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LITAR ELEKTRIK II EET 102/4 SILIBUS LITAR ELEKTRIK II Mutual Inductance Two port Network Pengenalan Jelmaan Laplace Kaedah Jelmaan Laplace Dlm Analisis Litar Sambutan Frekuensi Litar AC Siri Fourier Jelmaan Fourier MUTUAL INDUCTANCE Self inductance Concept of mutual inductance Dot convention Energy in a coupled circuit Linear transformer Ideal transformer MUTUAL INDUCTANCE INTRODUCTION magnetically coupled When two loops with or without contacts between them, affect each other through magnetic field generated by one of them – they are said to be magnetically coupled. Example of device using this concepttransformer. Transformer Use magnetically coupled coils to transfer energy from one circuit to another. Key circuit element where it is used for stepping down or up ac voltages or currents. Also used in electronic circuits such as radio and tv receiver. Consider a single inductor with N turns. When current i, flow through coil, magnetic flux is produced around it. Faraday’s Law Induced voltage, v in the coil is proportional to number of turns N and the time rate of change of magnetic flux, . d vN dt we know that the flux is produce by current i, thus any change in the current will change in flux as well. But d di di vN vL @ di dt dt The inductance L of the inductor is thus given by d LN di Self-Inductance Self Inductance Inductance that relates the induced voltage in a coil with a time-varying current in the same coil. Mutual Inductance When two inductors or coils are in close proximity to each other, magnetic flux caused by current in one coil links with the other coil, therefore producing the induced voltage. Mutual Inductance Magnetic flux 1 originating from coil 1 has 2 components: 1 11 12 Since entire flux 1 links coil 1, the voltage induced in coil 1 is: d1 v1 N1 dt Only flux 12 links coil 2, so the voltage induced in coil 2 is: d12 v2 N 2 dt As the fluxes are caused by current i1 flowing in coil 1, equation v1 can be written as: Self d1 di1 di1 v1 N1 L1 inductance of di1 dt dt coil 1 Similarly for equation v2: di1 d12 di1 M 21 v2 N 2 dt di1 dt Mutual inductance of coil 2 With respect to coil 1 Coil 2 Magnetic flux 2 comprises of 2 components: 2 21 22 entire flux 2 links coil 2, so the voltage induced in coil 2 is: The d2 d2 di2 di2 v2 N 2 N2 L2 dt di2 dt dt Self-inductance of coil 2 Since only flux 21 links with coil 1, the voltage induced in coil 1 is: d21 d21 di2 di2 v1 N1 N1 M 12 dt di2 dt dt Mutual inductance of coil 1 with respect to coil 2 For simplicity, M12 and M21 are equal: M12 M 21 M Mutual inductance between two coils Reminder Mutual coupling exists when inductors or coils are in close proximity and circuit are driven by time-varying sources. Mutual inductance is the ability of one inductor to induce voltage across a neighboring inductor, measured in henrys (H). Dot Convention A dot is placed in the circuit at one end of each of the two magnetically coupled coils to indicate the direction of magnetic flux if current enters that dotted terminal of the coil. Dot convention is stated as follows: If a current enters the dotted terminal of one coil, the reference polarity of mutual voltage in second coil is positive at dotted terminal of second coil. If a current leaves the dotted terminal of one coil, the reference polarity of mutual voltage in second coil is negative at dotted terminal of second coil. Dot convention for coils in series L L1 L2 2M L L1 L2 2M Example 1 Example 1 Coil 1: di1 di2 v1 i1 R1 L1 M dt dt Coil 2: di2 di1 v2 i2 R2 L2 M dt dt In frequency domain.. V1 ( R1 jL1 ) I1 jMI 2 V2 jMI1 ( R2 jL2 ) I 2 Example 2 Example 2 V ( Z1 jL1 ) I1 jMI 2 0 jMI1 ( Z L jL2 ) I 2 Example 3 Solution.. For coil 1, we use KVL: 12 ( j 4 j5) I1 j3I 2 0 jI1 j3I 2 12 For coil 2, j3I1 (12 j 6) I 2 0 (12 j 6) I 2 I1 (2 j 4) I 2 j3 Substitute equation 1 into 2: ( j 2 4 j3) I 2 (4 j) I 2 12 12 o I2 2.9114.04 A 4 j Solve for I1: I1 (2 j 4) I 2 (4.372 63.43 )( 2.9114.04 ) o 13.01 49.39 A o o Energy in a coupled circuit Energy stored in an inductor is given by: 1 2 w Li 2 Now, we want to determine energy stored in magnetically coupled coils. Circuit for deriving energy stored in a coupled circuit Power in coil 1: di1 p1 (t ) v1i1 i1 L1 dt Energy stored in coil 1: w1 p1dt L1 I1 0 1 2 i1di1 L1 I1 2 Maintain i1 and we increase i2 to I2. So, the power in coil 2 is: p2 (t ) i1v1 i 2v 2 di2 i2 v2 i1M 12 dt di2 di2 i 2 L2 I1M 12 dt dt Energy stored in coil 2: w2 p2 dt I2 I2 M 12 I1 di2 L2 i2 di2 0 1 2 M 12 I1 I 2 L2 I 2 2 0 Total energy stored in the coils when both i1 and i2 have reached constant values is: w w1 w2 1 1 2 2 L1 I1 L2 I 2 M 12 I1I 2 2 2 Since M12=M21=M, thus 1 1 2 2 w L1 I1 L2 I 2 MI1 I 2 2 2 Generally, energy stored in magnetically coupled circuit is: 1 1 2 2 w L1 I1 L2 I 2 MI1 I 2 2 2 Coupling coefficient, k A measure of the magnetic coupling between two coils; 0 ≤ k ≤ 1 M k L1 L2 Linear Transformer Transformer is generally a four-terminal device comprising two or more magnetically coupled coils. Coil that is directly connected to voltage source is primary winding. Coil connected to the load is called secondary winding. R1 and R2 included to calculate for losses in coils. Linear Transformer Primary winding Secondary winding Obtain input impedance, Zin as seen from source because Zin governs the behaviour of primary circuit. Apply KVL to the two loops: V ( R1 jL1 ) I1 jMI 2 0 jMI1 ( R2 jL2 Z L ) I 2 Input impedance Zin: V Z in I1 M R1 jL1 R2 jL2 Z L 2 ZR 2 Equivalent circuit of linear transformer Equivalent T circuit Equivalent ∏ circuit Voltage-current relationship for primary and secondary coils give the matrix equation: V1 jL1 V jM 2 jM I1 jL2 I 2 By matrix inversion, this can be written as: L2 I1 j ( L1 L2 M 2 ) I M 2 j ( L1 L2 M 2 ) M 2 V j ( L1 L2 M ) 1 L1 V2 2 j ( L1 L2 M ) Matrix equation for equivalent T circuit: V1 j ( La Lc V jL c 2 jLc I1 j ( Lb Lc ) I 2 If T circuit and linear circuit are the same, then: La L1 M Lb L2 M Lc M For ∏ network, nodal analysis gives the terminal equation as: 1 1 I1 jLA jLC I 1 2 jLC 1 jLC V1 1 1 V2 jLB jLC Equating terms in admittance matrices of above, we obtain: L1 L2 M LA L2 M 2 L1 L2 M LB L1 M L1 L2 M LC M 2 2 IDEAL TRANSFORMER Properties of ideal transformer: Coils have very large reactances (L1, L2, M→∞) Coupling coefficient is equal to unity (k=1) Primary and secondary winding are lossless (R1=0=R2) Ideal transformer is a unity-coupled, lossless transformer where primary and secondary coils have infinite self-inductance. Transformation ratio We know that: d v1 N1 dt d v2 N 2 dt Divide v2 with v1, we get: v2 N 2 n v1 N1 Energy supplied to the primary must equal to energy absorbed by secondary since no losses in ideal transformer. v1i1 v2i2 Transformation ratio is: I1 V2 n I 2 V1 Types of transformer: Step-down transformer One whose secondary voltage is less than its primary voltage. Step-up transformer One whose secondary voltage is greater than its primary voltage. Typical circuits in ideal transformer Complex Power From: Complex power in primary winding for ideal txt: V2 V1 @ I1 nI 2 n V2 * * S1 V I (nI 2 ) V2 I 2 S 2 n * 1 1 Input impedance We know that: Since V2 / I2 = ZL , thus V1 1 V2 Z in 2 I1 n I 2 ZL Z in 2 n Reflected impedance Example Find I1 dan I2 for given circuit: Solution… 1st: Find input impedance 2 Z in 10 2 n ZR 1 n 3 Z in 10 18 28 Therefore, solve for I1 140 I1 0.5 A 28 o I1 0.5 I2 1.5 A n ( 13 ) THE END