Toroidal coil

A toroidal coil , also known as a circular ring coil , ring coil or toroidal core coil , is a specially shaped coil in electrical engineering that consists of a core in the form of a circular ring (so-called toroidal core ) around which the electrical conductor is wound. The specialty of this design is that the magnetic flux spreads almost exclusively in the circular core and the mostly disruptive stray field in the outer space of the annulus coil is comparatively weak. Tokamaks for fusion research and the ATLAS detector at CERN are prominent examples of the large-scale application of toroidal coils.

Embodiments and applications

Toroidal coil with two windings.

Annular coils are mainly used in passive electrical filters to suppress unwanted high-frequency interference. It can be designed as a classic coil with only one conductor; but two or more conductors on the bobbin are also possible. In order to avoid magnetic saturation of the core, either appropriate materials are necessary as core material or an air gap is artificially built into the annulus . However, if a choke with two or more windings is operated so that the sum of all currents is zero, the individual magnetic fields cancel each other out, saturation is avoided and one speaks of a current-compensated choke . While a toroidal core choke without an air gap (powder core chokes are not included) goes into saturation with small currents, a current-compensated choke can achieve high inductances for EMC filtering against common-mode interference without the core becoming saturated. Only the leakage inductance of the choke is visible in the useful signal or circuit, but it is only a fraction of the nominal inductance.

Toroidal coils with two or more windings are also used as an essential component in residual current circuit breakers for detecting a residual current.

Another area of application is as a transformer . The voltage is transferred from one winding, the primary side, to the second winding, the secondary side. In this application, the core must not have an air gap. See toroidal transformer .

Calculating the inductance

The inductance L of a toroidal coil with a winding with N turns and a rectangular core of width b , the inner radius r and the outer radius R can be approximated for thin wire with the formula

{\ displaystyle L = N ^ {2} \ cdot {\ frac {\ mu _ {0} \ mu _ {r} b} {2 \ pi}} \ cdot \ ln {\ frac {R} {r}} }

to calculate. Here, μ _{0 is} the magnetic field constant and μ _{r is} the permeability number of the core material. Instead of the radii, the corresponding diameters can also be used.

If the relative difference between the outer and inner radius of the ring is small, the mean radius is denoted by and the cross-sectional area of the ring is denoted by A , the inductance of the ring coil can be approximated ${\ displaystyle r _ {\ mathrm {m}} = (R + r) / 2}$

{\ displaystyle L = N ^ {2} \ cdot {\ frac {\ mu _ {0} \ mu _ {r} A} {2 \ pi \, r _ {\ mathrm {m}}}}}

to calculate.

If the coil is also interrupted by an air gap of the length , the following applies ${\ displaystyle l}$

${\ displaystyle L = N ^ {2} \ cdot {\ frac {\ mu _ {0} \ mu _ {r} A} {2 \ pi r_ {m} + l (\ mu _ {r} -1) }}}$

Magnetic fields of the toroidal coil

Magnetic field inside the coil

If you consider the magnetic field inside a toroidal coil with a small diameter in relation to its radius , this can be derived using Ampère's law . Consider a toroidal coil with circumference , number of turns and current strength : ${\ displaystyle r_ {m}}$ ${\ displaystyle U}$ ${\ displaystyle N}$ ${\ displaystyle I}$

{\ displaystyle \ oint \ limits _ {U} {\ vec {H}} \ {\ text {d}} {\ vec {s}} = I \ N}

Since the H field always runs parallel to the integration path (circular shape through the inside of the coil), the scalar product here is equal to the product of the amounts.

{\ displaystyle \ Rightarrow H \ cdot U = H \ cdot 2 \ pi r_ {m} = I \ N}

With the mean radius of the coil. Solving for gives: ${\ displaystyle r_ {m}}$ ${\ displaystyle H}$

{\ displaystyle \ Rightarrow H = {\ frac {I \ N} {2 \ pi r_ {m}}}}

or if you use.

{\ displaystyle B = \ mu _ {0} \ mu _ {r} {\ frac {I \ N} {2 \ pi r_ {m}}}}

{\ displaystyle B = \ mu _ {0} \ mu _ {r} H}

Magnetic field inside with air gap

If the toroidal coil is interrupted by an air gap the length of the above connection, also with Ampère's law , becomes the following: ${\ displaystyle l}$

{\ displaystyle \ Rightarrow H_ {S} \ cdot U '+ H_ {L} \ cdot l = I \ N}

where the field in the coil describes the length and the field in the air gap. ${\ displaystyle H_ {S}}$ ${\ displaystyle U '= 2 \ pi r_ {m} -l}$ ${\ displaystyle H_ {L}}$

If the stray fields at the ends of the coil are now and are neglected, it can be set because the normal component of the B field does not change at the transition between materials . This results in: ${\ displaystyle l << U '}$ ${\ displaystyle B_ {S} \ approx B_ {L}}$

{\ displaystyle H_ {S} (2 \ pi r_ {m} -l) + H_ {L} \ cdot l = B_ {L} ({\ frac {2 \ pi r_ {m} -l} {\ mu _ {0} \ mu _ {r}}} + {\ frac {l} {\ mu _ {0}}}) = IN}

and thus for the magnetic flux density in the air gap: ${\ displaystyle B_ {L}}$

{\ displaystyle B_ {L} = {\ frac {\ mu _ {0} IN} {{\ frac {2 \ pi r_ {m} -l} {\ mu _ {r}}} + l}} = { \ frac {\ mu _ {0} \ mu _ {r} IN} {2 \ pi r_ {m} + l (\ mu _ {r} -1)}}}

Magnetic field outside the coil

Outside the coil, the toroidal coil can be viewed in simplified form as a conductor loop with a radius because of its circular shape . ${\ displaystyle r_ {m}}$

For a straight line that is perpendicular to the circular area around which the toroidal coil runs and runs through its center point:

{\ displaystyle B (z) = {\ frac {\ mu _ {0} I} {2}} \, {\ frac {r_ {m} ^ {2}} {\ left (r_ {m} ^ {2 } + z ^ {2} \ right) ^ {3/2}}}}

where the distance with respect to the -axis describes if the toroidal coil is at the origin, in the - -plane of a 3-dimensional Cartesian coordinate system . ${\ displaystyle z}$ ${\ displaystyle Z}$ ${\ displaystyle X}$ ${\ displaystyle Y}$

In particular, the following applies to the center point : ${\ displaystyle z = 0}$

{\ displaystyle B (0) = {\ frac {\ mu _ {0} I} {2r_ {m}}}}

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↑ EPCOS AG, "Power line chokes: Current-compensated ring core chokes", Data Book "Inductors" 2008 (English)

^ Karl Küpfmüller : Introduction to Theoretical Electrical Engineering. 13th edition, 1990, Springer-Verlag.

^ N. Fliege, University of Mannheim: Lecture Electrical Engineering I , Chapter 2: Electrical Components and Networks ( Memento from May 4, 2006 in the Internet Archive ) (PDF, 1.5 MB).

↑ P. Weiß, University of Kaiserslautern: Script for the lecture Fundamentals of Electrical Engineering ( Memento of the original from June 13, 2007 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF, 4.9 MB). @1@ 2

↑ The magnetic field of a toroid. Retrieved July 20, 2020 .

^ ^A ^b Wolfgang Demtröder: Experimentalphysik 2. Electricity and optics . 7th edition. Springer-Verlag, 2017, ISBN 978-3-662-55789-1 , pp. 110 .

[1] EPCOS AG, "Power line chokes: Current-compensated ring core chokes", Data Book "Inductors" 2008 (English)

[2] Karl Küpfmüller : Introduction to Theoretical Electrical Engineering. 13th edition, 1990, Springer-Verlag.

[3] N. Fliege, University of Mannheim: Lecture Electrical Engineering I , Chapter 2: Electrical Components and Networks ( Memento from May 4, 2006 in the Internet Archive ) (PDF, 1.5 MB).