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**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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**Forces on Current-Carrying Wires**

Force on straight wire of length l in a constant magnetic field

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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 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.

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Lenz’s Law

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**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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**1 Coil A coil with N turns produced = magnetic flux**

only has self inductance, L

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1 Coil

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

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1 Coil

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**Mutual inductance of M21 of coil 2 with respect to coil 1**

2 coils Mutual inductance of M21 of coil 2 with respect to coil 1 Coil 1 has N1 turns and Coil 2 has N2 turns produced 1 = 11 + 12 Magnetically coupled

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**Mutual voltage (induced voltage)**

Voltage induced in coil 1: Voltage induced in coil 2 : M21 : mutual inductance of coil 2 with respect to coil 1

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

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**Mutual inductance of M12 of coil 1 with respect to coil 2**

2 coils Mutual inductance of M12 of coil 1 with respect to coil 2 Coil 1 has N1 turns and Coil 2 has N2 turns produced 2 = 21 + 22 Magnetically coupled

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**Mutual voltage (induced voltage)**

Voltage induced in coil 2: Voltage induced in coil 1 : M12 : mutual inductance of coil 1 with respect to coil 2

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**Dot Convention Not easy to determine the polarity of mutual voltage –**

4 terminals involved Apply dot convention

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Dot Convention

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Dot Convention

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**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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Example 1 Calculate the phasor current I1 and I2 in the circuit

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Exercise 1 Determine the voltage Vo in the circuit

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**Energy In A Coupled Circuit**

Energy stored in an inductor: Unit : Joule 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

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**Energy In A Coupled Circuit**

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**Energy In A Coupled Circuit**

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

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**Energy In A Coupled Circuit**

The coupling coefficient k is a measure of the magnetic coupling between two coils or Where: or

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**Energy In A Coupled Circuit**

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

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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 R1 and R2 are winding Zin resistances. k < 0.5**

The coils are wound on a magnetically linear material (air, plastic, wood) Zin R1 and R2 are winding resistances.

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**Example 3 Calculate the input impedance and current I1.**

Take Z1 = 60 − j100 Ω , Z2 = 30 + j40 Ω, and ZL = 80 + j60 Ω

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**Ideal Transformers (1/3)**

When Coils have very large reactance (L1, L2, M ~ ) Coupling coefficient is equal to unity (k = 1) Primary and secondary are lossless (series resistances R1= R2= 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 (V2 > V1). If n < 1 , we have a step-down transformer (V2 < V1).

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**Dot convention for Ideal Xformers**

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

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**Impedance Transformation**

Zin

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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)**

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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)**

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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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**Example 3 Calculate the input impedance and current I1.**

Take Z1 = 60 − j100 Ω , Z2 = 30 + j40 Ω, and ZL = 80 + j60 Ω

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

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