Three-Phase Asynchronous Motor Part 1 PDF

## Macchina asincrona trifase The three-phase asynchronous machine is mainly used as a motor and will be presented in this unit as such. However, since it operates reversibly like all electric machines, its operation as a generator will also be mentioned. ### B2.1 Rotating Three-Phase Magnetic Fie...

## Macchina asincrona trifase The three-phase asynchronous machine is mainly used as a motor and will be presented in this unit as such. However, since it operates reversibly like all electric machines, its operation as a generator will also be mentioned. ### B2.1 Rotating Three-Phase Magnetic Field The operation of the three-phase asynchronous motor is based on a special type of induction field, known as a rotating field, discovered by the Italian scientist Galileo Ferraris. To obtain a rotating three-phase magnetic field, the following two conditions must be met: * There are three fixed windings in space, equal to each other, with the same number of turns and arranged with axes at 120°. * Three sinusoidal alternating magnetizing currents circulate through the windings, having the same frequency, the same RMS value and shifted by 120° in time, forming a balanced set of currents. Consider (Figure B2.1) three coils of N turns, arranged at 120° to each other and influenced by a balanced set of three currents (Figure B2.2); in this initial phase, the coils are represented schematically and in a general way, regardless of their practical implementation. Taking as reference phase zero the current I1, the sinusoidal expressions of the three currents are: * i₁ = IM sen(wt) * i₂ = IM sen(wt - π/2) * i₃ = IM sen(wt + π/3) Each coil creates a magnetomotive force (MMF) Fm = Ni; therefore, there will be three sinusoidal MMFs, having the same peak value FmM = NIM, in phase with the respective currents and shifted by 120° in time: * Fm1 = Ni₁ = NIM sen(wt) = FmM sen(wt) * Fm2 = Ni₂ = NIM sen(wt - π/3) = FmM sen(wt - π/3) * Fm3 = Ni₃ = NIM sen(wt + π/3) = FmM sen(wt + π/3) Each MMF acts, in space, in the direction of the coil that produces it and with a direction depending on the sign of the MMF at the instant considered. In Figure B2.1, the three radial directions are indicated with r1, r2 and r3. The magnetic field that develops depends on the total MMF that exists at any given time. Suppose then to analyze the situations that arise at times t1, t2, t3 and t4, chosen in such a way as to have ωt1 = 0, ωt2 = π/6, ωt3 = π/3, ωt4 = π/2, where ω = 2πf and is, in this context, the angular frequency of the sinusoidal current. * For ωt1 = 0 (t1 = 0), the three MMFs are: * Fm1 = 0 * Fm2 = FmM sen(-π/3) = -√3/2 FmM * Fm3 = FmM sen(π/3) = √3/2 FmM The MMFs of coils 2 and 3 can be represented in space with two vectors of magnitude 0.866 FmM (√3/2 = 0.866), directed respectively in the direction opposite to r2 due to the negative sign of the second, and in the direction of r3, obtaining the diagram of Figure B2.3. From their composition, a vector is obtained, oriented perpendicular to r1, and whose magnitude is: * FmT = √2 FmM COS30° = √2 * √3/2 FmM = √3/2 FmM * For ωt2 = π/6 (t2 = π / 6ω) we have: * Fm1 = FmM sen(π/6) = √3 / 2 FmM * Fm2 = FmM sen(π/6 - π/3) = - 1/2 FmM * Fm3 = FmM sen(π/6 + π/3) = FmM sen(π/2) = FmM The diagram of the arrangement in space of the MMFs becomes that of Figure B2.4. By composing the three vectors, the resulting MMF at the instant considered is obtained, rotated by 30° with respect to the position at time t1 and with a value still equal to: * FmT = √3/2 FmM as can be verified graphically and demonstrated analytically. * For ωt3 = π/3 (t3 = π / 3ω) we have: * Fm1 = FmM sen(π/3) = √3 / 2 FmM * Fm2 = FmM sen(π/3 - π/3) = 0 * Fm3 = FmM sen(π/3 + π/3) = FmM sen(π) = 0 The diagram of the arrangement in space of the MMFs becomes that of Figure B2.5, in which, by composing the vectors, the resulting MMF at the instant considered is obtained, rotated by 60° with respect to its position at time t1, and with a value still equal to: * FmT = √3/2 FmM * For ωt4 = π/2 (t4 = π / 2ω) we have: * Fm1 = FmM sen(π/2) = FmM * Fm2 = FmM sen(π/2 - π/3) = FmM sen(π/6) = √3/2 FmM * Fm3 = FmM sen(π/2 + π/3) = FmM sen(5π/6) = FmM sen(-π/6) = -√3/2 FmM The diagram of the arrangement in space of the MMFs is that of Figure B2.6; by composing the three vectors, the resulting MMF at the instant considered is obtained, rotated by 90° with respect to its position at time t1 and with a value always equal to: * FmT = √3 /2 FmM This discussion can be repeated for other instants, always arriving at the following conclusion: * The resulting MMF of the three coils has, at every instant, the same value, equal to 1.5 times the peak value of the phase MMF, and rotates in space with a constant angular velocity, equal to the angular frequency of the magnetizing currents. The equality between angular velocity and frequency is deduced by observing that in the time corresponding to 90° of the sinusoid, an angular displacement of 90° has occurred. Now make the following considerations: * The MMF produces a magnetic flux linked to it by Hopkinson's law Fm = RP; * The magnetic flux is associated with a magnetic induction whose intensity is proportional to B = Φ/Σ. It can be deduced that the MMF generates a rotating magnetic field whose N-S polarities move continuously in space, as if it were a magnet placed in rotation by a mechanical system (Figure B2.7). The great importance of the rotating field lies precisely in this: with a static electrical system (three fixed coils), a moving magnetic field is obtained, which magnetically alternates with N-S polarities the surrounding space. This phenomenon can be demonstrated even with more rigorous methods than the one used, which, however, has the merit of being intuitive; in any case, the following general rule is reached, known as the **Galileo Ferraris theorem**: * The superposition of n alternating magnetic fields, produced by a balanced polyphase system of n currents circulating in n coils arranged at (360/n)°, gives rise to a rotating magnetic field of constant amplitude equal to n/2 the amplitude of each component, rotating in space with a speed equal to the angular frequency of the magnetizing current. The three-phase rotating field is, therefore, a special case, as it is also possible to have six-phase, twelve-phase, etc. fields. ### B2.2 Rotating Magnetic Field in the Asynchronous Three-Phase Machine In the three-phase asynchronous machine, the rotating magnetic field is created by the stator windings which, connected to the electrical power supply, absorb a balanced set of alternating magnetizing currents. The magnetic field that is created along the air gap of the machine has some important characteristics, which can be highlighted by representing the inner surface of the cylindrical stator magnetic circuit, as if it were flat, imagining to "cut" the surface itself and to lay it out on a plane, as indicated in Figures B2.8 a, b, and B2.9 a, b, relative to a stator with 12 slots. The particularities of the magnetic field concern both its distribution in space and the time variability of the induction value at a given point of the machine and are the following: * The distribution in space of the magnetic induction is sinusoidal, although with a certain approximation, in every space coil; this means that at a given time, the distribution of the magnetic induction value can be considered sinusoidal, even if with an approximation, in the space occupied by the conductors. * Due to the rotation of the magnetic field, the curve of the induction is shifted in time. This means that each conductor, positioned in a specific slot and exposed to a given induction value, sees a value that changes in time, as can be seen from figures B2.8 a, b and B2.9 a, b by considering, for example, the conductors in slots 1 and 2.