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Chapter 15: Problem 41

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 inductance will increase to 200 mH with the ferrite core.

Step by step solution

01

Understanding the Problem

We need to find the new inductance of a toroidal coil when the core material is changed from a plastic core with relative permeability, 03 0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0Building the Formula0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0A0Final Calculation0solution_stepsalletPneumapneuma80Modelif80H80Tpermeability80A80nderstanding80 80an80 800

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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 type of inductor where the wire is wound in a donut-shaped form. This distinctive design offers several advantages compared to traditional coils, especially when used in electronic circuits. The main feature of a toroidal coil is its closed-loop design, which helps contain the magnetic field within the core.

The closed path minimizes electromagnetic interference, making toroidal coils efficient in minimizing energy losses. These coils are often used in transformers, filters, and inductors due to their compact nature and high inductance relative to their size.
  • Advantages: Good energy efficiency, reduced electromagnetic interference, and high inductance.
  • Applications: Often used in power supply modules, RF signal applications, and audio devices.
Relative Permeability
Relative permeability is a measure of how easy it is for a material to support the formation of a magnetic field within itself compared to a vacuum. It is denoted by the symbol \( \mu_{r} \).

In the context of inductors, higher relative permeability means the material can support more magnetic field lines, leading to higher inductance.
\[L = \frac{\mu_{r} \cdot N^{2} \cdot A}{l}\]
This formula is crucial as it helps us understand the contribution of the core material to the coil's overall inductance. Here \( L \) is the inductance, \( N \) is the number of turns, \( A \) is the cross-sectional area of the core, and \( l \) is the magnetic path length.
  • Key point: The relative permeability of the material directly influences the inductance of the coil.
Ferrite Core
Ferrite cores are commonly used in inductors and transformers due to their high magnetic permeability and low electrical conductivity. They are made from a mixture of iron oxide and other metals, making them particularly effective at combating electromagnetic interference.

The high \( \mu_{r} \) of ferrite cores, often ranging from 50 to 3000, allows these cores to increase the inductance of coils significantly. \[\text{Inductance with Ferrite Core} = \mu_{r} \times \text{Inductance with Air Core} \]
This principle is critical when designing circuits that require high inductive values without significantly increasing the size of the coil.
  • Benefits: Ferrite cores enhance magnetic field intensity while decreasing size.
  • Applications: Used in RF circuits, signal transformers, and power conversion systems.
Magnetic Path
The concept of the magnetic path is essential in understanding how magnetism flows within a coil. In a toroidal coil, the magnetic path is the circular loop formed by the core material.

A continuous and closed magnetic path, as seen in toroidal coils, reduces the chances of leakage and maintains high inductance efficiency.

The length of the magnetic path affects the inductance, as represented in the following formula: \[L = \frac{\mu_{r} \cdot N^2 \cdot A}{l}\]
The magnetic path length \( l \) inversely affects the value of inductance; a shorter path leads to higher inductance if other factors are constant.
  • Important note: A shorter magnetic path can potentially increase the efficiency of magnetic flux linkage and boost inductance.

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Most popular questions from this chapter

Two coils having inductances \(L_{1}\) and \(L_{2}\) are wound on a common core. The fraction of the flux produced by one coil that links the other coil is called the coefficient of coupling and is denoted by \(k\). Derive an expression for the mutual inductance \(M\) in terms of \(L_{1}\), \(L_{2}\), and \(k\).

Two very long parallel wires are \(1 \mathrm{~cm}\) apart and carry currents of \(10 \mathrm{~A}\) in the same direction. The material surrounding the wires has \(\mu_{r}=1\). Determine the force on a \(0.5-\mathrm{m}\) section of one of the wires. Do the wires attract or repel one another?

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?

A transformer is needed that will cause an actual load resistance of \(25 \Omega\) to appear as \(100 \Omega\) to an ac voltage source of \(240 \mathrm{~V} \mathrm{rms}\). Draw the diagram of the circuit required. What turns ratio is required for the transformer? Find the current taken from the source, the current flowing through the load, and the load voltage.

The magnetic field of the earth is approximately \(3 \times 10^{-5} \mathrm{~T}\). At what distance from a long straight wire carrying a steady current of \(10 \mathrm{~A}\) is the field equal to 10 percent of the earth's field? Suggest at least two ways to help reduce the effect of electrical circuits on the navigation compass in a boat or airplane.
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