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Transformers A transformer consists of several independent windings (N 1, N 2, N 3, …) which link the same, low reluctance, magnetic path. One terminal of each winding is identified (usually with a dot) indicating that currents entering the dotted terminal of any winding will produce flux in the same direction in the shared magnetic path. Typical application involves applying a source of voltage to one winding (primary), and connecting loads to the other windings (secondaries). Current entering the primary produces an amount of flux in the magnetic path, m, which links the secondaries, and a leakage flux l. The leakage flux may be accounted for by a series leakage inductance L l, which we will ignore for the present discussion. We are interested only in the effects of the linked flux. Since the same flux, m, links all windings, Faraday’s and Ampere’s laws requires that Faraday Ampere For each winding: The e k are the potentials at the dotted terminals with respect to un-dotted terminals The i k are currents entering the dotted terminals.

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Transformers (cont) Arbitrarily choose N 1 as the primary and apply a voltage e 1. Faraday’s law requires Any load connected to a secondary will result in a current flowing out of its dotted terminal in response to the induced secondary voltage. These currents will tend to reduce the flux in the core (i k 1), thus all of the magnetization of the core must arise from the primary current. The primary current must generate all of the magnetizing flux (I m1 ), plus provide enough additional current (I 1 / ) to zero out the reduction in flux due to secondary currents. Negative sign denotes that these currents are < 0, i.e. flowing out of the dotted terminals.

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Transformers (cont) The additive current component I m represents a magnetizing current flowing through a parallel path, generating a flux m in a reluctance R m through N 1 flux linkages. By the definition of inductance, …represents the transformer magnetizing inductance. The transformer action is defined by

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+v2-+v2- Transformer Models +v1-+v1- +v2-+v2- i1i1 i2i2 Ideal Impedance Transformation! +v1-+v1- i2i2 Ideal Lossless i1i1 k = 1 k < 1 + kv 1 -

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CIRCUITS and SYSTEMS – part I

CIRCUITS and SYSTEMS – part I

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