 # CHAPTER 5: TRANSFORMER AND MUTUAL INDUCTANCE

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

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

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.

Right-Hand Rule

Forces on Charges Moving in Magnetic Fields

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

Force on a Current Carrying Wire

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:

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

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.

Lenz’s Law

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

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

1 Coil

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

1 Coil

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

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

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.

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

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

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

Dot Convention

Dot Convention

Frequency Domain Circuit
For coil 1 : For coil 2 :

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:

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

Exercise 1 Determine the voltage Vo in the circuit

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

Energy In A Coupled Circuit

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.

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

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

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

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.

Example 3 Calculate the input impedance and current I1.
Take Z1 = 60 − j100 Ω , Z2 = 30 + j40 Ω, and ZL = 80 + j60 Ω

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)

Ideal Transformers (2/3)

Ideal Transformers (3/3)

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).

Dot convention for Ideal Xformers

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

Impedance Transformation
Zin

Application: Impedance Matching
Linear network

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

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

Ideal Transformer Circuit (2/3)
1

Ideal Transformer Circuit (3/3)

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

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.

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

Applications: Power Distribution

Determine the voltage Vo. (20∠-90° V)

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

Example 3 Calculate the input impedance and current I1.
Take Z1 = 60 − j100 Ω , Z2 = 30 + j40 Ω, and ZL = 80 + j60 Ω

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