Magnet Design

Formulas for Magnet Design

Note these formulas are mostly related to conventional (iron dominated) electro-magnets for accelerator desgin. Eventually, more formulas will be added.

Excitation Formulas

Dipole Excitation
Quadrupole Excitation
Sextupole Excitation

Quad Conformal Mapping

Quadrupole to Dipole Conformal Map
Dipole to Quadrupole Conformal Map

Overhang Formulas

Unoptimized Overhang
Optimized Overhang

Time Constant Formula

DISCLAIMER: The material presented here is for reference purposes only. The author makes no claim as to the accuracy or applicability of these formulas for any purpose. If you intend to use these formulas, please verify them for yourself and use them wisely.

Excitation Formulas

These formulas can be used to calculate the desired amp turns to achieve a specified field requirement or determine the expected field or gradient expected for a given excitation and efficiency of a magnet.

Dipole: Where:

B is the dipole central field in Tesla.
Mu-not is permeability of air (4 Pi x 10^-7).
Ada is the efficiency.
N is the number of linked turns per pole.
I is the current in each turn in Amps.
And g is the half-gap of the dipole in meters.

Quadrupole: Where:

B-prime is the quadrupole gradient in Tesla per meter.
Mu-not is permeability of air (4 Pi x 10^-7).
Ada is the efficiency.
N is the number of linked turns per pole.
I is the current in each turn in Amps.
R-not is the aperture radius of the poletip in meters.

Sextupole: Where:

B-double-prime is the sextupole gradient in Tesla per meter squared.
Mu-not is permeability of air (4 Pi x 10^-7).
Ada is the efficiency.
N is the number of linked turns per pole.
I is the current in each turn in Amps.
R-not is the aperture radius of the poletip in meters.

Quadrupole Conformal Mapping

These formulas are used to map between quadrupole and dipole spaces. The purpose is to analyze quadrupole geometry designs in dipole space where the perfect field is simplified to a constant. This improves the accuracy of the solutions in codes like POISSON where only linear approximations are made in the mesh.

Quadrupole to Dipole Conformal Map: Where:

X and Y are the input coordinates in quadrupole space.
Ro is the bore radius in the quadrupole space.
U and V are the output coordinates in the dipole space.

Dipole to Quadrupole Conformal Map: Where:

U and V are the input coordinates in the dipole space.
Ro is the bore radius in the quadrupole space.
X and Y are the output coordinates in quadrupole space.

Pole Overhang Formulas

These formulas apply to dipole design but can be used for quadrupole and sextupole geometries which are conformally mapped to the dipole domain. The formulas for an optimized pole don't give the optimization, just the expected required overhang for a properly done optimization. Overhang is defined as the horizontal distance between the edge of the good field region at the midplane and the pole corner location.

Unoptimized Dipole: Where:

Delta-B/B is the desired field error at some good field half-width.
X is the normalized overhang. X = overhang / half-gap.

Optimized Dipole: Where:

Delta-B/B is the desired field error at some good field half-width.
X is the normalized overhang. X = overhang / half-gap.

Time Constant Formula

This formula predicts a rough time constant for a magnet to damp to steady state when the excitation is changed. For a solid magnet, the dimension which controls the constant will typically be the pole width or return leg width. For a laminated magnet, the controlling dimension will typically be the lamination thickness. The formula assumes constant permeability (no saturation effects).

Time Constant: Where:

Tau is the predicted time constant in seconds for the magnet.
Mu is the relative permeability for the magnet core material. For unsaturated steel, 10000 is an approximate value.
Mu-not is the absolute permeability of air (4 Pi x 10^-7).
The controlling dimension is d and equals the half-width of the smallest dimension of the largest enclosed flux path in meters.
Rho is the resistivity of the magnet core material in ohm-meters. For steel, 10 x 10^-8 ohm-meters is an approximate value.

Dipole:	Where:

	B is the dipole central field in Tesla. Mu-not is permeability of air (4 Pi x 10^-7). Ada is the efficiency. N is the number of linked turns per pole. I is the current in each turn in Amps. And g is the half-gap of the dipole in meters.

Quadrupole:	Where:

	B-prime is the quadrupole gradient in Tesla per meter. Mu-not is permeability of air (4 Pi x 10^-7). Ada is the efficiency. N is the number of linked turns per pole. I is the current in each turn in Amps. R-not is the aperture radius of the poletip in meters.

Sextupole:	Where:

	B-double-prime is the sextupole gradient in Tesla per meter squared. Mu-not is permeability of air (4 Pi x 10^-7). Ada is the efficiency. N is the number of linked turns per pole. I is the current in each turn in Amps. R-not is the aperture radius of the poletip in meters.

Quadrupole to Dipole Conformal Map:	Where:

	X and Y are the input coordinates in quadrupole space. Ro is the bore radius in the quadrupole space. U and V are the output coordinates in the dipole space.
Dipole to Quadrupole Conformal Map:	Where:

	U and V are the input coordinates in the dipole space. Ro is the bore radius in the quadrupole space. X and Y are the output coordinates in quadrupole space.

Unoptimized Dipole:	Where:

	Delta-B/B is the desired field error at some good field half-width. X is the normalized overhang. X = overhang / half-gap.

Optimized Dipole:	Where:

	Delta-B/B is the desired field error at some good field half-width. X is the normalized overhang. X = overhang / half-gap.

Time Constant:	Where:

	Tau is the predicted time constant in seconds for the magnet. Mu is the relative permeability for the magnet core material. For unsaturated steel, 10000 is an approximate value. Mu-not is the absolute permeability of air (4 Pi x 10^-7). The controlling dimension is d and equals the half-width of the smallest dimension of the largest enclosed flux path in meters. Rho is the resistivity of the magnet core material in ohm-meters. For steel, 10 x 10^-8 ohm-meters is an approximate value.