Ci it IICircuits IIEE221EE221Unit 10Instructor: Kevin D DonohueInstructor: Kevin D. DonohueMagnetically Coupled Circuits, Linear gypTransformers, Transformer CircuitsCoupling Coefficient The coupling coefficient is a measure of the percentage of flux from one coil that links pganother (mutual inductance) coil. The coupling coefficient for 2 mutual inductors is given by: M 12LLMkIf k> 0.5 , then most of the flux from the one coil links the other and the coils are said to be ihl ld If k 0 5 h f h fl 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. ypLinear 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 t t lt ( itii t t ) f t current to voltage (maintaining constant power) from a source to a load. R1R2 +Vs-ZLR1R2Primary Coil Secondary CoilLinear Transformers Equivalent (T Network)A z parameter model of a T (or Y) network of inductors is given by:LaLb2i1iLc+-+-2v1v1212221121121iizzzzvv2121ˆˆ)()(ˆˆIILLjLjLjLLjVVcbcccaFind the equivalent T network for a transform circuit given by a mutual inductance pair. Show equivalent T network requires:MI1I2+V1+V2L1L2MMLLMLac1 --12MLLb2Linear Transformers Equivalent (T Network)By setting the corresponding y-parameter model elements equal, the transformer can be mapped to the following equivalent T network.LMLM2iL1-ML2-MM++2vv2i1iy-parameter equivalent model for transformer is given by:--21vyparameter equivalent model for transformer is given by:122122121ˆ)()(ˆVMLLjMMLLjLI2221122121212ˆ)()()()(ˆVMLLjLMLLjMjjIA similar equivalence can be obtained for the or network.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 directionthe voltages and currents result in flux going in the same direction. Then the result is:nNNVV22ˆˆddNvddNv2211 and And for the ideal transformer assume no power is lost in the magnetic circuit (infinite permeability) or the coils (no wireNV11dtdt2211magnetic circuit (infinite permeability) or the coils (no wire resistance), which implies:*22*11ˆˆˆˆIVIV nNNVVII1221ˆˆˆˆNVI112Proper Polarity for Ideal TransformersProper Polarity for Ideal TransformersExampleExampleFind withZ1=10,Z2=-j50, andZL=20VˆFind with Z1 10, Z2 j50, and ZL20 oV+Z2Z1+1:41200-Vo-ZLCircuit TransformationsCircuit TransformationsThe primary circuit can be transformed into the secondary byThe primary circuit can be transformed into the secondary by performing a Thévenin equivalent at the terminals of the transformer secondary.Z2Z1 +Z2Z1+1:nVs1-Vs2-Primary Coil Secondary CoilCircuit TransformationsCircuit TransformationsThe result is:The result is:Z2n2Z1 +Z2 nZ1+nVs1-Vs2-ExampleExampleFind withZ1=10,Z2=-j50, andZL=20VˆFind with Z1 10, Z2 j50, and ZL20 oV+Z2Z1+1:41200-Vo-ZLCircuit TransformationsCircuit TransformationsThe result is:The result is:Z2/n2Z1 +Z2 / nZ1+Vs1-Vs2 / n-Circuit TransformationsCircuit TransformationsPerform a Norton transformation on the above circuits to show how aPerform 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 di i i b2lt di i i bdtdivision by n2, voltage source division by n, and current source multiplication by n.Reflecting the primary circuit to the secondary side requires impedance multiplication by n2, voltage source multiplication by n, and current source division by n.Circuit TransformationsCircuit TransformationsUse the formulae derived from Thévenin and Norton to compute the equivalent impedance.1:5 4:16248-j10j2Zeq Show that Zeq= 12.8 +
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