9. Operation Principle of the Electric Machines

Transcription

1 9. Operation Principle of the Electric Machines J Electric Machines Operation Principle Ø An electric achine consists of two electric circuits coupled by eans of a agnetic flu, that is linked with both of the. ϑ i i Ø A tie varying linked agnetic flu, generated by the current i, induces a tension (e..f.) on the circuit. The current i at the sae tie generates a agnetic flu which interacts with the flu produced by i (electrical or/and echanical interaction). i (t) (t) v i = + Ø The agnetic flu linked with the circuits can vary due to the tie variation of the currents by which it is produced or/and due to the variation of the reciprocal position of the circuits.

2 For the - th winding it is c v = R i + t where c is the flu linked with the - th winding. When assuing to have n windings, and thus of currents, is: = ( i, i,..,i,...i, J ) c c Electric Machines k n c ϑ i i d di dj L di d J (linear assuption) n n c c c c = å k + = å k k + k= ik J k= J di v R i + L ω n k c = å k + k= dt J w = d J /dt is the angular speed of the agnetic field in respect to the - th winding. Electric Machines ϑ i i v n = R i + å k= L k di dt k + J c ω n å k= L k di dt Transforer EMF k J c ω Motional EMF

3 Electric Machines The Transforer: - ω = 0 - i, i sinusoidal Rotating Electric Machines (Rotating EM): - ω 0 ω i i - Induction EM (i stator: three phase; i rotor: three phase) - ynchronous EM (i stator: three phase; i rotor: DC current) - DC EM (stator and rotor: DC current) v v v = R i + n å k= L di di = R i + L + L dt dt di di = R i + L + L dt dt k The Transforer di k dt If the input (v ) is sinusoidal, i, v and i are sinusoidal at the sae frequency. Moreover it is L =L =M. v (t) i (t) ϕ(t) v i V! = V! = ( R ) I! I! + w L + w M w M I! ( ) I! + R + w L 3

4 The Transforer V! V! = = ( R ) I! I! + w L + w M w M I! ( ) I! + R + w L In the ideal assuption the ideal transforer is perfectly agnetically coupled (the flu generated by one coil is copletely linked to the other coil) and is lossless. For the ideal transforer it is: dφ V = dt, V = dφ dt V V = = K; p = p, V I =V I I I = V V = = K; where K = / is the turn ratio and V, I, V, I are the tensions and currents of the priary and secondary windings. on-linear Materials: Ferroagnetic Materials Energy losses The losses in agnetic aterials are: Ø The losses in copper - Ohic losses Ø The losses in iron eddy current losses and hysteresis loses Iron losses hysteresis losses The electrical power lost per unit of volue of iron, due to the hysteresis, depends on the area of the hysteresis cycle, and is proportional to the nuber of cycles per unit of tie. A sei- epirical forulation states that the dissipated power du to hysteresis is: -M M W h = k h f M a M is the aial value of the agnetic flu density in the hysteresis cycle. The constant α is approiately. w = 0 d 4

5 on-linear Materials: Ferroagnetic Materials Iron losses eddy current losses In iron cores, due to varying agnetic flu density, e..f. are induced. ence, owing to the high iron electrical conductivity, eddy currents are generated and energy is dissipated due to Joule effect. The induction field in iron is (t) = M cosω t Energy losses Assuing that is surface which border is the circuit of the eddy current and R is the iron resistance, the power lost P d is P d = R i = v R = dφ R dt, where Φ = M cosω t, hence P d = M ω sen ωt, and on average P R d = M ω R on-linear Materials: Ferroagnetic Materials Energy losses Iron losses eddy current losses ence the dissipated power due to eddy currents Is W e = k e f M In order to reduce the eddy current losses iron with an high resistance laination is utilized. Laination of an iron core 5

6 Rotating Electric Machines v n = R i + å k= L k di k dt + J c ω ϑ i i The echanic torque corresponds to the oveent of the winding and can be derived through the energy balance. dl = p dt = v i dt = F ds = C d J = C w dt where v is due to the flu variation as the position of a coil in respect to the other varies: c c i ωdt = C w dt Þ C = i J J Rotating Electric Machines A rotating electric achine consists of two series of windings ounted on two iron structures ade of steel sheets (electric laination). The eternal structure is usually stationary and is said stator on which the priary winding is ounted. The other rotates and is said rotor on which the secondary winding is ounted. The two iron structures are separated by an air gap. Φ a Is Air Gap Iron 6

7 Induction Motor The Rotating Field Magnetic lines of the agnetic field generated by the current in the stator winding (I a ) Φa Is Ais of syetry of the agnetic field distribution, generated by a stator winding 7

8 The Rotating Field generated by a winding along the air gap Φ a For a DC current the agnetic field along the air gap is steady. The Rotating Field generated by a winding along the air gap Φ a For an AC current the agnetic field along the air gap is stationary. 8

9 The Rotating Field generated by a winding along the air gap Φ a For an AC current the agnetic field along the air gap is stationary. The Rotating Field generated by a winding along the air gap Φ a For an AC current the agnetic field along the air gap is stationary. 9

10 The Rotating Field generated by a winding along the air gap Φ a For an AC current the agnetic field along the air gap is stationary. The Rotating Field generated by a winding along the air gap Φ a For an AC current the agnetic field along the air gap is stationary. 0

11 The Rotating Field generated by a winding along the air gap Φ a For an AC current the agnetic field along the air gap is stationary. A stationary alternated field is the coposition of two rotating field in the clockwise direction and in the conterclockwise direction. The Rotating Field Φa a Φb 0 c 0 b Φc The stator is realized by a balanced three phase windings. Each phase occupies one third of the stator slots. Φa, Φb, Φc are the ais of syetry of the agnetic field coponents generated by the currents I a, I b, I c of the three phase windings.

12 The Rotating Field a Φ b c ω c Φ c b Around the stator each phase winding is shifted of 0 with respect to the other. Each of the produce a stationary field with two coponents rotating in the clockwise direction, the other in the counterclockwise direction. Moreover an AC three phase balanced syste (the phases are 0 out of tie to each other) is used. The result is a single rotating agnetic field. b Φ a c a If p is the nuber of polar pair per phase (in the figure the agnetic poles per phase are two, hence p = ), it is ω c = ω/p ω ω c electric radian frequency angular speed of the field (In the figure ω c = ω) The Rotating Field v c of the total field generated by the stator windings along the air gap. The agnetic field rotates along the air gap with a speed ω C. The agnetic field generated by the stator interacts with the field generated by the rotor. As a consequence, an electroechanic torque is produced.

13 The Rotating Field v c of the total field generated by the stator windings along the air gap. The agnetic field rotates along the air gap with a speed ω C. The agnetic field generated by the stator interacts with the field generated by the rotor. As a consequence, an electroechanic torque is produced. The Rotating Field v c of the total field generated by the stator windings along the air gap. The agnetic field rotates along the air gap with a speed ω C. The agnetic field generated by the stator interacts with the field generated by the rotor. As a consequence, an electroechanic torque is produced. 3

14 The Rotating Field v c of the total field generated by the stator windings along the air gap. The agnetic field rotates along the air gap with a speed ω C. The agnetic field generated by the stator interacts with the field generated by the rotor. As a consequence, an electroechanic torque is produced. The Rotating Field of the total field generated by the stator windings along the air gap. v c The agnetic field rotates along the air gap with a speed ω C. The agnetic field generated by the stator interacts with the field generated by the rotor. As a consequence, an electroechanic torque is produced. 4

15 Induction Machine tator Rotor winding tator: three phase Rotor: three phase slot Induction Machine c a A b C C A b a c Induction otor 3 rotor circuits 3 stator circuits The rotor currents are sustained by the ef induced by the stator field. The rotor windings are short circuited (squirrel cage). 5

16 Induction Machine (a) quirrel- cage induction otor; (b) conductors in rotor; (c) photo of squirrel- cage induction otor; (d) views of a rotor, stator, and stator cross section. Induction Machine The rotating stator field oin the rotating rotor field and drag it together with the rotor (ω cs = ω cr ). The coupling between rotor and stator fields can be seen as the coupling of two agnets which oin eerting a force on each other. 6

17 Induction Machine a c A b ω cr angular speed of the rotor agnetic field ω cs angular speed of the stator agnetic field C A b a c C ω cs = ω cr If the stator and rotor fields would not have the sae speed, could not oin. Therefore the stator could not drag the rotor. Induction Machine c a A b C C A b a c Therefore in an induction achine it is: ω cs = ω cr = ω c = ω/p The angular velocity of the rotor is: ω = (- s) ω c = (- s)ω/p where s = (w -w C )/w C is the slip. ence in an induction achine, asynchronous achine, the velocity of the rotor and of the rotating echanical shaft is different fro the field velocity. 7

18 Induction Motor ynchronous Machine tator: three phase Rotor: DC current alient pole rotor Cylindrical rotor 8

19 ω = ω c ynchronous Machine Alternator (three phase voltage generator) In the generator the rotor field rotes with the rotor and drags the stator field which induces a three phase syste in the stator. The rotor is supplied by a DC current through sliding contact on the rotating echanical shaft. In the stator a three phase winding syste, as in induction achines, is present. y eans of the rotation of the rotor the DC agnetic field rotates and its linking with the three phase stator windings changes in tie. This induces a three phase voltage syste and, if the stator windings are connected to a load, an electrical current flows on the. The echanical energy supplied to the rotor shaft is converted into the electrical power supplied to the load. rushless ynchronous Machine Peranent Magnet rushless Motor The rotor field is produced by a set of peranent agnets (and not by a DC current) 3 9

20 Peranent Magnet rushless Motor The agnetic flu generated by the stator DC currents, is linked to the rotor windings (lines a and b with and ). As it appears fro the figure, when the rotor rotates, the flu linked to the rotor windings varies and an e..f. is generated between the A - terinals (sliding contacts). DC Electric Machine If fro A to through the pole the voltage is positive, the voltage fro A to through the pole is also positive (due to the rotation the variation of the agnetic flu has the sae sign). If the sliding contacts in A and would not be present, the total voltage in the whole winding fro the position A to A (or fro to ) would be equal to zero. With the sliding contacts on the points A and, which are syetric with respect to the rotor diagonal, the voltage fro A to along the left part of the rotor will be added to the voltage induced fro A to in the right part of the rotor, which are of the sae sign. 0

21 DC Electric Machine In the figures reported below two different realizations of DC achines are shown. J a ia e ia DC Electric Machine EMF generated in a turn dϕc dϕc e = - = - dt dϑ ω If the sliding contacts would not be present, the total e..f. on the whole winding would be equal to zero as seen on the previous slide. When the two contacts A and are present: $/& $ V g = ' e - $/& e A J ia where is the total nuber of turns and Vg in the voltage generated.

22 DC Electric Machine teel sheet of a stator with salient poles for the ecitation winding in a four poles DC achine DC Electric Machine DC achine with peranent agnets in a two pole configuration

23 Terinology air gap DC achine eddy current induction achine laination otional EMF peranent agnet brushless otor priary winding rotating agnetic field rotor salient pole rotor secondary winding traferro acchina a corrente continua corrente parassita acchina asincrona lainazione f.e.. trasforatorica acchina a agneti peranenti avvolgiento priario capo agnetico rotante rotore rotore a poli salienti avvolgiento secondario slot squirrel- cage induction otor stator synchronous achine transforer EMF cava acchina asincrona a gabbia di scoiattolo statore acchina sincrona ω, angular frequency pulsazione ω C, angular velocity of the agnetic field ω, angular velocity of the achine velocità angolare del capo agnetico velocità angolare della acchina 45 3