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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 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: 6 Dot Convention II 7 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. 8 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. 9 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 10 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. 11 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 12 Ideal Transformers II • Iron core transformers are close to ideal. • The voltages are related to each other by the turns ration 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 13 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 14 Reflected Impedance • The input impedance that appears at the source is: 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. 15 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. 16 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. 17