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1 CHAPTER 5: TRANSFORMER AND MUTUAL INDUCTANCE Review of Magnetic Induction Mutual Inductance Linear & Ideal Transformers

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Magnetic Field Lines Magnetic fields can be visualized as lines of flux that form closed paths The flux density vector B is tangent to the lines of flux

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Magnetic Fields Magnetic flux lines form closed paths that are close together where the field is strong and farther apart where the field is weak. Flux lines leave the north-seeking end of a magnet and enter the south-seeking end. When placed in a magnetic field, a compass indicates north in the direction of the flux lines.

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Right-Hand Rule

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Forces on Charges Moving in Magnetic Fields

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Force on straight wire of length l in a constant magnetic field Forces on Current-Carrying Wires

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Force on a Current Carrying Wire

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Flux Linkages and Faraday’s Law Magnetic flux passing through a surface area A: For a constant magnetic flux density perpendicular to the surface: The flux linking a coil with N turns:

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Faraday’s Law Faraday’s law of magnetic induction: The voltage induced in a coil whenever its flux linkages are changing. Changes occur from: Magnetic field changing in time Coil moving relative to magnetic field

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Lenz’s law states that the polarity of the induced voltage is such that the voltage would produce a current (through an external resistance) that opposes the original change in flux linkages. Lenz’s Law

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12 Introduction 1 coil (inductor) –Single solenoid has only self-inductance (L) 2 coils (inductors) –2 solenoids have self-inductance (L) & Mutual- inductance

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13 1 Coil A coil with N turns produced = magnetic flux only has self inductance, L

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14 1 Coil

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15 Self-Inductance Voltage induced in a coil by a time-varying current in the same coil (two derivations): either: or:

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16 1 Coil

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17 2 coils Mutual inductance of M 21 of coil 2 with respect to coil 1 Coil 1 has N 1 turns and Coil 2 has N 2 turns produced 1 = 11 + 12 Magnetically coupled

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18 Mutual voltage (induced voltage) Voltage induced in coil 1: Voltage induced in coil 2 : M 21 : mutual inductance of coil 2 with respect to coil 1

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19 Mutual Inductance When we change a current in one coil, this changes the magnetic field in the coil. The magnetic field in the 1 st coil produces a magnetic field in the 2 nd coil EMF produced in 2 nd coil, cause a current flow in the 2 nd coil. Current in 1 st coil induces current in the 2 nd coil. Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H)

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20 2 coils Mutual inductance of M 12 of coil 1 with respect to coil 2 Coil 1 has N 1 turns and Coil 2 has N 2 turns produced 2 = 21 + 22 Magnetically coupled

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21 Mutual voltage (induced voltage) Voltage induced in coil 2: Voltage induced in coil 1 : M 12 : mutual inductance of coil 1 with respect to coil 2

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22 Dot Convention Not easy to determine the polarity of mutual voltage – 4 terminals involved Apply dot convention

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23 Dot Convention

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24 Dot Convention

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25 Frequency Domain Circuit For coil 1 : For coil 2 :

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Use of the Dependent Source Model for Magnetically Coupled Circuits Draw dependent sources in each circuit with + in same orientation as the dot in that circuit's coil. If the other circuit's current is entering its dot terminal then the induced voltage of the dependent source is positive, otherwise: negative We'll redraw the previous circuit to show how this works:

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28 Example 1 Calculate the phasor current I 1 and I 2 in the circuit

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30 Exercise 1 Determine the voltage V o in the circuit

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32 Energy In A Coupled Circuit Energy stored in an inductor: Energy stored in a coupled circuit: Positive sign: both currents enter or leave the dotted terminals Negative sign: one current enters and one current leaves the dotted terminals Unit : Joule

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33 Coupled Circuit Energy In A Coupled Circuit

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34 Energy stored must be greater or equal to zero. or Mutual inductance cannot be greater than the geometric mean of self inductances. Energy In A Coupled Circuit

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35 The coupling coefficient k is a measure of the magnetic coupling between two coils or Where: or Energy In A Coupled Circuit

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36 Perfectly coupled : k = 1 Loosely coupled : k < 0.5 - Linear/air-core transformers Tightly coupled : k > 0.5 - Ideal/iron-core transformers Coupling coefficient is depend on : 1. The closeness of the two coils 2. Their core 3. Their orientation 4. Their winding Energy In A Coupled Circuit

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37 Example 2 Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductor at time t=1s if

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Linear Transformers Z in R 1 and R 2 are winding resistances. 1.k < 0.5 2.The coils are wound on a magnetically linear material (air, plastic, wood)

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42 Example 3 Calculate the input impedance and current I 1. Take Z 1 = 60 − j100 Ω, Z 2 = 30 + j40 Ω, and Z L = 80 + j60 Ω

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Ideal Transformers (1/3) 1.When Coils have very large reactance (L 1, L 2, M ~ ) 2.Coupling coefficient is equal to unity (k = 1) 3.Primary and secondary are lossless (series resistances R 1 = R 2 = 0)

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Ideal Transformers (2/3)

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Ideal Transformers (3/3)

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Types of IDEAL Transformers When n = 1, we generally call the transformer an isolation transformer. If n > 1, we have a step-up transformer (V 2 > V 1 ). If n < 1, we have a step-down transformer (V 2 < V 1 ).

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Dot convention for Ideal Xformers 48

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Find I1, V1, I2, V2 and Zin 49 I1= 100<-16.26 A, V1 = 2427<-4.37 V, I2 = 1000<-16.26, V2 = 242.71<-4.37

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Impedance Transformation Z in

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Application: Impedance Matching Linear network

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a) Find n so that max power is delivered to load b) compare power to load with and w/o xformer

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Ideal Transformer Circuit (1/3) Linear network 1 Linear network 2

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Ideal Transformer Circuit (2/3) 1

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Ideal Transformer Circuit (3/3) cc

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Applications of Transformers To step up or step down voltage and current (useful for power transmission and distribution) To isolate one portion of a circuit from another As an impedance matching device for maximum power transfer Frequency-selective circuits

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Applications: Circuit Isolation When the relationship between the two networks is unknown, any improper direct connection may lead to circuit failure. This connection style can prevent circuit failure.

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Applications: DC Isolation Only ac signal can pass, dc signal is blocked.

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Applications: Load Matching

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Applications: Power Distribution

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Determine the voltage Vo. (20 ∠ -90° V) 61

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62 Exercise 2 For the circuit below, determine the coupling coefficient and the energy stored in the coupled inductors at t=1.5s.

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63 Example 3 Calculate the input impedance and current I 1. Take Z 1 = 60 − j100 Ω, Z 2 = 30 + j40 Ω, and Z L = 80 + j60 Ω

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Find I1, V1, I2, V2 and Zin 64 I1= 100<-16.26 A, V1 = 2427<-4.37 V, I2 = 1000<-16.26, V2 = 242.71<-4.37

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