Presentation on theme: "1 CHAPTER 5: TRANSFORMER AND MUTUAL INDUCTANCE Review of Magnetic Induction Mutual Inductance Linear & Ideal Transformers."— Presentation transcript:
1 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.
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:
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
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
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
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
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)
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
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
22 Dot Convention Not easy to determine the polarity of mutual voltage – 4 terminals involved Apply dot convention
25 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:
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
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
35 The coupling coefficient k is a measure of the magnetic coupling between two coils or Where: or Energy In A Coupled Circuit
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
37 Example 2 Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductor at time t=1s if
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)
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 ).
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.