Physics
Mutual Inductance
Mutual inductance is a phenomenon in which a changing current in one coil induces a voltage in an adjacent coil. This occurs due to the magnetic field produced by the changing current, which links with the adjacent coil and induces an electromotive force. Mutual inductance is a key principle in the operation of transformers and is fundamental to the functioning of many electrical devices.
Written by Perlego with AI-assistance
Related key terms
1 of 5
10 Key excerpts on "Mutual Inductance"
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
14.1 | Mutual Inductance Learning Objectives By the end of this section, you will be able to: • Correlate two nearby circuits that carry time-varying currents with the emf induced in each circuit • Describe examples in which Mutual Inductance may or may not be desirable Inductance is the property of a device that tells us how effectively it induces an emf in another device. In other words, it is a physical quantity that expresses the effectiveness of a given device. When two circuits carrying time-varying currents are close to one another, the magnetic flux through each circuit varies because of the changing current I in the other circuit. Consequently, an emf is induced in each circuit by the changing current in the other. This type of emf is therefore called a mutually induced emf, and the phenomenon that occurs is known as Mutual Inductance (M). As an example, let’s consider two tightly wound coils (Figure 14.2). Coils 1 and 2 have N 1 and N 2 turns and carry currents I 1 and I 2 , respectively. The flux through a single turn of coil 2 produced by the magnetic field of the current in coil 1 is Φ 21 , whereas the flux through a single turn of coil 1 due to the magnetic field of I 2 is Φ 12 . Figure 14.2 Some of the magnetic field lines produced by the current in coil 1 pass through coil 2. The Mutual Inductance M 21 of coil 2 with respect to coil 1 is the ratio of the flux through the N 2 turns of coil 2 produced by the magnetic field of the current in coil 1, divided by that current, that is, (14.1) M 21 = N 2 Φ 21 I 1 . Similarly, the Mutual Inductance of coil 1 with respect to coil 2 is (14.2) M 12 = N 1 Φ 12 I 2 . Like capacitance, Mutual Inductance is a geometric quantity. It depends on the shapes and relative positions of the two coils, and it is independent of the currents in the coils.
eBook - PDFGrounds for Grounding
A Handbook from Circuits to Systems
- Elya B. Joffe, Kai-Sang Lock(Authors)
- 2022(Publication Date)
- Wiley-IEEE Press(Publisher)
Diminution of this current would result in a subsequent decrease of the magnetic field, inducing an opposing emf (i.e. voltage). 2.3.3 Mutual Inductance Mutual Inductance refers to magnetic flux penetrating a conducting loop, C 2 , produced by current, I 1 , flowing through another conducting loop, C 1 , causing induction of an emf in the second circuit (Figure 2.25). Observe that the voltage source representing the induced emf in loop 2 has a polarity, according to Faraday’s Law, such that it tends to induce a current in the second loop, which produces a magnetic flux that opposes the original magnetic flux passing through its surface, which is due to the current in loop 1. Mutual Inductance is an important mechanism by which transformers function, but it can also result in unwanted coupling between conductors in a circuit. For the common and important case of electrical circuits consisting of thin wires (or traces), the derivation of mutual induc- tance is straightforward. In a system consisting of K wire loops, each with one or more wire turns, the flux linkage of loop m, λ m , is given by: λ m = N m Φ m = K n = 1 L m,n i n 2 101 where N m denotes the number of turns in loop m, Φ m represents the magnetic flux through loop m and the constants L m,n are described below. This equation follows from Ampere’s Law. Applying Faraday’s Law of Induction to Equation (2.101) results in E m = dλ m dt = N m dΦ m dt = K n = 1 L m,n di n dt 2 102 where E m denotes the emf induced in circuit m, which is consistent with the common above definition of inductance if the constant coefficients L m,n are identified with the coefficients of inductance. Since the total currents N n i n contribute to the flux Φ m , it also follows that L m,n is proportional to the product of turns N m N n .
eBook - PDF- Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
C H A P T E R 15 Mutual Inductance AND TRANSFORMERS From the foregoing facts, it appears that a current of electricity is produced, for an instant, in a helix of copper wire surrounding a piece of soft iron whenever magnetism is induced in the iron; also that an instantaneous current in one or the other direction accompanies every change in the magnetic intensity of the iron. Joseph Henry, 1831, American Physicist Some History Behind This Chapter The discovery of electromagnetic induction led Michael Faraday to wind two separate insulated wires around an iron ring. One winding was connected to a battery via a switch and the other to a galvanometer. He observed that the current in the first winding did indeed induce a current in the second, but only during the instant when the switch was opened or closed. From this, he correctly deduced that it was the change in current that produced the inductive effect. Faraday’s coils and iron ring were actually a crude transformer. Practical ac power transformers were perfected by the British engineers Lucien Gauland and Josiah Gibbs in the early 1880s. Why This Chapter Is Important Today Coupled coils are found in applications such as power systems, communications, and high-quality audio. In this chapter, you will learn how to model the magnetic coupling between coils using the concept of Mutual Inductance. This concept leads to a new circuit element called a transformer, which provides impedance matching, electrical isolation, and changes in the voltage level in power systems. The large power transformers used in such systems are very efficient, with internal losses often less than 1%. Chapter Sections 15–1 Coupled Inductors 15–2 The Dot Convention 15–3 Energy Analysis 15–4 The Ideal Transformer 15–5 Linear Transformers Chapter Learning Objectives 15-1 Mutual Inductance (Sects. 15–1 to 15–3) Given the current through or voltage across two coupled inductors, find other currents or voltages.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
This mutual induction is described by ℰ 2 = −M di 1 ___ dt and ℰ 1 = −M di 2 ___ dt , where M (measured in henries) is the Mutual Inductance. LEARNING OBJECTIVES Mutual Induction In this section we return to the case of two interacting coils, which we first dis- cussed in Module 30.1, and we treat it in a somewhat more formal manner. We saw earlier that if two coils are close together as in Fig. 30.1.2, a steady current i in one coil will set up a magnetic flux Φ through the other coil (linking the other coil). If we change i with time, an emf ℰ given by Faraday’s law appears in the sec- ond coil; we called this process induction. We could better have called it mutual induction, to suggest the mutual interaction of the two coils and to distinguish it from self-induction, in which only one coil is involved. Let us look a little more quantitatively at mutual induction. Figure 30.9.1a shows two circular close-packed coils near each other and sharing a common central axis. With the variable resistor set at a particular resistance R, the battery produces a steady current i 1 in coil 1. This current creates a magnetic field repre- sented by the lines of B → 1 in the figure. Coil 2 is connected to a sensitive meter but contains no battery; a magnetic flux Φ 21 (the flux through coil 2 associated with the current in coil 1) links the N 2 turns of coil 2. We define the Mutual Inductance M 21 of coil 2 with respect to coil 1 as M 21 = N 2 Φ 21 ______ i 1 , (30.9.1) which has the same form as Eq. 30.4.1, L = NΦ/i, (30.9.2) the definition of inductance. We can recast Eq. 30.9.1 as M 21 i 1 = N 2 Φ 21 . (30.9.3) If we cause i 1 to vary with time by varying R, we have M 21 di 1 ___ dt = N 2 d Φ 21 _____ dt . (30.9.4) 896 CHAPTER 30 Induction and Inductance The right side of this equation is, according to Faraday’s law, just the magnitude of the emf ℰ 2 appearing in coil 2 due to the changing current in coil 1.
eBook - PDF- David Wyatt, Mike Tooley(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
1.30. This effect is called self-inductance (or just induct-ance ) which has the symbol L . Self-inductance is measured in henries (H) and is calculated from: e L di dt where L is the self-inductance, di dt / is the rate of change of current and the minus sign indicates that the polarity of the generated e.m.f. opposes the change (you might like to compare this relationship with the one shown earlier for electromagnetic induction). The unit of inductance is the henry (H) and a coil is said to have an inductance of 1 H if a voltage of 1V is induced across it when a current changing at the rate of 1 A/s is flowing in it. Example 1.18 A coil has a self-inductance of 15 mH and is subject to a current that changes at a rate of 450 A/s. What e.m.f. is produced? Now e L d i d t / and hence: e 1 5 10 4 50 6 7 3 . V 7 5 Note the minus sign. This reminds us that a back e.m.f. of 6.75V is induced. Example 1.19 A current increases at a uniform rate from 2 A to 6 A in a time of 250 ms. If this current is applied to an inductor determine the value of inductance if a back e.m.f. of 15V is produced across its terminals. Now e L di dt / and hence L e di dt / Thus L ( ) ( ) . . 250 10 15 62 5 1 0 937 5 1 0 0 9 . 4 3 3 3 H Finally, when two inductors are placed close to one another, the flux generated when a changing current flows in the first inductor will cut through the other inductor (see Fig. 1.31). This changing flux will, in turn, induce a current in the second inductor. This effect is known as Mutual Inductance and it occurs whenever two inductors are inductively coupled. This is the principle of a very useful component, the trans-former , which we shall meet later. Figure 1.30 Self-inductance 19 Electrical fundamentals 1.6.4 Inductors Inductors provide us with a means of storing electri-cal energy in the form of a magnetic field. Typical applications include chokes, filters, and frequency selective circuits.
- John Bird(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
9. The Mutual Inductance between two coils, when a current changing at 20 A/s in one coil induces an e.m.f. of 10 mV in the other, is: (a) 0.5 H; (b) 200 mH; (c) 0.5 mH; (d) 2 H. 9.9 Short answer questions on electromagnetic induction 1. What is electromagnetic induction? 2. State Faraday's laws of electromagnetic induction. 3. State Lenz's law. 4. Explain briefly the principle of the generator. 5. The direction of an induced e.m.f. in a gener-ator may be determined using Fleming's rule. 6. The e.m.f. E induced in a moving conductor may be calculated using the formula E = Blv. Name the quantities represented and their units. 7. What is self-inductance? State its symbol. 8. State and define the unit of inductance. 9. When a circuit has an inductance L and the current changes at a rate of ( // ) then the induced e.m.f. E is given by E = volts. 10. If a current of / amperes flowing in a coil of N turns produces a flux of Φ webers, the coil inductance L is given by L = henrys. 11. The energy W stored by an inductor is given by W = joules. 12. What is Mutual Inductance? State its symbol. 13. The Mutual Inductance between two coils is M. The e.m.f. E 2 induced in one coil by the current changing at ( ^/ ) in the other is given by E 2 = volts. 9.10 Further questions on electromagnetic induction Induced e.m.f. 1. A conductor of length 15 cm is moved at 750mm/s at right angles to a uniform flux density of 1.2 T. Determine the e.m.f. induced in the conductor. [0.135 V] 2. Find the speed that a conductor of length 120 mm must be moved at right angles to a magnetic field of flux density 0.6 T to induce in it an e.m.f. of 1.8 V. [25 m/s] 3. A 25 cm long conductor moves at a uniform speed of 8 m/s through a uniform magnetic field of flux density 1.2 T. Determine the current flowing in the conductor when (a) its ends are open-circuited, (b) its ends are connected to a load of 15 ohms resistance.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
We could better have called it mutual induction, to suggest the mutual interaction of the two coils and to distinguish it from self-induction, in which only one coil is involved. Let us look a little more quantitatively at mutual induction. Figure 30.9.1a shows two circular close-packed coils near each other and sharing a common central axis. With the variable resistor set at a particular resistance R, the battery produces a steady current i 1 in coil 1. This current creates a magnetic field repre- sented by the lines of B → 1 in the figure. Coil 2 is connected to a sensitive meter but contains no battery; a magnetic flux Φ 21 (the flux through coil 2 associated with the current in coil 1) links the N 2 turns of coil 2. Checkpoint 30.8.1 The table lists the number of turns per unit length, current, and cross-sectional area for three solenoids. Rank the solenoids according to the magnetic energy density within them, greatest first. Turns per Solenoid Unit Length Current Area a 2n 1 i 1 2A 1 b n 1 2i 1 A 1 c n 1 i 1 6A 1 30.9 MUTUAL INDUCTION Learning Objectives After reading this module, you should be able to . . . 30.9.1 Describe the mutual induction of two coils and sketch the arrangement. 30.9.2 Calculate the Mutual Inductance of one coil with respect to a second coil (or some second current that is changing). 30.9.3 Calculate the emf induced in one coil by a sec- ond coil in terms of the Mutual Inductance and the rate of change of the current in the second coil. Key Idea ● If coils 1 and 2 are near each other, a changing current in either coil can induce an emf in the other. This mutual induction is described by ℰ 2 = −M di 1 ___ dt and ℰ 1 = −M di 2 ___ dt , where M (measured in henries) is the Mutual Inductance. 944 CHAPTER 30 INDUCTION AND INDUCTANCE We define the Mutual Inductance M 21 of coil 2 with respect to coil 1 as M 21 = N 2 Φ 21 ______ i 1 , (30.9.1) which has the same form as Eq. 30.4.1, L = NΦ/i, (30.9.2) the definition of inductance.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
The magnetic field creates a magnetic flux Φ through the coil and as the current increases, the magnetic flux does, too. Thus, there is an induced emf in the coil that, according to Lenz’s law, opposes the battery. Because of the opposing emf, the current increases gradually (rather than jumping quickly to some final value). This effect is called self-inductance (or simply inductance). When a coil of wires is used as a circuit element for its self-inductance, it is called an inductor. An inductor resists changes in current by virtue of the changing mag- netic flux it creates within its own coils. Inductors are often used to smooth out current changes in electric circuits. 22.7 SELF-INDUCTANCE AND RL CIRCUITS Learning Objective I N T E R A C T I V E F E A T U R E I N T E R A C T I V E F E A T U R E Self-Inductance and RL Circuits | 629 Inductance If an inductor has N turns and the magnetic flux through each turn is Φ, then the total flux through the inductor is N Φ. The magnetic field created by the each turn of the loop is pro- portional to the current I, so the total flux must also be proportional to the current—that is, N I Φ ∝ . The proportionality constant is called the inductance, L: N LI Φ = (22.7.1) The SI unit of inductance is the henry (H), named after American scientist Joseph Henry, who discovered electromagnetic induction independently of Michael Faraday. The SI unit of magnetic flux is the weber, where 1 Wb 1 T m 2 = ⋅ . Using Equation 22.7.1, then, we can show that / 1 H 1 T m A 2 = ⋅ . If the current in the inductor changes by I ∆ , resulting in a change in magnetic flux of ∆Φ, then, according to Equation 22.7.1, N L I Φ ∆ = ∆ . Now we apply Faraday’s law to the inductor as follows: N t L I t = ∆Φ ∆ = ∆ ∆ E (22.7.2) The inductance L depends only on the geometry (shape, number of windings, etc.) and the type of core material used. Recall that a solenoid is a tightly wound coil in which the mag- netic field is uniform.
eBook - PDF- Bhag Singh Guru, Hüseyin R. Hiziroglu(Authors)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
These equations are rarely used to determine the Mutual Inductance between any two coils because of the double integration. It is much easier to calculate the self-and Mutual Inductances on the basis of flux linking the coils. EXAMPLE 7.8 A toroidal coil of 2000 turns is wound over a magnetic ring with inner radius of 10 mm, outer radius of 15 mm, height of 10 mm, and relative permeability of 500. A very long, straight conductor passing through the center of the toroid carries a time-varying current. Determine the Mutual Inductance between the toroid and the straight conductor. Solution A toroid with inner radius a , outer radius b , and height h with a very long conductor carrying current i ( t ) passing through its center is shown Figure 7.15 Mutual Inductance between a straight current-carrying conductor and a toroid in Figure 7.15. From the application of Amp` ere’s law, the magnetic flux density at any radius ρ within the toroid is B 1 = µ i 2 πρ a φ Thus, the flux linking the toroid is 21 = s 2 B 1 · d s 2 = µ i 2 π b a 1 ρ d ρ h 0 dz = µ i 2 π ln( b / a ) h 298 7 Time-varying electromagnetic fields Hence, the Mutual Inductance is L 21 = N 2 d 21 di = µ 2 π hN 2 ln( b / a ) Substituting the values, we obtain L 21 = 500 × 4 π × 10 − 7 2 π × 0 . 01 × 2000 × ln(15 / 10) = 0 . 81 mH 7.7 Inductance of coupled coils ................................. Two magnetically coupled coils can be connected either in series or in parallel. In each case, the effective inductance of the coupled coils depends upon the orientation of the coils and the direction of the flux produced by each coil. We now discuss series and parallel connections and in turn their aiding and opposing connections. 7.7.1 Series connection When the two coils are connected in tandem (end to end), they are said to be connected in series. When two series-connected coils produce a flux in the same direction (Figure 7.16a) they are said to be connected in series aiding .
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
550 Chapter 22 | Electromagnetic Induction 22.1 | Induced Emf and Induced Current There are a number of ways a magnetic field can be used to generate an electric current, and Figure 22.1 illustrates one of them. This drawing shows a bar magnet and a helical coil of wire to which an ammeter is connected. When there is no relative motion between the magnet and the coil, as in part a of the drawing, the ammeter reads zero, indicating that no current exists. However, when the magnet moves toward the coil, as in part b, a current I appears. As the magnet approaches, the magnetic field B B that it creates at the location of the coil becomes stronger and stronger, and it is this changing field that produces the current. When the magnet moves away from the coil, as in part c, a current is also pro- duced, but with a reversed direction. Now the magnetic field at the coil becomes weaker as the magnet moves away. Once again it is the changing field that generates the current. A current would also be created in Figure 22.1 if the magnet were held stationary and the coil were moved, because the magnetic field at the coil would be changing as the coil approached or receded from the magnet. Only relative motion between the magnet and the coil is needed to generate a current; it does not matter which one moves. The current in the coil is called an induced current because it is brought about (or “in- duced”) by a changing magnetic field. Since a source of emf (electromotive force) is always needed to produce a current, the coil itself behaves as if it were a source of emf. This emf is known as an induced emf. Thus, a changing magnetic field induces an emf in the coil, and the emf leads to an induced current. 22 | Electromagnetic Induction © Jeff Greenberg/Age Fotostock Electric guitars are famous for their amplified and manipulatable sound. To produce this sound, virtually all of them have one or more electro- magnetic pickups located beneath the strings (see Section 22.6).
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
