15 Problem 1. 3 a Draw the equivalent circuit diagram of the synchronous machine. 2 b What is the expected synchronous speed of the machine?

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1 Exam Electrical Machine and Drive (ET4117) 6 November 009 from 9.00 to Thi exam conit of 4 problem on 4 page. Page 5 can be ued to anwer problem quetion b. The number before a quetion indicate how many credit you can earn by anwering that quetion. A partly correct anwer may give a part of the credit. Thi examination ha to be made without uing a book, old examination, note, dictionarie or programmable calculator; a pocket calculator may be ued. 15 Problem 1 A ynchronou machine ha 3 electrical phae and 4 magnetic pole. It i uitable to be connected to the 50Hz grid with a phae voltage of 30 V. The tator reitance can be neglected. 3 a Draw the equivalent circuit diagram of the ynchronou machine. b What i the expected ynchronou peed of the machine? The ynchronou machine i connected to the upply grid and rotate at the ynchronou peed. The figure how the phae terminal voltage (or grid voltage) and the phae excitation voltage (or back emf). Both voltage have an RMS value of 30V. 4 c Sketch the phaor diagram (with the phaor for the terminal voltage, the back emf and the current) for thi ituation. 4 d Calculate how much power i flowing between the grid and the machine if the ynchronou reactance i 1Ω. e Doe the machine operate a a generator or a a motor? Amplitude (V) Grid Back EMF Time () 1

2 30 Problem The figure below on the left depict an axial flux permanent magnet machine. The rotor conit of two dic with permanent magnet. The picture on the right depict a linearized cro ection of two pole pitche. The arrow depict the direction of the magnetiation of the magnet. The tator conit of tator iron with a three-phae winding wound around it. The machine ha an air gap winding; the tator ha no lot. The machine ha the following characteritic (mot are in the figure): - The magnet length in the direction of magnetization i l m = 5 mm. - The air-gap length (including the winding) i l g = 5 mm. - The machine ha 8 pole. - The width of a pole i 80% of the width of a pole pitch ( bm = 0.8τ p). - The outer radiu of the magnet and the tator iron i r o = 50 mm. - The inner radiu of the magnet and the tator iron i r i = 30 mm. - Each phae winding conit of 8 coil; each coil ha 50 turn, o the number of turn of a phae winding i N = The remanent flux denity of the magnet i B rm = 1.T - The relative recoil permeability of the magnet μ rm = 1 - The BH characteritic in the econd quadrant of the BH plane i a traight line In the calculation, it may be aumed that: - The magnetic permeability of iron i infinite - The flux denity croe the magnet and the air gap perpendicularly - The wire of the turn are very thin In the quetion a through i, the tator current are zero. a Write down Ampere law (a implified form of the firt of Maxwell equation). b Sketch a contour and a urface in the cro ection to which you can apply Ampere law to calculate the magnet flux denity in the air gap. c Give an equation for the BH characteritic of the magnet in the econd quadrant. 4 d Derive an expreion for the air gap flux denity B g uing Ampere law. e Write down the econd of Maxwell equation or Faraday law. f Decribe the contour and the urface to which you can apply the econd of Maxwell equation to derive the voltage equation of thi winding. In the ret of thi problem, ue that the air gap flux denity i B g = 0.6 T.

3 3 g Calculate the flux of one pole. In the ret of the problem, ue that the flux of one pole i 0.3 mwb. h Calculate the maximum flux linkage of a turn (of phae c in the left figure). i Calculate the maximum flux linkage of a phae winding. Thi machine i operated a a bruhle DC machine. The current i A. 4 j Calculate the torque. You are allowed to ue the power balance. You are alo allowed to ue the Lorentz force. 3 k How mut the tator be laminated in order to limit the eddy current loe? One of the rotor dic i rotated over one pole pitch (45 degree), while the other tay in poition. l What i the effect of thi change on the motor performance (e.g. the induced voltage)? 0 Problem 3 A DC machine with independent electrical excitation i connected to a DC voltage ource. Armature reaction i neglected. The voltage i V t, the motor contant i K, the armature reitance i R, the pole flux i Φ 0. 3 a Draw the equivalent circuit and give the (teady tate) voltage equation. b Give an expreion for the no-load peed. c Give an expreion for the tall torque (torque at zero peed with locked rotor). d Sketch the torque-peed characteritic including the no-load peed and the tall torque. e Now ketch the torque-peed characteritic including the no-load peed and tall torque if the pole flux i reduced to Φ 0 /. When DC machine are loaded, there i an armature reaction. Problem due to armature reaction can be reduced by adding interpole and ometime compenation winding. Including the armature reaction and: 4 f Sketch a DC machine cro ection. Include the interpole and the compenating winding. Indicate current direction with dot and croe. 3 g Explain which problem may arie due to armature reaction. h Why i it dangerou to diconnect the field winding of a eparately excited DC machine running under no-load condition? 3

4 35 Problem 4 Thi problem deal with a 4-pole three-phae induction machine connected to a ymmetrical three-phae upply with a frequency of 50 Hz and a phae voltage of 30 V. The parameter of thi equivalent circuit are given by L = 81 mh; L σ = 9 mh; R R = 0.3 Ω; R = 0. Iron loe are neglected. The depicted circle diagram give the tator current phaor trajectory a a function of the lip. a In which of the operating point A to L i the machine in generator operation? b Calculate the no-load current phaor (the rotor rotate ynchronouly). 16 c For point D, calculate - the lip, - the angular frequency of the electrical quantitie in the rotor, - the mechanical peed (in rpm), - the current through the rotor reitance, - the tator power, - the copper loe in the rotor, - the efficiency, and - the electromagnetic torque. Note: in point D, the reitance and the reactance of the rotor branch of the equivalent circuit are equal. 3 d Sketch the torque-peed characteritic of a three-phae induction machine. e Indicate the three mode of operation in the torque-peed characteritic. 5 f Draw the electrical diagram (diode and active witche) of the peed controlling mechanim that can be ued to control the peed of three-phae AC motor. 5 g Decribe how a ingle phae induction machine can be made to rotate. 4

5 Examination Electrical Machine and Drive ET4117 Friday, November 6, 009 from 9.00 to 1.00 Name: Student number: Anwer to quetion b 5

6 Anwer to the exam Electrical Machine and Drive ET Problem 1 3 a 60 f b n = = rpm = 1500rpm p 4 4 c 4 d If the terminal voltage i choen in the real axi, the excitation voltage can be written a E = E (coδ + jin δ ) f f * f δ + δ t t f δ t f δ t (co jin ) 3 in 3 j( co ) * E V VE V E V S = P+ jq = 3VtI = 3Vt = + j X X X The angle between excitation voltage (or back emf) and terminal voltage i 45. Therefore, VE t f 30 P = 3 inδ = 3 in 45 W = 11.kW X 1 e The grid voltage i lagging the machine back emf, therefore the machine i operating a a generator and the machine i delivering power to the grid. 30 Problem a H τ d = J nda Cm Sm The magnetic field intenity H, the unit vector in the direction of the contour τ, the current denity J, and the normal vector n are vector. The dot mean that the inproduct of thee vector are taken. b A poible contour and a poible urface are indicated. The boundary of the urface i the contour. The contour hould approximately follow the field line. c Bm = μ0μrmhm + Brm 4 d Hl g g+ Hl m m= 0 Bl g g ( Bm Brm) lm Uing the contitutive relation for magnet and air: + = 0 μ0 μ0μrm Flux continuity: B nda 0. Therfore, BA g g = BA m m Bg = Bm A 6

7 Bl g g ( Bg Brm) lm Subtitution give + = 0 μ0 μ0μrm lm Bg = Brm lg + lm e d E τ d= B nda d t Ce Se f The contour i choen in the electrical circuit, in the wire. The contour i the boundary of the urface. 3 g 0.8 π ( ro ri ) Am = = 50.6 mm p Φ = = mwb p AB m g h The flux linkage of a tator turn i equal to the pole flux. Half of the flux of a magnet goe to the left and the other half to the right, o half of the pole flux of one pole link with a turn on the tator. However, the machine i double ided, o the magnet on the other ide alo give a contribution of half the pole flux. λ turn =Φ p = AB m g = mwb i λ max = N Φ p = 0.10 Wb 4 j The force on a conductor i contant, independent from the radiu. The torque depend on the radiu. We can calculate the torque uing the average radiu: ( ro + ri ) ( ro + ri ) T = Fr = F = 4 NBg I( ro ri ) = NBg I( ro ri ) = 1.536Nm The factor 4 come from the fact that every turn ha two turn ide that make a torque and from the fact that in a bruhle DC machine, alway phae are conducting. The other way of calculating the torque i via the power balance: Pem EmaxI T = = ωm ωm In 80% of half an electrical period, the flux linkage change from the maximum flux linkage to the minimum flux linkage. Therefore: d λ 4 f λmax Emax = = = 5 fλmax dt 0.8 Pem 10ωeλmaxI 5pλmaxI T = = = = 1.58 Nm ωm πωm π The reult are lightly different, becaue the firt i baed on an air gap flux denity of 0.6 T, while the econd i baed on a maximum flux linkage of 0.1 Wb. 3 k The lamination hould be wound like tape on a roll. l The machine will not work anymore, the induced voltage become zero becaue the flux linkage become zero 0 Problem 3 3 a V t = RaI + Ea b In no-load, the current i zero, therefore V = E = KΦω t a V Therefore, ω = t K Φ 7

8 c The tall torque i the torque when the peed i zero, therefore, Vt = RaI. Vt The torque i then T = KΦI = KΦ Ra d e 4 f The direction of the current in the compenating winding and the current around the interpole i oppoite to the direction of the current in the armature. Therefore, in the right half of the machine, there hould be dot in the compenating winding and around the interpole, and in the left half of the machine there hould be tar. 3 g 1. Saturation, which lead to a decreae of the pole flux.. Commutation problem due to the fact that the commutating coil i not in the fieldfree zone while commutating, which lead to park and additional wear. h When the field winding i diconnected, the machine peed will increae, theoretically until infinity. In practice, motly omething ele happen before infinity i reached. Motly that i damaging and dangerou. 35 Problem 4 a Motor convention i ued in the equivalent circuit, o in the point H, I, J, K, and L, the power i negative, o the machine i in generator operation. b U I = jωl = j9.038a 16 c In operating point D, R R = ωl σ Therefore, R R = = ωl σ R R ω = ω = = 33.33rad/ r L σ n= (1 ) n = 1341rpm U U I R = = = R R ω Lσ + ( ωlσ ) 57.5 A 8

9 3 d e P Cur = 3RRIR =.978 kw 3RI R R P = = 8.06 kw If there are no other loe, the efficiency can be calculated a η = 1 = The electromagnetic torque can be calculated a Pm p (1 ) P p R RI R p U R R Tmech = = 3 = 3 = 3 ωm (1 ) ω ω ω R R + ( ωlσ ) p U R R = 3 = Nm ω ( ω L ) σ 5 f Figure depict an inverter. 5 g When connected to an AC ource, the tator of a ingle-phae induction machine create a pulating field. The pulating magnetic field of a ingle phae mut get a 9

10 rotating component. Thi can be done by adding an auxiliary winding. The axi of thi winding i hifted 90 degree with repect to the axi of the main winding. The current in thi winding ha a phae hift with repect to the current in the main winding becaue it ha a different impedance or becaue a capacitor i added. Another way of creating a rotating component in a pulating field i by making a plit phae induction motor. Here the tator ha an even number of pole with the ingle phae AC winding around them. Around a part of the pole, there i a hort-circuited copper ring, which delay the change of flux compared to the ret of the pole, thu giving a rotating component. 10