Magnetically Coupled Circuits Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Contents Introduction Mutual Inductance Energy in a Coupled Circuit Linear Transformers Ideal Transformers Applications

Introduction Conductively coupled circuit means that one loop affects the neighboring loop through current conduction. Magnetically coupled circuit means that two loops, with or without contacts between them, affect each other through the magnetic field generated by one of them. Based on the concept of magnetic coupling, the transformer is designed for stepping up or down ac voltages or currents.

Self Inductance

Mutual Inductance (1/5)

Mutual Inductance (2/5)

Mutual Inductance (3/5) We will see that M 12 =M 21 =M. Mutual coupling only exists when the inductors or coils are in close proximity, and the circuits are driven by time-varying sources. Mutual inductance is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H). i1i1 +_+_ The dot convention states that a current entering the dotted terminal induces a positive polarity of the mutual voltage at the dotted terminal of the second coil.

Mutual Inductance (4/5)

Mutual Inductance (5/5) i1i1 +_+_ i1i1 +_+_ i2i2 +_+_ i2i2 +_+_

Series-Aiding Connection + _ v1v1 + _ v2v2

Series-Opposing Connection + _ v1v1 + _ v2v2

Example 1

Circuit Model for Coupled Inductors

Example 2

Example 3

Energy in a Coupled Circuit (1/4) i1i1 i2i2 I1I1 I2I2 III t

Energy in a Coupled Circuit (2/4) i1i1 i2i2 I1I1 I2I2 III t

Energy in a Coupled Circuit (3/4)

Energy in a Coupled Circuit (4/4)

More about k

Coupling vs. Winding Style Loosly coupled k < 0.5 Tightly coupled k > 0.5

Example

Linear Transformers Z in R 1 and R 2 are winding resistances.

T (or Y) Equivalent Circuit

П (or  ) Equivalent Circuit

Ideal Transformers (1/3) 1.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)

Ideal Transformers (2/3)

Ideal Transformers (3/3)

Types of 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 ).

Impedance Transformation Z in

How to make a transformer ideal? Z in The linear transformer model

Impedance Matching Linear network

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

Ideal Transformer Circuit (2/3) 1

Ideal Transformer Circuit (3/3) cc

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: Load Matching

Applications: Power Distribution