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Chapter 13 Magnetically Coupled Circuits Chapter Objectives: Understand magnetically coupled circuits. 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 Eeng224 Circuit Theory II Department of Electrical and Electronic Engineering Eastern Mediterranean University Eeng 224 ‹#› Mutual Inductance Transformers are constructed of two coils placed so that the charging flux developed by one will link the other. The coil to which the source is applied is called the primary coil. The coil to which the load is applied is called the secondary coil. Three basic operations of a transformer are: Step up/down Impedance matching Isolation Eeng 224 ‹#› Mutual Inductance Devices Eeng 224 ‹#› Mutual Inductance 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. Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor. v1 N1 d1 d (11 21 ) N1 dt dt v2 N 2 d 2 d (12 22 ) N2 dt dt Eeng 224 ‹#› Mutual Inductance v2 M 21 a) Magnetic flux produced by a single coil. v1 M 12 di1 dt b) Mutual inductance M21 of coil 2 with respect to coil 1. di2 dt c) Mutual inductance of M12 of coil 1 with respect to coil 2. Eeng 224 ‹#› Mutual Inductance Mutual inductances M12 and M21 are equal. They are referred as M. We refer to M as the mutual inductance between two coils. M is measured in Henry’s. Mutual inductance exists when two coils are close to each other. Mutual inductance effect exist when circuits are driven by time varying sources. Recall that inductors act like short circuits to DC. M12 M 21 M Eeng 224 ‹#› Dot Convention 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. v2 M di1 dt v2 M di1 dt v1 M v1 M di2 dt di2 dt Eeng 224 ‹#› Dot Convention Eeng 224 ‹#› Coils in Series 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=L1+L2+2M b) Series-opposing connection. L=L1+L2-2M Eeng 224 ‹#› Time-domain and Frequency-domain Analysis jM V1 I1 jL1 jL2 I2 V2 b) Frequency-domain circuit a) Time-domain circuit Time Domain di1 di2 v1 i1 R1 L1 M dt dt di di v2 i2 R2 L2 2 M 1 dt dt Frequency Domain V1 ( R1 j L1 ) I1 j MI 2 V2 j MI1 ( R2 j L2 ) I 2 Eeng 224 ‹#› Induced mutual voltages Eeng 224 ‹#› Induced mutual voltages Eeng 224 ‹#› P.P.13.2 Determine the phasor currents Mesh 1 1260=(5+j2+j6-j3 2)I1 j 6I2 j3I2 j3I1 + - + - j3I2 + - Mesh 2 0=(j6-j4)I2 j 6I1 j3I1 j3I1 Eeng 224 ‹#› Mutually Induced Voltages To find I0 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. -j50 Io I3 j20Ic j40 + j10Ib + + Ia j60 j30Ic + + Ic j30Ib j20Ia 500 V + j80 I1 Ib + I2 100 I a = I1 – I3 I b = I2 – I1 I c = I3 – I2 Io = I3 Blue Voltage due to Ia Red Voltage due to Ic Green Voltage due to Ib j10Ia Eeng 224 ‹#› Mutually Induced Voltages To find I0 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 ‹#› Energy in a Coupled Circuit The total energy w stored in a mutually coupled inductor is: Positive sign is selected if both currents ENTER or LEAVE the dotted terminals. Otherwise we use Negative sign. 1 2 1 w L1i1 L2i2 2 Mi1i2 2 2 Eeng 224 ‹#› Coupling Coefficient The Coupling Coefficient k is a measure of the magnetic coupling between two coils 0 k 1 k 1 Perfect Coupling k 0.5 Loosly Coupling k 0.5 Tightly Coupling a) Loosely coupled coil b) Tightly coupled coil 0 k 1 k M L1 L2 Eeng 224 ‹#› Linear Transformers A transformer is generally a four-terminal device comprising two or more magnetically coupled coils. The transformer is called LINEAR if the coils are wound on magnetically linear material. For a LINEAR TRANSFORMER flux is proportional to current in the windings. Resistances R1 and R2 account for losses in the coils. The coils are named as PRIMARY and SECONDARY. Eeng 224 ‹#› Reflected Impedance for Linear Transformers Let us obtain the input impedance as seen from the source, ZR V ( R1 j L1 ) I1 j MI 2 0 j MI1 ( R2 j L2 Z L ) I 2 V 2M 2 Zin R1 j L1 R1 j L1 Z R I1 R2 j L2 Z L 2M 2 ZR R2 j L2 Z L REFLECTED IMPEDANCE • Secondary impedance seen from the primary side is the Reflected Impedance. Eeng 224 ‹#› Eeng 224 ‹#› Equivalent T Circuit for Linear Transformers 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 La L1 M , Lb L2 M , Lc M Eeng 224 ‹#› Equivalent П Circuit for Linear Transformers 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 L1L2 M 2 L1L2 M 2 L1L2 M 2 LA , LB , LC L2 M L1 M M Eeng 224 ‹#› La L1 M Lb L2 M Lc M Eeng 224 ‹#› Homework 2 Problem 13.79 X23 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 X23=j0, j10, j15 Ω Eeng 224 ‹#› Homework 2 Schematic Diagram Eeng 224 ‹#›