CHAPTER V. METHODS OF APPROXIMATION. 124. The investigation even of apparently simple engineer- ing problems frequently leads to expressions which are so complicated as to make the numerical calculations of a series of values very cumbersonme and almost impossible in practical work. Fortunately in many such cases of engineering prob- lems, and especially in the field of electrical engineering, the different quantities which enter into the problem are of very different magnitude. Many apparently compHcated expres- sions can frequently be greatly simplified, to such an extent as to permit a quick calculation of numerical values, by neglect- ing terms which are so small that their omission has no appre- ciable effect on the accuracy of the result; that is, leaves the result correct within the limits of accuracy required in engineer- ing, which usually, depending on the nature of the problem, is not greater than from 0.1 per cent to 1 per cent^;^ Thus, for instance, the voltage consumed by the resistance of an alternating-current transformer is at full load current only a small fraction of the supply voltage, and the exciting current of the transformer is only a small fraction of the full load current, and, therefore, the voltage consumed by the exciting current in the resistance of the transformer is only a small fraction of a small fraction of the supply voltage, hence, it is negligible in most cases, and the transformer equations are greatly simplified by omitting it. The power loss in a large generator or motor is a small fraction of the input or output, the drop of speed at load in an induction motor or direct- current shunt motor is a small fraction of the speed, etc., and the square of this fraction can in most cases be neglected, and the expression simplified thereby. Frequently, therefore, in engineering expressions con- taining small quantities, the products, squares and higher 187 188 ENGINEERING MATHEMATICS. powers of such quantities may be dropped and the expression thereby simplified; or, if the quantities are not quite as small as to permit the neglect of their squares, or where a high accuracy is required, the first and second powers may be retained and only the cubes and higher powers dropped. The most common method of procedure is, to resolve the expression into an infinite series of successive powers of the small quantity, and then retain of this series only the first term, or only the first two or three terms, etc., depending on the smallness of the quantity and the required accuracy^ 125. The forms most frequently used in the reduction of expressions containing small quantities are multipHcation and division, the binomial series, the exponential and the logarithmic series, the sine and the cosine series, etc. Denoting a small quantity by s, and where several occur, by Si, S2, S3 . . . the following expression may be written: / (,0±Sl)(l±S2)=l±S]±S2±SlS2, ; and, since S1S2 is small compared with the small quantities Si and S2, or, as usually expressed, S1S2 is a small quantity of r* fjj higher order (in this case of second order), it may be neglectod, '-^ f and the expression written: X (l±Si)(l±S2)=l±Si±S2 (1) This is one of the most useful simplifications : the multiplica- tion of terms containing small quantities is replaced by the simple addition of the small quantities. If the small quantities Si and S2 are not added (or subtracted) to 1, but to other finite, that is, not small quantities a and h, a and h can be taken out as factors, thus, X (a±si)(6±S2)=a6/l±^Vl±|Va/l±^-±|Y . (2) where — and -j- must be small quantities. As seen, in this case, s\ and S2 need not necessarily be abso- lutely small quantities, but may be quite large, provided that a and h are still larger in magnitude; that is, Si must be small compared with a, and S2 small compared with h. For instance. METHODS OF APPROXIMATION. 189 in astronomical calculations the mass of the earth (which absolutely can certainly not be considered a small quantity) is neglected as small quantity compared with the mass of the sun. Also in the effect of a lightning stroke on a primary distribution circuit, the normal line voltage of 2200 may be neglected as small compared with the voltage impressed by lightning, etc. 126. Example. In a direct-current shunt motor, the im- pressed voltage is eo = 125 volts; the armature resistance is ro = 0.02 ohm; the field resistance is ri = 50 ohms; the power consumed by friction is pf=^300 watts, and the power consumed by iron loss is pi= iOO watts. What is the power output of the motor at ^o = 50, 100 and 150 amperes input? The power produced at the armature conductors is the product of the voltage e generated in the armature conductors, and the current i through the armature, and the power output at the motor pulley is, p = ei-pf-pi. ....... (3) The current in the motor field is — , and the armature current n therefore is, ^ = ^0--, (4) where — is a small quantity, compared with 2*0. The voltage consumed by the armature resistance is roi, and the voltage generated in the motor armature thus is: e = eo — roi, (5) where roi is a small quantity compared with eo. Substituting herein for i the value (4) gives. eo- ro{io-'f) (6) Since the second term of (6) is small compared with eo, and in this second term, the second term — is small com- ri pared with io, it can be neglected as a small term of highc^r 190 ENGINEERING MATHEMATICS. order; that is, as small compared with a small term, and expression (6) simplified to e = eo-roio (7) Substituting (4) and (7) into (3) gives, p = (eo - ^o^o) [io - yj - Vf- Pi --(•-?)('-5)-'"-'" w Expression (8) contains a product, of two terms with small quantities, which can be multipHed by equation (1), and thereby gives. p = eo^o[l—---JJ-pf-pi = eoio-roto^-- — Pf-pi (9) Substituting the numerical values gives, p = 125^o-0.02^o2-562.5-300-400 = 125^0 - Omio^ - 1260 approximately ; thus, for 10=50, 100, and 150 amperes; p = 4940, 11,040, and 17,040 watts respectively. 127. Expressions containing a small quantity in the denom- inator are frequently simplified by bringing the small quantity in the numerator, by division as discussed in Chapter II para- graph 39, that is, by the series, :; = 1^X+X^TX^-^X^^X^+ . . .; . . . (10) l±X which series, if a; is a small quantity s, can be approximated by: 1 1 — « (11) METHODS OF APPROXIMATION. 191 or, where a greater accuracy is required, 1 1+s 1-s+s^ ^^ — - = 1+S+S2, 1 — s (12) By the same expressions (11) and (12) a small quantity contained in the numerator may be brought into the denominator where this is more convenient, thus : i+.s=j--; l — s = rr--r-', etc. l+s' (13) More generally then, an expression like -— , where s is small compared with a, may be simplified by approximation to the form, h b _b/ s\ (14) QTj^wli^ a greater exactness is required, by taking in the second term, ■^V T±s--ai}^-a^-^ '15) 128. Example. AVhat is the current input to an induction motor, at impressed voltage eo and slip s (given as fraction of synchronous speed) if ro — jxo is the impedance of the primary circuit of the motor, and ri — jxi the impedance of the secondary circuit of the motor at full frequency, and the exciting current of the motor is neglected; assuming s to be a small quantity; that is, the motor running at full speed? Let E be the e.m.f. generated by the mutual magnetic flux, that is, the magnetic flux which interlinks with primary and with secondary circuit, in the primary circuit. Since the fre- quency of the secondary circuit is the fraction 8 of the frequency 192 ENGINEERING MATHEMATICS. of the primary circuit, the generated e.m.f. of the secondary circuit is sE. Since x\ is the reactance of the secondary circuit at full frequency, at the fraction s of full frequency the reactance of the secondary circuit is sxi, and the impedance of the sec- ondary circuit at slip s, therefore, is ri — jsx\] hence the secondary current is, • ri-]sxi If the exciting current is neglected, the primary current equals the secondary current (assuming the secondary of the same number of turns as the primary, or reduced to the same number of turns) ; hence, the current input into the motor is sE (16) ri-]sxi The second term in the denominator is small compared with the first term, and the expression (16) thus can be approximated bv '■'"h^) The voltage E generated in the primary circuit equals the impressed voltage eo, minus the voltage consumed by the current / in the primary impedance; to—jxq thus is E = eo-I{ro-jXo) (IS) Substituting (17) into (18) gives E = eo-'Aro-jxo)(lH'^) (19) In expression (19), the second term on the right-hand side, which is the impedance drop in the primary circuit, is small compared with the first term eo, and in the factor (■<■) of this small term, the small term j^— can thus be neglected METHODS OF APPROXIMATION. 193 as a small term of higher order, and equation (19) abbreviated to E = eo---{ro-]Xo). ^1 (20) From (20) it follows that E = and from (13), l+-{ro-jxo) ^1 E = eo \ l-—(ro-jxo) (21) Substituting (21) into (17) gives / seo and from (1), (•OI'-^<-«>l seo\ ^ ro . Xo + .Ti = — \ 1 — S— + 1S ri ri ' ri j (22) If then, /oo^^'o+ytV is the exciting current, the total current input into the motor is, approximately, /o=(+|oo =7f|l+«^+J«^77-^)+to+no'. ■ ■ . (23) 129. One of the most important expressions used for the J^ ^^^^ r(Hkiction of small terms is the binomial series: 1 (l±.x)« = l±nx + n{n—l') ^ n(n—\){n—2) -x^± 13 n{n-lMn-2)(n-S) , + -^- ^±. (24) If X is a small term s, this gives the approximation, l±s)" = l±ns; (25) 194 ENGINEERING MATHEMATICS. or, using the second term also, it gives y^ (l±s)« = l±ns+^^^^^^s2 (26) In a more general form, this expression gives ^(a±s)- = a"(l±^)" = a-(l±^'j; etc. . . (27) By the binomial, higher powers of terms containing small quantities, and, assuming n as a fraction, roots containing small quantities, can be eliminated; for instance. \/a±.s = (a±s)«=a«f l±-j '' = v^fl±— j; 1 11 1 / s\-^_l^/ ns\ 1 1 (a±s) «=a «(1± m nn ^ . X- -/ s\« n/ — / ^A V(a±s)^ = (a±s)«=a«l 1 ±- I =va»»M±— I; etc. One of the most common uses of the binomial series is for the elimination of squares and square roots, and very fre- quently it can be conveniently applied in mere numerical calcu- lations; as, for instance, (201)2 = 2002(1 +-J-J = 40,000(1 +j-y =40,400; 29.92 = 302(l-3i3)^900(l-j^^)=000-6 = 894; vmS = 10\/l-0.02 = 10(1 -0.02)2 =10(1-0.01) = 9.99; 1 1 1 XOS (1+0.03)1/2 1.015 = 0.985; etc. METHODS OF APPROXIMATION. 195 130. Example i. If r is the resistance, x the reactance of an alternating-current circuit with impressed voltage e, the current is 1 = r2+x2 If the reactance x is small compared with the resistance r, as is the case in an incandescent lamp circuit, then, ._ _ _ _ef /xV] ~2 m i^' e r If the resistance is small compared with the reactance, as is the case in a reactive coil, then, e e £' r /r \- 1 2 Vf^+x^ TrV ^ -,V^)-"-'© <^-w, (28) Example 2. How does the short-circuit current of an alternator vary with the speed, at constant field excitation? When an alternator is short circuited, the total voltage generated in its armature is consumed by the resistance and the synchronous reactance of the armature. The voltage generated in the armature at constant field excitation is proportional to its speed. Therefore, if eo is the voltage generated in the armature at some given speed So, for instance, the rated speed of the machine, the voltage generated at any other speed S is 196 ENGINEERING MATHEMATICS. S or, if for convenience, the fraction -^ is denoted by a, then a = -^ and e = aeo, oo where a is the ratio of the actual speed, to that speed at which the generated voltage is eo- If r is the resistance of the alternator armature, xq the synchronous reactance at speed So, the synchronous reactance at speed Sh x = axo, and the current at short circuit then is i=^^=^ , "■"> (29) Usually r and xo are of such magnitude that r consumes at full load about 1 per cent or less of the generated voltage, while the reactance voltage of xq is of the magnitude of from 20 to 50 per cent. Thus r is small compared with xq, and if a is not very small, equation (29) can be approximated by aeo eo 1 = \ \axoJ \ 2 xo\ 2 \axoj axo^ /I 4-1 — 1 '-.Ki)T ■ ■ ™ ^ )/ Then if a:o = 20r, the following relations exist: a= 0.2 0.5 1.0 2.0 i = -X0.9688 0.995 0.99875 0.99969 That is, the short-circuit current of an alternator is practi- cally constant independent of the speed, and begins to decrease only at very low speeds. 131. Exponential functions, logarithms, and trigonometric functions are the ones frequently met in electrical engineering. The exponential function is defined by the series, ^ ^ x^ 7^ x^ xP ^> ^^ '— '— '— I— METHODS OF APPROXIMATION. 197 and, if x is a small quantity, s, the exponential function, may be approximated by the equation, €±^ = l±s; (32) or, by the more general equation, and, if a greater accuracy i.s required, the second term may be included, thus. ?2 £±«=i±s+^,, {m and then ^±as^lj^as+'^'-. ...... (35) — ; hence, /^o^^*'*^^'^* )<^log.(l±x)=±Jl-^-. Resolving -r-TT ^^^^ ^ series, by (10), and then integrating, gives log£ (l±x)=/± I (lTx+a:2q=r'^ + . . .)dx\ ( y? x^ x^ x^\ ,_ ■ [y^[jj{. This logarithmic series (36) leads to the apj^mximaiion, log£(l±5)=±s; (37) or, including the second term, it gives )( loge(l±s)=±s-A^ (38) and the more generaLexpression is, respectively, S^log. (a±s) = log a\\ ±^- j =log a+log \\ ±^) =log a±^,-i^(39) 198 ENGINEERING MATHEMATICS and, more accurately, s s^ loge (a±s) = loga±---2 a a^ (40) Since logio iV = logio eXloge N = OASiS loge A/", equations (39) and (40) may be written thus, logio(l±s)=±0.4343s; ■logio(a±s)=logio a ±0.4343- \' ' ' ' (^1) 132. The trigonometric functions are represented by the infinite series : ^ x^ x^ ofi cosx=l-|2+|4-J6+..-; X^ X^ x^ (42) which w^hen s is a small quantity, may be approxima^eiby 1 cos s = \ and sin ,s = s; . . . or, they may be represented in closer approximation b^ - cos s = l — — ; sm s==s(^— g-j; or, by the more general expressions, A ( 1 «'^') cos as = 1 and \cosa.s = l — ^y JL Sin as = as and \ sm as =-as[ — -^ j . (43) (44) u (45) 133. Other functions containing small terms may frequently be approximated by Taylor's series, or its special case, MacLaurin's series. MacLaurin's series is written thus : /(.t) =/(0) +xf'm +pf"(0) +p"'(o) +, (46) METHODS OF APPROXIMATION. 199 where /', /", /'", etc., are respectively the first, second, third, etc., differential quotient of/; hence, /(a)^/(0)+s/'(0); j /(as)=/(0)+as/'(0). J Taylor's series is written thus, (47^ (49) fib+x) =f{b) +xf'{b) +^/"(b) +T^f"'(b) +..., . (48) and leads to the approximations : f{h±s)=f{b)±sf{b); f(b±as)=f(b)±a.sr(b). J Many of the previously discussed approximations can be considered as special cases of (47) and (49). 134. As seen in the preceding, convenient equations for the approximation of expressions containing small terms are derived from various infinite series, which are summarized below : 1 l±x l=Fx+x2=Fx3+x*qF. . . X . n(n-l) ^ n(n-l)(n-2) , (l±.x)" = l±nx + — j^ — -x^±- r^ -x^ + . /V.2 nf3 /Tr«4 /y»2 QcS 'Y'4 loge {l±x)=±x--±-^-j±. . . ; /y^ y»4 /Y»t) cosx = l-i2+jj-j^+...; x^ x^ x"^ ,,nx = x-^+^-i^ + ...; .Ax) =/(0) +x/'(0) +£r(0) +i^/"'(0) + . . . ; fib ±x) =f{b) ±xf'(b) +|/"(6) ±|/"'(« + . . . (SO) 200 ENGINEERING MATHEMATICS. The first approximations, derived by neglecting all higher terms but the first power of the small quantity a; = s in these series, are: il±s>=---l±ns; e±^ = l±S', log5(l±s)= ±s: ■ .S2" _ ~2r cos s = 1 ; - .S2" ^ 2J' sin s = s; /(s)=/(0)+s/'(0); +fr(0)]; f{h±s)==fib)±sf'ih); +Jr(^^)]; (51) and, in addition hereto is to be remembered the multiplication rule, (l±si)(l±82) = l±.si±S2; [±8182]. . . (52) 135. The accuracy of the approximation can be. estimated by calculating the next term beyond that which is used. This term is given in brackets in the above equations (50) and (51). Thus, when calculating a series of numerical values by approximation, for the one value, for which, as seen by the nature of the problem, the approximation is least close, the next term is calculated, and if this is less than the permissible limits of accuracy, the approximation is satisfactory. For instance, in Example 2 of paragraph 130, the approxi- mate value of the short-circuit current was found in (30), as % = ■ eo Xo 2\axo) 1 1-^1- METHODS OF APPROXIMATION. 201 The next term in the parenthesis of equation (30), by the binomial, would have been H ^^ — s^; substituting n=— J: s = ( — ) , the next becomes +'5-1 — ) . The smaller the a, the Ma^ \axo/ ' 8 \axo/ ' *^^^ less exact is the approximation. ^ The smallest value of a, considered in paragraph 130, was a = 0.2. For Xo = 20r, this gives +o( — =0.00146, as the value of the first neglected term, and in the accuracy of the result this is of the magnitude of - X 0.00146, out of - X 0.9688, the value given in paragraph 130; that is, the approximation gives the result correctly within n Qfion =0.0015 or within one- sixth of one per cent, which is sufficiently close for all engineer- ing purposes, and with larger a the values are still closer approximations. 136. It is interesting to note the different expressions, which are approximated by (1+s) and by (1 — s). Some of them are given in the following : 1+. 1 1-s' -r^.v-v^- 202 ENGINEERING MATHEMATICS. ^ {n—m)s' etc. 2-s-'; l+log. (1+s); l-log.(l-s); 1 + 1-n 1 + nlog=(l+^); etc. 1 +sin s] l+nsm — : VI +2s; VT-2s; 1 1 Vl-2s' \/r+2i' 1 1 •v^l — ns' v^l+ns' 4-ms / 1 — ws \ l + (n-m)s' etc. £" -3- 2- -C; l+loge(l-s); -hged+s); l+nlog.(l-0; -nhg,(l+~); 1+log.^;^; -iog.^i_,: etc. -sins; -71 sin ^; METHODS OF APPROXIMATION. 203 , 1 . , 1 . IH — sin ns: 1 — sin ns; cos \/ — 2s; cos v^; etc. etc. 137. As an example may be considered the redaction to its simplest form, of the expression : 2si /9T" Va- /,= /o(l+^)±y^o, (16) and the error made hereby is of the magnitude of less than — of the line impedance voltage and line charging current. 141. Example. Assume 200 miles of 60-cycle line, on non- inductive load of ^0 = 100,000 volts; and io = 100 amperes. The line constants, as taken from tables are Z = 104 — 140/ ohms and F=— 0.0013/ ohms; hence, Zy=- (0.182 +0.136/); ^1 = 100000(1-0.091-0.068/) +100(104-104/) = 101400 - 20800/, in volts ; fi = 100(1 -0.091 -0.068/) -0.0013/ + 100000 = 91 — 136.8/, in amperes. _, . zy 0.174X0.0013 0.226 ^^_ The error is -^ = ^ = — ^ — = 0.038. 6 6 6 Neglecting the second term of Ei, ^7o = 17,400, the error is 0.038 X 17400 = 660 volts = 0.6 per cent. Neglecting the second term of h, yEo = lSO, the error is 0.038 X 130 = 5 amperes = 3 per cent. ' Although the charging current of the line is 130 per cent of output current, the error in the current is only 3 per cent. Using the equations (15), which are nearly as simple, brings the error down to -^= '^. =0.0021, or less than one-quarter per cent. Hence, only in extreme cases the equations (14) need to be used. Their error would be less than i=^ = 3.6x10"^, or one three-thousandth per cent. 208 ENGINEERING MATHEMATICS. The accuracy of the preceding approximation can be esti- mated by considering the physical meaning of Z and Y: Z is the hne impedance; hence Zl the impedance voltage, and zi u=--^, the impedance voltage of the line, as fraction of total voltage; Y is the shunted admittance; hence YE the charging YE current, and v=—j-, the charging current of the line, as fraction of total current. Multiplying gives uv = ZY; that is, the constant ZY is the product of impedance voltage and charging current, expressed as fractions of full voltage and full current, respectively. In any economically feasible power transmission, irrespective of its length, both of these fractions, and especially the first, must be relatively small, and their product therefore is a small quantity, and its higher powers negligible. In any economically feasible constant potential transmission line the preceding approximations are therefore permissible.