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Chapter 6: Problem 43

The self-inductances of two magnetically coupled coils are \(L_{1}=400 \mu \mathrm{H}\) and \(L_{2}=900 \mu \mathrm{H} .\) The coupling medium is nonmagnetic. If coil 1 has 250 turns and coil 2 has 500 turns, find \(\mathscr{P}_{11}\) and \(\mathscr{P}_{21}\) (in nanowebers per ampere) when the coefficient of coupling is 0.75

Short Answer

Expert verified
\(\mathscr{P}_{11} = 281.25\, \text{nWb/A}\), \(\mathscr{P}_{21} = 250\, \text{nWb/A}\)

Step by step solution

01

Understanding the Problem

We are given two coils with self-inductances \(L_1 = 400\, \mu \text{H}\) and \(L_2 = 900\, \mu \text{H}\). The number of turns for coil 1 is 250, and for coil 2 is 500. The coefficient of coupling \(k\) is 0.75. We need to find the mutual inductance \(M\) first, and then use it to find \(\mathscr{P}_{11}\) and \(\mathscr{P}_{21}\).
02

Calculate Mutual Inductance (M)

The formula for mutual inductance \(M\) between two inductors is given as \(M = k \sqrt{L_1 L_2}\). Substituting the given values:\[M = 0.75 \sqrt{400 \times 900} \] Simplifying under the square root, we have:\[M = 0.75 \sqrt{360000} \]\[M = 0.75 \times 600 \]\[M = 450 \, \mu \text{H}\]
03

Calculate \(\mathscr{P}_{11}\)

The mutual flux linkage per ampere for coil 1, \(\mathscr{P}_{11}\), can be calculated using the formula:\[\mathscr{P}_{11} = \frac{M N_1}{L_1}\]Substituting the given values along with the calculated \(M\):\[\mathscr{P}_{11} = \frac{450 \times 250}{400}\]\[\mathscr{P}_{11} = \frac{112500}{400}\]\[\mathscr{P}_{11} = 281.25\, \text{nWb/A}\]
04

Calculate \(\mathscr{P}_{21}\)

The mutual flux linkage per ampere for coil 2, \(\mathscr{P}_{21}\), can be calculated using the formula:\[\mathscr{P}_{21} = \frac{M N_2}{L_2}\]Substituting the given values and \(M\):\[\mathscr{P}_{21} = \frac{450 \times 500}{900}\]\[\mathscr{P}_{21} = \frac{225000}{900}\]\[\mathscr{P}_{21} = 250\, \text{nWb/A}\]

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Key Concepts

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

Magnetic coupling
Magnetic coupling occurs when two coils are positioned in such a way that the magnetic field created by the current in one coil induces a voltage in the second coil. This phenomenon is foundational in the study of inductance. When two coils are magnetically coupled, energy can efficiently transfer from one coil (the primary) to another (the secondary) by creating a mutual magnetic field between them.
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  • Magnetic coupling is vital in transformers, where it is used to transfer electrical energy from one circuit to another through coils.
  • The strength of magnetic coupling is influenced by factors like the physical closeness of the coils, the medium between them, and the orientation of the coils.
  • Magnetic coupling can be quantified using the coefficient of coupling, which indicates how well the magnetic lines of one coil link to the other coil.
Understanding magnetic coupling allows engineers to design circuits that maximize energy transfer and minimize loss, making systems like power transmission more efficient.
Self-inductance
Self-inductance is a property of a coil that describes its ability to induce an electromotive force (EMF) on itself when there is a change in current through it. It is denoted by the symbol \(L\). This concept is similar to how the coil resists changes in current flow due to its own generated magnetic field. Here are some key points:
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  • Self-inductance depends on the coil's physical characteristics, such as the number of turns, area, and the type of core material used.
  • A larger inductance means the coil can store more magnetic energy per ampere of current flowing through it.
  • Self-inductance is measured in henries (H) and is a significant factor in AC circuits where it opposes changes in current.
The formula for self-inductance \(L\) can be affected by a coil's dimensions and the medium in which it is placed, which is why it's crucial in designing inductors and transformers.
Mutual flux linkage
Mutual flux linkage describes the connection between two coils where the magnetic field created by one coil links with the other. This linkage can produce a mutual inductance \(M\), leading to energy transfer between the coils. Understanding mutual flux linkage is fundamental in studying how energy can be transferred through magnetic fields.
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  • The mutual flux linkage depends on the orientation and proximity of the coils, as well as the coefficient of coupling \(k\).
  • Calculating the mutual flux linkage involves understanding the mutual inductance and the number of turns in each coil, as shown by the formulas \(\mathscr{P}_{11} = \frac{M N_1}{L_1}\) and \(\mathscr{P}_{21} = \frac{M N_2}{L_2}\).
Mutual flux linkage is paramount in applications such as transformers, where it ensures the efficient transfer of electrical power between coils.
Coefficient of coupling
The coefficient of coupling \(k\) is a dimensionless factor that measures how effectively two coils are magnetically coupled. It ranges from 0 to 1, where a value of 1 indicates perfect coupling, meaning all the magnetic flux of one coil links to the other. This coefficient helps to gauge the efficiency of energy transmission between magnetically linked coils.
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  • The value of \(k\) is influenced by factors such as the distance between coils, the type of medium, and the alignment of the coils.
  • A higher coefficient of coupling implies a more effective transmission of energy, benefiting power and signal transfer applications.
  • In our exercise, a \(k\) value of 0.75 indicates strong, but not perfect, coupling between the coils.
Understanding and calculating the coefficient of coupling is crucial for designing efficient electric circuits and ensuring proper functionality in devices that rely on magnetic induction.

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Most popular questions from this chapter

Derive the equivalent circuit for a series connection of ideal capacitors, Assume that each capacitor has its own initial voltage. Denote these initial voltages as \(v_{1}\left(t_{0}\right), v_{2}\left(t_{0}\right),\) and so on. (Hint: Sum the voltages across the string of capacitors, recognizing that the series connection forces the current in each capacitor to be the same.)

The voltage across the terminals of a \(0.4 \mu \mathrm{F}\) capacitor is $$v=\left\\{\begin{array}{ll} 25 \mathrm{V}, & \mathrm{t} \leq 0 \\ A_{1} t e^{-1500 t}+A_{2} e^{-1500 t} \mathrm{V}, & t \geq 0 \end{array}\right.$$ The initial current in the capacitor is 90 mA. Assume the passive sign convention. a) What is the initial energy stored in the capacitor? b) Evaluate the coefficients \(A_{1}\) and \(A_{2}\) c) What is the expression for the capacitor current?

A \(0.5 \mu \mathrm{F}\) capacitor is subjected to a voltage pulse having a duration of 2 s. The pulse is described pully the following equations: $$v_{c}(t)=\left\\{\begin{array}{ll} 40 t^{3} \mathrm{V}, & 0 \leq t \leq 1 \mathrm{s} \\ 40(2-t)^{3} \mathrm{V}, & 1 \mathrm{s} \leq t \leq 2 \mathrm{s} \\ 0 & \text { elsewhere } \end{array}\right.$$ Sketch the current pulse that exists in the capacitor during the 2 s interval.

Two magnetically coupled coils have self-inductances of \(52 \mathrm{mH}\) and \(13 \mathrm{mH}\), respectively. The mutual inductance between the coils is \(19.5 \mathrm{mH}\). a) What is the coefficient of coupling? b) For these two coils, what is the largest value that \(M\) can have? c) Assume that the physical structure of these coupled coils is such that \(\mathscr{P}_{1}=\mathscr{P}_{2}\). What is the turns ratio \(N_{1} / N_{2}\) if \(N_{1}\) is the number of turns on the \(52 \mathrm{mH}\) coil?

The current in a \(100 \mu \mathrm{H}\) inductor is known to be $$i_{\mathrm{L}}=20 \mathrm{te}^{-5 \mathrm{t}} \mathrm{A} \quad \text { for } t \geq 0$$ a) Find the voltage across the inductor for \(t>0\) (Assume the passive sign convention.) b) Find the power (in microwatts) at the terminals of the inductor when \(t=100 \mathrm{ms}\) c) Is the inductor absorbing or delivering power at \(100 \mathrm{ms} ?\) d) Find the energy (in microjoules) stored in the inductor at \(100 \mathrm{ms}\) e) Find the maximum energy (in microjoules) stored in the inductor and the time (in microseconds) when it occurs.
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