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Fundamentals of Electric Circuits Chapter 13 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Overview • This chapter introduces the concept of mutual inductance. • The general principle of magnetic coupling is covered first. • This is then applied to the case of mutual induction. • The chapter finishes with coverage of linear transformers. 2 Inductance • When two conductors are in close proximity to each other, the magnetic flux due to current passing through will induce a voltage in the other conductor. • This is called mutual inductance. • First consider a single inductor, a coil with N turns. • Current passing through will produce a magnetic flux, . 3 Self Inductance • If the flux changes, the induced voltage is: vN d dt • Or in terms of changing current: vN d di di dt • Solved for the inductance: LN d di • This is referred to as the self inductance, since it is the reaction of the inductor to the change in current through itself. 4 Magnetic Coupling • Now consider two coils with N1 and N2 turns respectively. • Each with self inductances L1 and L2. • Assume the second inductor carries no current. • The magnetic flux from coil 1 has two components: 1 11 12 • 11 links the coil to itself, 12 links both coils. 5 Magnetic Coupling II • Even though the two coils are physically not connected, we say they are magnetically coupled. • The entire flux passes through coil 1, thus the induced voltage in coil 1 is: v1 N1 d1 dt • In coil 2, only 12 passes through, thus the induced voltage is: v2 N 2 d12 dt 6 Magnetic Coupling III • These can be expressed in terms of the current through coil 1. di1 v1 L1 dt di1 v2 M 21 dt • Where M21 is the mutual inductance of coil2 with respect to coil 1. • A similar coupling exists for coil1 with respect to coil 2 7 Mutual Inductance • Mutual inductance, measured in Henries, is always positive. • But the induced voltage does not need to be positive. • Unlike self inductance though, the sign of the voltage is not exclusively determined by the direction of the current flow. • We need to know the orientation of the two coils with respect to each other. 8 Mutual Inductance II • This is inconvenient to show in a circuit diagram. • Therefore, the dot convention is used. • A dot is placed in the circuit at one end of each of the two magnetically coupled coils. • The dot indicates the direction of the flux if current enters the dotted terminal. 9 Dot Convention • If a current enters the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coil. • If a current leave the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is negative at the dotted terminal of the second coil. • See the examples in the next slide: 10 Dot Convention II 11 Coils in Series • The coupled coils can be connected in series in two different ways. • The total induction is: – Series aiding connection: L L1 L2 2M – Series opposing connection L L1 L2 2M 12 Series-aiding • Knowing the dot convention, we can analyze the series aiding connection. • Applying KVL to coil 1: v1 i1 R1 L1 di1 di M 2 dt dt • For coil 2: v2 i2 R2 L2 di2 di M 1 dt dt • In the frequency domain: V1 R1 j L1 I1 j MI 2 V2 j MI1 R2 j L2 I 2 13 Series-opposing • Now looking at the series-opposing connection. • Applying KVL to coil 1 gives: V Z1 j L1 I1 jMI 2 • Applying KVL to coil 2 gives: 0 jMI1 Z L j L2 I 2 14 Problem Solving • Mutually coupled circuits are often challenging to solve due to the ease of making errors in signs. • If the problem can be approached where the value and the sign of the inductors are solved in separate steps, solutions tend to be less error prone. • See the illustration for the proposed steps. 15 Problem Solving II • Each inductor will be represented as an inductor and a dependent voltage source. • It is possible to calculate the values of the induced voltages first, without determining the signs of the voltages. • Next, noting the direction of current flow into the dotted terminal, the sign of the dependent source on the opposite coupled inductor can be determined. 16 Energy in a Coupled Circuit • We saw previously that the energy stored in an inductor is: w 1 2 Li 2 • For coupled inductors, the total energy stored depends on the individual inductance and on the mutual inductance. w 1 2 1 L1i1 L2i2 2 Mi1i2 2 2 • The positive sign is selected when the currents both enter or leave the dotted terminals. 17 Limit on M • With the total energy established for the mutual inductors, we can establish an upper limit on M. • The system cannot have negative energy because the system is passive. 1 2 1 L1i1 L2i2 2 Mi1i2 0 2 2 • From this we get: M L1L2 18 Coupling Coeffcient • We can describe a parameter that described how closely the value of M approaches the upper limit. k M L1 L2 • This is called the coupling coefficient • k can range from 0 to 1 • It is determined by the physical configuration of the coils. 19 Linear Transformers • A transformer is a magnetic device that takes advantage of mutual inductance. • It is generally a four terminal device comprised of two or more magnetically coupled coils. • The coil that is connected to the voltage source is called the primary. • The one connected to the load is called the secondary. • They are called linear if the coils are wound on a magnetically linear material. 20 Transformer Impedance • An important parameter to know for a transformer is how the input impedance Zin is seen from the source. • Zin is important because it governs the behavior of the primary circuit. • Using the figure from the last slide, if one applies KVL to the two meshes: 2M 2 ZR R2 j L2 Z L • Here you see that the secondary impacts Zin 21 Equivalent circuits • We already know that coupled inductors can be tricky to work with. • One approach is to use a transformation to create an equivalent circuit. • The goal is to remove the mutual inductance. • This can be accomplished by using a T or a network. • The goal is to match the terminal voltages and currents from the original network to the new network. 22 Equivalent Circuits II • Starting with the coupled inductors as shown here: • Transforming to the T network the inductors are: V1 j L1 V j M 2 j M I1 j L2 I 2 • Transforming to the network the inductors are: L1 L2 M 2 LA L2 M L1 L2 M 2 LB L1 M L1 L2 M 2 LC M 23 Ideal Transformers • An ideal transformer is one with perfect coupling (k=1). • It has two or more turns with a large number of windings on a core of high permeability. • The ideal transformer has: 1. Coils with very large reactance (L1, L2, M →) 2. Coupling coefficient is equal to unity. 3. Primary and secondary coils are lossless 24 Ideal Transformers II • Iron core transformers are close to ideal. • The voltages are related to each other by the turns ration n: V N 2 V1 2 N1 n • The current is related as: I 2 N1 1 I1 N 2 n • A step down transformer (n<1) is one whose secondary voltage is less than its primary voltage. • A step up (n>1) is the opposite 25 Ideal Transformers III • There are rules for getting the polarity correct from the transformer in a circuit: • If V1 and V2 are both positive or both negative at the dotted terminal, use +n otherwise use –n • If I1 and I2 both enter or leave the dotted terminal, use -n otherwise use +n • The complex power in the primary winding is: V2 * S1 V I nI 2 V2 I 2* S 2 n * 1 1 26 Reflected Impedance • The input impedance that appears at the source is: V1 V2 1 Z in I1 n nI 2 ZL Z in 2 n • This is also called the reflected impedance since it appears as if the load impedance is reflected to the primary side. • This matters when one considers impedance matching. 27 Removing the transformer • We can remove the transformer from the circuit by adding the secondary and primary together by certain rules: • The general rule for eliminating the transformer and reflecting the secondary circuit to the primary side is: Divide the secondary impedance by n2, divide the secondary voltage by n, and multiply the secondary current by n. • The rule for eliminating the transformer and reflecting the primary circuit to the secondary side is: Multiply the primary impedance by n2, multiply the primary voltage by n, and divide the primary current by n. 28 Ideal Autotransformer • An autotransformer uses one winding for primary and secondary • It can do step-down and stepup. • The voltage relationship is: V1 N1 V2 N1 N 2 • It does not offer isolation! 29 Three Phase Transformer • When working with three phase power, there are two choices for transformers: – A transformer bank, with one transformer per phase – A three phase transformer • The three phase transformer will be smaller and less expensive. • The same connection permutations of Delta and Wye hold as discussed previously. 30