# Presentation: 333452

##                                       Magnet    Design

-                                       Magnet    Design

-               W. Meng
-             10 May 2001

## Magnet Design

- ----- overview

-                    Steps in Designing a Magnet
- (A) Define Specifications ---
-           B,  or G (dB/dr, d2B/dr2 ), and integral parameters
-           uniformity, tolerances, ...
- (B) Field Analysis ---
-          Determine ampere-turn, pole shape, steel length ...
-          Compute analytic parameters to match (A), estimate
-          field qualities
- (C) Engineering Design ---
-          Detailed design on mechanical structure, construction
-          techniques, conductor cooling, resistance, voltage, ...
-          estimate assembling error, cost ...

## Magnet Design

- -----

- Non-linear Materials --- steel, ferrite
-     B = o (H +M)  (si)             B = H + 4  M (cgs)
- Finite Element Programs ---
-      Poisson Group Code (2d: **A**)
-      Opera (2d: **A**;  3d:  or **A**)
-      MAFIA (finite integration)
-      Ansys ( or **A)**

## Magnet Design

- ----

-                   Dipole Magnet      B = Bo (constant)
- Inside Steel: H=B/(o r) 0

- NI=**H•dl**
-     = (Bo/o) (gap/2)
-    (Bt  Bo)

## Magnet Design

- ----

-                    Quadrupole Magnet      G = dB/dr = constant
-                                                           B(r) =  G dr = G r
- Inside Steel: H=B/(o r) 0

- NI=**H•dl**
-     = B/odr
-     = G Ro2/(2 o)
-     = Bt Ro/(2 o)
-           (Bt = G Ro)

## Magnet Design

- ----

-                   Sextupole Magnet      G = d2B/dr2 = constant
-                                                       B(r) =  (G dr) dr = G r2/2
- Inside Steel: H=B/(o r) 0

- NI=**H•dl**
-     = B/odr
-     = G Ro3/(6 o)
-     = Bt Ro/(6 o)
-           (Bt = G Ro2)

-   **2d** Multipole Expansion of  Complex Potential --- F(z) = A(x,y) + i(x,y)
-              z = x + i y;                                                        = (z - zo)n = (x+iy)n
-                A(x,y) = const (Flux lines)                (x,y) = const (Equ pot lines)
-                Re                                                  Im

- Magnet Design

- n=1       x                                             y
- n=2       x2 - y2                                     2xy
- n=3      x3 -3xy2                                 3x2y - y3
- n=4    ......

##  B(z)  = By + iBx = [bn + i an] (z/Rref)n-1             n = 1, 2...

-    Multipole Expansion of  Complex Field ---
-              z = x + i y;                        z = r (cos  + i sin)

-  B(z)  = By + iBx = [bn + i an] (z/Rref)n-1             n = 1, 2...
-                                                                                      D,Q...
-          Bx = [an cos(n-1) + bnsin(n-1)](r/Rref)n-1
-          By = [bn cos(n-1) + ansin(n-1)](r/Rref)n-1
- Br(r,) = [bn sin(n-1) + ancos(n-1)](r/Rref)n-1
- B(r,) = [bncos(n-1) + ansin(n-1)](r/Rref)n-1
-          an = (1/)(Rref/r)n-1  Br(r,)cos(n)d    (skew)
-            bn = (1/)(Rref/r)n-1  Br(r,)sin(n)d    (normal)

- Magnet Design

## Magnet Design

- -----

- Edge Chamfers (to reduced 12-pole component)

## Magnet Design

-    Use edge chamfers to minimize integrated b6 (12-pole) component :
-                                           21Q40 Prototype

##          Shimming Tools --- an example   (dB/Bo)dl 1ppm

- Magnet Design

- ----

-          Shimming Tools --- an example   (dB/Bo)dl 1ppm
- Dipole correction: top/bottom plates, wedge translation (b1constant)
- Quad correction: pole tilt angle, tapered wedge angle (b2 r)
- Sextupole correction: pole chamfers, rose shim thickness (b3 r2)
- ...... and more: azimuthal correction, surface coils fine tuning ...

- Z (cm)

- R(cm)

- Injection Kicker Inconel Vac Chamber
-      Eddy Current Heating      Ceramic Vac Chamber
- rise time constant** ******** **= 200 s
- eddy current plot at  t = 100 s