Toroid

A toroid is a ring-shaped, three-dimensional geometric figure resembling a doughnut or inner tube. In engineering and physics, toroids are widely used in the design of inductors, transformers, and magnetic coils due to their ability to efficiently contain magnetic fields. Mathematically, a toroid is formed by revolving a circle around an external axis, creating a unique structure with both practical and theoretical significance.

1.0Definition of Toroid

• A toroid can be considered a ring-shaped closed solenoid. Hence, it is like an endless cylindrical solenoid.

Consider a toroid having n turns per unit length. Magnetic field at a point P in the figure is given as

$B=\frac{{\mu }_{0}Ni}{2\pi r}={\mu }_{0}ni$ where $n=\frac{N}{2\pi r}$

2.0Magnetic Field Due To Toroid

• A solenoid bent into the form of a closed ring is called a toroidal solenoid. Alternatively,it is an anchor ring (torous) around which a large number of turns of a metallic wire are wound. We shall see that the magnetic field B has a constant magnitude everywhere inside the toroid, while it is zero in the open space interior point P and exterior point Q to the toroid.
• A toroidal solenoid is formed by bending a straight solenoid into the shape of a closed ring.
• It can also be described as an anchor ring around which a large number of wire turns are uniformly wound.
• The magnetic field $(\vec{B})$inside the toroid is uniform in magnitude at all points along its circular path.

The magnetic field is zero at any point:
(a) Inside the central empty space of the ring (interior point P)

(b) Outside the toroid (exterior point Q)

The figure shows a cross-sectional view of a toroidal solenoid.

The magnetic field inside the toroid is clockwise, following the right-hand thumb rule. Three circular Amperean loops (shown as dashed lines) are used to analyze the field.
By symmetry:

• The magnetic field is tangential to each loop.
• The field has a constant magnitude along each loop.

Case(1): Magnetic Field Inside the Hollow of the Toroid

Let B₁ represent the magnitude of the magnetic field along Amperian Loop 1, which has a radius of r₁.

Length of the Loop 1, $L_1=2\pi r_1$

As the loop encloses no current, I=0

By using Ampere Law, $B_1{L}_1={\mu }_{0}I$

$B_1\times 2\pi r_1={\mu }_0\times 0$

$B_1=0$

Magnetic field at any point P in the open space interior to the toroid is zero.

Case(2): Magnetic Field Inside the Toroid

Let B denote the magnitude of the magnetic field along an Amperian loop with radius r

Length of the Loop 2, $L_2=2\pi r$

If N is the total number of turns in the toroid and I  is the current flowing through it, then the total current enclosed by Loop 2 is NI

By using Ampere Law, $B\times 2\pi r={\mu }_0\times NI$

$B=\frac{{\mu }_{0}NI}{2\pi r}$

If r is the average radius of the toroid and n represents the number of turns per unit length, then

$\begin{array}{rl}& N=2\pi rn\\ & B={\mu }_{0}nI\end{array}$

Case(3): Magnetic Field Outside the Toroid:

• Each turn of the toroid intersects the area enclosed by Amperian Loop 3 twice.
• For every turn, the current flowing out of the plane is cancelled by the current flowing into the plane.
• Therefore, the net current enclosed by Loop 3 is zero, I=0
• As a result, the magnetic field along Loop 3 is zero, B₃=0

3.0Types of Toroid

1. Round Toroids: Donut-shaped and common; ideal for low-profile applications; available in various sizes and materials.
2. Square Toroids: Square-shaped; used when a high Q factor is needed; less suitable for low-profile designs.
3. Rectangular Toroids: Designed for high flux density; perfect for limited space with high magnetic performance requirements.
4. Elliptical Toroids: Oval-shaped; provide high inductance and low profile, often used in power supplies and audio equipment.
5. Multisection Toroids: Made of multiple sections with separate windings; used in broadband transformers and filters.

Materials of Toroids

1. Ferrite Toroids: Most common; high magnetic permeability, low loss, and ideal for high Q factor applications.
2. Iron Powder Toroids: Made from iron powder; suited for high flux density but have higher losses and lower Q factor.
3. Amorphous Toroids: Made from amorphous metals; offer low core loss and high efficiency for demanding applications.

4.0Essential Key Points of Toroid

• The magnetic field inside a toroid is independent of radius.
• It depends only on the current (I) and number of turns per unit length (n).
• Inside the toroid, the field is of constant magnitude and tangential at every point.
• In an ideal toroid, coils are perfectly circular, and the external magnetic field is zero.
• In a real toroid, turns form a helical shape, causing a small external magnetic field to appear.