Transformers

Transformers - 1 EE 4420 Electric Machine Analysis Fall...

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1 1 EE – 4420 Electric Machine Analysis Fall 2005 Instructor: Ernest Mendrela TRANSFORMERS 1. MAGNETIC CIRCUIT EXCITED BY ALTERNATING CURRENT According to the Faraday’s experiment the voltage e induced in a loop linking a changing magnetic field (see Fig.1) is proportional to the time rate of change of flux Φ : e d dt = Φ (1) The polarity of the induced voltage can be determined by the Lenz’s law that says: “The induced voltage is always in such a direction as to tend to oppose the change in flux linkage that produces it” Φ e Φ i i i Fig.1 Explanation to equation (1) This is shown in Fig.1. For multitern coil the induced voltage is: eN d dt d dt == λ (2) where: N – is the number of coil turns and λ – is the flux linkage in Weber turns. Suppose we have a coil wound on one leg of a closed iron core as shown in Fig.2. To draw an equivalent circuit of such a device called inductor, and then to analyze its behavior under variable supply condition let us consider first an ideal inductor.
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2 2 1.1. An ideal inductor An ideal inductor is defined by the following assumptions: The coil of inductance L has the resistance R equal to zero, There is an ideal magnetic circuit of the iron core with no power losses in it, There is no leakage flux, what means that the whole magnetic flux is within the iron core. e v i Φ Ν Fig.2 Scheme of the inductor supplied from the ac. source Assuming a linear relation between current and flux, the sinusoidal current i I t m = sin( ) ω (3) produces the sinusoidal flux Φ = m t sin( ) (4) The voltage induced in N-turn coil is eN d dt Nt E t mm == = ωω cos( ) cos( ) (5) The effective value of this voltage is: E E N Nf Nf = = 22 2 2 444 ωΦ π ΦΦ . (6) The voltage expressed in terms of current flowing trough the coil is: eL di dt = (7)
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3 3 For sinusoidal current: eL I t E t mm = = ω cos cos (8) The effective value of the voltage expressed in the complex form is: E jX I = µ (9) where X L = is the magnetizing reactance. For an ideal inductor, the induced voltage ( emf – E ) is equal to the supply voltage E V = . The equivalent circuit of such an inductor is shown in Fig.3. The phasor diagram of voltages and current is shown in Fig.4. E Φ V I Φ E = V I Fig.3 Inductor equivalent circuit 1.2 A real inductor A real inductor has a real coil and the real magnetic circuit. This magnetic circuit is described by the hysteresis loop of B-H characteristic shown in Fig.5. During the process of magnetization by the alternating flux the energy is lost due to the hysteresis loop. This energy loss, called the hysteresis loss is proportional to the area closed by the hysteresis loop. That means it depends on the material the inductor core is made of. The empirical formula for this loss is: PK f B hhm n =⋅ (10) where the constant K h and n vary with the core material. n is often assumed to be 1.6 – 2.
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This note was uploaded on 02/26/2012 for the course EE 4150 taught by Professor Wu during the Fall '10 term at LSU.

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Transformers - 1 EE 4420 Electric Machine Analysis Fall...

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