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10 CORNERSTONES OF POWER 8 S S INDUCTOR PRINCIPLE,ELECTRONICS Hoft. Under steady state conditions the average voltage,1 KVL across an inductor is zero. Kirchoff s Voltage Law The sum of the changes in, voltage around a circuit loop is equal to zero This is 9 S S CAPACITOR PRINCIPLE. true in both the instantaneous and average integrate. over one cycle sense Under steady state conditions the average current. through a capacitor is zero,10 FOURIER SERIES, Kirchoff s Current Law The current entering a node is. equal to the current leaving the node This is also true In the 1820s Fourier came out with a 1 page paper on. in both the instantaneous and average integrate over his Fourier series A periodic function may be. one cycle sense described as an infinite sum of sines and cosines. 3 vi RESISTOR v t Vavg ak cos k 0t bk sin k 0t,The voltage to current relationship in a resistor.

v ir See p4,DISTORTION,4 vi CAPACITOR, The voltage to current relationship in a capacitor Distortion is the degree to which a signal differs from. its fundamental frequency,dv RMS value of harmonics for k 1. dt RMS value of fundamental frequency k 1,5 vi INDUCTOR Vdis. The voltage to current relationship in a inductor Vrms1. Vrms Vrms1,Use the polar form of the Fourier Series see p4. 6 AVERAGE DC AND RMS Vdis rms voltage distortion V. Average and dc will be synonymous in this class but Vrms1 fundamental frequency rms voltage V. are not the same as rms Vrms rms voltage V,THD Total Harmonic Distortion V.

1 t0 T 1 t0 T 2,v t dt v t dt,CREST FACTOR no units. The crest factor quantifies the smoothness of the, waveform and is related to the weight of its impact on. components For DC and a square wave the crest, We are concerned with both instantaneous and factor is 1 for a sine wave it is 1 414 A large crest. average power As with rms values power is related factor means the wave is not as efficient at delivering. to heating energy,1 t0 T Vpeak,Pavg p t dt p t v t i t CF. p t instantaneous power W, Tom Penick tom tomzap com www teicontrols com notes PowerElectronics pdf 8 18 2003 Page 2 of 21.

PF POWER FACTOR no units C CAPACITANCE F, The power factor is the ratio of true power the power i t I f I o I f e t. consumed ignoring the reactive factor to apparent v C R. v t V f Vo V f e t,power the total power consumed Also the power. factor is the cosine of the angle by which the current where RC. lags the voltage assuming an inductive load 1 t,PF cos v i. L INDUCTANCE H,DECIBELS dB,i t I f I o I f e t, A log based unit of energy that makes it easier to v L. v t V f Vo V f e t R,describe exponential losses etc The decibel means.

10 bels a unit named after Bell Laboratories where L R. voltage or current 1 t,reference voltage or current. v L t L dtdi I L t,power of an inductor L,L 10 log I e 10. reference power,L inductance H,permeability H cm,UNITS electrical N number of turns. Ae core cross section cm2, I current in amps q W J N m V C Ie core magnetic path length cm. s V V s V s s,q charge in coulombs I s V C N m W s.

LC TANK CIRCUIT,C capacitance in farads q q q J I s. Resonant frequency,V J N m V 2 V C L 1,H inductance in henrys V s note that H F s 2 2 LC. J energy in joules N m V q W s I V s C V 2 q PARALLEL RESISTANCE. I never can remember the,J q V W s kg m R1 R2, N force in newtons 2 formula for two resistances in. m m m s R1 R2,parallel I just do it the hard R1 R2. Wb V s H I way,T magnetic flux density in teslas,V electric potential in volts.

W J J W s N m q,I q I s q q C,W power in watts,J N m q V C V 2 1. s s s s 746,Wb magnetic flux in webers H I V s J,Temperature C or K 0 C 273 15K. where s is seconds, Tom Penick tom tomzap com www teicontrols com notes PowerElectronics pdf 8 18 2003 Page 3 of 21. FOURIER SERIES FOURIER SERIES and Symmetry, When the function f t is symmetric certain shortcuts. The Fourier Series is a method of describing a,can be taken.

complex periodic function in terms of the frequencies. and amplitudes of its fundamental and harmonic When f t is an even function i e. frequencies f t f t bk is zero The Fourier,series becomes. Let f t f t T any periodic signal,f t Favg ak cos k 0t. where T the period k 1,If there is also half wave,symmetry then. f t cos n t dt,t When f t is an odd function i e,f t f t ak is zero The Fourier. series becomes,f t Favg bk sin k 0t, Then f t Favg ak cos k 0t bk sin k 0t If there is also half wave.

symmetry then,0 the fundamental frequency k 1 in radians sec 4. f t sin n t dt, k 0 the harmonic frequencies k 2 3 4 in radians sec bk. k denotes the fundamental k 1 or harmonic frequencies. k 2 3 4 not the wave number or propagation When f t has half wave. constant symmetry i e f t f t T 2 there, Favg the average value of f t or the DC offset are only odd harmonics. f t cos n t dt,ak twice the average value of f t cos k 0t ak. f t cos k t dt,bk f t sin n t dt,bk twice the average value of f t sin k 0t T t0.

bk f t sin k t dt,T t0 FOURIER SERIES Polar Form,t0 an arbitrary time. f t Favg Fk sin k 0t k,Fk ak2 bk2 k tan 1,f t cos n t dt. bk f t sin n t dt, Tom Penick tom tomzap com www teicontrols com notes PowerElectronics pdf 8 18 2003 Page 4 of 21. FOURIER SERIES OF A SQUARE WAVE SINGLE PHASE RECTIFIERS. A 50 duty cycle square wave can be represented as HALF WAVE RECTIFIER. an infinite sum of a fundamental sine wave and, As the supply voltage begins it s positive sinusoidal. smaller odd harmonics, excursion the diode conducts and current begins to.

flow in the inductor When the voltage crosses zero. sin 0t sin 3 0t sin 5 0t sin 7 0t L the current continues to flow through the inductor for a. 3 5 7 short period due to its stored energy and the diode. conducts until the inductor current flow has halted. 4A This point is called extinction and occurs at the angle. A where 2 At this time vd which has, followed the supply voltage into the negative region. becomes zero discontinuous vd and the inductor,current remain at zero until the next cycle. Vm sin t vd RL, While current is flowing through the diode there is zero. voltage across the diode When current flow stops at angle. the voltage across the diode becomes negative,discontinuous Since the average dc voltage at the. source is zero and the average voltage across an inductor is. zero the average voltage across RL is the negative of the. average voltage across the diode,Vdiode Vm sin d VR avg.

The average voltage across RL can also be expressed as. the product of the average short circuit current and RL. Under short circuit conditions the average voltage can be. found by integrating of a half period,VR avg I sc RL VR avg. Vm sin t dt, Now there is enough information here to find iteratively. Tom Penick tom tomzap com www teicontrols com notes PowerElectronics pdf 8 18 2003 Page 5 of 21. BRIDGE RECTIFIER CONSTANT LOAD u COMMUTATION INTERVAL. VOLTAGE Commutation is the transfer of the electrical source. The current id begins to flow when the magnitude of from one path to another For bridge rectifiers it. the supply voltage exceeds Vd The current peaks refers to the period of time when diodes from two. when the supply voltage magnitude returns to the sources are on simultaneously i e the delay interval. level of Vd As the supply voltage magnitude associated with a reverse biased diode turning off. continues to fall id rapidly returns to zero The commutation interval is usually expressed as an. id angle u, The commutation interval is associated with rectifier. circuits having a constant current load inductance. L dominates load and a finite inductance Ls in the. supply The interval begins when the source voltage. vL crosses zero going positive or when the thyristor gate. Vm sin t Vd,is triggered,No trigger cos u 1,Vd Vm sin b 2 Ls I d. With trigger cos u cos,vL L Vm sin t Vd the supply frequency rad sec.

dt Ls the supply inductance H,0 Vm sin t Vd d t Id the constant load current A. b Vm the peak input voltage V,in area vs,Vd POWER AND COMMUTATION. In order to have power the commutation interval must. not be zero,b p f t the supply frequency rad sec,Ls the supply inductance H. Vd the voltage at the output V Id the constant load current A. Vm the peak input voltage V Vm the peak input voltage V. b the angle at which an increasing supply voltage, waveform reaches Vd and current begins to flow in the. inductor radians VOLTAGE AND COMMUTATION, p the angle at which id peaks radians p b The average voltage output of a full wave bridge.

f the angle at which the current id returns to zero radians rectifier is. Vd cos u 1,SINGLE PHASE BRIDGE RECTIFIER,id iL the supply frequency rad sec. Ls the supply inductance H,Id the constant load current A. L Vm the peak input voltage V,Vm sin t C RL, Tom Penick tom tomzap com www teicontrols com notes PowerElectronics pdf 8 18 2003 Page 6 of 21. TH VENIN EQUIVALENT KIMBARK S EQUATIONS with, The Th venin equivalent for a single phase full wave When the trigger angle is included Kimbark s. bridge rectifier equations become,2Vm RTH Id VLLp,I DC cos cos u.

2 Ls VTH Vd 3VLLp,the supply frequency rad sec 2,Pdc cos 2 cos 2 u. Ls the supply inductance H 3,Id the constant load current A 4 L. Vm the peak input voltage V VLLp peak line to line voltage V. Vd the average output voltage V the supply frequency rad sec. the angle t at which the thyristor is triggered degrees. THREE PHASE RECTIFIERS L the per phase inductance H. u the commutation interval degrees,THREE PHASE RECTIFIER. This circuit is known as 3 phase 6 pulse line, commutated converter When the load is dominated 3 PHASE POWER AND COMMUTATION. by inductance it is modeled as a current source In order to have power the commutation interval must. constant current as shown below If the load is not be zero In commercial systems the commutation. capacitive it is modeled as a voltage source interval is typically 4 to 5 degrees but may be as high. as 20 in special high power converters The,TRANSFORMER theoretical maximum is u 60.

3 PHASE VOLTAGE AND COMMUTATION, B Lb The average voltage output of a three phase rectifier. Idc can be found by integrating over the first 60 In the. formula below the limits of integration have been, shifted to make the function fit the cosine function. Vab Vac Vbc Vba Vca Vcb,KIMBARK S EQUATIONS, Kimbark s equations give the average current voltage. and power of a 3 phase rectifier as a function of the. commutation interval u,I dc 1 cos u,2 Vdc 1 cos u 1 u 3. 2 Vdc 0 V cos d 6,VLLp cos d,3 Pdc 2 60,VLLp peak line to line voltage V.

the supply frequency rad sec,L the load inductance H. u the commutation interval degrees, Tom Penick tom tomzap com www teicontrols com notes PowerElectronics pdf 8 18 2003 Page 7 of 21. PLOTTING Vdc constant current WITH PLOTTING iA constant current. Vdc is periodic at 60 intervals During the iA is the current through the A phase supply of a 3. commutation interval from to u Vdc follows the phase rectifier iA is periodic at 360 intervals The. 3 2 Vbn curve This curve is centered between the Vcb plot of iA consists of the plot of I1 and the inverse plot. and Vab curves For the remainder of its period Vdc of I4 A graph sheet is provided on page 21 for. follows the Vab curve A graph sheet is provided on practice. page 21 for practice,Vab 0 180 t,2 Vbn INFLUENCE OF TRANSFORMER TYPE. u ON iA WAVEFORM, iA is the current through the A phase supply of a 3. VLLp peak line to line voltage V,phase rectifier, the angle t at which the thyristor is triggered degrees DELTA WYE OR WYE DELTA DELTA DELTA OR WYE WYE.

TRANSFORMER TRANSFORMER,PLOTTING V1 constant current. V1 is the voltage across diode D1 V1 is more, complicated and is periodic at 360 intervals Refer to. the circuit entitled Three Phase Rectifier on page 7. Plot Vdc first While D1 is on V1 is zero When D1, goes off V1 briefly follows Vab then Vdc After that it. tracks Vac and then repeats the cycle A graph sheet. 3 PHASE TH VENIN EQUIVALENT, is provided on page 21 for practice The Th venin equivalent for a 3 phase full wave. bridge rectifier,to 120 u V1 is zero,120 u to 180 V1 tracks Vab 3VLLp RTH Id.

180 to 300 u V1 is Vdc,300 u to V1 tracks Vac 3 Ls VTH Vd. The 0 reference is 30 before Vab peak voltage RTH,VLLp peak line to line voltage V. PLOTTING I1 constant current the supply frequency rad sec. I1 is the current through diode D1 of a 3 phase Ls the supply inductance H. rectifier I1 is periodic at 360 intervals A graph sheet Id the constant load current A. is provided on page 21 for practice Vd the average output voltage V. to u I1 rises from 0 to I1 max, u to 120 I1 is constant 3 PHASE TH VENIN EQUIVALENT. 120 to 120 u I1 falls to zero with, The 0 reference angle is 30 before Vab peak voltage The Th venin equivalent for a 3 phase full wave. The curvature of the rising and falling diode current plots is bridge rectifier. related to the trigger angle 3VLLp RTH Id,90 VTH cos.

0 t 3 Ls VTH,0 VLLp peak line to line voltage V,90 the supply frequency rad sec. the angle t at which the thyristor is triggered degrees. PF POWER FACTOR no units The power factor is the ratio of true power the power consumed ignoring the reactive factor to apparent power the total power consumed Also the power factor is the cosine of the angle by which the current lags the voltage assuming an inductive load PF cos vi 0 DECIBELS dB

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