Engineering Circuit Analysis CH7 Magnetically Coupled Circuits 7.1 Self-inductance and mutual inductance 7.2 Dot convention 7.3 Transformer

Ch7 Magnetically Coupled Circuits Physical phenomenon: By having two closely located coils, the magnetic flux created by one coil will affect the other – leading to the mutual inductance. Voltage across a coil is introduced by the self inductance and mutual inductance. Application: The transformer – coverts the ac voltage to a higher or lower value required by the load.

Ch7 Magnetically Coupled Circuits 7.1 Self-inductance and mutual inductance Coupled Circuits and v ~ i relationship Magnetic flux: 1 ＝ f(i1) (1 = N11) The flux is proportional to the current in linear inductor: 1(t) ＝ L1i1(t) L is a lumped element abstraction for the coil.  i1 v1  + -

Ch7 Magnetically Coupled Circuits 7.1 Self-inductance and mutual inductance Coupled Circuits and v ~ i relationship  v1 + -  i1 v2  i2 ——Ideal Coupled Circuits’ v ~ i relationship L1、L2、M represent Ideal Coupled Inductor, Self inductance Mutual inductance

Ch7 Magnetically Coupled Circuits 7.1 Self-inductance and mutual inductance Coupled Circuits and v ~ i relationship  v1 + -  i1  i2  v1 + -  i1 v2  i2 v2

Ch7 Magnetically Coupled Circuits  i1  i2 v1 v2 • Ch7 Magnetically Coupled Circuits 7.2 Dot Convention  v1 + -  i1 v2  i2  v1 + -  i1 v2  i2  i1  i2 v1 v2 • A current entering the dotted terminal of one coil produces an open circuit voltage with a positive voltage reference at the dotted terminal of the second coil.

Ch7 Magnetically Coupled Circuits 7.2 Dot Convention Example Apply dot convention , M creates a negative potential at the dot position of the primary mesh Apply dot convention , M creates a negative potential at the dot position of the secondary mesh

Ch7 Magnetically Coupled Circuits 7.2 Dot Convention Question：The terminal is dotted，how can we get v ~ i equations to coupled inductor？ u2  i1  i2 v1 • v2  i1  i2 v1 v2 • Suppose direction of the i and is consistent with Dot convention! Steps to determine the coupled circuit voltage : 2.For mutual-inductance voltage + - 1.For self-inductance voltage

Ch7 Magnetically Coupled Circuits 7.2 Dot Convention P4.16，For the circuit shown in following figures, determine v1and v2. u2  i2 v1 • v2 i1 L1 L2 M ＋ －

Ch7 Magnetically Coupled Circuits 7.2 Dot Convention Coupled Circuits and v ~ i relationship  v1 + -  i1 v2  i2 ——Ideal Coupled Circuits’s v ~ i relationship For sinusoidal circuit,

Ch7 Magnetically Coupled Circuits 7.2 Dot Convention + - Example (Practice 10.2, p267) Primary mesh : Secondary mesh :

Ch7 Magnetically Coupled Circuits 7.2 Dot Convention Example10.2 (KVLS for the three meshes) ~ ( Mesh 1 : Mesh 2 : Mesh 3 :

Ch7 Magnetically Coupled Circuits 7.3 Transformer Equivalent networks---- equivalent network In the equivalent network, mutual inductance no longer exists. And the dot convention has been removed. , and are also treated as self-inductance.

Ch7 Magnetically Coupled Circuits 7.3 Transformer A B C D Example10.4 (P274) One the equivalence : Let Let Apply the equivalent network: Apply the original transformer:

Ch7 Magnetically Coupled Circuits 7.3 Transformer ---- equivalent network (※)

Ch7 Magnetically Coupled Circuits 7.3 Transformer -- ideal transformer Turn ratio Number of turns of wire forming the coil

Ch7 Magnetically Coupled Circuits 7.3 Transformer KVL for both primary and secondary meshes V1 = jwL1I1 – jwMI2 0 = -jwMI1 + (ZL + jwL2)I2 With

Ch7 Magnetically Coupled Circuits 7.3 Transformer Impedance Matching Condition of maximizing the power that is being transferred One can apply the ideal transformer to have the above matching to be achieved. if if Current and voltage adjustment

Ch7 Magnetically Coupled Circuits 7.3 Transformer Thevenin equivalent for mesh 1 : Example 10.6 (P281) Determine Thevenin equivalent for mesh 2 : Knowing , the equivalent impedance of the resistor in the primary mesh is : Hence ,