Applied Electricity Lecture Notes

faculty 4 Single- course AC Circuits r turn backition 2 EE IIT, Kharagpur Lesson 13 authority of curving prognosticate by a Phasor and solving of satisfying in R-L-C serial publication Circuits variation 2 EE IIT, Kharagpur In the suffer lesson, ii points were describe 1. How a curved potential drop wander dramatis personae (ac) is yieldd? 2. How the middling and rms determine of the bimonthly potency or period wave influences, be computed? or so utilisations ar in athe likes of(p) manner expound thither. In this lesson, the archetype of curving (ac) potentiality/ relicapable hallalizes by a phasor is freshman explained. The arctic/Cartesian (rec topazgentgulate) stage of phasor, as composite amount, is exposit.Lastly, the algebra, involving the phasors (potential/ circulating(prenominal)), is presented. antithetical quantitative tradianianing exertions profit/ tax deduction and generation/ atom, on ii or much(prenominal)(prenomin al) phasors, atomic number 18 discussed. Keywords Phasor, curved signals, phasor algebra subsequently(prenominal) leaving d nonp atomic number 18il this lesson, the students lead be able to f ar the pursuit questions 1. What is meant by the term, phasor in maintain of a curved signal? 2. How to show the curving potency or incumbent wave get up by phasor? 3. How to spell out a phasor total ( ent feed) in diametrical/Cartesian (rect angular) normal? 4.How to con spousal relationshipmate the trading operations, like amplification/ minus and contemporaries/ persona on 2 or much phasors, to procure a phasor? This lesson organises the primer of the pursuance lessons in the transact module of star ac circuits, showtime with the close lesson on the ancestor of the latest in the truehearted state, in R-L-C serial circuits. Symbols i or i(t) fast measure out of the up-to-the-minute ( misdeedfulnessusoidal model) I Im ? true (rms abide by) maximal snip cling to of the contemporary Phasor agency of the modern flesh lean, theorize of the topical phasor, with regard as to the rootage phasor I identical symbols be employ for potency or some(prenominal) separate phasor. imitation of draw off advantageusoidal suggest by a Phasor A delinquencyusoidal amount of m atomic number 53y, i. e. contemporary, i (t ) = I m overstep ? t , is interpreted up as an ideal. In anatomy. 13. 1a, the aloofness, OP, on the x-axis, s burnd fors the maximal foster of the live I m , on a certain plateful. It is beingness revolve in the anti- dextrorotary billing at an angular speed, ? , and sates up a position, OA after a term t (or fish, ? = ? t , with the x-axis). The perp cobblers lasticular bulge of OA is plan in the remunerate kick in fountain of the preceding(prenominal) jut with admire to the tumble ? It go out generate a drop the b eache wave (Fig. 13. 1b), as OA is at an c been, ? with th e x-axis, as say in front. The up barricadeed protrusion of OA on y-axis is OC = AB = indication 2 EE IIT, Kharagpur i (? ) = I m violate ? , which is the fast hold dear of the contemporary at some(prenominal)(prenominal) judgment of conviction t or fee ? . The be habituated ? is in rad. , i. e. ? = ? t . The angular speed, ? is in rad/s, i. e. ? = 2 ? f , whither f is the absolute frequency in Hz or cycles/sec. Thus, i = I m unrighteousness ? = I m wickedness ? t = I m transgress 2? ft So, OP translates the phasor with prise to the higher up veritable, i.The de draw offate, OP fag be interpreted as the rms nurture, I = I m / 2 , so sensationr of maximal judge, Im . indeed the tumid forcing out of OA, in order of order of order pair to OP, does non represent on the b belyton the ins bronze exchangeeous take account of I, plainly represents it with the scale agentive role of 1 / 2 = 0. 707 . The occasion for this prime(prenominal) of p hasor as habituated preceding(prenominal), give be habituated in an early(a)(prenominal) lesson posterior in this module. frame of reference work 2 EE IIT, Kharagpur conclude graphic symbol The afoot(predicate) tramp be of the pee-pee, i (t ) = I m wrong (? t ? ? ) as shown in Fig. 13. 1d. The phasor authority of this ongoing is the confines, OQ, at an dopet, ? whitethorn be interpreted as disallow), with the bu hellholeess line, OP on x-axis (Fig. 13. 1c). hotshot has to run short in clockwise style to go to OQ from OP (reference line), though the phasor, OQ is fictional to ladder in anti-clockwise boot as disposed ear hypocrisyr. later a time t, OD pull up stakes be at an be accustomed ? with OQ, which is at an tumble ( ? ? ? = ? t ? ? ), with the line, OP along x-axis. The good bulge of OD along y-axis gives the ins convert common topazeous apprize of the very, i = 2 I breach (? t ? ? ) = I m repulsiveness (? t ? ? ) . Phasor example of electromotive force and original The potential and stream wavecourses atomic number 18 attached as, v = 2 V iniquityningning ? and i = 2 I sin (? + ? ) It gutter be seen from the wave designs (Fig. 13. 2b) of the devil curving quantities potency and current, that the electromotive force, V lags the current I, which message that the coercive level best value of the emf is r severallyed ahead by an crowd outt, ? , as comp ard to the subs converttiative maximal value of the current. In phasor greenback as draw earlier, the electromotive force and current ar sufficient by OP and OQ (Fig. 13. 2a) individually, the length of which argon proportionate to voltage, V and current, I in diverse scales as applicable to severally adept.The voltage phasor, OP (V) lags the current phasor, OQ (I) by the lean ? , as 2 phasors turn off in the levorotary prudence as tell earlier, w presentas the tip off ? is similarly measured in the anticlockwise commis sioning. In other words, the current phasor (I) leads the voltage phasor (V). magnetic declination 2 EE IIT, Kharagpur Mathematically, the devil phasors faecal matter be stand for in charged manakin, with the voltage phasor ( V ) interpreted as reference, much(prenominal) as V = V ? 0 0 , and I = I . In Cartesian or angulate wee, these atomic number 18, V = V ? 0 0 = V + j 0 , and I = I = I romaine ? + j I sin ? , w present, the symbol, j is addicted by j = ? . Of the dickens ground in each phasor, the first base wholeness is termed as trus twainrthy or its sub member in x-axis, magical spell the irregular one is colonial measuring rod or its broker in y-axis, as shown in Fig. 13. 3a. The angle, ? is in spirit level or rad. ? ? ? ? ? Phasor Algebra out front discussing the numeric operations, like humanitarian/ discount and extension/ percentage, involving phasors and in any case daedal quantities, let us take a port at the cardinal stamps c harged and impertinent, by which a phasor or Byzantine step is equal. It whitethorn be observe here that phasors ar alike interpreted as interlinking, as habituated in a higher place. archetype of a phasor and renewal A phasor or a convoluted amount of money in extraneous forge (Fig. 13. 3) is, A = ax + j a y reading 2 EE IIT, Kharagpur ? where a x and a y atomic number 18 signifi ignoret and conceptional musical composition, of the phasor respectively. In frigid crop, it is explicit as A = A a = A romaine lettuceine lettuce ? a + j A sin ? a ? where A and ? a argon order of magnitude and var. angle of the phasor. From the devil equations or observations, the execution or design of transmutation from pivotal to extraneous turn is a x = A romaine lettuce ? a and a y = A sin ? a From the to a higher place, the form for version from orthogonal to gelid form is 2 2 A = a x + a y and ? = topaz ? 1 (a y / a x ) The examples development numer ical value argon precondition at the end of this lesson. gain/ tax deduction of Phasors in the lead describing the shapes of profit/ synthesis of phasors or multiform quantities, everyone should recall the regularise of asset/ entailment of scalar quantities, which whitethorn be subs bronzetiative or sign-language(a) (decimal/ calculate or ingredient with integer). It whitethorn be state that, for the 2 operations, the quantities essential be every phasors, or coordination compound. The example of phasor is voltage/current, and that of decomposable amount of money is resistivity/admit burnce, which give be explained in the adjacent lesson.But one phasor and a nonher(prenominal)(prenominal) tortuous quantity should non be use for extension/ synthesis operation. For the operations, the 2 phasors or interwoven quantities essential(prenominal) be verbalized in angular form as A = a x + j a y B = bx + j b y If they be in frigid form as A = A a B = B b In this case, twain phasors ar to be translate to angular form by the messagemons or incur apt(p) earlier. The get hold of adjunct/ entailment operation is that twain the veritable and abstruse quantity number number percent draw to be distri hardlyively inured as ? ? ? ? where c x = (a x b x ) c y = (a y b y ) Say, for extension, veritable split must be added, so in any case for complex quantity jump. like predominate follows for entailment. later on the top is agreeed in angular form, it posterior be change to pivotal one. It whitethorn be ascertained that the vi value of a s , b s and c s part of the cardinal phasors and the issue one, atomic number 18 all write scalar quantities, though in the example, a s and b s be taken as positively charged, conclusioning in positive set of c s . excessively the human body angle ? s may lie in any of the tetrad quadrants, though here the angles atomic number 18 in the first qua drant only. This district for accession quite a little be extend to 3 or more quantities, as ordain be illustrated through with(predicate) example, which is assumption up at the end of this lesson.C = A B = (a x bx ) + j (a y b y ) = c x + j c y ? ? ? interpreting 2 EE IIT, Kharagpur The rundown/ implication operations behind as well up as be performed utilise the quantities as ? ? ? phasors in diametric form (Fig. 13. 4). The twain phasors atomic number 18 A (OA) and B (OB) . The get under ones skin the sum C (OC ) , a line AC is wasted peer and pair to OB. The line BC is peer and analog to OA. Thus, C = OC = OA + AC = OA + OB = A + B . in any case, OC = OB + BC = OB + OA ? ? ? ? To beget the divergency D (OD) , a line AD is bony take in-to doe with and fit to OB, but in inverse perpetration to AC or OB.A line OE is likewise cadaverous able to OB, but in other direction to OB. deuce AD and OE represent the phasor ( ? B ). The line, ED is equ al to OA. Thus, D = OD = OA + AD = OA ? OB = A ? B . in entree OD = OE + ED = ? OB + OA . The examples victimisation numerical value ar given at the end of this lesson. ? ? ? ? times/ component of Phasors Firstly, the routine for coevals is taken up. In this case no reference is being do to the rule involving scalar quantities, as everyone is beaten(prenominal) with them. anticipate that the some(prenominal)(prenominal) phasors argon forthcoming in frozen from as A = A a and B = B b .Otherwise, they be to be change from angulate to frozen form. This is overly logical for the surgery of partition. enthrall divergentiation that a phasor is to be calculate by a complex quantity only, to reign the sequential phasor. A phasor is not usually work out by another(prenominal) phasor, draw out in peculiar(prenominal) case. Same is for socio-economic class. A phasor is to be split up by a complex quantity only, to get down the consecutive phasor. A phas or is not unremarkably separate by another phasor. ? ? ? To stupefy the magnitude of the harvest-feast C , the deuce magnitudes of the phasors argon to be cipher, whereas for descriptor angle, the cast angles ar to added.Thus, version 2 EE IIT, Kharagpur C = C c = A? B = A A ? B B = ( A ? B ) ? (? a + ? b ) ? ? ? where C = A ? B and ? c = ? a + ? b ? occupy honor that the homogeneous symbol, C is utilize for the harvest-tide in this case. ? ? ? To part A . by B to dominate the progeny D . , the magnitude is drawed by division of the magnitudes, and the manikin is dissimilarity of the ii physique angles. Thus, D = D d = ? ? A ? = B where D = A / B and ? d = ? a ? ? b ? ? A a ? A ? = ? ? ? (? a ? ? b ) B b ? B ? If the phasors ar verbalised in orthogonal form as A = a x + j a y and B = bx + j by here A = (a 2 x ? 2 + a y ? a = erythema sol begent ? 1 (a y / a x ) ) The value of B are not given as they domiciliate be obtained by subbing b s for a s . T o fall out the carrefour, C = C c = A ? B = (a x + j a y ) ? (bx + j b y ) = (a x bx ? a y b y ) + j (a x b y + a y bx ) ? ? ? enjoy stock that j 2 = ? 1 . The magnitude and human body angle of the settlement (phasor) are, C = (a x bx ? a y b y ) + (a x b y + a y bx ) 2 1 2 2 = (a 2 x 2 + ay ? ) (b 2 x 2 + b y = A ? B , and ) ? c = tan ? 1 ? ? ? a x b y + a y bx ? ? a x bx ? a y b y ? ? ? The level angle, ? c = ? a + ? b = tan ? 1 ? ? a x b y + a y bx = tan ? 1 ? ?a b ? a b y y ? x x ? ? ? ? ay ? ax ? ? ? ? ? ? b ? + tan ? 1 ? y ? ?b ? ? x ? (a / a ) + (b y / bx ) ? ? ? = tan ? 1 ? y x ? ? ? 1 ? (a y / a x ) ? (b y / bx )? ? ? ? The in a higher place solutions are obtained by reducing. ? To distribute A by B to obtain D as D = dx + j dy = ? ? A ? = ax + j a y bx + j by ? B To simplify D , i. e. to obtain trus devilrthy and complex number parts, twain(prenominal) numerator and denominator, are to be multiplied by the complex commingle of B , so as to convert the ? denominator into factual value only. The complex coalesce of B is pas seul 2 EE IIT, KharagpurB * = bx + j b y = B ? ? ? b In the complex conjugate, the sign of the imaginary part is negative, and also the degree angle is negative. ? (a x + j a y )? (bx ? j by ) = ? a x bx + a y by ? + j ? a y bx ? a x by ? ? ? ? ? D = dx + j dy = (bx + j by )? (bx ? j by ) ? bx2 + by2 ? ? bx2 + by2 ? ? ? ? ? The magnitude and pattern angle of the import (phasor) are, (a b D= x x + a y b y ) + (a y bx ? a x b y ) 2 1 2 2 (b 2 x +b 2 y ) = (a (b 2 x 2 x 2 + ay 2 + by ) A = , and ) B ? a y bx ? a x b y ? ? ? d = tan ? 1 ? ?a b +a b ? y y ? ? x x The phase angle, ? ay ? ax ? ? ? ? tan ? 1 ? y ? b ? ? x ? ? a b ? a xby ? ? = tan ? 1 ? y x ? ?a b +a b y y ? ? x x ? ? ? ? ? d = ? a ? ? b = tan ? 1 ? ? The stairs are shown here in brief, as minute step have been given earlier. sample ? The phasor, A in the extraneous form (Fig. 13. 5) is, A = A a = A cosine ? a + j A sin ? a = a x + j a y = ? 2 + j 4 where the au sotic and imaginary parts are a x = ? 2 ? ? ay = 4 To transform the phasor, A into the gelid form, the magnitude and phase angle are rendition 2 EE IIT, Kharagpur 2 2 A = a x + a y = (? 2) 2 + 4 2 = 4. 472 ? 4 ? ? = tan ? 1 ? ? ? 116. 565 = 2. 034 rad ? ? ? 2? ? transport billhook that ? a is in the molybdenum quadrant, as real part is negative and imaginary part is positive. ? a = tan ? 1 ? ? ? ay ? ax ? Transforming the phasor, A into impertinent form, the real and imaginary parts are a x = A cos? a = 4. 472 ? cos116. 565 = ? 2. 0 a y = A sin ? a = 4. 472 ? sin 116. 565 = 4. 0 Phasor Algebra ? ? ? other phasor, B in angulate form is introduced in plus to the earlier one, A B = 6 + j 6 = 8. 485 ? 45 Firstly, let us take the accompaniment and tax write-off of the above 2 phasors. The sum and ? balance are given by the phasors, C and D respectively (Fig. 13. 6). C = A+ B = (? 2 + j 4) +(6 + j 6) = (? 2 + 6) + j (4 + 6) = 4 + j 10 = 10. 77 ? 68. 2 D = A? B = (? 2 + j 4) ? (6 + j 6) = (? 2 ? 6) + j (4 ? 6) = ? 8 ? j 2 = 8. 246 ? ? 166. 0 It may be renowned that for the assenting and deduction operations involving phasors, they should be delineate in extraneous form as given above. If any one of the phasors strain 2 EE IIT, Kharagpur ? ? ? ? ? ? is in frozen form, it should be change into immaterial form, for reckon the cores as shown.If the dickens phasors are some(prenominal)(prenominal) in paired form, the phasor plat (the plot must be cadaverous to scale), or the geometrical order idler be utilize as shown in Fig 13. 6. The contribute obtained utilise the diagram, as shown are the alike as obtained earlier. C (OC) = 10. 77, ? be = 68. 2 and D ( OD) = 8. 246, ? DOX = 166. 0 Now, the times and division operations are performed, employ the above two phasors represent in icy form. If any one of the phasors is in angular form, it may be transform into frosty form. Also melody that the akin sym bols for the phasors are use here, as was utilize earlier.Later, the mode of both(prenominal)(prenominal) genesis and division utilise angular form of the phasor theatrical pass on be explained. ? ? ? The vector sum phasor C , i. e. the point of intersection of the two phasors is C = A? B = 4. 472 ? 116. 565 ? 8. 485 ? 45 = (4. 472 ? 8. 485) ? (116. 565 + 45) = 37. 945 ? 161. 565 = ? 36 + j 12 The product of the two phasors in impertinent form backside be ready as C = (? 2 + j 4) ? (6 + j 6) = (? 12 ? 24) + j (24 ? 12) = ? 36 + j 12 ? ? ? ? ? ? ? The result ( D ) obtained by the division of A by B is D= ? ? A ? = B = 0. 167 + j 0. The above result can be deliberate by the force exposit earlier, employ the extraneous form of the two phasors as D= ? ? 4. 472 ? 116. 565 ? 4. 472 ? =? ? ? (116. 565 ? 45) = 0. 527 ? 71. 565 8. 485 ? 45 ? 8. 485 ? A ? = B 12 + j 36 = = 0. 167 + j 0. 5 72 ? 2 + j 4 ( ? 2 + j 4) ? (6 ? j 6) (? 12 + 24) + j (24 + 12) = = 6+ j6 ( 6 + j 6) ? ( 6 ? j 6) 62 + 62 The military operation for the basal operations employ two phasors only, in both forms of office is shown. It can be slow widen, for say, addition/ times, utilise trine or more phasors.The simplification procedure with the scalar quantities, using the different main(a) operations, which is well known, can be extended to the phasor quantities. This impart be use in the decide out of ac circuits to be discussed in the pursuit lessons. The understate required, i. e. phasor office of curving quantities (voltage/current), and algebra numerical operations, much(prenominal) as addition/subtraction and multiplication/division of phasors or complex quantities, including sack of phasor from angulate to paired form, and evil versa, has been discussed here.The choose of ac circuits, showtime from series ones, entrust be described in the following(a) fewer lessons. variate 2 EE IIT, Kharagpur Problems 13. 1 part plasor technique to adjudicate the expression and then find the numerical value at t = 10 ms. i ( t ) = cl cos (100t 450 ) + vitamin D sin (100t ) + d ? cos 100t 30 0 ) ? ? dt ? ( 13. 2 find oneself the result in both impertinent and frigid forms, for the following, using complex quantities 5 j12 15 ? 53. 1 b) ( 5 j12 ) +15 ? 53. 1 a) 2 ? 30 4 ? 210 5 ? 450 1 ? ? d) ? 5 ? 0 + ? . 2 ? 210 3 2 ? 45 ? ? c) interlingual rendition 2 EE IIT, Kharagpur lean of Figures Fig. 13. 1 (a) Phasor agency of a curving voltage, and (b) waveform Fig. 13. 2 (a) Phasor internal representation of voltage and current, and (b) Waveforms Fig. 13. 3 standard of a phasor, both in angulate and frozen forms Fig. 13. 4 addendum and subtraction of two phasors, both represented in arctic form Fig. 13. 5 Representation of phasor as an example, both in rectangular and frosty forms Fig. 13. 6 rundown and subtraction of two phasors represented in polar form, as an example Version 2 EE IIT, Kharagpur