Chapter 13 Magnetically Coupled Circuits Chapter Objectives Understand

Chapter 13 Magnetically Coupled Circuits Chapter Objectives: Ø Understand magnetically coupled circuits. Ø Ø Ø Ø Learn the concept of mutual inductance. Be able to determine energy in a coupled circuit. Learn how to analyze circuits involving linear and ideal transformers. Be familiar with ideal autotransformers. Learn how to analyze circuits involving three-phase transformers. Be able to use PSpice to analyze magnetically coupled circuits. Apply what is learnt to transformer as an isolation device and power distribution Huseyin Bilgekul Eeng 224 Circuit Theory II Department of Electrical and Electronic Engineering Eastern Mediterranean University Eeng 224 1

Mutual Inductance Ø Transformers are constructed of two coils placed so that the charging flux developed by one will link the other. Ø The coil to which the source is applied is called the primary coil. Ø The coil to which the load is applied is called the secondary coil. Ø Three basic operations of a transformer are: Ø Step up/down Ø Impedance matching Ø Isolation Eeng 224 2

Mutual Inductance Devices Eeng 224 3

Mutual Inductance Ø When two coils are placed close to each other, a changing flux in one coil will cause an induced voltage in the second coil. The coils are said to have mutual inductance M, which can either add or subtract from the total inductance depending on if the fields are aiding or opposing. Ø Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor. Eeng 224 4

Mutual Inductance a) Magnetic flux produced by a single coil. b) Mutual inductance M 21 of coil 2 with respect to coil 1. c) Mutual inductance of M 12 of coil 1 with respect to coil 2. Eeng 224 5

Mutual Inductance Ø Mutual inductances M 12 and M 21 are equal. Ø They are referred as M. Ø We refer to M as the mutual inductance between two coils. Ø M is measured in Henry’s. Ø Mutual inductance exists when two coils are close to each other. Ø Mutual inductance effect exist when circuits are driven by time varying sources. Ø Recall that inductors act like short circuits to DC. Eeng 224 6

Dot Convention Ø If the 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 the current LEAVES 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. Eeng 224 7

Dot Convention Eeng 224 8

Coils in Series Ø The total inductance of two coupled coils in series depend on the placement of the dotted ends of the coils. The mutual inductances may add or subtract. a) Series-aiding connection. L=L 1+L 2+2 M b) Series-opposing connection. L=L 1+L 2 -2 M Eeng 224 9

Time-domain and Frequency-domain Analysis j M V 1 a) Time-domain circuit I 1 j L 2 I 2 V 2 b) Frequency-domain circuit Eeng 224 10

Induced mutual voltages Eeng 224 11

Induced mutual voltages Eeng 224 12

j 3 I 1 + + - j 3 I 2 + - P. P. 13. 2 Determine the phasor currents j 3 I 1 Eeng 224 13

Mutually Induced Voltages Ø To find I 0 in the following circuit, we need to write the mesh equations. Let us represent the mutually induced voltages by inserting voltage sources in order to avoid mistakes and confusion. -j 50 Io I 3 j 20 Ic + j 40 j 10 Ib + Ia j 60 + j 30 Ic + Ic j 30 Ib + j 20 Ia 50 0 V + j 80 I 1 Ib + I 2 100 Ia = I 1 – I 3 Ib = I 2 – I 1 Ic = I 3 – I 2 Io = I 3 Blue Voltage due to Ia Red Voltage due to Ic Green Voltage due to Ib j 10 Ia Eeng 224 14

Mutually Induced Voltages Ø To find I 0 in the following circuit, we need to write the mesh equations. Let us represent the mutually induced voltages by inserting voltage sources in order to avoid mistakes and confusion. Eeng 224 15

Energy in a Coupled Circuit Ø The total energy w stored in a mutually coupled inductor is: Ø Positive sign is selected if both currents ENTER or LEAVE the dotted terminals. Ø Otherwise we use Negative sign. Eeng 224 16

Coupling Coefficient Ø The Coupling Coefficient k is a measure of the magnetic coupling between two coils a) Loosely coupled coil b) Tightly coupled coil Eeng 224 17

Linear Transformers Ø A transformer is generally a four-terminal device comprising two or more magnetically coupled coils. Ø The transformer is called LINEAR if the coils are wound on magnetically linear material. Ø For a LINEAR TRANSFORMER flux is proportional to current in the windings. Ø Resistances R 1 and R 2 account for losses in the coils. Ø The coils are named as PRIMARY and SECONDARY. Eeng 224 18

Reflected Impedance for Linear Transformers Ø Let us obtain the input impedance as seen from the source, ZR • Secondary impedance seen from the primary side is the Reflected Impedance. Eeng 224 19

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Equivalent T Circuit for Linear Transformers Ø The coupled transformer can equivalently be represented by an EQUIVALENT T circuit using UNCOUPED INDUCTORS. a) Transformer circuit b) Equivalent T circuit of the transformer Eeng 224 21

Equivalent П Circuit for Linear Transformers Ø The coupled transformer can equivalently be represented by an EQUIVALENT П circuit using uncoupled inductors. a) Transformer circuit b) Equivalent Π circuit of the transformer Eeng 224 22

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Homework 2 Problem 13. 79 X 23 Ø Submit your results by giving in the following results similar to the form given below by May 2, 2007. a) Your origiinal schematic diagram b) The print page of your results c) Repeat the calculation for 2 other values of X 23=j 0, j 15 Ω Eeng 224 24

Homework 2 Schematic Diagram Eeng 224 25