Magnetically Coupled Circuits PDF

Magnetically Coupled Circuits Electromagnetics is the branch of electrical engineering (or physics) that deals with the analysis and application of electric and magnetic fields. In electromagnetics, electric circuit analysis is applied at low frequencies. The principles o...

Magnetically Coupled Circuits Electromagnetics is the branch of electrical engineering (or physics) that deals with the analysis and application of electric and magnetic fields. In electromagnetics, electric circuit analysis is applied at low frequencies. The principles of electromagnetics (EM) are applied in various allied disciplines, such as electric machines, electromechanical energy conversion, radar meteorology, remote sensing, satellite communications, bio electromagnetics, electromagnetic interference and compatibility, plasmas, and fiber optics. EM devices include electric motors and generators, transformers, electromagnets, magnetic levitation, antennas, radars, microwave ovens, microwave dishes, superconductors, and electrocardiograms. The design of these devices requires a thorough knowledge of the laws and principles of EM. EM is regarded as one of the more difficult disciplines in electrical engineering. One reason is that EM phenomena are rather abstract. But if one enjoys working with mathematics and can visualize the invisible, one should consider being a specialist in EM, since few electrical engineers specialize in this area. Electrical engineers who specialize in EM are needed in microwave industries, radio/TV broadcasting stations, electromagnetic research laboratories, and several communications industries.  Introduction The circuits we have considered so far may be regarded as conductively coupled, because one loop affects the neighboring loop through current conduction. When two loops with or without contacts between them affect each other through the magnetic field generated by one of them, they are said to be magnetically coupled. The transformer is an electrical device designed on the basis of the concept of magnetic coupling. It uses magnetically coupled coils to transfer energy from one circuit to another. Transformers are key circuit elements. They are used in power systems for stepping up or stepping down ac voltages or currents. They are used in electronic circuits such as radio and television receivers for such purposes as impedance matching, isolating one part of a circuit from another, and again for stepping up or down ac voltages and currents. We will begin with the concept of mutual inductance and introduce the dot convention used for determining the voltage polarities of inductively coupled components. Based on the notion of mutual inductance, we then introduce the circuit element known as the transformer. We will consider the linear transformer, the ideal transformer, the ideal autotransformer, and the three-phase transformer. Finally, among their important applications, we look at transformers as isolating and matching devices and their use in power distribution.  Mutual Inductance When two inductors (or coils) are in a close proximity to each other, the magnetic flux caused by current in one coil links with the other coil, thereby inducing voltage in the latter. This phenomenon is known as mutual inductance. Let us first consider a single inductor, a coil with N turns. When current i flows through the coil, a magnetic flux Ø is produced around it (Fig. 1). According to Faraday’s law, the voltage V induced in the coil is proportional to the number of turns N and the time rate of change of the magnetic flux ; that is, Eq. (1) But the flux Ø is produced by current i so that any change in Ø is caused by a change in the current. Hence, Eq. (1) can be written as Magnetic flux produced by Eq. (2) a single coil with N turns Eq. (3)  which is the voltage-current relationship for the inductor. From Eq. (2) and (3), the inductance L of the inductor is thus given by Eq. (4) This inductance is commonly called self-inductance, because it relates the voltage induced in a coil by a time-varying current in the same coil. Now consider two coils with self-inductances L1 and L2 that are in close proximity with each other (Fig. 2). Coil 1 has N1 turns, while coil 2 has N2 turns. For the sake of simplicity, assume that the second inductor carries no current. The magnetic flux Ø1 emanating from coil 1 has two components: one component Ø11 links only coil 1, and another component Ø12 links both coils. Hence, Eq. (5) Although the two coils are physically separated, they are said to be magnetically coupled. Since the entire flux Ø1 links coil 1, the voltage induced in coil 1 is Eq. (6) Mutual inductance M12 of coil 2 with respect to coil 1. Only flux Ø12 links coil 2, so the voltage induced in coil 2 is Eq. (7) Again, as the fluxes are caused by the current i1 flowing in coil 1, Eq. (6) can be written as Eq. (8) where L1 = N1 dØ1/ di1 is the self-inductance of coil 1. Similarly, Eq. (7) can be written as Eq. (9) Eq. (10) M21 is known as the mutual inductance of coil 2 with respect to coil 1. Subscript 21 indicates that the inductance M21 relates the voltage induced in coil 2 to the current in coil 1. Thus, the open-circuit mutual voltage (or induced voltage) across coil 2 is Eq. (11) Suppose we now let current i2 flow in coil 2, while coil 1 carries no current (Fig. 3). The magnetic flux Ø2 emanating from coil 2 comprises Ø22 flux that links only coil 2 and flux Ø21 that links both coils. Hence, Eq. (12) Eq. (13) Eq. (14) Mutual inductance M12 of coil 1 with respect to coil 2. Eq. (15)  which is the mutual inductance of coil 1 with respect to coil 2. Thus, the open-circuit mutual voltage across coil 1 is Eq. (16) We will see in the next section that M12 and M21 are equal; that is, M =M =M 12 21 Eq. (17) and we refer to M as the mutual inductance between the two coils. Like self-inductance L, mutual inductance M is measured in henrys (H). Keep in mind that mutual coupling only exists when the inductors or coils are in close proximity, and the circuits are driven by time-varying sources. We recall that inductors act like short circuits to dc. From the two cases in Figs. 2 and 3, we conclude that mutual inductance results if a voltage is induced by a time-varying current in another circuit. It is the property of an inductor to produce a voltage in reaction to a time-varying current in another inductor near it. Thus, Although mutual inductance M is always a positive quantity, the mutual voltage M di/dt may be negative or positive, just like the self-induced voltage L di/dt. However, unlike the self-induced L di/dt whose polarity is determined by the reference direction of the current and the reference polarity of the voltage (according to the passive sign convention), the polarity of mutual voltage M di/dt is not easy to determine, because four terminals are involved. The choice of the correct polarity for M di/dt is made by examining the orientation or particular way in which both coils are physically wound and applying Lenz’s law in conjunction with the right-hand rule. Since it is inconvenient to show the construction details of coils on a circuit schematic, we apply the dot convention in circuit analysis. By this convention, a dot is placed in the circuit at one end of each of the two magnetically coupled coils to indicate the direction of the magnetic flux if current enters that dotted terminal of the coil. This is illustrated in Fig. 4. Given a circuit, the dots are already placed beside the coils so that we need not bother about how to place them. The dots are used along with the dot convention to determine the polarity of the mutual voltage. The dot convention is stated as follows: Illustration of the dot convention. Thus, the reference polarity of the mutual voltage depends on the reference direction of the inducing current and the dots on the coupled coils. Application of the dot convention is illustrated in the four pairs of mutually coupled coils in Fig. 5. For the coupled coils in Fig. 5(a), the sign of the mutual voltage v2 is determined by the reference polarity v2 for and the direction of i1. Since i1 enters the dotted terminal of coil 1 and v2 is positive at the dotted terminal of coil 2, the mutual voltage is + M di1/dt. For the coils in Fig. 5(b), the current i1 enters the dotted terminal of coil 1 and v2 is negative at the dotted terminal of coil 2. Hence, the mutual voltage is - M di1/dt. The same reasoning applies to the coils in Fig. 5(c) and 5(d). Examples illustrating how to apply the dot convention. Figure.6 shows the dot convention for coupled coils in series. For the coils in Fig. 6(a), the total inductance is Eq. (18) Eq. (19) Dot convention for coils in series; the sign indicates the polarity of the mutual voltage: (a) series-aiding connection, (b) series-opposing connection. Now that we know how to determine the polarity of the mutual voltage, we are prepared to analyze circuits involving mutual inductance. As the first example, consider the circuit in Fig. 7. Applying KVL to coil 1 gives Eq. (20) For coil 2, KVL gives Eq. (21) We can write Eq. (20 and 21) in the frequency domain as Eq. (22) Eq. (23) As a second example, consider the circuit in Fig. 8. We analyze this in the frequency domain. Applying KVL to coil 1, we get Eq. (24) For coil 2, KVL yields Eq. (25) Equations (24) and (25) are solved in the usual manner to determine the currents. At this introductory level we are not concerned with the determination of the mutual inductances of the coils and their dot placements. Like R, L, and C, calculation of M would involve applying the theory of electromagnetics to the actual physical properties of the coils. In this text, we assume that the mutual inductance and the placement of the dots are the “givens’’ of the circuit problem, like the circuit components R, L, and C. Example 1 Calculate the phasor currents I1 and I2 in the circuit of Fig. below: Solution: For coil 1, KVL gives or For coil 2, KVL gives. or Substituting these Eq. Example 2 Calculate the mesh currents in the circuit of Fig. below: Solution: The key to analyzing a magnetically coupled circuit is knowing the polarity of the mutual voltage. We need to apply the dot rule. Suppose coil 1 is the one whose reactance is 6Ω and coil 2 is the one whose reactance is 8Ω To figure out the polarity of the mutual voltage in coil 1 due to I2 current we observe that I2 leaves the dotted terminal of coil 2. Since we are applying KVL in the clockwise direction, it implies that the mutual voltage is negative, that is - j2 I2. Alternatively, it might be best to figure out the mutual voltage by redrawing the relevant portion of the circuit, as shown in Fig. (a), where it becomes clear that the mutual voltage is V1 = -2j I2. Thus, for mesh 1 Similarly, to figure out the mutual voltage in coil 2 due to current , consider the relevant portion of the circuit, as shown in Fig. (b). Applying the dot convention gives the mutual voltage as Also, current sees the two coupled coils in series in Fig. below; since it leaves the dotted terminals in both coils, Therefore, for mesh 2 in Fig. below , KVL gives Putting Eqs. in matrix form, we get