Electronic Power Conversion

Transcription

1 Electronic Power Conversion Review of Basic Electrical and Magnetic Circuit Concepts Challenge the future

2 3. Review of Basic Electrical and Magnetic Circuit Concepts Notation Electric circuits Steady state sinusoidal, non-sinusoidal Power apparent, real, reactive Power factor, THD, harmonics Magnetic circuits Ampere s law, Faraday s law Magnetic reluctance and magnetic circuits Inductor Transformer

3 Notation Notation: Lowercase instantaneous value e.g. v o (=v 0 (t) ) Uppercase average or rms value e.g. V o Bold phasor, vector Steady state: State of operation where waveforms repeat with a period T that is specific for the system. 3

4 A B C D 3 3 R Title Size Number Revision B Date: Sheet of File: Sheet.SchDoc Drawn By: 6 6 A B C D Instantaneous, Average Power and RMS current Instantaneous power Average power p= vi T P av = pdt= vidt T T T 0 0 Power to resistive load + v i R T P av,r =R i dt T 0 or P av,r P av,r = R I V = R - I RMS (Root-mean-square) T I= idt T 0 4

5 Example v= 0V + 5V sin( π 0 khz t) V V av rms = 0V = 0.607V R = 0Ω P av =.5W v( t) i( t) 0 v( t) i( t) V av V av V rms V rms p( t) 0 p( t) P av P av t 5

6 Steady-State ac Sinusoidal Waveforms Resistive/inductive load v(t) = V cosω t it () = I cos( ωt - φ) I p ω Phasor representation j0 V = V e I = -j I e φ Complex impedance Z = i p (t) = I p cosωt = ( I cosφ)cosωt i q (t) = j R+ jωl=ze φ V = I Z i () t = i () t + i () t p I q sinωt = ( I sinφ)sinωt q v( t) i( t) i p ( t) i q ( t) φ ji q t I V = V 0 = I -φ 6

7 Complex, Real and Reactive Power Complex power [VA] S * = VI = Se jφ Apparent power S = V I Real average power [W] [ S] P = Re =VIcosφ Reactive power [VAr] [ S] Q = Im =VIsinφ jq ji q S P S = P + I p jq ω V = V 0 I = I -φ Only I p is responsible for power transfer from source to load! 7

8 Complex, Real and Reactive Power Physical meaning Apparent power S influences cost and size: Insulation level and magnetic core size depend on V Conductor size depends on I Real power P useful work and losses; Reactive power Q preferably zero. 8

9 Power Factor (arbeidsfactor) Power factor How effectively is energy transferred between source and load PF For sinusoidal systems: S P V I cosφ PF = = = S V I cosφ 9

10 Power Factor Beer Mug Analogy Glass (over)sized to hold beer and foam Power wires (over)sized for Watts and VArs Reactive Power Active Power Apparent Power Source: Wikimedia Commons CC-BY-SA 0

11 Three Phase Circuits VLL P = VIcosϕ ph Sph = 3V = VI S3 ph = 3VI = 3VLLI P3 ph = 3VIcosϕ = 3VLLIcosϕ PF = cosφ

12 Steady-State Non-Sinusoidal Waveforms Periodic non-sinusoidal waveform Represented as sum of its harmonics st 3 rd 0 i h ( t) 5 i( t) t i( t) := 0A sin ω t ( ) 7 A i 3h ( t) i 5h ( t) i 7h ( t) 0 5 π + sin 3ω t 4 A 5 th sin( 5ω t) + A t ( ) sin 7ω t 7 th

13 Fourier Analysis of Non-Sinusoidal Waveforms Non-sinusoidal periodic waveform f(t) f( t ) = F 0+ fh( t) = a0 + { ah cos( hωt) + bh sin( hωt)} where h= h= π F0 = a0 = f() t d( ωt) f() t d() t π = T 0 0 T π ah = f( t)cos( hωt) d( ωt) π 0 π bh = f( t)sin( hωt) d( ωt) π 0 h =,..., h =,..., 3

14 Line Current Distortion 0 v () t = V sinω t s s v( t) i( t) v( t) i( t) i h ( t) i h ( t) i dis ( t) 0 0 i dis ( t) 0 s h = sh is t + id is i ( t) = i () t + i ( t ) = () ( t) s i ( t ) = I sin( ωt - φ ) + I sin( ω t - φ ) s s sh h h h= t current i = fundamental harmonic i s + distortion current i dis 4

15 Total Harmonic Distortion (THD) A measure of how much a composite current deviates from an ideal sine wave Caused by the way that electronic loads draw current i ( t ) = I sin( ωt - φ ) + I sin( ω t - φ ) I = s s s sh h h h= T i 0 s(t) dt T or I s= I s + I h = sh and: dis s s h = sh I = I I = I Total harmonic distortion: THD = I I sh h= s 5

16 Harmonics Negative effects of harmonics Conductor overheating Capacitors can be affected by heat rise increases due to power loss and reduced life time Distorted voltage waveform Increased losses (e.g. transformers overheating) Fuses and circuit breakers fault operation Harmonic standards International Electrotechnical Commission Standard IEC-555 IEEE/ANSI Standard 59 6

17 Power Factor PF S with and T T P = p() t dt = v () t i () t dt T S = V I T 0 0 s s P = V sin( ωt ) I sin( ω t - φ ) dt T T 0 s h = sh h h P = V t I t - dt = V I T 0 vs () t i () t T sin( ω ) sin( ω φ ) cos φ s s s s s s Real power (vermogen) Apparent power (schijnbaar vermogen) s VI s cos φ PF = VI = I s I s s s cosφ DPF 7

18 Inductor and Capacitor Phasor Diagrams Phasors are only applicable to sinusoidal steady state waveforms. i L jωl -/jωc + - v L + - v C ω I C ω V L V C I L 8

19 Inductor and Capacitor Response Time domain: di dvc L vl() t = L ic () t = C dt dt t t il() t =il( t ) + vl() t dt vc() t =vc( t ) + ic() t dt C L t t 9

20 Inductor in Steady State Volt-second Balance Steady state implies: v(t +T)= v(t) i(t +T)=i(t) i () t =i ( t ) + v () t dt L L L t L t t t + T v L () t dt = 0 Net change in flux is zero v Lav, = 0 0

21 Capacitor in Steady State Amp-second balance t vc() t =vc( t ) + ic() t dt C t t + T t i C () t dt = 0 i Cav, = 0 Net change in charge is zero Fig. 3-9 Note: error in fig 3-9 from textbook i C in the second interval should be constant, because v c has a constant derivative in that interval

22 Basic Magnetics Theory Ampere s law (4 th Maxwell s equation) c H d l = i c Kcore sections i Hl = N I k k m m M windings Hl + H l = Ni g g Right-hand rule:

23 H-field, B-field and Material Properties Relationship between B and H given by: material properties often approximated by: B= µ H Continuity of flux Flux (by def.): φ = B A d A φ closedarea = Continuity of flux: Gauss s law of magnetism ( nd Maxwell s equation) B da = 0 A 3

24 Magnetic Reluctance Ampere s law: lk lk lk N I = H l = ( B A ) = φ = φ = φ R µ A µ A µ A m m k k k k k k m k k k k k k k k k k B= µ H Magnetic reluctance: R k = lk µ A k k Magnetic R k φ Ni Electrical R i v 4

25 Magnetic Circuit Analysis m Ni Magnetic = φ R m m k k k Electrical v = i R N I = φ R v = m i R k φ k = 0 i k = 0 k m k k Ohm s law: Kirchhoff s voltage law Kirchhoff s current law 5

26 Faraday s law e = dψ d(n φ ) db = = NA dt dt dt 6

27 Inductor Coil self-inductance Nφ i di e = L d(n φ ) dt e = dt Nφ i L = = R φ R= Ni L = N µ A l R= µ A N l 7

28 Transformer Ideal transformer Faraday s law dφ dφ v = N v = N dt dt v N = Ampere s law v N Ni ii= φ R Ni Ni i=, Core reluctance = φ R l R= µa 0 in = in 8

29 Transformer Magnetising inductance R= l µa 0 Non-zero core reluctance Ni Ni = φ R dφ v = N dt N d N d v = ( i i ) = Lm im dt N dt R m N L m = R N = N i i i 9

30 Transformer Leakage flux φ l, leakage flux φ = φ + φ l φ = φ + φ l R, ohmic resistances of the windings dφ dφ l dφ v = Ri + N = Ri + N dt + dt dt dφ dφ l dφ v = Ri N = Ri N dt dt dt 30

31 Equivalent Transformer Circuit dφ dφ l dφ v = Ri + N = Ri + N L = N φ / i dt + dt dt dφ dφ l dφ v = Ri N = Ri N dt dt dt di dim dt dt di N di dt N dt v =Ri + Ll + Lm v = Ri Ll + Lm m i = i + N i m N di v =Ri + Ll + e dt di v = Ri L + e dt l 3

32 Image credits All uncredited diagrams are from the book Power Electronics: Converters, Applications, and Design by N. Mohan, T.M. Undeland and W.P. Robbins. All other uncredited images are from research done at the EWI faculty. 3