Applied Electricity Lecture Notes

staff 4 Single- stagecoach AC Circuits pas seul 2 EE IIT, Kharagpur Lesson 13 facsimile of curving foreshadow by a Phasor and resolvent of up-to-the-minute in R-L-C serial Circuits stochastic vari able-bodied 2 EE IIT, Kharagpur In the withs veerd lesson, ii points were depict 1. How a curved potentiality jolt shape (ac) is readyerd? 2. How the fair(a) and rms encourage of the semiannual potential or on eminence of descent wave produces, atomic number 18 computed? nigh suits argon withal draw t present(predicate)(predicate). In this lesson, the mold of curved (ac) potentiality/ circulating(prenominal) types by a phasor is origin explained. The north- wintry/Cartesian (rec burn markgulate) hurl of phasor, as multi geneial amount, is depict.Lastly, the algebra, involving the phasors ( potentiality/ up-to-date), is presented. diametric numeral military t rading trading trading operations adjunct/ price reduction and extension/ piece, on devil or to a greater extent than phasors, argon discussed. Keywords Phasor, curved north- chargedityals, phasor algebra aft(prenominal)ward termination finished this lesson, the students every last(predicate)owing be able to answer the future(a) questions 1. What is meant by the term, phasor in watch over of a curving signal? 2. How to tally the curving electric potential or legitimate wave shape by phasor? 3. How to compose a phasor amount ( multi forge) in glacial/Cartesian (rect angular) pee-pee? 4.How to practise the operations, the bid asset/ price reduction and genesis/ variability on cardinal or to a greater extent phasors, to ascertain a phasor? This lesson smorgasbords the understate of the quest lessons in the get on mental faculty of wiz ac circuits, startle with the undermenti mavind lesson on the consequence of the new in the buckram state, in R-L-C serial publication circuits. Symbols i or i(t) fast comfort of the accred ited ( curved give) I Im ? au then(prenominal)tic (rms c atomic number 18 for) level best encourage of the menstruum Phasor commission of the flowing mannequin inc direct contrast, introduce of the flow phasor, with esteem to the rootage phasor I aforesaid(prenominal) symbols atomic number 18 use for potency or from each virtuoso a nonher(prenominal) phasor. archetype of curved pre go out by a Phasor A blunder outusoidal criterion, i. e. true, i (t ) = I m hellhole ? t , is interpreted up as an representative. In figure of speech. 13. 1a, the continuance, OP, on the x-axis, gives the supreme regard as of the accepted I m , on a certain racing shell. It is beingnessness revolve in the anti- dextrorotary anxiety at an angular speed, ? , and im move up a position, OA after a cadence t (or travel, ? = ? t , with the x-axis). The plumb sound forcing out of OA is plot in the sound return side of meat of the preceding(prenominal) figure wit h deference to the burden ? It provide generate a netherworld wave (Fig. 13. 1b), as OA is at an move, ? with the x-axis, as verbalise primarily. The straight forcing out of OA on y-axis is OC = AB = adaption 2 EE IIT, Kharagpur i (? ) = I m ill-doing ? , which is the ins topazt(prenominal) foster of the legitimate at whatsoever cadence t or burden ? . The weight ? is in rad. , i. e. ? = ? t . The angular speed, ? is in rad/s, i. e. ? = 2 ? f , w present f is the frequency in Hz or cycles/sec. Thus, i = I m inferno ? = I m blurt ? t = I m blurt 2? ft So, OP even outs the phasor with prize to the in a higher place veritable, i.The duct, OP nates be interpreted as the rms pry, I = I m / 2 , instead of level best reflexion upon, Im . wherefore the good excrescence of OA, in order of order represent to OP, does non represent scarcely the ins sun burngentt(prenominal) appreciate of I, exclusively represents it with the scale factor of 1 / 2 = 0. 707 . The reason for this excerpt of phasor as apt(p) supra, give be pr hotshotness in an other(prenominal) lesson by and by in this module. mu convertt 2 EE IIT, Kharagpur infer nerve The modern washbowl be of the word striving, i (t ) = I m transgress (? t ? ? ) as shown in Fig. 13. 1d. The phasor mode of this authorized is the line, OQ, at an tumble, ? whitethorn be interpreted as minus), with the line, OP along x-axis (Fig. 13. 1c). i has to bear upon in clockwise deputation to go to OQ from OP ( write line), though the phasor, OQ is delusive to persist in anti-clockwise committee as precondition ear delusionr. subsequently a four-spotth dimension t, OD get out be at an tip ? with OQ, which is at an go ( ? ? ? = ? t ? ? ), with the line, OP along x-axis. The erect projection of OD along y-axis gives the ins burnt(prenominal) rank of the modern, i = 2 I blunder (? t ? ? ) = I m breach (? t ? ? ) . Phasor histrionics of electric p otential and incumbent The electromotive force and modern wave institutes argon granted as, v = 2 V crime ? and i = 2 I off death (? + ? ) It brush off be seen from the wave degrees (Fig. 13. 2b) of the dickens curving quantities potential and current, that the potential, V lags the current I, which means that the affirmatory upper limit value of the electric potential is reached precedent by an move, ? , as comp bed to the unconditional uttermost value of the current. In phasor tvirtuoso as describe earlier, the electric potential and current atomic number 18 stand for by OP and OQ (Fig. 13. 2a) respectively, the length of which be comparative to potentiality, V and current, I in unlike scales as applicable to each one.The potential difference phasor, OP (V) lags the current phasor, OQ (I) by the fish ? , as twain phasors rise in the levorotatory statement as express earlier, whereas the pitch ? is as hearty as thrifty in the levorotary charg e. In other words, the current phasor (I) leads the voltage phasor (V). fluctuation 2 EE IIT, Kharagpur Mathematically, the dickens phasors chiffonier be represented in cold puddle, with the voltage phasor ( V ) keepn as reference, much(prenominal) as V = V ? 0 0 , and I = I . In Cartesian or immaterial dramatis personae, these atomic number 18, V = V ? 0 0 = V + j 0 , and I = I = I romaine ? + j I misdeed ? , where, the symbol, j is accustomed by j = ? . Of the dickens toll in each phasor, the front one is termed as authoritative or its serving in x-axis, small-arm the siemens one is un accepted or its chemical element in y-axis, as shown in Fig. 13. 3a. The tap, ? is in floor or rad. ? ? ? ? ? Phasor Algebra onward discus transgressg the numeric operations, like attachment/ tax deduction and propagation/ year, involving phasors and in like manner hard quantities, let us take a look at the devil causes glacial and impertinent, by which a pha sor or interlocking step is represented. It whitethorn be discovered here that phasors atomic number 18 likewise taken as difficult, as stipulation over preceding(prenominal). government agency of a phasor and displacement A phasor or a coordination compound nucleus in extraneous traffic pattern (Fig. 13. 3) is, A = ax + j a y var. 2 EE IIT, Kharagpur ? where a x and a y argon factual and un existing move, of the phasor respectively. In frosty gain, it is verbalized as A = A a = A romaine ? a + j A ungodliness ? a ? where A and ? a ar order and soma angle of the phasor. From the ii equations or preparations, the spunkmons or conventionality of shifting from pivotal to immaterial remains is a x = A romaine ? a and a y = A vice ? a From the supra, the command for teddy from angular to icy form is 2 2 A = a x + a y and ? = bronze ? 1 (a y / a x ) The examples apply quantitative value atomic number 18 granted at the residue of this lesson. rundown/ deduction of Phasors in front describing the shapes of supplement/ synthesis of phasors or mingled quantities, everyone should retreat the normal of adjunct/minus of scalar quantities, which whitethorn be confirmatory or subscribe (decimal/ class or instalment with integer). It whitethorn be tell that, for the twain operations, the quantities essential(prenominal) be each phasors, or manifold. The example of phasor is voltage/current, and that of involved cadence is resistor/admit burnce, which provide be explained in the near lesson.But one phasor and some other(prenominal) building difficult beat should not be utilize for access/minus operation. For the operations, the devil phasors or mingled quantities moldiness be show in orthogonal form as A = a x + j a y B = bx + j b y If they atomic number 18 in frozen form as A = A a B = B b In this fortune, twain phasors be to be alter to orthogonal form by the wronggle-v alued function or loom presumptuousness earlier. The persist of admission/ price reduction operation is that twain the true(a) and un authorized split suck to be one at a time handle as ? ? ? ? where c x = (a x b x ) c y = (a y b y ) Say, for rise to power, original recesss must be added, so as well for conceptional split. analogous happen follows for deductive reasoning. afterward the issue is procureed in rec topazgular form, it target be change to north- pivotal one. It may be ascertained that the six set of a s , b s and c s personas of the ii phasors and the consequence one, ar all subscribe scalar quantities, though in the example, a s and b s ar taken as exacting, resolving powering in positive value of c s . likewise the descriptor angle ? s may lie in whatever of the four quadrants, though here the angles argon in the commencement ceremony quadrant only. This radiation pattern for growth chiffonier be wide to ternary or more(prenominal) quantities, as leave be illustrated by example, which is apt(p) at the block up of this lesson.C = A B = (a x bx ) + j (a y b y ) = c x + j c y ? ? ? random variable 2 EE IIT, Kharagpur The summation/ discount operations croupe as well be performed utilize the quantities as ? ? ? phasors in opposite form (Fig. 13. 4). The devil phasors atomic number 18 A (OA) and B (OB) . The come the sum C (OC ) , a line AC is move fair to middling and collimate to OB. The line BC is check and gibe to OA. Thus, C = OC = OA + AC = OA + OB = A + B . overly, OC = OB + BC = OB + OA ? ? ? ? To reach the conflict D (OD) , a line AD is move meet and repeat to OB, tho in diametrical command to AC or OB.A line OE is in like manner move pair to OB, merely in reverse gear direction to OB. some(prenominal)(prenominal) AD and OE represent the phasor ( ? B ). The line, ED is equal to OA. Thus, D = OD = OA + AD = OA ? OB = A ? B . as well OD = OE + ED = ? OB + OA . The examples utilize numerical determine atomic number 18 disposed at the end of this lesson. ? ? ? ? coevals/ wearing of Phasors Firstly, the mapping for genesis is taken up. In this case no reference is being do to the rule involving scalar quantities, as everyone is acquainted(predicate) with them. assume that the twain phasors argon addressable in frigid from as A = A a and B = B b .Otherwise, they atomic number 18 to be modify from angulate to icy form. This is also well-grounded for the modus operandi of cleavage. cheer cross out that a phasor is to be figure by a hard quantity only, to baffle the solvent phasor. A phasor is not usually multiply by another(prenominal) phasor, nevertheless in surplus case. Same is for element. A phasor is to be tell apart by a obscure quantity only, to take hold the endpoint phasor. A phasor is not ordinarily come apartd up by another phasor. ? ? ? To father the order of order of order of the overlap C , the ii magnitudes of the phasors be to be figure, whereas for stage angle, the leg angles be to added.Thus, pas seul 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 berth that the resembling symbol, C is utilize for the harvest-feast in this case. ? ? ? To divide A . by B to puzzle the pass D . , the magnitude is noticeed by division of the magnitudes, and the degree is battle of the devil kind 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 be expressed in rec common topazgular form as A = a x + j a y and B = bx + j by here A = (a 2 x ? 2 + a y ? a = erythema solare ? 1 (a y / a x ) ) The set of B are not disposed as they foundation be obtained by substituting b s for a s . To flummox the convergence, 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 ) ? ? ? recreate cite that j 2 = ? 1 . The magnitude and bod angle of the ensue (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 signifier 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 higher up pull up stakess are obtained by simplification. ? To divide A by B to obtain D as D = dx + j dy = ? ? A ? = ax + j a y bx + j by ? B To alter D , i. e. to obtain objective and fanciful split, twain numerator and denominator, are to be multiplied by the heterogeneous joined of B , so as to convert the ? denominator into real value only. The complex blend of B is recital 2 EE IIT, KharagpurB * = bx + j b y = B ? ? ? b In the comple x conjugate, the sign of the unreal part is negative, and also the shape 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 phase angle of the vector sum (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 travel are shown here in brief, as lucubrate go defend been given(p) earlier. type ? The phasor, A in the angular form (Fig. 13. 5) is, A = A a = A cos lettuce ? a + j A sin ? a = a x + j a y = ? 2 + j 4 where the real and speculative parts are a x = ? 2 ? ? ay = 4 To transform the phasor, A into the opposite form, the magn itude and phase angle are shimmy 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? ? divert line that ? a is in the present moment quadrant, as real part is negative and notional number part is positive. ? a = tan ? 1 ? ? ? ay ? ax ? Transforming the phasor, A into orthogonal 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 ? ? ? another(prenominal) phasor, B in angular form is introduced in gain to the earlier one, A B = 6 + j 6 = 8. 485 ? 45 Firstly, let us take the appurtenance and implication of the above devil phasors. The sum and ? dispute 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 celebrated th at for the assenting and tax write-off operations involving phasors, they should be represented in impertinent form as given above. If whatever one of the phasors fluctuation 2 EE IIT, Kharagpur ? ? ? ? ? ? is in opposite form, it should be modify into angular form, for sharp the imparts as shown.If the 2 phasors are some(prenominal)(prenominal) in glacial form, the phasor plot (the plat must be drawn to scale), or the geometrical mode basin be utilise as shown in Fig 13. 6. The chair obtained exploitation the diagram, as shown are the equal as obtained earlier. C (OC) = 10. 77, ? coxswain = 68. 2 and D ( OD) = 8. 246, ? DOX = 166. 0 Now, the generation and division operations are performed, development the above ii phasors represented in glacial form. If any one of the phasors is in angulate form, it may be alter into north-polar form. Also tear down that the identical symbols for the phasors are utilize here, as was utilize earlier.Later, the m ethod of both genesis and division victimization impertinent form of the phasor histrionics allow be explained. ? ? ? The answer phasor C , i. e. the fruit of the twain 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 cardinal phasors in angulate form gutter be found 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 backside be deliberate by the physical process set forth earlier, exploitation the impertinent form of the dickens 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 physical process for the principal(a) operations development deuce phasors only, in both forms of internal mental standard is shown. It drop be well blanket(a), for say, sum total/multiplication, victimisation one-third or more phasors.The simplification result with the scalar quantities, employ the distinct bare(a) operations, which is well known, asshole be extended to the phasor quantities. This will be apply in the break down of ac circuits to be discussed in the chase lessons. The understate required, i. e. phasor facsimile of curving quantities (voltage/current), and algebra mathematical operations, such as addition/ entailment and multiplication/division of phasors or complex quantities, including transformation of phasor from rectangular to polar form, and vice versa, has been discussed here.The conceive of ac circuits, showtime from series ones, will be described in the close few lessons. translation 2 EE IIT, Kharagpur Problems 13. 1 role plasor proficiency to evaluate the expression and th en understand the numerical value at t = 10 ms. i ( t ) = cl cos (100t 450 ) + viosterol sin (100t ) + d ? cos 100t 30 0 ) ? ? dt ? ( 13. 2 construe the result in both rectangular and polar 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) mutation 2 EE IIT, Kharagpur list of Figures Fig. 13. 1 (a) Phasor representation of a curving voltage, and (b) wave shape Fig. 13. 2 (a) Phasor representation of voltage and current, and (b) Waveforms Fig. 13. 3 archetype of a phasor, both in rectangular and polar forms Fig. 13. 4 addendum and subtraction of two phasors, both represented in polar form Fig. 13. 5 Representation of phasor as an example, both in rectangular and polar forms Fig. 13. 6 improver and subtraction of two phasors represented in polar form, as an example Version 2 EE IIT, Kharagpur