Grunntækni og rafmagn í riðstraums (AC) raforkukerfi

Transcription

1 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 1 Grunntækni og rafmagn í riðstraums (AC) raforkukerfi Haust 2016

2 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 2 Samræmi vélrænna og rafrænna stærða Afl = straumur * spenna U U V = Ex = gx = q m U = qe x U = F x = mg x du dq du dx p = = Ex = I V p = = F = F v dt dt dt dt dm = gx dt V = spenna, q= hleðsla, g = þyngdarhröðun, x = fjarlægð m= massi, U = orka, p = afl, t = tími, v = hraði, F = kraftur I = rafstraumur E = rafsvið, V = rafspenna

3 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 3 Power and Energy concepts Mechanical Energy Rotational Energy Electrical Energy Power Electrical Power

4 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 4 Instantaneous Sinusoidal Current/Voltage in Power Systems Einkennandi stærðir: (A) STÆRÐ sveiflunnar enna, i = 2I eða v = 2V (A) FASI eða fasahorn sveiflunnar v max i max φ Spenna 1/50 sek Straumur δ vt ( ) = v sinωt max it () = i sin( ωt φ) max tìmi max [%] ω = 2 π f = 49,86-49,90 49,90-49,94 49,94-49,98 49,98-50,02 max f tíðni í Hz Frequency Range (Hz) 50,02-50,06 50,06-50,10 50,10-50,14 Tíðni/Frequency: 50Hz in Europe (60 Hz in North America)

5 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 5 Power in AC Circuits Consider a simple AC circuit with voltage and current: i( t) = i sin( ) max ωt φ vt ( ) = v sinωt max it ( ) = i sin( ωt φ) max Ix Ir φ Ir I Ix V = I = V e I e V j 0! j φ v( t) = vmax sin ωt Use the trigonometric identity: 1 sin xsin y = cos( x y) cos( x+ y) 2 [ ] Instantaneous power will be current times voltage: pt () = vtit ()() = v i sinωtsin( ωt φ) max max vmax imax pt () = cosφ cos( 2ωt φ) 2

6 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 6 Current, Voltage and Power vmax imax pt () = cosφ cos( 2ωt φ) 2 Voltage, current or power Voltage v(t) Current i(t) Instantaneous power p(t) f time Real average power

7 Phasors and Instantaneous VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 7 Power Define: V I = = i v max 2 max 2 Then from the previous eq.: pt ( ) = V Icosφ V Icos(2 ωt φ) Using a trigonometric identity: cos( x y) = cos xcos y + sin xsin y pt ( ) = V Icosφ 1 cos 2ωt V I sinφ sin 2ωt we get: ( )

8 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 8 Real Power - Reactive Power Defining the following quantities P = V I cosφ (P is called Real Power) Q = V I sinφ (Q is called Reactive Power) We get: ( ) pt () = P1 cos2ωt Qsin2ωt

9 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 9 Real Power - Reactive Power From the last equation the instantaneous power is made up of 2 components: The first component is always positive with an average value P. Power is always transferred in the same direction and can do useful work The second component swings back and forth. The average = 0, while the amplitude is Q power swings back and forth and does NOT do useful work

10 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 10 Real and Reactive Power current voltage Average power or real power, P Instantaneous power, p(t) P(1-cos(2ω t)) Qsin(2 ω t))

11 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 11 The Power Triangle S = V I P = V I cosφ 2 2 S = P + Q Q = V I sinφ S f P Q

12 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 12 Apparent Power Apparent power is defined as the product of the RMS Voltage and RMS current: S = V I The unit of measurement is watt. However the power industry tradition is to use the unit VA (= volt - amperes), or kva or MVA

13 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 13 Definition of Power Concepts Power concept Units Formula Real power W, kw,mw P = V I cosφ Reactive power Apparent power Var, kvar,mvar VA, kva,mva Q= V S = I V I sinφ

14 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 14 Real Power The instantaneous power in an AC circuit oscillates, with a frequency double that of the voltage and current, around a certain average value. (In a 50 Hz system the frequencey of the power oscillation is thus 100 Hz) The real power is a measure of this average value (~ is this average value)

15 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 15 Reactive power The instantaneous power in an AC circuit oscillates, with a frequency double that of the voltage and current, around a certain average value. The reactive power is a measure of the amplitude of this oscillation (or a measure of the deviation of the instantaneous power from this average value)

16 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 16 The Apparent Power In an AC circuit of a specified voltage the apparent power is a measure of the magnitude (amplitude) of the alternating current (AC)

17 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 17 3 phase power systems (line to line/phase voltages/currents, 3 phase power, star/delta connections, 3 phase 3 wire/4 wire)

18 Advantages of three VÉL102M phase. - Orkufrek framleiðsluferli - Egill Benedikt Why Hreinsson 18 3 phases systems? Smooth flow of power (instantaneous power is a constant). Constant torque (reduced vibrations) The power delivery capacity is tripled (increased by 200%!) by increasing the number of conductors from 2 to 3 (increase by 50%) Reduced cost (same power less wire or more power same wire) Greater "power per kg" in motors, generators, and transformers.

19 3 separate identical VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 19 and simple 1 phase systems + V a I a V b + V c I b I c - 6 conductors can be reduced to 3! Generation Transmission Load

20 Waveforms: b- Phase VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 20 Different representation of 3 phase quantities : Voltage or current time v ( t) = 2 V sin( ωt) a v ( t) = 2 V sin ( ωt 120 ) b v ( t) = 2 V sin ( ωt+ 120 ) c Formulas: a- Phase c- Phase i ( t) = 2 I sin( ωt φ) a i ( t) = 2 I sin( ωt 120 φ) b i ( t) = 2 I sin( ωt+ 120 φ) c i () t + i () t + i () t = a b c [ 2 I sin( ωt ϕ) + sin( ωt 120 ϕ) + sin( ωt ϕ) = 0 The 3 return conductors can be thrown away!! ]

21 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 21 A 3 phase 3 wire system + V a I a V b + V c I b I c - Both neutrals may or may not be grounded

22 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 22 A mechanical analogy with a 3 phase hydraulic generator The total power delivery in a three phase system is smooth!

23 A mechanical analogy VÉL102M - Orkufrek with framleiðsluferli - Egill a Benedikt 1 Hreinsson 23 phase hydraulic generator Rafali Flutningur Stefna aflflæ is Álag Fyllt og loku vökvarás The power delivery in a one phase system is bumpy!

24 A 3 phase 4 wire VÉL102M system - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 24 (with ground wire) + V a I a V b + V c I b I c - Ground wire Both neutrals may or may not be grounded

25 A 2 phase 3 wire VÉL102M system - Orkufrek framleiðsluferli - Egill with Benedikt Hreinsson 25 ground wire) - + v a + v b i a i b - Ground wire v ( t) = 2 V cos( ωt) i ( t) = 2 I cos( ωt φ) a π π vb( t) = 2 V cos( ωt ) ib( t) = 2 I cos( ωt φ ) 2 2 a

26 A 1 phase system VÉL102M with - Orkufrek framleiðsluferli 2 - Egill Benedikt Hreinsson 26 wires i a v () t = 2V cos( ωt) 1 1 z= r+ jωl V 2 + L /2 L /2 -i a z= r+ jωl 25 km The system is balanced against the earth

27 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 27 How are the 3 phases labelled? North America: a, b, c Europe: (old) R,S,T Europe: (new) L1, L2, L3

28 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 28 Instantaneous power in a 3 phase system v ( t) = 2 V sin( ωt) a v ( t) = 2 V sin ( ωt 120 ) b v ( t) = 2 V sin ( ωt+ 120 ) c 3 phase voltage: i ( t) = 2 I sin( ωt φ) a i ( t) = 2 I sin( ωt 120 φ) b i ( t) = 2 I sin( ωt+ 120 φ) c 3-phase current: From the above formulas, we get the total instantaneous power: p () t = v () t i () t + v () t i () t + v () t i () t 3 phase a a b b c c

29 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 29 Instantaneous power in a 3 phase system(2) p () t = v () t i () t + v () t i () t + v () t i () t 3 phase a a b b c c By inserting the formulas, we get: [ = 2 I V sin( ωt)sin( ωt φ) + sin( ωt 120 )sin( ωt 120 φ) + sin( ωt+ 120 )sin( ωt+ 120 φ) ]

30 Instantaneous power VÉL102M in - Orkufrek framleiðsluferli a 3 - Egill phase Benedikt Hreinsson 30 system (3) Use the following trigonometric identities to simplify: 1 sin xsin y = cos( x y) cos( x+ y) 2 [ ] cos( x) + cos( x 120 ) + cos( x+ 120 ) = 0 And we get finally the following formula p () t = 3I V cosφ = 3 P 3phase 1phase P = I V 1phase cos φ

31 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 31 Total 3 phase power Therefore, the total instantaneous power in all 3 phases is constant - or - 3 times the real power in each phase The power oscillates in each phase (although the sum of power in the phases is constant) No reactive power appears in the formula!! Reactive power is, however, very much present in each individual phase Remember: 200% increase in power with only 50% more wires

32 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 32 Y - Delta Connections in 3 Phase Systems

33 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 33 A 3 phase system - drawn like phasors - Star (Y) configuration V cf Neutral grounding I c V af I a V bf I b Neutral grounding

34 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson A 3 phase system with neutral 34 return V cf + I c + V bf + n V 1 af I a n 2 I b The neutrals are labelled n 1 and n 2 respectively n I

35 VÉL102M - Orkufrek framleiðsluferli - Egill Benedikt Hreinsson 35 A 3 phase system - drawn like phasors - Delta (D) connection V cf Neutral grounding I c V af I a V bf I b The Delta connection has never any neutral!