1

1

2

3

1

3

1

1

3

11 2

11

( )( )

; sin( )( )

∆

Similarly, in a quadrupole field:

( )V N R V N R V N R4 4 42

5 5 52

3 3 322 2 2 2 2( ) ; ( ) ; ( ) sin∝ ∝ ∝ ∆

( )∴

=

=

RR

VV

NN

RR

VV

NN

5

4

5

4

4

5

1 23

4

23

4

4

3

22

22

2( )( )

; sin( )( )

/

∆

If the field strengths in the dipole and the quadrupole magnets are alsoknown, one can obtain the absolute values of R1, R2, R4, R5. From thesevalues, the absolute values of R3 and ∆ can also be determined. If the absolutestrengths are not known, one can only calculate the ratios of radii. Even then,we have only four equations in five unknowns and need more information.


Calibration of a Five-Winding Tangential Coil: Radii, ∆

Rc

∆

θ

ω

X

Y

R c – ε

Rc + ε

Vj(n) = Amplitude of n-th harmonic in the j-th winding.

From data in a Dipole field:

RR

VV

NN

RR

VV

NN

2

1

2

1

1

2

3

1

3

1

1

3

11 2

11

=

=

( )( )

; sin( )( )

∆

From data in a Quadrupole field:

( )RR

VV

NN

RR

VV

NN

5

4

5

4

4

5

1 23

4

23

4

4

3

22

22

2

=

=

( )( )

; sin( )( )

/

∆

To know relative radii, five Unknowns: (R2/R1), (R3/R1), (R4/R1), (R5/R1), ∆Need at least one more equation. If we use a sextupole field:

( )V N R V N R V N R1 1 13

2 2 23

3 3 333 3 3 3 2( ) ; ( ) ; ( ) sin /∝ ∝ ∝ ∆

( )∴

=

=

RR

VV

NN

RR

VV

NN

2

1

2

1

1

2

1 33

1

33

1

1

3

33

3 233

( )( )

; sin /( )( )

/

∆

These equations give the five unknowns required to calculate the fieldstrengths, namely, (R2/R1), (R3/R1), (R4/R1), (R5/R1) and ∆. We get oneredundant equation, which could be used for consistency check on (R2/R1).To know the absolute values of the radii, we need to determine just oneradius. This could be obtained easily if a reference field is available.Otherwise, the value of R1 (or any other winding) can be simply guessed frommechanical measurements, or calibrated against other known coils.


Calibration of a Five-Winding Tangential Coil: Angles and ε

Rc

∆

θ

ω

X

Y

R c – ε

Rc + ε

If reference dipole and quadrupole magnets are available with preciselyknown phase angles, the angles δ1, δ2, δ4, and δ5 can be easily determinedfrom the phases of the dipole and the quadrupole components of the measuredsignals. Also, we can get δ3(1) in a dipole magnet and δ3(2) in a quadrupolemagnet. These can be used to obtain δ3

0 and (ε/R3) using the relations:

δ δ λ3 30( )n n= + ; λ

εn

cnnR

n=

−12

1tan cot∆

If the absolute phase angles of both the dipole and the quadrupole fields arenot known, we make use of a sextupole field also. We can determine:

In Dipole field: δ2 – δ1 and δ3(1) – δ1

In Quadrupole field: δ5 – δ4 and δ3(2) – δ4

In Sextupole field: δ2 – δ1 and δ3(3) – δ1

Combining the data from the dipole and the sextupole fields, we get thequantity δ3(3) – δ3(1), which depends on (ε/R3) and the opening angle, ∆.Since ∆ is obtained from calibration of the radii, the parameter (ε/R3) isdetermined. Knowing (ε/R3), one can calculate δ3(2)–δ1, which can becombined with the data in a quadrupole field to get δ4 and δ5 also relative toδ1. All angles are thus known relative to one of the windings. For measuringcoils equipped with a gravity sensor, the absolute angles can be obtained bymaking measurements from the lead and the non-lead ends of a magnet. Forother systems, absolute values of angles are often unnecessary.

measurements of field quality using harmonic coils · a “dipole coil” is therefore sensitive to...

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