CHAPTER VI. EMPIRICAL CURVES. A. General. 142. The results of observation or tests usually are plotted in a curve. Such curves, for instance, are given by the core loss of an electric generator, as function of the voltage; or, the current in a circuit, as function of the time, etc. When plotting from numerical observations, the curves are empirical, and the first and most important problem which has to be solved to make such curves useful is to find equations for the same, that is, find a function, y=f{x), which represents the curve. As long as the equation of the curve is not known its utihty is very limited. While numerical values can be taken from the plotted curve, no general conclusions can be derived from it, no general investigations based on it regarding the conditions of efficiency, output, etc. An illustration hereof is afforded by the comparison of the electric and the magnetic circuit. In the electric circuit, the relation between e.m.f. and e current is given by Ohm's law, i = -, and calculations are uni- versally and easily made. In the magnetic circuit, however, the term corresponding to the resistance, the reluctance, is not a constant, and the relation between m.m.f. and magnetic flux cannot be expressed by a general law, but only by an empirical curve, the magnetic characteristic, and as the result, calcula- tions of magnetic circuits cannot be made as conveniently and as general in nature as calculations of electric circuits. If by observation or test a number of corresponding values of the independent variable x and the dependent variable y are determined, the problem is to find an equation, y=f{x), which represents these corresponding values: xi, X2, xz . , , Xn, and 2/1, 2/2, 2/3 .. . yn, approximately, that is, within the errors of observation. 209 210 ENGINEERING MATHEMATICS. The mathematical expression which represents an empirical curve may be a rational equation or an empirical equation. It is a rational equation if it can be derived theoretically as a conclusion from some general law of nature, or as an approxima- tion thereof, but is an empirical equation if no theoretical reason can be seen for the particular form of the equation. For instance, when representing the dying out of an electrical current in an inductive circuit by an exponential function of time, we have a rational equation: the induced voltage, and therefore, by Ohm's law, the current, varies proportionally to the rate of change of the current, that is, its differential quotient, and as the exponential function has the characteristic of being proportional to its differential quotient, the exponential function thus rationally represents the dying out of the current in an inductive circuit. On the other hand, the relation between the loss by magnetic hysteresis and the magnetic density: W=-q(^^'^, is an empirical equation since no reason can be seen for this law of the 1.6th power, except that it agrees with the observa- tions. A rational equation, as a deduction from a general law of nature, applies universally, within the range of the observa- tions as well as beyond it, while an empirical equation can with certainty be relied upon only within the range of observation from which it is derived, and extrapolation beyond this range becomes increasingly uncertain. A rational equation there- fore is far "preferable to an empirical one. As regards the accuracy of representing the observations, no material difference exists between a rational and an empirical equation. An empirical equation frequently represents the observations with great accuracy, while inversely a rational equation usually does not rigidly represent the observations, for the reason that in nature the conditions on which the rational law is based are rarely perfectly fulfilled. For instance, the representation of a decaying current by an exponential fimction is based on the assumption that the resistance and the inductance of the cu'cuit are constant, and capacity absent, and none of these conditions can ever be perfectly satisfied, and thus a deviation occurs from the theoretical condition, by what is called " secondary effects." 143. To derive an equation, which represents an empirical curve, careful consideration should first be given to the physical EMPIRICAL CURVES. 211 nature of the phenomenon which is to be expressed, since thereby the number of expressions which may be tried on the empirical curve is often greatly reduced. Much assistance is usually given by considering the zero points of the curve and the points at infinity. For instance, if the observations repre- sent the core loss of a transformer or electric generator, the curve must go through the origin, that is, y = 0 for x = 0, and the mathematical expression of the curve y =f(x) can contain no constant term. Furthermore, in this case, with increasing x, i/must continuously increase, so that for x = 00, y = Qc. Again, if the observations represent the dying out of a current as function of the time, it is obvious that for x = oo, y=0. In representing the power consumed by a motor when running without load, as function of the voltage, for x = 0, y cannot be =0, but must equal the mechanical friction, and an expression like y = Axf^ cannot represent the observations, but the equation must contain a constant term. Thus, first, from the nature of the phenomenon, which is represented by the empirical curve, it is determined (a) Whether the curve is periodic or non-periodic. (6) Whether the equation contains constant terms, that is, for x = 0, 2/7^0, and inversely, or whether the curve passes through the origin: that is, y = 0 for a: = 0, or whether it is h3^erbolic ; that is, y= 00 for x = 0, or x = 00 for y = 0. (c) What values the expression reaches for 00. That is, whether for x = oo, 2/ = ^, or 2/ = 0, and inversely. (d) Whether the curve continuously increases or decreases, or reaches maxima and minima. (e) Whether the law of the curve may change within the range of the observations, by some phenomenon appearing in some observations which does not occur in the other. Thus, for instance, in observations in which the magnetic density enters, as core loss, excitation curve, etc., frequently the curve law changes with the beginning of magnetic saturation, and in this case only the data below magnetic saturation would be used for deriving the theoretical equations, and the effect of magnetic saturation treated as secondary phenomenon. Or, for instance, when studying the excitation current of an induction motor, that is, the current consumed when running light, at low voltage the current may increase again with decreasing voltage, 212 ENGIN^EERING MATHEMATICS. instead of decreasing, as result of the friction load, when the voltage is so low that the mechanical friction constitutes an appreciable part of the motor output. Thus, empirical curves can be represented by a single equation only when the physical conditions remain constant within the range of the observations. From the shape of the curve then frequently, with some experience, a guess can be made on the probable form of the equation which may express it. In this connection, therefore, it is of the greatest assistance to be familiar with the shapes of the more common forms of curves, by plotting and studying various forms of equations y=f{x). By changing the scale in which observations are plotted the apparent shape of the curve may be modified, and it is therefore desirable in plotting to use such a scale that the average slope of the curve is about 45 deg. A much greater or much lesser slope should be avoided, since it does not show the character of the curve as well. B. Non-Periodic Curves. 144. The most common non-periodic curves are the potential series, the parabolic and hyperbolic curves, and the exponential and logarithmic curves. The Potential Serie^. Theoretically, any set of observations can be represented exactly by a potential series of any one of the following forms : y = ao + aiX-\-a2X^-{-a3X^-\- . . . ; .... (1) y = aiX+a2X^-haix^ + . . . ; (2) cii 0,2 CLS /^s 2/=a„+-+-+-3 + ...: (3) ai 02 as , ,., 2'=^+F^ + ? + W if a sufficiently large number of terms are chosen. For instance, if n corresponding numerical values of x and y are given, Xi, 2/1; X2, 2/2; ... x„, yn, they can be represented EMPIRICAL CURVES. 213 by the series (1), when choosing as many terms as required to give n constants a: y = ao+aix+a2X^-{-. . .+an_in"~i. (5) By substituting the corresponding values Xi, yi] X2, 2/2, •• . into equation (5), there are obtained n equations, which de- termine the n constants ao, ai, a2, . . . a„_i. Usually, however, such representation is irrational, anc; therefore meaningless and useless. Table I. e 100 ""^ Pi=y -0.5 + 2x + 2.5x2 -1.5x3 + 1.5x< -2x6 + X6 0.4 0.6 O.S 0.63 1.36 2.18 -0.5 -0.5 -0.5 +0.8 + 1.2 + 1.6 + 0.4 +0.9 + 1.6 -0.10 -0.32 -0.77 + 0.04 +0.19 + 0.61 - 0.02 - 0.16 - 0.65 0 + 0.05 + 0.26 1.0 1.2 1.4 3.00 3.93 6.22 -0.5 -0.5 -0.5 + 2.0 + 2.4 + 2.8 + 2.5 +3.6 +4.9 -1.50 -2.59 -4.12 + 1.50 + 3.11 + 5.76 - 2.00 - 4.98 -10.76 + 1.00 + 2.89 + 6.13 1.6 8.59 -0.5 +3.2 + 6.4 -6.14 +9.83 -20.97 + 16.78 Let, for instance, the first column of Table I represent the voltage, YrjQ = ^j in hundreds of volts, and the second column the core loss, Pi=y, in kilowatts, of an 125- volt 100-h.p. direct- current motor. Since seven sets of observations are given, they can be represented by a potential series with seven con- stants, thus, y = ao+aix-\-a2X^+. . .-\-aex^, .... (6) and by substituting the observations in (6), and calculating the constants a from the seven equations derived in this manner, there is obtained as empirical expression of the core loss of the motor the equation. t/= -0.5 +2X+2.5X2- 1.5x3 + 15^4_2a;5+x6. (7) This expression (7), however, while exactly representing the seven observations, has no physical meaning, as easily seen by plotting the individual terms. In Fig. 60, y appears 214 ENGINEERING MATHEMATICS. as the resultant of a number of large positive and negative terms. Furthermore, if one of the observations is omitted, and the potential series calculated from the remaining six values, a series reaching up to x^ would be the result, thus, 2/ = ao4-aix+a2x2 + . . .+a5X^, .... (8) 16 1 J 12 / r b / V 8 i ^ r i -^ rt y^ ^ ^ ^ €1 'i^X. D _ _-3 = ^ ^ "~^~ !». ■^ :'--^ -fl .5 -i N s ^^ ^^. ??- ^ \ ^ ^ -8 \ 0^ ^' ^ -12 \ V \ -ic \ \ x = \ •SO 0 2 0 4 0 6 0 8 1 0 1 2 1 4 ) 6 Fig. 60. Terms of Empirical Expression of Excitation Power. but the constants a in (8) would have entirely different numer- ical values from those in (7), thus showing that the equation (7) has no rational meaning. 145' The potential series (1) to (4) thus can be used to represent an empirical curve only under the following condi- tions': 1. If the successive coefficients ao, ai, a2, ... decrease in value so rapidly that within the range of observation the higher terms become rapidly smaller and appear as mere secondary terms. EMPIRICAL CURVES. 215 2. If the successive coefficients a follow a definite law, indicating a convergent series which represents some other function, as an exponential, trigonometric, etc. 3. If all the coefficients, a, are very small, with the exception of a few of them, and only the latter ones thus need to be con- sidered. Table II. X 1/ v^ 1/1 0.4 0.6 0.8 0.89 1.35 1.96 0.88 1.34 1.94 0.01 0.01 0.02 1.0 1.2 1.4 2.72 3.62 4.63 2.70 2.59 4.59 0.02 0.03 0.04 1.6 5.76 5.65 0.11 For instance, let the numbers in column 1 of Table II represent the speed x of a fan motor, as fraction of the rated speed, and those in column 2 represent the torque y, that is, the turning moment of the motor. These values can be represented bj' the equation, 2/ = 0.r)+0.02x + 2.5.r2-0.3r^ +0.015x^-0.02x5 +0.01.r6. (9) In this case, only the constant term and the terms with x2 and x^ have appreciable values, and the remaining terms probably are merely the result of errors of observations, that is, the approximate equation is of the form. y = ao+a2X^ + asX^. .... Using the values of the coefficients from (9), gives y = 0.5+2.5x^-0.3x^ \ . . (10) (11) The numerical values calculated from (11) are given in column 3 of Table II as y', and the difTerence between them and the observations of column 2 are given in column 4, as yi. 216 ENGINEERING MATHEMATICS. The values of column 4 can now be represented by the same form of equation, namely, 2/1 = &o + &2x2 +63^3, (12) in which the constants ho, 62, hs are calculated by the method of least squares, as described in paragraph 120 of Chapter IV, and give 2/1= 0.031 -0.093:r2+ 0.076x3 (13) Equation (13) added to (11) gives the final approximate equation of the torque, as, 2/0 = 0.531 +2.407.T2- 0.224x3 (14) The equation (14) probably is the approximation of* a rational equation, since the first term, 0.531, represents the bearing friction; the second term, 2.407x^ (which is the largest), represents the work done by the fan in moving the air, a resistance proportional to the square of the speed, and the third term approximates the decrease of the air resistance due to the churning motion of the air created by the fan. In general, the potential series is of limited usefulness; it rarely has a rational meaning and is mainly used, where the curve approximately follows a simple law, as a straight line, to represent by small terms the deviation from this simple law, that is, the secondary effects, etc. Its use, thus, is often temporary, giving an empirical approximation pending the derivation of a more rational law. The Parabolic and the Hyperbolic Curves. 146. One of the most useful classes of curves in engineering are those represented by the equation, y = ax^; (15) or, the more general equation, y-h = a{x-cy (16) Equation (16) differs from (15) only by the coastant terms h and c; that is, it gives a different location to the coordinate EMPIRICAL CURVES. 217 center, but the curve shape is the same, so that in discussing the general shapes, only equation (15) need be considered. If n is positive, the curves y = ax'^ are 'parabolic curves, passing through the origin and increasing with increasing x, li n>\,y increases with increasing rapidity, if ns s s \ \ < 5 a t t C ) < § i o ? — ' -= ' ■ ^ ^ ^^ ^ S^ \ V \ \ \ 5 f c T r I I H : I i 5 ? § 3 5 c c J c5 PL, « CD o O -^ ^ ^ ^^ •^ ^ •^ ^ V, V N \ \ s \ \, s \ \ \ \ i c ^ r 5 ii ft C ^ < > 0 ■; L. i 5 \ i U r4 « II 5si 00 O 'o EMPIRICAL CURVES. 219 ^ IaI ^ "^ ^ ^ "^ .^ x^ ,y y / y j\.a— / / / / J\-A / / / ( J 1 0 2 0 4 p 6 0 8 1 0 1 % I 4 1 6 i 8 2 9 Fig. 64. Parabolic Curve. y = \/x. tf'S- ^~~" -— ■ =^ liO- -^ -^" ^ -^ ' 0-8- ^^ -^ -^ ^ n-fi-1 / / /. r n-^- _ 0 2 0 1 0 6 0 8 1 0 1 2 1 4 1 6 1 8 2 0 Fig. 65. Parabolic Curve. y= -yix. 220 ENGINEERING MATHEMATICS. . ^- ■ n fi > / z' L r 0 3 0 4 0 6 0 8 1 0 1 2 1 i 1 5 I 3 2 ) Fig. 66. Parabolic Curve. y = 's/x. ' I \ \ \ « -1 a \ V \ 1 O \ \ \ f\ Q \ V \ v^ l\-A ^-- ^ ■ i 0 4 0 8 1 2 1 6 2 « 2 * 2 8 3 2 3 6 4 0 Fig, 67. Hyperbolic Curve (Equilateral Hyperbola), y EMPIRICAL CURVES. 221 jfcan n O Q_ ' I \ o_/\_ \ \ \ \ \ 1^ \ /\_Q_ \ \ s A_i4 N V, "^ "~~~ — 1 — 0.4 0.8 1.3 1.6 .2.0 2.4 2.8 3.2 3.6 4.0 4.4 Fig. 68. Hyperbolic Curve. y X 9^ . Ol * 1-R- hSi- \ \ \ \ \ \, \ v^ "^^ 1 ____ __ 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 Fig. 69. Hyperbolic Curve. y = —^. 7 222 ENGINEERING MATHEMATICS. i:k{p n >/ll i 4 4iO \ \ \ \ \ s N ^ ^^ ^■ . — - ■ — ■ 0 i 0 8 1 2 1 6 2 0 2 i 2 8 3 2 3 6 A 0 4 4 Fig. 70. Hyperbolic Curve. y=-—p. r 0 Q 2v4 s. \ s V ^ ^ "^ ^- . "~ "^~~ 0 i 0 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 i 0 4 4 Fig. 71. Hyperbolic Curve. y = EMPIRICAL CURVES. 223 In Fig. 72, sixteen different parabolic and hyperbolic curves are drawn together on the same sheet, for the following values : n = l, 2, 4, 8, ^; i i *, 0; -1, -2, -4, -8; -i, -{, -i 147. Parabolic and hyperboHc curves may easily be recog- nized by the fact that ifx is changed by a constant factor, y also changes by a constant factor. Thus, in the curve y = x~, doubling the x increases the y fourfold; in the curve y = x^'^^, doubling the x increases the y threefold, etc.; that is, if in a curve, f(qx) "iTT-T- = constant, for constant g, . . . (17) the curve is a parabolic or hyperbolic curve, y = ax'^, and fiqx) a{qxY f{x)^~l^^'^^ ^^^^ If q is nearly 1, that is, the x is changed onjy by a small value, substituting ^' = 1+8, where s is a small quantity, from equation (18), f{x-\-sx) ^^ . hence, f{x-\-sx)-f{x)=ns; (19) that is, changing x by a small percentage sx, y changes by a pro- portional small percentage nsy. Thus, parabolic and hyperbolic curves can be recognized by a small percentage change of x, giving a proportional small percentage change of y, and the proportionaUty factor is the exponent n; or, they can be recognized by doubling x and seeing whether y hereby changes by a constant factor. As illustration are shown in Fig. 73 the parabolic curves, which, for a doubling of .r, increase y: 2, 3, 4, 5, 6, and 8 fold. Unfortunately, this convenient way of recognizing parabolic and hyperbolic curves applies only if the curve passes through the origin, that is, has no constant term. If constant terms exist, as in equation (16), not x and y, but (x—c) and (y—b) follow the law of proportionate increases, and the recognition 224 ENGINEERING MATHEMATICS. becomes more difficult; that is, various values of c and of h are to be tried to find one which gives the proportionality. Z^i \ \ \, 8 fi' •y 2.0 \ \ \ \ I 1 II ^ 1 1 / \ y=x ■\ \ \ 1 1 ^ / / / - L.8 \ \ \ \ \ 1 H / f / ) \ \ \ \ \ \ -/ 1 / \ \ \ \ \ \ \ \ \ / / / \ \ \ \ \ \ \ / / / '^^ .f/ \ \ \ V \ \ \ \ \ \ i ' 1 / / \ \, \ N, \ \ \ \ \ 1 / 1 o N ■^ N \, \ \ V \ jl 1 / % ^ \ \, \ .,\ % /// / ^ ^ ^^^ .^'^ Hyp erbo ic Cu rves ^^^ s^ \ \\\ 1 ::i!ir_ "^^ V Fill ^1 i.U Pari ibolic Curv es ^ ^ '^ '*^. n:^ ■:^ y i^IZj ^ ^ .'^' y X \\\ s '■•- y *i*?-c / / y /" X ^ / / / \\ \ \ \ ^< '*-.' / / / y /- / / \ // y \ \^ "^ ^ '/ f / / / / 1 1 \ \ \ \ / / / / / / 1 , \ \ ^^ 0.4 / / / / / 1 / \ V \ »i 7 / / / . / / \ \ V "\^ 1 / / y // 'A y / y << d y Z ^ ^ ^ y 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Fig. 73. Parabolic Curves. y=xn. 1.6 This again applies only if the curve contain no constant term. If constant terms exist, the logarithmic line is curved. Therefore, by trying different constants c and 5, the curvature of the logarithmic line changes, and by interpolation such constants can be found, which make the logarithmic Hne straight, and in this way, the constants c and h may be evaluated. If only one constant exist, that is, only h or only c, the process is relatively simple, but it becomes rather complicated with both 226 ENGINEERING MATHEMATICS. constants. This fact makes it all the more desirable to get from the physical nature of the problem some idea on the existence and the value of the constant terms. Exponential and Logarithmic Curves. 149. A function, which is very frequently met in electrical engineering, and in engineering and physics in general, is the exponential function, 2/ = a£"^; (21) which may be written in the more general form, . I/— ?> = a£«^^-^^ (22) Usually, it appears with negative exponent, that is, in the form. y = ae (23) Fig. 74 shows the curve given by the exponential function (23) for a = l; n^l] that is. 2/=£- (24) as seen, with increasing positive x, y decreases to 0 at x= + 00, and with increasing negative x, y increases to 00 at a: = — 00. The curve, y=z^'', has the same shape, except that the positive and the negative side (right and left) are interchanged. Inverted these equations (21) to (24) may also be written thus, 1 y ^ a' n{x—c) = \og '-h nx= — log y x=-hgy] that is, as logarithmic curves. (25) EMPIRICAL CURVES. 227 150. The characteristic of the exponential function (21) is, that an increase of x by a constant term increases (or, in (23) and (24), decreases) y by a constant factor. Thus, if an empirical curve, y^f{x), has such characteristic that f(x + q) fix) = constant, for constant q, (26) \1 r^ -1t4 o) \ \ \ \ \ ■onj- \ \ ItO \ \ \ \ D-A V \ N \ \ s. \^ s. 0 n \ \ \ \^ n A N N s \ in \ ^ ^ -0.-JJ ^" -^ "^ -^ .,.. -2.0 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 Fi G. 74 . Exponential Function . y^e-x. 1.6 2.0 the curve is an exponential function, y^ae^"", and the following equation may be written : f(x-\-q) a£"(^+g> fix) ~ as"^ = enq (27) Hereby the exponential function can easily be recognized, and distinguished from the parabolic curve; in the former a constant terw,^ in the latter a constant factor of x causes a change of 2/ by a constant factor. As result hereof, the exponential curve with negative exponent vanishes, that is, becomes negligibly small, with far greater rapidity than the hyperboHc curve, and the exponential 228 ENGINEERING MATHEMATICS. function with positive exponent reaches practically infinite values far more rapidly than the paraboHc curve. This is illustrated in Fig. 75, in which are shown superimposed the exponential curve, 2/=^"^ and the hyperbolic curve, 2.4 2/ = 7 — , -, r-\oy which coincides with the exponential curve at :r = 0 and at x = l. Taking the logarithm of equation (21) gives log 2/=^ log a + nx log £, that is, log 2/ is a linear function of x, and plotting log y against x gives a straight line. This is characteristic of V 1 -1:0 \ \ -U;o ^ s5 ' xt 2.4 ^. 0;6 (a J+LS 5)=^S ^ V V H 0:4 ^ "^ 2 .4 2 ^r^ rl.55 £■ ^ ^ 0;2 ■" — ■ ■^ 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 P'iG. 75. Hyperbolic and Exponential Curves C'omparison. 4.0 the exponential functions, and a convenient method of recog- nizing them. However, both of these characteristics apply only if x and y contain no constant terms. With a single exponential function, only the constant term of y needs consideration, as the constant term of x may be eliminated. Equation (22) may be written thus: 2/— 6 = a£"^^""''^ =^At^^ (28) where ^ = as"* is a constant. An exponential function which contains a constant term h would not give a straight line when plotting log 2/ against x. EMPIRICAL CURVES. 229 but would give a curve. In this case then log (y—h) would be plotted against x for various values of b, and by interpolation that value of b found which makes the logarithmic curve a straight line. 151. While the exponential function, when appearing singly, is easily recognized, this becomes more difficult with com- i_i (1) 2/=£-a?+o.5^"2«' (5) 2/=r«^-o.5£~2« (6) y=e-X_Q^^s-2X (7) y = e-X-£-'^^ (8) y= e'a'-i.ss-saj 1 0 \^ (1) \ \ -i-.o Vs) l\ \ \ -0.-8 \* ,v v\ \ nN \ -A r; \^ [^1 \ ^ \ n\ ^ V -0.-4- (Ql ^ ^ N / ^ -~-^ "^ ^ ^ -0.2 ^~--i:i^^ / <-) fe=^ ^ L^ ' ' ^k= / / -==^ ==- / 1 0 8 1 2 1 6 2 " 2 4 2 8 — 0- A8) -0.2 / / -0.4 / L. Fig. 76. Exponential Functions. binations of two exponential functions of different coefficients in the exponent, thus, 2/ = ai£-''»^±a2£"'^, (29) since for the various values of ai, 0.2, c\, C2, quite a number of various forms of the function appear. As such a combination of two exponential functions fre- quently appears in engineering, some of the characteristic forms are plotted in Figs. 76 to 78. 230 ENGINEERING MATHEMATICS. r (1) 2/=£-«^+0.6£"10* f3) 2/=£~^-o.l£'ioa; (4) 2/=£-a;-o.5£-io^ (5) 2/=£-a;-,£-ioa; (6) 2/=£-«^-U£-10» V ) \ i\ <-^ \ 'V- \ W ?^ K // \ \ f ) \| \ s \ \ ^ ■^ ^^ ■■ ■— 0 i 0 8 1 2 1 6 2 0 2 4 2 8 L4 1^ 1.0 0.8 0.6 0.4 0.2 -0.2 -0.4 Fig. 77. Exponential Functions. Fig. 76 gives the following combinations of £~^ and £~2'; (1) l/=£-^+0.5£-2^; (2) 2/=£-^+0.2£-2^; (3) y=e--] (4) y=£-^-0,2e-2'; (5) i/=£-^-0.5£-2x; (6) 2/=^"^-O.S£-2^; (7) 2/=£---£-2- (8) 2/=£""'^-1.5£"2^ EMPIRICAL CURVES. 231 ■■ ~1 / ■fJOi / / -2;0 / / ■4 / / / / / / / / / . / / / .^^ r / y / / / / _, / .^7 / • -0.8* / / y / (he / /■ ni / / / / 0 2 0 4 » 6 0 8 1 0 1 2 1 4 Fig. 78. Hyperbolic Functions. Fig. 77 gives the following combination of t~^ and £~^®^: (1) 2/=£--+0.5£-i<^^; (2) 2/=^"^; (3) ^=£-^-O.U-io^; (4) i/=£-^-0.5£-i<^^; (5) |/=.-x_,-10x. (6) '^=£-^-1.5£-i^^ Fig. 78 gives the hyperbolic functions as combinations cf f^' and £~^ thus, 2/ = cosh a: = i( e"^ =^ + £~ ^) ; 2/ = sinh a: = J(£''"^— £~^). 232 ENGINEERING MATHEMATICS. C. Evaluation of Empirical Curves. 152. In attempting to solve the problem of finding a mathe- matical equation, y=f{x), for a series of observations or tests, the corresponding values of x and y are first tabulated and plotted as a curve. From the nature of the physical problem, which is repre- sented by the numerical values, there are derived as many data as possible concerning the nature of the curve and of the function which represents it, especially at the zero values and the values at infinity. Frequently hereby the existence or absence of constant terms in the equation is indicated. The log X and log y are tabulated and curves plotted between X, y, log X, log y, and seen, whether some of these curves is a straight line and thereby indicates the exponential function, or the parabolic or hyperbolic function. If cross-section paper is available, having both coordinates divided in logarithmic scale, and also cross-section paper having one coordinate divided in logarithmic, the other in common scale, the tabulation of log x and log y can be saved and x and y directly plotted on these two forms of logarithmic cross- section paper. If neither of the four curves: x, y; x, log y; hgx, y; log x, log 2/ is a straight line, and from the physical condition the absence of a constant term is assured, the function is neither an exponential nor a parabolic or hyperbolic. If a constant term is probable or possible, curves are plotted between x, y—b, log X, log (y—h) for various values of 6, and if hereby one of the curves straightens out, then, by interpolation, that value of h is found, which makes one of the curves a straight line, and thereby gives the curve law. In the same manner, if a constant term is suspected in the x, the value (x—c) is used and curves plotted for various values of c. Frequently the existence and the character of a constant term is indicated by the shape of the curve ; for instance, if one of the curves plotted between x, y, log x, log y approaches straightness for high, or for low values of the abscissas, but curves considerably at the other end, a constant term may be suspected, which becomes less appreciable at one end of the range. For instance, the efTect of the constant c in (x—c) decreases with increase of x. EMPIRICAL CURVES. 233 Sometimes one of the curves may be a straight Hne at one end, but curve at the other end. This may indicate the presence of a term, which vanishes for a part of the observations. In this case only the observations of the range which gives a straight line are used for deriving the curve law, the curve calculated therefrom, and then the difference between the calculated curve and the observations further investigated. Such a deviation of the curve from a straight line may also indicate a change of the curve law, by the appearance of secondary phenomena, as magnetic saturation, and in this case, an equation may exist only for that part of the curve where the secondary phenomena are not yet appreciable. If neither the exponential functions nor the parabolic and hyperbolic curves satisfactorily represent the observations, X further trials may be made by calculating and tabulating - y X XI and -, and plotting curves between x, y,-, -. Also expressions X y X ' as x^+hy^, and {x—aY+h(y—c)^, etc., may be studied. Theoretical reasoning based on the nature of the phenomenon represented by the numerical data frequently gives an indi- cation of the form of the equation, which is to be expected, and inversely, after a mathematical equation has been derived a trial may be made to relate the equation to known laws and therebj^ reduce it to a rational equation. In general, the resolution of empirical data into a mathe- matical expression largely depends on trial, directed by judg- ment based on the shape of the curve and on a knowledge of the curve shapes of various functions, and only general rules can thus be given. A number of examples may illustrate the general methods of reduction of empirical data into mathematical functions. 153. Example 1. In a 118-volt tungsten filament incan- descent lamp, corresponding values of the terminal voltage e and the current i are observed, that is, the so-called '' volt- ampere characteristic '' is taken, and therefrom an equation for the volt-ampere characteristic is to be found. The corresponding values of e and i are tabulated in the first two columns of Table III and plotted as curve I in Fig. 79. In the third and fourth column of Table III are given log e 234 ENGINEERING MATHEMATICS. and logi. In Fig. 79 then are plotted e, logi, as curve II log e, i, as curve III; log e, log i, as curve IV. As seen from Fig. 79, curve IV is a straight line, that is, log i = A+7iloge] or, i = ae^., which is a parabolic curve. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2:0 2.2 2A=log 6 2 p 4 ) e p 8 ) 160 1^0 \' 0 1 K) 1^0 2( K) 2^=e logi ^.-^ -J 9.6 y- J ^ ^ ^ as jsr ^ >^ ,S.^ l^ 'y l^ 9:4. xoS X^ / r/ V 9.3 / / IV / / 9.2 9.1 / / / / / t / i F.o / fy^ 1 0.45 8.9 f .S y " / 040 "5a / ^ i^ ^ / 0.35 8.7 0.30 8.6 025 8 <% / ^^ ^ y'^ / 1 / < ^ y i / / y y / .^ 0.20 8.4 0.15 8.3 0.10 8.2 0 05 8.1 / / / 4 / / ^^ A .0** / ^ ^ 1 - -^ Fig. 79. Investigation of Volt-ampere Characteristic of Tungsten Lamp Filament. The constants a and n may now be calculated from the numerical data of Table III by the method of least squares, as discussed in Chapter IV, paragraph 120. While this method gives the most accurate results, it is so laborious as to be seldom EMPIRICAL CURVES. 235 used in engineering; generally, values of the constants a and n, sufficiently accurate for most practical purposes, are derived by the following method : Table III. VOLT-AMPERE CHARACTERISTIC OF 118-VOLT TUNGSTEN LAMP. e i log e log i 8211 +0-6 log 6 A 2 00245 0-301 8-392 8-389 -0.003 4 0-037 0-602 8-568 5-572 -0004 8 00568 0-903 8-754 g-753 + 0001 16 0-0855 1-204 8-932 8- 933 -0.001 25 0-1125 1398 9-051 9-050 + 0.001 32 0-1295 1-505 9-112 9-114 -0002 50 64 01715 0-200 1.699 1.806 §.234 9-230 9-295 + 0004 + 0006 9.301 100 0-2605 2-000 9-416 9-411 + 0005 125 0-2965 2-097 9-472 9-469 + 0-003 150 03295 2-176 9-518 §-518 0 180 0-3635 2-255 9-561 9-564 -0-003 200 03865 2-301 9- 587 9-592 -0005 218 0407 2-338 9-610 9-614 -0-004 17= 7. 612 2-040 avg. ±0.003 = 0 . 7 per cent r= 14-973 6-465 f= 7-361 4-425 4-425 n= = 7381 0-6011~0 -6 i'14= 22-585 8- 505 06X22-585 = = 13. 551 4 = 8.505-13 ■551 = 4.954 1-954-14 = 8-211 logt = g.211+0.6l og e and i = 0-01625e''« The fourteen sets of observations are divided into two groups of seven each, and the sums of log e and log i formed. They are indicated as 27 in Table III. Then subtracting the two groups 27 from each other, oliminates A, and dividing the two differences J, gives the exponent, n = 0.6011; this is so near to 0.6 that it is reasonable to assume that n = 0.6, and this value then is used. 236 ENGINEERING MATHEMATICS. Now the sum of all the values of log e is formed, given as 214 in Table II, and multiplied with n = 0.6, and the product subtracted from the sum of all the logl The difference J then equals 14^, and, divided by 14, gives A = log a = 8.211; hence, a = 0.01625, and the volt-ampere characteristic of this tungsten lamp thus follows the equation, logt = 8.211+0.61oge; ^ = 0.01625e0•6. From e and i can be derived the power input p = ei, and the resistance r = — : I - p = 0.016256i-6; ,0-4 ^ 0.01625' and, eliminating e from these two expressions, gives p = 0.01625V = 1.135r4xl0-i^ that is, the power input varies with the fourth power of the resistance. Assuming the resistance r as proportional to the absolute temperature T, and considering that the power input into the lamp is radiated from it, that is, is the power of radiation P^, the equation between p and r also is the equation between P^ and T, thus, P, = A:T4; that is, the radiation is proportional to the fourth power of the absolute temperature. This is the law of black body radiation, and above equation of the volt-ampere characteristic of the tungsten lamp thus appears as a conclusion from the radiation law, that is, as a rational equation. 154. Example 2. In a magnetite arc, at constant arc length, the voltage consumed by the arc, e, is observed for different EMPIRICAL CURVES. 237 values of current i. To find the equation of the volt-ampere characteristic of the magnetite arc : Table IV. VOLT-AMPERE CHARACTERISTIC OF MAGNETITE ARC. e log i log e (e-40) log (e- 40) (e-30) log (e-30) ec 160 9-699 2-204 120 2- 079 130 2.114 158 120 0000 2-079 80 1-903 90 1954 120-4 94 0301 1-973 54 1-732 64 1806 94 75 0-602 1-875 35 1-544 45 1-653 75-2 62 0-903 1.792 22 1-342 32 1505 62 56 1079 1.748 16 1-204 26 1.415 56-2 0.5 1 2 4 8 12 -2 + 0-4 0 + 0-2 0 + 0-2 ^■3 = 0-000 5874 ^■3 = 2.584 4. 573 ^ = 2-584 -1301 -1301 a = = - 0 - 5034 ~ - 0 - 5 2. 584 - i"6 = 2.584 10-447 2. 584 X -0-5 = -1292 ^= 11-739 11-739-^6= 1956 log(e-30) = 1.956-0.5logi 90- 4 e— 30 =90-4*""* and e= '■ — =- 30 + ^i The first four columns of Table IV give i, e, log i, log e. Fig. 80 gives the curves: i, e, as I; i, hge, as II; hgi, e, as III; log i, log e, as IV. Neither of these curves is a straight line. Curve IV is relatively the straightest, especially for high values of e. This points toward the existence of a constant term. The existence of a constant term in the arc voltage, the so-called " counter e.m.f. of the arc " is physically probable. In Table IV thus are given the values (e— 40) and log (e— 40), and plotted as curve V. This shows the opposite curvature of IV. Thus the. constant term is less than 40. Estimating by interpolation, and calculating in Table IV (e—SO) and log (e— 30), the latter, plotted against log i gives the straight line VI. The curve law thus is log (e-30) =A+a log ^. 238 ENGINEERING MATHEMATICS. Proceeding in Table IV in the same manner with logi and log (6—30) as was done in Table III with loge and logt, gives n=-0.5; A = loga = 1.956; and a = 90.4; Fig. 80. Investigation of Volt-ampere Characteristic of Magnetite Arc. hence log (e-30) = 1.956-0.5 log i"; e-30-90.4i;-o-5; 90.4 e = 30+- \% EMPIRICAL CURVES, 239 which is the equation of the magnetite arc volt-ampere charac- teristic. 155. Example 3. The change of current resulting from a change of the conditions of an electric circuit containing resist- ance, inductance, and capacity is recorded by oscillograph and gives the curve reproduced as I in Fig. 81. From this curve log ■\ "^ _^ ^ H V i \ \ s 0 A 1 \ \^ N \ \ \ \ \, \ \ N \^ ] r \ V \ II \ 1 n —) V*- \ N k y \, "< K iiA^ \ \, \. y:o \ \ \, \ \ i\ <.• N ^ \ ^ 0 4 0 8 1 2 t 1 6 2 0 2 4 2 8 Fig. 81. Investigation of Curve of Current Change in Electric Circuit. are taken the numerical values tabulated iis t and i in the first two columns of Table V. In the third and fourth columns are given log^ and \ogi, and curves then plotted in the usual manner. Of these curves only the one between t and logi is shown, as II in Fig. 81, since it gives a straight line for the higher values of t. For the higher values of t, therefore, \ogi = A~ht', or, ^ = a£~"*; that is, it is an exponential function. 240 ENGINEERING MATHEMATICS. Table V. TRANSIENT CURRENT CHARACTERISTICS. t i log< logi i 11 = 4.94£-1.07< i' = J log i' 12 = 2.85£-3.84« V = ii — 12 1 0 2.10 — 0.322 0 4.94 2.84 0461 2.85 2. 09 -0-01 0.1 2.48 9.000 0394 0.1 4.44 1.96 0.292 1.94 2.50 + 002 0.2 2-66 5.301 0.425 02 3.98 1.32 0.121 1.32 2.66 0 0.4 0.8 2.58 2.00 9.602 9. 903 0.412 0.301 0 8 3.21 2.09 0.63 009 9.799 0.61 0.13 2-60 1.96 + 0.02 -0-04 8-954 1.2 1.36 0079 0134 1.2 1.36 0 — 0.03 1.33 -0.03 1.6 090 0.204 9. 954 1.6 0.89 -0.01 — 0.01 0-88 -0.02 2.0 0.58 0301 9. 763 2.0 0.58 0 — — 0-58 0 2.5 0.34 0398 9.531 2.5 0.34 0 — — 0-34 0 3.0 0.20 0.477 9.301 3.0 0.20 0 ~ " 0.20 0 13 =4.8 ^3 = 9.851 ^2 = 0.1 0.753 48 9-851 — ,= 1.6 ^-^^ = 9.950 ^■2 = 0. 6 9. 920 3 ' 3 ^2 =5.5 ^"2 = 9. 832 J ^ = 0.5 -0.83; 5.5 „ „, 9. 832 ^ „„ = 2.75 = 9. 416 0.5Xlog £ = 0.217 2 . 2 ^ = 1.15 ^ = -0.534 n=— 3.J )4 1.15Xlog £ = 0.499; n=-1.07 ^"4 = 0. 7 0653 ^"5 =10.3 i'5 = 8.683 0.7Xlog£ = 0.304 10. 3Xlog £ = 4.473 O.S04X-3.84 = -1.16' 1 3 5 4. 473 X -1.107 = -4. 784 f = 1.82( ^ = 3. 467 1.820-4 = 045 8.467-*-5 = 0.693 log M = 0. 693-1 07< log £ log t2 = 0 . 455 - 3 . 84« log e ti=^4.94£-1.07f 12= 2. 85 E-3.84< ic = = 4.94e-1.07«-2.85s-3.84< To calculate the constants a and n, the range of values is used, in which the curve II is straight; that is, from t = \.2 to ^ = 3. As these are five observations, they are grouped in two pairs, the first 3, and the last 2, and then for t and log i, one- third of the sum of the first 3, and one-half of the sum of the last 2 are taken. Subtracting, this gives, ii = 1.15; J log?: =-0.534. Since, however, the equation, i = a£~^\ when logarithmated, gives log i = log a—nt log s, thus A log i=—n log eAt, EMPIRICAL. CURVES. 241 it is necessary to multiply M by log £ = 0.4343 before dividing it into log i to derive the value of n. This gives n== 1.07. Taking now the sum of all the five values of ty multiplying by log £, and subtracting from the sum of all the five values of log 1, 5A= 3.467; hence, A = log a =0.693, and log ii= 0.693 -1.07nog£; The current ii is calculated and given in the sixth column of Table V, and the difference ^' = J = ^l — ^ in the seventh column. As seen, from t = 1.2 upward, ii agrees with the observations. Below t = 1.2, however, a difference i' remains, and becomes considerable for low values of t. This difference apparently is due to a second term, which vanishes for higher values of t. Thus, the same method is now applied to the term i'; column 8 gives log^', and in curve III of Fig. 81 is plotted logi' against t. This curve is seen to be a straight line, that is, i' is an exponential function of t. Resolving t' in the same manner, by using the first four points of the curve, from ^ = 0 to ^ = 0.4, gives log 12 = 0.455 -3.84nog e; l2 = 2.85£-3-84^ and, therefore, ^=^l_^2 = 4.94£-l•07^-2.85£-3•84« is the equation representing the current change. The numerical values are calculated from this equation and given under ic in Table V, the amount of their difference from the observed values are given in the last column of this table. A still greater approximation may be secured by adding the calculated values of 12 to the observed values of i in the last five observations, and from the result derive a second approximation of ii, and by means of this a second approxi- mation of ^2. 242 ENGINEERING MATHEMATICS. 156. As further example may be considered the resolution of the core loss curve of an electric motor, which had been expressed irrationally by a potential series in paragraph 144 and Table I. Table VI. CORE LOSS CURVE. e Volts. Pi kw. log e log Pi 1.6 log e A=logPi -1.6 log e Pc J 40 0.63 1.602 9.799 2.563 7.236 0.70 -0.07 60 1.36 1778 0.134 2.845 7.289 7.293 I avg. 1.34 + 0.02 80 2.18 1.903 0.338 3.045 2.12 + 0.06 100 3. 00 2.000 0.477 3.200 7.277 7.282 3. 03 -0.03 120 3.93 2.079 0.594 3.326 7.268 J 4.05 -0.12 140 6.22 2.146 0.794 3.434 7. 360 5.20 + 1.02 160 8.59 2.204 0.934 3.526 7-408 6.43 + 2.16 i-a = 5.283 0.271 lcgPt = 7.282 + 1.6loge ^■3-3 = 1.761 0.090 i't = 1.914e»«, in watts i-s = 4.079 1.071 2'2-f-2 = 2.0395 0.535 A = 02785 0.445 r 0.445 = 1.598- 1.6 0.2785 4 The first two columns of Table VI give the observed values of the voltage e and the core loss Pi in kilowatts. The next two columns give log e and log Pi. Plotting the curves shows that loge, log Pi is approximately a straight Hne, as seen in Fig. 82, with the exception of the two highest points of the curve. Excluding therefore the last two points, the first five obser- vations give a parabolic curve. The exponent of this curve is found by Table VI as 71=1.598; that is, with sufficient approximation, as n=1.6. To see how far the observations agree with the curve, as given by the equation, Pi=ae^'^ in the fifth column 1.6 log e is recorded, and in the sixth column, A = loga = logPi— 1.61oge. As seen, the first and the last two values of A differ from the rest. The first value corre- EMPIRICAL CURVES. 243 spends to such a low value of Pi as to lower the accuracy of the observation. Averaging then the four middle values, gives A = 7.282 ; hence, log Pf= 7.282 + 1.6 log e, Pi=1.914ei-»- in watts. 1.6 1.7 1.8 1.9 2.0 2.1 2.2 log Pi lo ^e ■ -J ( Px / kw. 0^8 / 6 jd^ / 0:6 / / ,9\ / f 70 0r4 r \o>| hy ( . / Q Q_ / / / ^-fv / /< » ^ 1 r 4 1 3 2 3 2 0 ^ 0 ^r 0 e D 7 > sb Fig. 83. Investigation of Magnetization Curve, and herefrom, 5C (B = 0.211 +0.0507 5C is the equation of the magnetic characteristic for values of 3C from eight upward. The values calculated from this equation are given as (B in Table VII. 246 ENGINEERING MATHEMATICS. nn The difference between the observed values of-^, and the value given by above equation, which is appreciable up to JC=-6, could now be further investigated, and would be found to approximately follow an exponential law. D. Periodic Curves. 158. All periodic functions can be expressed by a trigo- nometric series, or Fourier series, as has been discussed in Chapter III, and the methods of resolution and arrangements devised to carry out the work rapidly have also been dis- cussed in Chapter III. The resolution of a periodic function thus consists in the determination of the higher harmonics, which are super- imposed on the fundamental wave. As periodic curves are of the greatest importance in elec- trical engineering, in the theory of alternating-current phe- nomena, a familiarity with the wave shapes produced by the different harmonics is desirable. This familiarity should be sufficient to enable one to judge immediately from the shape of the wave, as given by oscillograph, etc., which harmonics are present. The effect of the lower harmonics, such as the third, fifth, seventh, etc. (or the second, fourth, etc., where present), is to change the shape of the wave, make it differ from sine shape, and in the '' Theory and Calculation of Alternating- current Phenomena," 4th. Ed., Chapter XXX, a number of characteristic distortions, such as the flat top, peaked wave, saw tooth, double and triple peaked, sharp zero, flat zero, etc., have been discussed with regard to the harmonics that enter into their composition. 159. High harmonics do not change the shape of the wave much, but superimpose ripples on it, and by counting the number of ripples per half wave, or per wave, the order of the harmonic can rapidly be determined. For instance, the wave shown in Fig. 84 contains mainly the eleventh harmonic, as there are eleven ripples per wave (Fig. 84). Very frequently high harmonics appear in pairs of nearly the same frequency and intensity, as an eleventh and a thir- EMPIRICAL CURVES. 247 teenth harmonic, etc. In this case, the ripples in the wave shape show maxima, where the two harmonics coincide, and nodes, where the two harmonics are in opposition. The presence of nodes makes the counting of the number of ripples per complete wave more difficult. A convenient method of procedure in this case is, to measure the distance or space Fig. 84. Wave in which Eleventh Harmonic Predominates. between the maxima of one or a few ripples in the .range where they are pronounced, and count the number of nodes per cycle. For instance, in the wave. Fig. 85, the space of two ripples is about 60 deg., and two nodes exist per complete wave. 60 deg. for two ripples, gives 2 X-:^= 12 ripples per Fig. 85. Wave in which Eleventh .and Thirteenth Harmonics Predominate. complete wave, as the average frequency of the two existing harmonics, and since these harmonics differ by 2 (since there are two nodes), their order is the eleventh and the thirteenth harmonics. This method of determining two similar harmonics, with a little practice, becomes very convenient and useful, and may 248 ENGINEERING MATHEMATICS. frequently be used visually also, in determining the frequency of hunting of synchronous machines, etc. In the phenomenon of hunting, frequently two periods are superimposed, forced frequency, resulting from the speed of generator, etc., and the natural frequency of the machine. Counting the number of impulses, /, per minute, and the number of nodes, n, gives the 71/ Tl two frequencies :/+- and/— -; and as one of these frequencies is the impressed engine frequency, this affords a check. Not infrequently wave-shape distortions are met, which are not due to higher harmonics of the fundamental wave, but are incommensurable therewith. In this case there are two entirely unrelated frequencies. This, for instance, occurs in the secondary circuit of the single-phase induction motor; two sets of currents, of the frequencies /« and (2/—/^) exist (where / is the primary frequency and /s the frequency of slip). Of this nature, frequently, is the distortion produced by surges, oscillations, arcing grounds, etc., in electric circuits; it is a combination of the natural frequency of the circuit with the impressed frequency. Telephonic currents commonly show such multiple frequencies, which are not harmonics of each other.