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radial current segments cancel. The situation for axial currents is different. The axial current on the outside of the toroid is pointed down and the axial current on the inside of the toroid is pointed up. Each axial current segment on the outside of the toroid can be matched with an equal but oppositely directed segment on the inside of the toroid. The segments on the inside are closer than the segments on the outside to the axis, therefore there is a net upward component of the
28:
44:
1357:
146:
245:
363:
However, at points a distance of several times the winding spacing, the toroid does look symmetric. There is still the problem of the circumferential current. No matter how many times the winding encircles the core and no matter how thin the wire, this toroidal inductor will still include a one coil loop in the plane of the toroid. This winding will also produce and be susceptible to an
36:
1109:
350:
field). Because of the symmetry, the lines of B flux must form circles of constant intensity centered on the axis of symmetry. The only lines of B flux that encircle any current are those that are inside the toroidal winding. Therefore, from Ampere's circuital law, the intensity of the B field must
656:
Representing the magnetic vector potential (A), magnetic flux (B), and current density (j) fields around a toroidal inductor of circular cross-section. Thicker lines indicate field lines of higher average intensity. Circles in the cross-section of the core represent B flux coming out of the picture.
1407:
field sourced from the primary currents is shown as green ellipses. The secondary winding is shown as a brown line coming directly down the axis of symmetry. In standard practice, the two ends of the secondary are connected with a long wire that stays well away from the torus, but to maintain the
644:
field (magnetic vector potential) is not confined. Arrow #1 in the picture depicts the vector potential on the axis of symmetry. Radial current sections a and b are equal distances from the axis but pointed in opposite directions, so they will cancel. Likewise, segments c and d cancel. All the
230:
Due to the symmetry of a toroid, little magnetic flux escapes from the core (leakage flux). Thus, a toroidal inductor/transformer, radiates less electromagnetic interference (EMI) to adjacent circuits and is an ideal choice for highly concentrated environments. Manufacturers have adopted toroidal
448:
field in the plane of the toroid, as shown in figure 7. This can be mitigated by using a return winding, as shown in Figure 8. With this winding, each place the winding crosses itself; the two parts will be at equal and opposite polarity, which substantially reduces the E field generated in the
1352:{\displaystyle \mathbf {EMF} =\oint _{path}\mathbf {E} \cdot {\rm {d}}l=-\oint _{path}{\frac {\partial \mathbf {A} }{\partial t}}\cdot {\rm {d}}l=-{\frac {\partial }{\partial t}}\oint _{path}\mathbf {A} \cdot {\rm {d}}l=-{\frac {\partial }{\partial t}}\int _{surface}\mathbf {B} \cdot {\rm {d}}s}
1385:
This figure shows the half section of a toroidal transformer. Quasi-static conditions are assumed, so the phase of each field is the same everywhere. The transformer, its windings and all things are distributed symmetrically about the axis of symmetry. The windings are such that there is no
362:
Figure 3 of this section shows the most common toroidal winding. It fails both requirements for total B field confinement. Looking out from the axis, sometimes the winding is on the inside of the core and sometimes on the outside of the core. It is not axially symmetric in the near region.
341:
The absence of circumferential current (the path of circumferential current is indicated by the red arrow in figure 3 of this section) and the axially symmetric layout of the conductors and magnetic materials are sufficient conditions for total internal confinement of the
206:
In general, a toroidal inductor/transformer is more compact than other shaped cores because they are made of fewer materials and include a centering washer, nuts, and bolts resulting in up to a 50% lighter weight design. This is especially the case for power devices.
226:
In addition, because the windings are relatively short and wound in a closed magnetic field, a toroidal transformer will have a lower secondary impedance which will increase efficiency, electrical performance and reduce effects such as distortion and fringing.
1408:
absolute axial symmetry, the entire apparatus is envisioned as being inside a perfectly conductive sphere with the secondary wire "grounded" to the inside of the sphere at each end. The secondary is made of resistance wire, so there is no separate load. The
100:
Although closed-core inductors and transformers often use cores with a rectangular shape, the use of toroidal-shaped cores sometimes provides superior electrical performance. The advantage of the toroidal shape is that, due to its symmetry, the amount of
828:
field around a toroidal inductor. The thicker lines indicate paths of higher average intensity (shorter paths have higher intensity so that the path integral is the same). The lines are just drawn to look good and impart general look of the
1390:
field due to the primary current. The core and primary winding are represented by the gray-brown torus. The primary winding is not shown, but the current in the winding at the cross-section surface is shown as gold (or orange) ellipses. The
222:
coils). This is because most of the magnetic field is contained within the core. By comparison, with an inductor with a straight core, the magnetic field emerging from one end of the core has a long path through air to enter the other end.
322:
Fig. 1. Coordinate system. The Z-axis is the nominal axis of symmetry. The X-axis chosen arbitrarily to line up with the starting point of the winding. ρ is called the radial direction. θ is called the circumferential direction.
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627:
751:
794:
1025:
702:
914:
370:
Figures 4-6 show different ways to neutralize the circumferential current. Figure 4 is the simplest and has the advantage that the return wire can be added after the inductor is bought or built.
1051:
is responsible for the undesirable electric field coupling between primary and secondary. Transformer designers attempt to minimize the electric field coupling. For the rest of this section,
1395:
field caused by the primary current is confined to the region enclosed by the primary winding (i.e. the core). Blue dots on the left-hand cross-section indicate that lines of
1073:
1049:
560:
872:
518:
381:
Fig. 4. Circumferential current countered with a return wire. The wire is white and runs between the outer rim of the inductor and the outer portion of the winding.
231:
coils in recent years to comply with increasingly strict international standards limiting the amount of electromagnetic field consumer electronics can produce.
919:
475:. See Feynman page 15-11 for a diagram of the magnetic vector potential around a long thin solenoid which also exhibits total internal confinement of the
167:
1376:
In this figure, blue dots indicate where B flux from the primary current comes out of the picture and plus signs indicate where it goes into the picture.
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field (sourced from the secondary currents) forms the
Poynting vector, which points from the primary toward the secondary.
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1399:
flux in the core come out of the left-hand cross-section. On the other cross-section, blue plus signs indicate that the
1364:
which says the EMF is equal to the time rate of change of the B flux enclosed by the winding, which is the usual result.
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1651:
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193:
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Toroidal transformer
Poynting vector coupling from primary to secondary in the presence of total B field confinement
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428:
2232:
2114:
1964:
1784:
171:
418:
259:
633:
Number 4 will be presumed for the rest of this section and may be referred to the "quasi-static condition".
2023:
1892:
1424:
field from the primary to the secondary, if the secondary is not open-circuited. The cross product of the
110:
1027:
is responsible for the desirable magnetic field coupling between primary and secondary while the quantity
303:
2028:
1912:
1857:
1852:
1420:
field fills space, including inside the transformer core, so in the end, there is a continuous non-zero
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2001:
636:
Although the axially symmetric toroidal inductor with no circumferential current totally confines the
1979:
472:
458:
397:
1054:
1030:
2006:
156:
128:, which in turn are used in the vast majority of electrical equipment: TVs, radios, computers, and
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2104:
2011:
1957:
294:
In some circumstances, the current in the winding of a toroidal inductor contributes only to the
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71:
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2045:
1952:
1496:
657:
Plus signs on the other cross-section of the core represent B flux going into the picture. Div
538:
387:
210:
Because the toroid is a closed-loop core, it will have a higher magnetic field and thus higher
467:
Showing the development of the magnetic vector potential around a symmetric toroidal inductor.
463:
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2040:
2018:
1907:
1897:
856:
489:
1449:"What Separates Toroidal Coil Transformers From The Other Transformers? | Custom Coils Blog"
652:
377:
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1709:
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74:
1412:
field along the secondary causes current in the secondary (yellow arrows), which causes a
8:
2191:
2109:
2099:
2055:
2050:
2035:
1867:
1386:
circumferential current. The requirements are met for full internal confinement of the
1101:
along the secondary winding gives the secondary's induced EMF (Electro-Motive Force).
824:
field around a loop of current. The figure to the left is an artist's depiction of the
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2119:
2089:
2084:
2067:
1922:
1902:
1872:
1372:
1690:
968:{\displaystyle \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}}
309:
1988:
1887:
1737:
1719:
1713:
1696:
1667:
1661:
1647:
318:
116:
Toroidal inductors and transformers are used in a wide range of electronic circuits:
820:
flux (as would be produced in a toroidal inductor) is qualitatively the same as the
2186:
1472:
90:
1641:
440:
There will be a distribution of potential along the winding. This can lead to an
1996:
1947:
1927:
1917:
109:) can be made low, potentially making it more efficient and making it emit less
2155:
2150:
2140:
1862:
1842:
1827:
1079:
432:
Fig. 8. Voltage distribution with return winding. ±100 Volt excitation assumed.
333:
Fig. 2. An axially symmetric toroidal inductor with no circumferential current.
86:
1756:
452:
2226:
2165:
525:
422:
Fig. 7. Simple toroid and the E-field produced. ±100 Volt excitation assumed.
102:
82:
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63:
19:
1969:
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532:
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27:
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1937:
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1807:
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43:
837:
Toroidal transformer action in the presence of total B field confinement
2160:
2145:
1837:
211:
622:{\displaystyle {\frac {1}{c^{2}}}{\frac {\partial \phi }{\partial t}}}
401:
Fig. 6. Circumferential current countered with a split return winding.
1776:
125:
145:
2196:
1882:
310:
Sufficient conditions for total internal confinement of the B field
302:
field outside the windings. This is a consequence of symmetry and
254:
provides insufficient context for those unfamiliar with the subject
219:
215:
121:
55:
1732:
Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993),
444:-Field in the plane of the toroid and also a susceptibility to an
298:
field inside the windings. It does not contribute to the magnetic
234:
47:
Interior of a linear power supply with toroidal mains transformer.
2206:
1932:
746:{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} \ }
23:
Medium power toroidal mains transformer with laminated iron core.
391:
Fig. 5. Circumferential current countered with a return winding.
1640:
Feynman, Richard P; Leighton, Robert B; Sands, Matthew (1964),
575:
is used and a non-zero frequency that is low enough to neglect
67:
1822:
1770:
Industrial study material: Ferrite Toroid
Transformers Design
789:{\displaystyle {\frac {\partial E}{\partial t}}\rightarrow 0}
975:
and so even if the region outside the windings is devoid of
2211:
1020:{\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}}
453:
Toroidal inductor/transformer and magnetic vector potential
94:
39:
Traditional transformers wound on rectangular-shaped cores.
35:
1416:
field around the secondary (shown as blue ellipses). This
471:
See
Feynman chapter 14 and 15 for a general discussion of
2181:
697:{\displaystyle \nabla \times \mathbf {A} =\mathbf {B} \ }
909:{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} .}
218:
than an inductor of the same mass with a straight core (
520:. This would be true under the following assumptions:
1718:, Berkeley Physics Course, vol. II, McGraw-Hill,
796:) have the same form, then the lines and contours of
358:
Fig. 3. Toroidal inductor with circumferential current
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1057:
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994:
922:
883:
859:
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710:
671:
581:
541:
492:
409:
1731:
1534:
1075:
will assumed to be zero unless otherwise specified.
1757:Inductor and Transformer Design Guides - Magnetics
1351:
1094:is equal to a constant times the enclosed current
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1043:
1019:
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866:
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696:
621:
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512:
135:
2224:
535:is used and there is no distribution of charge,
77:, consisting of a circular ring or donut shaped
1764:includes formula, but assumes circular windings
1679:
1549:
235:Total B field confinement by toroidal inductors
1608:
1606:
1518:
1516:
1473:"Toroidal Transformers - Agile Magnetics, Inc"
1428:field (sourced from primary currents) and the
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1545:
1543:
1768:Design Considerations of Toroid Transformers
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1579:
1567:
1380:
486:field is accurate when using the assumption
1603:
1513:
174:. Unsourced material may be challenged and
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1785:
1618:
1540:
31:Small toroidal inductors with ferrite core
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1522:
1064:
1040:
863:
551:
282:Learn how and when to remove this message
194:Learn how and when to remove this message
2095:Rotary variable differential transformer
2075:Linear variable differential transformer
1643:The Feynman Lectures on Physics Volume 2
1371:
753:(assuming quasi-static conditions, i.e.
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346:field. (Some authors prefer to use the
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18:
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1573:
1497:"How Does a Toroidal Transformer Work?"
479:field, at least in the infinite limit.
2225:
1806:
1082:applies, so that the path integral of
1780:
1734:Foundations of Electromagnetic Theory
568:is used and zero frequency is assumed
264:providing more context for the reader
1848:Condition monitoring of transformers
1688:
1561:
367:field in the plane of the inductor.
238:
172:adding citations to reliable sources
139:
1943:Toroidal inductors and transformers
874:(scalar electric potential) fields
52:Toroidal inductors and transformers
13:
1762:Approximate inductance of a toroid
1535:Reitz, Milford & Christy (1993
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979:field, it is filled with non-zero
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771:
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649:field along the axis of symmetry.
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410:E field in the plane of the toroid
14:
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1750:
70:(ring or donut) shape. They are
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1332:
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1120:
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1090:flux, just as the path integral
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924:
899:
885:
849:fields can be computed from the
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718:
687:
679:
243:
144:
1663:Introduction to Electrodynamics
804:like the lines and contours of
640:field within the windings, the
136:Advantages of toroidal windings
105:that escapes outside the core (
2115:Variable-frequency transformer
1965:Transformer utilization factor
1555:
1528:
1489:
1465:
1441:
1068:{\displaystyle \nabla \phi \,}
1044:{\displaystyle \nabla \phi \,}
780:
351:be zero outside the windings.
1:
1695:(5th ed.), McGraw-Hill,
1633:
2024:Energy efficient transformer
1692:Engineering Electromagnetics
1550:Halliday & Resnick (1962
812:. Thus, a depiction of the
111:electromagnetic interference
7:
2029:Amorphous metal transformer
1913:Resonant inductive coupling
1853:Electrical insulation paper
10:
2249:
1680:Halliday; Resnick (1962),
456:
2174:
2128:
1987:
1976:
1815:
1715:Electricity and Magnetism
1660:Griffiths, David (1989),
1403:flux enters there. The
1381:Explanation of the figure
1086:is equal to the enclosed
555:{\displaystyle \rho =0\,}
473:magnetic vector potential
459:Magnetic vector potential
16:Type of electrical device
2007:Distribution transformer
1435:
2105:Solid-state transformer
2012:Pad-mounted transformer
1958:Transformer oil testing
1684:, John Wiley & Sons
867:{\displaystyle \phi \,}
816:field around a loop of
513:{\displaystyle bf{A}=0}
2080:Parametric transformer
2046:Instrument transformer
2002:Buck–boost transformer
1953:Dissolved gas analysis
1689:Hayt, William (1989),
1377:
1353:
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1045:
1021:
969:
910:
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623:
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402:
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382:
359:
334:
324:
304:Ampère's circuital law
48:
40:
32:
24:
2233:Electric transformers
2063:Isolation transformer
2041:Grounding transformer
2019:Delta-wye transformer
1898:Pressure relief valve
1588:, p. 15_1-15_16)
1576:, p. 14_1-14_10)
1375:
1354:
1097:The path integral of
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1046:
1022:
970:
911:
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791:
748:
699:
661:= 0 has been assumed.
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75:electronic components
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1477:Agile Magnetics, Inc
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665:Since the equations
579:
539:
490:
168:improve this section
2192:Mitsubishi Electric
2110:Trigger transformer
2100:Scott-T transformer
2056:Voltage transformer
2051:Current transformer
2036:Flyback transformer
1868:Induction regulator
260:improve the article
2202:Schneider Electric
2120:Zigzag transformer
2090:Rotary transformer
2085:Planar transformer
2068:Austin transformer
1923:Short-circuit test
1903:Quadrature booster
1873:Leakage inductance
1736:, Addison-Wesley,
1710:Purcell, Edward M.
1646:, Addison-Wesley,
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89:, iron powder, or
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33:
25:
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1888:Open-circuit test
1725:978-0-07-004859-1
1682:Physics, part two
1666:, Prentice-Hall,
1453:Custom Coils Blog
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85:material such as
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1982:
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1615:, p. 15_15)
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1600:, p. 15_11)
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1997:Autotransformer
1983:
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1948:Transformer oil
1928:Stacking factor
1918:Severity factor
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2156:Repeating coil
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2151:Polyphase coil
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2141:Induction coil
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1649:
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1625:Purcell (1965
1621:
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1613:Feynman (1964
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252:This article
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184:November 2016
177:
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158:
153:This section
151:
147:
142:
141:
133:
131:
130:audio systems
127:
123:
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114:
112:
108:
104:
103:magnetic flux
98:
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92:
88:
84:
83:ferromagnetic
80:
79:magnetic core
76:
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61:
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53:
45:
37:
29:
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1970:Vector group
1942:
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1733:
1714:
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1593:
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1569:
1557:
1530:
1504:. Retrieved
1500:
1491:
1480:. Retrieved
1476:
1467:
1456:. Retrieved
1452:
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664:
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573:Lorenz gauge
566:Lorenz gauge
533:Lorenz gauge
483:
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445:
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369:
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258:Please help
253:
229:
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166:Please help
154:
115:
107:leakage flux
99:
60:transformers
51:
50:
2136:Hybrid coil
1938:Tap changer
1878:Magnet wire
1808:Transformer
916:and :
629:is assumed.
2161:Tesla coil
2146:Oudin coil
1838:Center tap
1634:References
1562:Hayt (1989
1506:2018-04-03
1482:2018-04-03
1458:2018-04-03
808:relate to
800:relate to
212:inductance
126:amplifiers
97:is wound.
62:which use
1501:Sciencing
1337:⋅
1305:∫
1295:∂
1291:∂
1286:−
1270:⋅
1247:∮
1237:∂
1233:∂
1228:−
1212:⋅
1203:∂
1193:∂
1172:∮
1168:−
1152:⋅
1129:∮
1062:ϕ
1059:∇
1038:ϕ
1035:∇
1009:∂
999:∂
957:∂
947:∂
941:−
938:ϕ
935:∇
932:−
896:×
893:∇
861:ϕ
781:→
772:∂
764:∂
727:μ
715:×
712:∇
704:, and
676:×
673:∇
611:∂
606:ϕ
603:∂
543:ρ
272:June 2019
155:does not
122:inverters
56:inductors
2227:Category
2197:ProlecGE
1908:Resolver
1893:Polarity
1883:Metadyne
1712:(1965),
220:solenoid
216:Q factor
113:(EMI).
68:toroidal
2207:Siemens
1933:Synchro
1858:Growler
1833:Bushing
833:field.
571:4. the
564:3. the
531:2. the
528:is used
524:1. the
449:plane.
176:removed
161:sources
91:ferrite
72:passive
66:with a
1816:Topics
1810:topics
1740:
1722:
1699:
1670:
1650:
983:field.
741:
692:
124:, and
2129:Coils
1989:Types
1823:Balun
1436:Notes
2212:TBEA
1738:ISBN
1720:ISBN
1697:ISBN
1668:ISBN
1648:ISBN
853:and
845:and
841:The
482:The
214:and
159:any
157:cite
95:wire
58:and
54:are
2182:ABB
262:by
170:by
81:of
2229::
1605:^
1542:^
1515:^
1499:.
1475:.
1451:.
306:.
132:.
120:,
1800:e
1793:t
1786:v
1509:.
1485:.
1461:.
1430:B
1426:E
1422:B
1418:B
1414:B
1410:E
1405:E
1401:B
1397:B
1393:B
1388:B
1347:s
1342:d
1333:B
1327:e
1324:c
1321:a
1318:f
1315:r
1312:u
1309:s
1298:t
1283:=
1280:l
1275:d
1266:A
1260:h
1257:t
1254:a
1251:p
1240:t
1225:=
1222:l
1217:d
1206:t
1197:A
1185:h
1182:t
1179:a
1176:p
1165:=
1162:l
1157:d
1148:E
1142:h
1139:t
1136:a
1133:p
1125:=
1121:F
1118:M
1115:E
1099:E
1092:B
1088:B
1084:A
1012:t
1003:A
981:E
977:B
960:t
951:A
929:=
925:E
904:.
900:A
890:=
886:B
851:A
847:B
843:E
831:A
826:A
822:B
818:B
814:A
810:j
806:B
802:B
798:A
784:0
775:t
767:E
737:j
731:0
723:=
719:B
688:B
684:=
680:A
659:A
647:A
642:A
638:B
614:t
593:2
589:c
585:1
549:0
546:=
508:0
505:=
501:A
497:f
494:b
484:A
477:B
446:E
442:E
365:E
348:H
344:B
300:B
296:B
285:)
279:(
274:)
270:(
266:.
256:.
197:)
191:(
186:)
182:(
178:.
164:.
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