Toroidal Transformers - A Detailed Analysis:      

In these days of shrinking package size, the transformer is usually the largest component in a given circuit. The geometry of a standard transformer limits how small we can make it as size is proportional to the square root of the total transformer power. However, the familiar E-l or cut-C cores are not the only ones that can be used. For instance, using a toroidal, or donut-shaped core makes it possible to reduce the size and weight of a transformer by 20% to 50% compared with conventional cores with-out sacrificing performance.
That's possible because core losses in toroids ore typically 10% to 20% of the total power loss, with the balance being lost in the windings. That compares with core losses totaling 50% of the total power loss for conventional transformers. Lower core losses provide cooler operating temperatures and low magnetizing current
The toroidal transformer design best uses the high permeability and low-loss characteristics of a modern transformer core. Toroids are commonly used for current and instrument transformers, where low losses are extremely important. other advantages are higher efficiency lower operating temperature, before regulation, and lower noise. The disadvantage is somewhat higher cost, although improved production techniques are making toroidal transformers more cost competitive.

Transformer Basics.
The core and coils of a conventional transformer are shown in Fig. IA, Its schematic symbol is shown In Fig. lB; In its simplest form, a transformer is an electrical device that, by mutual electromagnetic induction, transfers electrical energy from one isolated circuit to another by means of coils called windings. The left, or primary winding is connected to on Ac voltage source. The right or secondary winding is connected to a load. The two windings are wound on a common iron core. Ac current in the primary winding provides the time-varying magnetomotive force (that creates an Ac magnetic flux in the core. The iron core provides a high-permeability magnetic path that ensures that a high percentage of the flux will be linked to the secondary winding. The voltage induced in the secondary winding and delivered to the load by the Ac core flux variations Is given by Faraday's law:
.e=kNd~/dt
In other words, the voltage induced in the secondary winding (e) is proportional to the number of turns (N) and the time rate of change of flux (~. Thus, the voltage can easily be changed from one level to another, lower. or higher, by properly selecting the number of primary and secondary turns. Also, at low frequency more turns are required to develop a given flux than at high frequency. Transformers only work with Ac voltages be-cause a time-varying AC flux is required.
In terms of real-world transformer design, Faraday's low becomes:
e = 4.44NBfA x 10^8
where the voltage in a winding is proportional to the number of turns (N), the flux density 'fl gauss (B), the Ac
-frequency-/ (I), and the core cross-sectional area in square centimeters (A).
Transformer Losses.
when a volt-age Is applied to the transformer primary ("~), the largest part of the current entering the primary winding (l~) goes toward creating the mmf that produces the mutual core flux, but not all of that current produces useful out-put power at the secondary The remaining portion of the primary current is required to provide the core exciting current 1E' as shown in Fig. 2. the equivalent circuit for a real transformer. The polarities shown in Fig. 2 are the instantaneous polarities of the primary and secondary waveforms of on Ac transformer.
Exciting current is the penalty to be paid for the significant advantages of using iron transformer cores. The existing current is divided Into magnetizing current (l~) and core-loss current (1~) Those loss components cause a volt-age division to occur due to the volt-age drop in the primary-winding Impedance (R~ and xp). That reduces the voltage available at the input to the Ideal (no-loss) transformer portion of the equivalent circuit (E~) and reduces the useful-primary current (Ii) Thus, additional primary voltage is required to establish the desired secondary voltage, E5. That additional primary voltage Is normally provided for by adding primary turns so the final turns ratio Is larger than the desired transformer voltage ratio.
Magnetizing current is the primary current that flows when the secondary is open-circuit and has no load. It is determined by:
1M = 0.794HM1/N where the magnetizing current is proportional to the mmf (H) times the magnetic path length (M1) divided by the number of turns (N). Transformer designers attempt to reduce the magnetizing current by minimizing the magnetic path length. That allows the fewest turns for a given primary voltage and has the added benefit of reducing the leakage flux.
The core-loss current is divided into  hysteresis and eddy current (EDDY) components. Hysteresis loss is the power required to magnetize the core and produce the Ac flux. The core consists of tiny magnetic dipoles or domains. Those domains align themselves and rotate with the Ac magnetic field. The hysteresis losses can be thought of as the friction caused by those rotating domains.
Eddy-current loss is power wasted by current flow in the conductive Iron core. Current Is induced in any conductor that is perpendicular to the flux path, including the windings and the lateral dimension of the iron core. For that reason, transformer cores, rather than being a single mass of iron, are laminated into thin strips whose long dimension Is parallel to the core flux. Doing that reduces the length of the conductive path in which the eddy current can flow. In addition, silicon is added to the iron to reduce its conductivity and further reduce the eddy currents.
Once a load is connected to the secondary winding, the secondary current flow creates a magnetic field whose back emf (electromotive force) opposes the primary f'.'x. The primary current Increases by the amount necessary to restore the original flux density supported by the primary voltage.
As the load current increases, the secondary impedance (R5 and Xs) causes a further reduction in the secondary voltage. The ratio of the difference between the no-load secondary voltage (\(~ and the full-load secondary voltage ("~~) for a fixed primary voltage is coiled the regulation.

Construction. Figure 3 illustrates the three common types of transformer cores. Unlike conventional transformers, which use stacked E-l ~ig. ~) or cut C-core (Fig. 3B) laminations, the toroidal transformer In Fig. 3C uses a tape-wound core with essentially no air gap. The stacked-lamination types are generally cheaper to manufacture because the windings are easily wound on separate bobbins or coil forms. Toroidal transformers require special winding equipment that first winds the wire on a circular shuttle inserted through the core center, then 'unwinds" the wire from the shuttle onto the core. A tensioned slider on the circumference of the shuttle controls the wire feed. while rollers rotate the core to evenly distribute the winding.
The magnetic core material for most power transformers Is made of grain-oriented, cold-rolled, 3% silicon steel that is coated and Insulated. That material has lower exciting current and core losses than regular steel. It also has relatively high saturation flux density with a high degree of squareness. Squareness is the ratio of residual flux density (remanence) to the maximum flux density (saturation), or Br/Bs. The oriented grain allows the steel to be operated at a higher saturation-flux density than non-oriented steel.
The core material is annealed at high temperature in a dry hydrogen furnace to remove impurities and relieve the material stresses. Annealing also develops the desired magnetic properties, such as high squareness and low core loss. The steel strip Is coated with a chemical finish to ensure high resistance between laminations. Finally the annealed cores are varnish impregnated, cured. and painted or epoxy coated.

Toroidal Transformer Advantages.
Higher flux density Is possible In a toroidal transformer because the windings are wound symmetrically Fig. 4. If an air gap is present, more magnetizing force (H) is needed to produce a given core flux (B). I

Fig. 5. Influx fringing. a small amount of core flux escapes the core at the air gap. decreasing the useful flux path and increasing core inductance,

over the gapless core. That symmetry results In smaller size and weight of the iron core. Also, because the windings completely enclose the core flux, stray magnetic fields that could Interfere with other circuitry within the en-closure are greatly reduced. Much less shielding Is required for use with sensitive or high-gain electronics.
The noise (hum) In a conventional transformer Is due to core magnetostriction, which is a very small deformation of the core iron under the influence of the magnetic field induced by the AC primary current. Be-cause the windings completely envelope the core in a toroidal transformer, audible hum Is reduced to 10% to 15% of that of a conventional transformer.
Because It has lower losses, the toroid transformer Is more efficient and runs at a lower operating temperature. It also has better load legulatlon than a conventional transformer of the same power rating.

The Effects of Air Gap. Much of the lower loss can be attributed to the reduced air gap of a toroidal transformer core. While intentional air gaps are designed into Inductors to pre-vent DC saturation, air gaps produce a number of undesirable effects In power transformers. Par the typical E-l-or C-core power-transformer laminations, the air gap is 0.002 Inches. In a toroidal transformer the effective air gap is extremely small (typically less than 0.00001 Inch) and can be Ignored for design purposes. The lack of a discrete air gap minimizes losses, leakage. and-flux-wave -distortion, and decreases the mmf needed to produce a given level of flux In the transformer.
The Ac inductance Is determined by the number of turns, the Impressed voltage, and the core cross-sectional area. The magnetic flux path has two components, the core magnetic length and the air-gap length. Those two components are not equal, be-cause air and Iron have vastly different permeability's.
Permeability Is the ratio of the change In magnetic induction (B) to the change In magnetizing force (H), and is equal to BIH. The permeability of air is constant at 1, while the permeability of silicon Iron depends on the degree of saturation in the core. At 80% saturation, silicon iron has a
permeability of about 4000. Because air offers 4000 times more reluctance to flux changes, a very small air gap has a great effect on the magnetizing volt-amperes needed to produce a given output power. An air gap In-creases the effective length of the magnetic path, reduces the inductance and, as shown in Fig. 4, causes more slope In the B-H curve. That re-quires more magnetizing force, and thus more primary current, to generate a given core flux. Once saturation Is reached, no further Increase in flux can occur even If the magnetizing current Is increased by raising the primary voltage.
Another disadvantage of an air gap Is flux fringing. Not all of the core flux remains within the core cross-section adjacent to the gap. A small per-centage curves outward near the edges of the core as shown in Fig. 5. causing core flux fringing. That fringing decreases the useful flux path area and increases the core Inductance 'Eddy currents are Induced
LEAKAGE FLUX Paths MAIN FLUX Paths

Fig. 6. The main (solid line) and leakage (dashed line) flux paths for the three most
common types of transformer cores are shown here. An E4 core is shown in A, a cut-C 56 core in B, and a toroid core is shown in C.
where the fringing flux returns to the core perpendicular to the desired magnetic path. causing additional losses. Fringing effect Is lower In a C-core than It is In an E-l core because one of the air gaps is enclosed by the windings. The magnetizing force-set up near the gap reduces the flux fringing In that gap. However. even If the air gap effect could be eliminated, the core halves cannot be perfectly aligned during assembly so same fringing will also occur due to misalignment.
Coupling and Leakage Rux~ In a
toroidal transformer. the entire magnetic path Is contained within the winding. which Is designed to be evenly distributed around the core.
That maximizes the coupling between windings and minimizes leakage flux. That Improved coupling Increases transformer efficiency Figures M and 6B show the main and leakage flux paths for E-l and C cores, where the leakage flux paths are completely outside the windings. All flux In a toroidal design is contained within the windings, as shown In Fig. 6C.
Toroidal cores generate a very small flux In the axial direction, but If necessary that flux can be contained by ring laminations assembled to the top and bottom of the core.

Limitations.
Because toroidal transformers have Insufficient air gaps to tolerate DC currents. only Ac currents can be impressed on the windings. To use a toroidal transformer as a push-pull tube-amplifier output transformer, for Instance, would require that perfectly matched tubes with balanced DC plate currents be used.
Toroidal power transformers should not be half-wave rectified. if It is, the core will become polarized and saturated in one direction.
The lack of an air gap can cause high inrush currents during energizing. limited only by the primary winding resistance. For larger transformers, a slow-blow fuse or soft-start Inrush-Ilmlt-mg circuit should be used.
The mounting bolt is usually placed through the center of a toroidal transformer. Care must be token that a difficult is not inadvertently completed between the top and bottom of that bolt. That would result In a shorted turn, causing excess heat and damage to the transformer.
Summary. The ideal transformer, which introduces no additional losses or voltage drops in the circuit. was characterized by MIT in 1943 as follows: ~ would have no lasses due to winding resistance and perfect coup-ling so there is no leakage flux. The care would have infinite permeability so there would be no saturation. It would also have infinite resistively and therefore no hysteresily or eddy-cur-rent loss. The care B-H curve would be a vertical line through zero salt would not take. any mmf to produce the flux. Finally, there would be no winding capacitance.
While far from that ideal, the toroidal transformer does a better Job of reducing those losses to their mini-mum practical values than any other production core form. Why not try to use one in your next project?
(source: Charles Hansen - All about Transformers)

Updated 10 March 2002 12:00 AM