]> Flux linkage in a radially thick coil

Flux linkage in a radially thick coil

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Figure RTZ: A loop of wire, radius r, sitting in a homogenous magnetic field To calculate the flux linkage in a plain, circular loop of wire immersed in a homogenous magnetic field (figure RTZ) you take as the area a value of π times the square of the loop radius, r -

λ = π r 2 B
Equation RTD



This result you may easily extend to the case of a 'single layer' solenoid coil by multiplying by N.



Figure RTY: A radially thick multilayer solenoid For a 'multi layer' coil the situation is more complicated because turns near the outside of the winding have a greater area and will link more flux than turns on the inside.



Figure RTX: An Archimedean spiral A little thought suggests that the geometry is closely represented by an Archimedean spiral, in which the radius, r, increases linearly as a function of azimuthal angle, φ, -

r=mφ+ r1
Equation RTA



where r 1 is the radius of the inside of the coil, and

m= r 2 - r 1 2π
Equation RTB




where r 2 is the radius of the outside of the coil.

You could analyze a 100 turn coil by integrating the area, A, swept out by the radius vector over the interval φ = 0 to φ = 200 π. However, it is simpler to consider the area of a one turn (0 to 2π) spiral and then multiply by N afterwards.

A= 1 2 r( rφ )= 1 2 r 2 φ
Equation RTC


Substituting equation RTA -

A= 1 2 ( mφ+ r 1 ) 2 φ
Equation RTE


= 1 2 m 2 φ 2 +2m r 1 φ+ r 1 2 φ
Equation RTF


A= 1 2 0 2π m 2 φ 2 +2m r 1 φ+ r 1 2 φ
Equation RTG


= 1 2 [ m 2 φ 3 3 +m r 1 φ 2 + r 1 2 φ ] 0 2π
Equation RTH


= 4 π 3 m 2 3 +2 π 2 m r 1 +π r 1 2
Equation RTI


Substituting equation RTB -

= π ( r 2 - r 1 ) 2 3 +π( r 2 - r 1 ) r 1 +π r 1 2
Equation RTJ


A= π 3 ( r 2 2 + r 2 r 1 + r 1 2 )
Equation RTK


For the degenerate case of r 1 = r 2 this gives A=π r 2

Example

Taking R1 = 5 mm and R2 = 11 mm (as figure RTX) you have

A= π 3 ( 11 2 +11×5+ 5 2 )=210 mm2
Equation RTL


Had you taken the arithmetic mean radius ((5+11)/2 = 8 mm) then π×82 gives 201 mm2 : an error of -4.3%.






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E-mail:R.Clarke@surrey.ac.uk
Last modified: 2008 May 11th.