Circuits II EE221 Unit 10 Instructor: Kevin D. Donohue Magnetically Coupled Circuits, Linear Transformers, Transformer Circuits

Coupling Coefficient The coupling coefficient is a measure of the percentage of flux from one coil that links another (mutual inductance) coil. The coupling coefficient for 2 mutual inductors is given by: If k > 0.5, then most of the flux from the one coil links the other and the coils are said to be tightly coupled. If k < 0.5, then most of the flux is not shared between the 2 coils and in this case the coils are said to be loosely coupled.

Linear Transformers A transformer is a 4-terminal device comprised of 2 or more magnetically coupled coils. In a typical application they are used the change the ratio of current to voltage (maintaining constant power) from a source to a load. +Vs-+Vs- ZLZL R1R1 R2R2 Primary CoilSecondary Coil

Linear Transformers Equivalent (T Network) A z parameter model of a T (or Y) network of inductors is given by: Find the equivalent T network for a transform circuit given by a mutual inductance pair. Show equivalent T network requires: LaLa LbLb LcLc I1I1 I2I2 +V1-+V1- +V2-+V2- L1L1 L2L2 M

Linear Transformers Equivalent (T Network) By setting the corresponding y-parameter model elements equal, the transformer can be mapped to the following equivalent T network. y-parameter equivalent model for transformer is given by: A similar equivalence can be obtained for the  or  network. L 1 -ML 2 -M M

Ideal Transformers For ideal transformers assume that coupling coefficient is 1. If this is the case, then both coils have the same flux. Assume the polarities of the voltages and currents result in flux going in the same direction. Then the result is: And for the ideal transformer assume no power is lost in the magnetic circuit (infinite permeability) or the coils (no wire resistance), which implies:

Proper Polarity for Ideal Transformers

Example Find with Z 1 = 10, Z 2 = -j50, and Z L =  0 - Z2Z2 Z1Z1 +Vo-+Vo- 1:4 ZLZL

Circuit Transformations The primary circuit can be transformed into the secondary by performing a Thévenin equivalent at the terminals of the transformer secondary. + V s1 - Z2Z2 Primary CoilSecondary Coil Z1Z1 + V s2 - 1:n

Circuit Transformations The result is: + nV s1 - Z2Z2 n2Z1n2Z1 + V s2 -

Example Find with Z 1 = 10, Z 2 = -j50, and Z L =  0 - Z2Z2 Z1Z1 +Vo-+Vo- 1:4 ZLZL

Circuit Transformations The result is: + V s1 - Z 2 / n 2 Z1Z1 + V s2 / n -

Circuit Transformations Perform a Norton transformation on the above circuits to show how a current source transforms to an equivalent on the primary or secondary circuits. In summary: Reflecting the secondary circuit to the primary side requires impedance division by n 2, voltage source division by n, and current source multiplication by n. Reflecting the primary circuit to the secondary side requires impedance multiplication by n 2, voltage source multiplication by n, and current source division by n.

Circuit Transformations Use the formulae derived from Thévenin and Norton to compute the equivalent impedance. Show that Z eq = j5.6 1:5 4:1 -j10  j2  66 24  88 Z eq