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Chapter 15: Problem 40

A symmetrical toroidal coil is wound on a plastic core \(\left(\mu_{r} \cong 1\right)\) and is found to have an inductance of \(1 \mathrm{mH}\). What inductance will result if the core material is changed to a ferrite having \(\mu_{r}=200 ?\) Assume that the entire magnetic path is composed of ferrite

Short Answer

Expert verified
The new inductance is 200 mH.

Step by step solution

01

Understand the Relationship

The inductance of a coil is directly proportional to the relative permeability of the core material. The formula is given by \( L = L_0 \cdot \mu_r \), where \( L_0 \) is the inductance with the initial core material, and \( \mu_r \) is the relative permeability.
02

Identify the Given Values

We know that the initial inductance \( L_0 \) is 1 mH with \( \mu_{r} \approx 1 \). The new \( \mu_{r} \) for the ferrite is 200.
03

Apply the Formula

Substitute \( L_0 = 1 \text{ mH} \) and \( \mu_r = 200 \) into the formula: \[ L = 1 \text{ mH} \times 200 \]
04

Calculate the New Inductance

Perform the multiplication to find the new inductance: \[ L = 200 \text{ mH} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Toroidal Coil
A toroidal coil is a circular-shaped coil that looks like a doughnut or a ring. This coil design is unique because the wire is wound around a core that forms a closed-loop path. This arrangement helps to contain the magnetic field inside the core, reducing the loss of magnetic lines and minimizing external interference. It鈥檚 widely used in electronic circuits like transformers and inductors because it effectively confines the magnetic field. When you see the term "toroidal," it's usually referring to something that鈥檚 shaped like a torus. These coils are great in applications where compact size and efficiency are crucial.
  • Reduces magnetic losses
  • Minimizes electromagnetic interference
  • Increases efficiency in electronic devices
Inductance Calculation
Inductance is a measure of an inductor's ability to store energy in its magnetic field. Calculating inductance involves understanding how the magnetic field lines form around a coil. The basic formula for inductance is linked to the number of turns of wire and the permeability of the core material. For a simple inductor, the formula is given by \( L = \frac{N^2 \mu A}{l} \), where:
  • \( L \) is the inductance
  • \( N \) is the number of turns
  • \( \mu \) is the permeability of the core
  • \( A \) is the cross-sectional area
  • \( l \) is the length of the coil
In cases where the coil is toroidal, like in this exercise, the formula is slightly simplified as seen in the solution: \( L = L_0 \cdot \mu_r \). By changing the core to a material with higher permeability, the inductance increases significantly.
Relative Permeability
Relative permeability \( \mu_r \) is a dimensionless measure of how much more (or less) a material concentrates magnetic field lines compared to vacuum (or air). In simpler terms, it's how well a material can "support" magnetic fields within it. A material with a high relative permeability will concentrate more magnetic lines, making it ideal for use in electromagnets or transformers. In the exercise given, the initial core had a relative permeability of approximately 1 (like air), while the new ferrite material had a \( \mu_r \) of 200. This change dramatically increased the coil鈥檚 inductive capability.
  • \( \mu_r = 1 \) is like having air or plastic as core material
  • Higher \( \mu_r \) means a stronger magnetic field within the coil
  • Used extensively in inductors, transformers, and electromagnets
Magnetic Core Materials
Magnetic core materials play a crucial role in determining the efficiency and effectiveness of a coil. Different materials have varying levels of permeability, which affects how they enhance magnetic fields. Common materials include plastics, iron, and ferrites, each serving different purposes. While plastic has permeability close to that of air, making it less effective at enhancing magnetic fields, materials like ferrites can significantly increase inductance due to their high permeability. Ferrites are ceramic compounds containing iron oxide combined with other metallic elements, giving them unique magnetic properties suitable for numerous electronics applications.
  • Plastic cores are often used in lower frequency applications
  • Ferrites enhance inductance by concentrating magnetic lines
  • Choice of core material depends on the desired application and frequency

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Most popular questions from this chapter

Two coils wound on a common core have $$L_{1}=1 \mathrm{H}, L_{2}=2 \mathrm{H}, \text { and } M=0.5 \mathrm{H}$$ The currents are \(l_{1}=1\) A and \(i_{2}=0.5 \mathrm{A}\) If both currents enter dotted terminals, find the flux linkages of both coils Repeat if \(i_{1}\) enters a dotted terminal and \(t_{2}\) leaves a dotted terminal

What are two causes of core loss for a coil with an iron core excited by an ac current? What considerations are important in minimizing loss due to each of these causes? What happens to the power loss in each case if the frequency of operation is doubled while maintaining constant peak flux density?

What characteristic is desirable in the \(B-\) \(H\) curve for a prospective material to be used in a permanent magnet? In a motor or transformer? Explain

Suppose that, in designing an electrical generator, we need to produce a voltage of \(120 \mathrm{V}\) by moving a straight conductor through a uniform magnetic field of \(0.5 \mathrm{T}\) at a speed of \(30 \mathrm{m} / \mathrm{s}\) The conductor, its motion, and the field are mutually perpendicular. What is the required length of the conductor? It turns out that in generator design, a conductor of this length is impractical, and we must use \(N\) conductors of length 0.1 m. However, by connecting the conductors in series, we can obtain the required 120 V. What is the number \(N\) of conductors needed?

Draw an equivalent circuit for a real transformer. Briefly discuss the reason that each element appears in the equivalent circuit.
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