Magnetic circuit

This page provides an overview of magnetic circuits for modelizing the reluctance motor.

The reluctance motor

The force exercised by a electrical reluctance motor is due to the minimization of the magnetic energy of the magnetic circuit.

Advantages of the reluctance motor:

no permanent magnet
consists mostly of soft iron and wiring
simple mechanics
also efficient at high speed
only limited by the electrical power source and electrical switching speed

Disadvantages of the reluctance motor:

Advanced electronics
Requires current sensors and position sensor

Magnetic circuits

Physical laws

Definitions:

$ℱ$ : magnetomotive force (unit: $A$ )
$𝛷$ : magnetic flux (unit: $W b$ or $H . A$ or $k g . m^{2} . s^{- 2} . A^{- 1}$ )
$ℛ$ : reluctance (unit: $H^{- 1}$ or $k g^{- 1} . m^{- 2} . s^{2} . A^{2}$ )
$N$ : number of wire loops
$i$ : electric current in one wire loops (unit: $A$ )
$L$ : length of the magnetic circuit (unit: $m$ )
$S$ : area of a section of the magnetic circuit (unit: $m^{2}$ )
$H$ : magnetizing field (unit: $A . m^{- 1}$ )
$B$ : magnetic flux density (unit: $T$ or $k g . s^{- 2} . A^{- 1}$ )
$μ$ : magnetic permeability (unit: $H . m^{- 1}$ or $k g . m . s^{- 2} . A^{- 2}$ )
$μ_{0}$ : vacuum magnetic permeability : $μ_{0} = 1.256 \times 10^{- 6} H . m^{- 1}$

Laws at macroscopic scale (from integral equations):

$ℱ = N i$
$𝛷 = \frac{ℱ}{ℛ}$
$ℛ = \frac{L}{μ S} = \frac{L}{μ_{r} μ_{0} S}$

Laws at microscopic scale (from differential equations):

$H = \frac{ℱ}{L} = \frac{𝛷}{μ S}$
$B = \frac{𝛷}{S} = μ H = μ_{r} μ_{0} H$
$μ = μ_{r} μ_{0} = \frac{B}{H}$

Energy:

$u_{m}$ : magnetic energy density (unit: $J . m^{- 3}$ or $k g . m^{- 1} . s^{- 2}$ )
$u_{m} = \frac{B H}{2} = \frac{B^{2}}{2 μ} = \frac{B^{2}}{2 μ_{r} μ_{0}}$
$E_{m}$ : energy of a magnetic circuit (unit: $J$ or $k g . m^{2} . s^{- 2}$ )
$E_{m} = \int_{V} u_{m}$

Electrical circuit:

$e$ : electromotive force of a turn (unit: $V$ or $k g . m^{2} . s^{- 3} . A^{- 1}$ )
$u$ : electromotive force of the winding (unit: $V$ or $k g . m^{2} . s^{- 3} . A^{- 1}$ )
$ℒ$ : inductance of a solenoid (unit: $H$ or $k g . m^{2} . s^{- 2} . A^{- 2}$ )
$e = - \frac{d 𝛷}{d t}$
$u = N e = - N \frac{d 𝛷}{d t}$
if ℛ constant over time
- $u = - \frac{N^{2}}{ℛ} \frac{d i}{d t}$
- Let's define $ℒ = \frac{N^{2}}{ℛ}$
- $ℒ = \frac{N^{2}}{ℛ} = \frac{N 𝛷}{i} = \frac{μ N^{2} S}{L}$
- $u = - ℒ \frac{d i}{d t}$
- $N 𝛷 = ℒ i$
- $E_{m} = \int_{T i m e} i u = \int_{T i m e} i ℒ \frac{d i}{d t} = \frac{ℒ i^{2}}{2}$

Regular torus

$L = 2 π R$ (length of the torus)
$ℱ = N i$
$ℛ = \frac{L}{μ S}$
$𝛷 = \frac{ℱ}{ℛ} = \frac{μ S N i}{L}$
$B = \frac{𝛷}{S} = \frac{μ N i}{L}$
$E_{m}$ $= \int_{V} u_{m}$ $= \int_{V} \frac{B^{2}}{2 μ}$ $= \frac{B^{2}}{2 μ} L S$ $= \frac{μ S N^{2} i^{2}}{2 L}$ $= \frac{ℒ i^{2}}{2}$
$ℒ = \frac{μ S N^{2}}{L} = \frac{μ_{r} μ_{0} S N^{2}}{L}$

Material	Relative permeability
Air	1
Iron 99.95	200 000
Iron 99.8	5000
Soft iron	5000
Cobalt	250
Nickel	600
Cobalt-iron	18000
Mu-matierial	50 000
Permalloy (nickel-iron)	1000 000

Symbol	Parameter	Value
$μ_{r}$	Relative permeability
R	Torus radius (mm)
$S$	Torus section area ( $m m^{2}$ )
$N$	Number of turns
$i$	Current in the winding (A)
$L$	Torus length (mm)	94.2 mm
$ℱ$	Magnetomotive force (A)	6.000e+3 A
$ℛ$	Reluctance ( $H^{- 1}$ )	1.500e+5 $H^{- 1}$
$𝛷$	Magnetic flux ( $H . A$ )	4.000e-2 H.A
$B$	Magnetic field ( $T$ )	4.000e+2 T
$E_{m}$	Magnetic energy ( $J$ )	1.200e+2 J
$ℒ$	Inductance ( $H$ )	1.067e+2 H

Torus with swelling

$ℱ = N i$
$ℛ = ℛ_{1} + ℛ_{2} = \frac{L_{1}}{μ S_{1}} + \frac{L_{2}}{μ S_{2}}$ $= \frac{L_{1} S_{2} + L_{2} S_{1}}{μ S_{1} S_{2}}$
$𝛷 = \frac{ℱ}{ℛ} = \frac{μ S_{1} S_{2} N i}{L_{1} S_{2} + L_{2} S_{1}}$
- $B_{1} = \frac{𝛷}{S_{1}} = \frac{μ S_{2} N i}{L_{1} S_{2} + L_{2} S_{1}}$
- $B_{2} = \frac{𝛷}{S_{2}} = \frac{μ S_{1} N i}{L_{1} S_{2} + L_{2} S_{1}}$
$E_{m}$ $= \int_{V} \frac{B^{2}}{2 μ}$ $= \int_{V_{1}} \frac{B_{1}^{2}}{2 μ} + \int_{V_{2}} \frac{B_{2}^{2}}{2 μ}$ $= \frac{B_{1}^{2}}{2 μ} L_{1} S_{1} + \frac{B_{2}^{2}}{2 μ} L_{2} S_{2}$ $= \frac{μ N^{2} i^{2}}{2} \frac{S_{1} S_{2}}{L_{1} S_{2} + L_{2} S_{1}}$
$ℒ = \frac{μ N^{2} S_{1} S_{2}}{L_{1} S_{2} + L_{2} S_{1}}$

Symbol	Parameter	Value
$L_{2}$	Percentage of torus with L2 (%)
$S_{2}$	Percentage of S2 compare to S1 (%)
$L_{1}$	Length of L1 ( $m m$ )	66.0 mm
$S_{1}$	Area of S1 ( $m m^{2}$ )	100.0 $m m^{2}$
$L_{2}$	Length of L2 ( $m m$ )	28.3 mm
$S_{2}$	Area of S2 ( $m m^{2}$ )	200.0 $m m^{2}$
$ℱ$	Magnetomotive force (A)	6.000e+3 A
$ℛ$	Reluctance ( $H^{- 1}$ )	1.275e+5 $H^{- 1}$
$𝛷$	Magnetic flux ( $H . A$ )	4.706e-2 H.A
$B_{1}$	Magnetic field ( $T$ )	4.706e+2 T
$B_{2}$	Magnetic field ( $T$ )	2.353e+2 T
$E_{m}$	Magnetic energy ( $J$ )	1.412e+2 J
$ℒ$	Inductance ( $H$ )	1.255e+2 H

Torus with air gap

$ℱ = N i$
$ℛ = ℛ_{L} + ℛ_{G} = \frac{L}{μ_{r} μ_{0} S} + \frac{G}{μ_{0} S}$ $= \frac{L + μ_{r} G}{μ_{r} μ_{0} S}$
$𝛷 = \frac{ℱ}{ℛ} = \frac{μ_{r} μ_{0} S N i}{L + μ_{r} G}$
$B = \frac{𝛷}{S} = \frac{μ_{r} μ_{0} N i}{L + μ_{r} G}$
$E_{m}$ $= \int_{V} \frac{B^{2}}{2 μ}$ $= \int_{V_{1}} \frac{B^{2}}{2 μ_{r} μ_{0}} + \int_{V_{2}} \frac{B^{2}}{2 μ_{0}}$ $= \frac{B^{2}}{2 μ_{r} μ_{0}} L S + \frac{B^{2}}{2 μ_{0}} G S$ $= \frac{B^{2} S}{2 μ_{r} μ 0} (L + μ_{r} G)$ $= \frac{μ_{r} μ_{0} S N^{2} i^{2}}{2 (L + μ_{r} G)}$
$ℒ = \frac{μ_{r} μ_{0} S N^{2}}{L + μ_{r} G}$

Symbol	Parameter	Value
$G$	The thickness of air-gap ( $m$ )
$ℱ$	Magnetomotive force (A)	6.000e+3 A
$ℛ$	Reluctance ( $H^{- 1}$ )	8.108e+6 $H^{- 1}$
$𝛷$	Magnetic flux ( $H . A$ )	7.400e-4 H.A
$B_{L}$	Magnetic field ( $T$ )	7.400e+0 T
$B_{G}$	Magnetic field ( $T$ )	7.400e+0 T
$E_{m}$	Magnetic energy ( $J$ )	2.220e+0 J
$ℒ$	Inductance ( $H$ )	1.973e+0 H

Torus with shuttle

$ℱ = N i$
- $ℛ = ℛ_{L} + \frac{1}{\frac{1}{ℛ_{G 1}} + \frac{1}{ℛ_{G 2}}}$
- $ℛ_{L} = \frac{L}{μ_{r} μ_{0} A B}$
- $ℛ_{G 1} = \frac{G}{μ_{0} x B}$
- $ℛ_{G 2} = \frac{G}{μ_{r} μ_{0} (A - x) B}$
- $ℛ = \frac{x L (1 - μ_{r}) + μ_{r} A (L + G)}{μ_{r} μ_{0} A B (x (1 - μ_{r}) + μ_{r} A)}$
- $ℛ_{x = 0} = \frac{L + G}{μ_{r} μ_{0} A B}$
- $ℛ_{x = A} = \frac{L (1 + μ_{r} G)}{μ_{r} μ_{0} A B} > ℛ_{x = 0}$
$𝛷 = \frac{ℱ}{ℛ}$ $= \frac{μ_{r} μ_{0} A B N i (x (1 - μ_{r}) + μ_{r} A)}{x L (1 - μ_{r}) + μ_{r} A (L + G)}$
- $B_{L} = \frac{𝛷}{A B}$ $= \frac{μ_{r} μ_{0} N i (x (1 - μ_{r}) + μ_{r} A)}{x L (1 - μ_{r}) + μ_{r} A (L + G)}$
- $B_{G 1} ≃ B_{L}$
- $B_{G 2} ≃ B_{L}$
$E_{m}$ $= \int_{V_{L}} \frac{B_{L}^{2}}{2 μ_{r} μ_{0}}$ $+ \int_{V_{G 1}} \frac{B_{G 1}^{2}}{2 μ_{0}}$ $+ \int_{V_{G 2}} \frac{B_{G 2}^{2}}{2 μ_{r} μ_{0}}$
$ℒ = \frac{N^{2}}{ℛ}$
$F_{x} = - \frac{\partial E_{m}}{\partial x}$

Symbol	Parameter	Value
$μ_{r}$	Relative permeability of the shuttle
$A$	Length of the air-gap ( $m m$ )
$B$	Width of the air-gap ( $m m$ )
$x$	Shuttle position (%)
$S$	Torus section area ( $m m^{2}$ )	100.0 $m m^{2}$
$S_{a i r}$	Air area ( $m m^{2}$ )	50.0 $m m^{2}$
$S_{s h u t t l e}$	Shuttle area ( $m m^{2}$ )	50.0 $m m^{2}$
$ℱ$	Magnetomotive force (A)	6.000e+3 A
$ℛ$	Reluctance ( $H^{- 1}$ )	1.532e+5 $H^{- 1}$
$𝛷$	Magnetic flux ( $H . A$ )	3.917e-2 H.A
$B_{L}$	Magnetic field in torus ( $T$ )	3.917e+2 T
$B_{G a i r}$	Magnetic field in air-gap ( $T$ )	3.917e+2 T
$B_{G s h u t t l e}$	Magnetic field in shuttle ( $T$ )	3.917e+2 T
$E_{m}$	Magnetic energy ( $J$ )	3.168e+3 J
$ℒ$	Inductance ( $H$ )	1.045e+2 H
$F_{x}$	Force ( $N$ )	5.840e+1 N

Torus with realistic shuttle

$ℱ = N i$
- $ℛ = ℛ_{L} + ℛ_{H} + \frac{1}{\frac{1}{ℛ_{G 1}} + \frac{1}{ℛ_{G 2}}}$
- $ℛ_{L} = \frac{L}{μ_{r} μ_{0} A B}$
- $ℛ_{G 1} = \frac{G}{μ_{0} x B}$
- $ℛ_{G 2} = \frac{G}{μ_{r} μ_{0} (A - x) B}$
- $ℛ_{H} = \frac{H}{μ_{0} A B}$
$𝛷 = \frac{ℱ}{ℛ}$
- $B_{L} = B_{H} = \frac{𝛷}{A B}$
- $B_{G 1} ≃ B_{L}$
- $B_{G 2} ≃ B_{L}$
$E_{m}$ $= \int_{V_{L}} \frac{B_{L}^{2}}{2 μ_{r} μ_{0}}$ $+ \int_{V_{G 1}} \frac{B_{G 1}^{2}}{2 μ_{0}}$ $+ \int_{V_{G 2}} \frac{B_{G 2}^{2}}{2 μ_{r} μ_{0}}$ $+ \int_{V_{H}} \frac{B_{H}^{2}}{2 μ_{0}}$
$ℒ = \frac{N^{2}}{ℛ}$
$F_{x} = - \frac{\partial E_{m}}{\partial x}$

Symbol	Parameter	Value
$H$	Thickness of slack ( $m m$ )
$x$	Shuttle position (%)
$ℱ$	Magnetomotive force (A)	6.000e+3 A
$ℛ$	Reluctance ( $H^{- 1}$ )	1.745e+6 $H^{- 1}$
$𝛷$	Magnetic flux ( $H . A$ )	3.439e-3 H.A
$B_{L}$	Magnetic field in torus ( $T$ )	3.439e+1 T
$E_{m}$	Magnetic energy ( $J$ )	3.383e+1 J
$ℒ$	Inductance ( $H$ )	3.007e+1 H
$F_{x}$	Force ( $N$ )	4.680e-1 N

Parametrix