Magnetically Coupled Circuits Chapter 13 Magnetically Coupled Circuits Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Mutual Inductance The magnetic flux created in one inductive coil can couple into a nearby coil, a process called mutual inductance. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Mutual Inductance The double headed arrow indicates that these inductors are coupled. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The Dot Convention 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. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Dot Convention: Four Cases Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Combined Self- and Mutual-Induction Voltages Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Combined Self- and Mutual-Induction Voltages Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Physical Basis for Dot Convention The assumed currents i1 and i2 produce additive fluxes. Dots may be placed either on the upper terminal of each coil or on the lower terminal of each coil. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example: Voltage Gain Show that V2/V1 =6.88e -j16.70◦ Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Energy in Coupled Inductors This equation implies a limit on M: Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The Coupling Coefficient The coupling coefficient k measures how tightly coupled the two inductors are: where 0 ≤ k ≤ 1 Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The Linear Transformer Transformers have a primary (source side) and a secondary (load side). Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Transformer: Reflected Impedance The impedance Zin seen by the source is Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The “T” Equivalent Network Consider the mesh-current equations to show that these circuits are equivalent. Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The “Delta” Equivalent Network Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

The Ideal Transformer In an ideal transformer, k=1 and the inductances are assumed large in comparison to the other impedances. The turns ratio a is defined as Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Transformer Applications Step Up Step Down Electronics Power Step Down to Household Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Transformer Applications Impedance matching: Current adjustment: Voltage adjustment: Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example: Transformer Calculations Determine the average power dissipated in the 10 kΩ resistor. Answer: 6.25 W Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Thévenin Equivalent Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example: Thévenin Equivalent Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.