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Magnetically Coupled Autotransformer

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Please refer to attached diagrams for complete questions and mentioned figures.

1. The diagram below shows an autotransformer represented as two separate inductors, L1 and L2, magnetically coupled with a coupling coefficient k.
a) Apply Kirchhoff's voltage law to the input and output circuits (to loops ABC and DBC) to obtain equations for V1 and V2.
b) Show that the open-circuit (I2 = 0) voltage ratio is given by:

2. In the practical autotransformer k=1 and the voltage ratio can be shown to become:

V1/V2 = 1 + SQRT(L1/L2)

Show that V1/V2 = 1 + SQRT(L1/L2) is the equivalent of V1/V2 = 1 + (N1/N2), where the windings 1 and 2 have N1 and N2 turns respectively.

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