CHAPTER II. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. A. GENERAL. 39. An expression such as y-xk w represents a fraction; that is, the result of division, and hke any fraction it can be calculated; that is, the fractional form eliminated, by dividing the numerator by the denominator, thus : l-x l = l+x + x2 + a:3 + . . . l-x x—x^ - x-—x^ -^x\ Hence, the fraction (1) can also be expressed in the form: ( 2/=TX~^-'^"^^ + ^^'^^'^' • • (2) This is an infinite series of successive powers of x, or a poten- tial series. In the same manner, by dividing through, the expression y^ih' ■ ^^^ can be reduced to the infinite series, y=j^ = l-x-hx^-x^+- |(4) 52 POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 53 The infinite series (2) or (4) is another form of representa- tion of the expression (1) or (3), just as the periodic decimal fraction is another representation of the common fraction (for instance 0.6363 = 7/11). 40. As the series contains an infinite number of terms, in calculating numerical values from such a series perfect exactness can never be reached: since only a finite number of terms are calculated, the result can only be an approximation. By taking a sufficient number of terms of tlie series, however, the approximation can be made as close as desired; that is, numerical values may be calculated as exactly as necessary, so that for engineering purposes the infinite series (2) or (4) gives just as exact numerical values as calculation by a finite expression (1) or (2), provided a sufficient number of terms are used. In most engineering calculations, an exactness of 0.1 per cent is sufficient; rarely is an exactness of 0.01 per cent or even greater required, as the unavoidable variations in the nature of the materials used in engineering structures, and the accuracy of the measuring instruments impose a hmit on the exactness of the result. For the value x = 0.5, the expression (1) gives y = z. — p-^ = 2; while, its representation by the series (2) gives 2/ = 1+0.5 + 0.25+0.125+0.0625 + 0.03125 + ... (5) and the successive approximations of the numerical values of y then are : xising one term: y=l =1; error: —1 " tw^o terms: i/=l + 0.5 =1.5; " -0.5 " three terms: ?/= 1 + 0.5+ 0.25 =1.75" '" -0.25 " four terms: 2/=l + 0.5+0.25+0.125 =1.875; " -0.125 '' five terms: ?/ = 1 + 0.5+0.25+0.125+0.0625= 1.9375 " -0.0625 It is seen that the successive approximations come closer and closer to the correct value, y = 2, but in this case always remain below it; that is, the series (2) approaches its limit from below, as shown in Fig. 24, in which the successive approximations are marked by crosses. For the value re = 0.5, the approach of the successive approximations to the Hmit is rather slow, and to get an accuracy of 0.1 per cent, that is, bring the error down to less than 0.002, requires a considerable number of terms. 54 ENGINEERING MATHEMATICS. For a: = 0.1 the series (2) is 2/ = l +0.1 +0.01 +0.001 +0.0001+ (6) and the successive approximations thus are 1: 2: 3: 2/ = l.ll; 4: 2/=i.ni; 5: 2/=i.iiii; and as, by (1), the final or Umiting value is y=Y 1 10 1.1111 ... -0.1 9 j^ + + 4 3 * "6 /^ < V' ■-, Fig. 24. Direct Convergent Series with One-sided Approach. the fourth approximation already brings the error well below 0.1 per cent, and sufficient accuracy thus is reached for most engineering purposes by using four terms of the series. 41. The expression (3) gives, for x = 0.5, the value. Represented by series (4), it gives 2/ = l-0.5 + 0.25 -0.125+0.0625 -0.03125+ - (7) the successive approximations are; 1st: 2/=l =1; error: +0.333... 2d: 2/=l-0.5 =0.5; " -0.1666... 3d: i/=l-0.5+0.25 =0.75; " +0.0833... 4th: 2/- 1-0.5+0.25-0.125 =0.625; " -0.04166... oth: 2/= 1-0.5+0.25-0.125+0.0625 = 0.6875; " +0.020833... As seen, the successive approximations of this series come closer and closer to the correct value ?/ = 0.6666 . . . , but in this case are alternately above and below the correct or limiting POTENTIAL SERIES AND EXPONENTIAL FUNCTION, 55 value, that is, the series (4) approaches its limit from both sides, as shown in P'ig. 25, while the series (2) approached the Umit from below, and still other series may approach their limit from above. With such alternating approach of the series to the limit, as exhibited by series (4), the limiting or final value is between any tw^o successive approximations, that is, the error of any approximation is less than the difference betw^een this and the next following approximation. 42. Substituting x = 2 into the expressions (1) and (2), equation (1) gives \ 3 + 5 i ^~l+x Fig. 25. Alternating Convergent Series. while the infinite series (2) gives 2/ = l+2-r4+8 + 16+32 + ...; and the successive approximations of the latter thus are 1; 3; 7; 15; 31; 63...; that is, the successive approximations do not approach closer and closer to a final value, but, on the contrary, get further and further aw^ay from each other, and give entirely wrong results. They give increasing positive values, which apparently approach oo for the entire series, while the correct value of the expression, by (1), is2/= -1. Therefore, for x = 2, the series (2) gives unreasonable results, and thus cannot be used for calculating numerical values. The same is the case with the representation (4) of the expression (3) for x = 2. The expression (3) gives J/=Y^ = 0.3333...; 56 ENGINEERING MATHEMATICS. while the infinite series (4) gives 2/ = l -2+4-8 + 16-32+ -.. ., and the successive approximations of the latter thus are 1; -1; +3; -5; +11: -21; . . .: hence, while the successive values still are alternately above and below the correct or limiting value, they do not approach it with increasing closeness, but more and more diverge there- from. Such a series, in which the values derived by the calcula- tion of more and more terms do not approach a final value closer and closer, is called divergent, while a series is called convergent if the successive approximations approach a final value with increasing closeness. 43 • While a finite expression, as (1) or (3), holds good for all values of x, and numerical values of it can be calculated whatever may be the value of the independent variable x, an infinite series, as (2) and (4), frequently does not give a finite result for every value of x, but only for values within a certain range. For instance, in the above series, (for —1 tnat is, approaches with increasing number of terms a finite^^ limiting value, s woiiM In* ( -) . the errui \/7 ' made by the simpler expression (13) is less than ( — ) . Thus, if To is 3 per cent of r, which is a fair average in interior light- ing circuits, {-) =0.032 = 0.0009, or less than 0.1 percent; hence, is usuall^^ negligible. 46. If an expression in its finite form is moi*e complicated and thereby less convenient for numerical calculation, as for instance if it contains roots, development into an infinite series frequently simplifies the calculation. Very convenient for development into an infinite series of powers or roots, is the binomial theorem, (14) X * n(n-l) _ n(n-l)(n-2) ^ If II 4 where |w«-lX2x3X. . .Xm. Thus, for instance, in an alternating-current circuit of resistance r, reactance x, and supply voltage e, the curi-ent is. ^■v^T7^ "^) 60 ENGINEERING MATHEMATICS. If this circuit is practically non-inductive, as an incandescent lighting circuit; that is, if x is small compared with r, (15) can be written in the form, ._ e e h©T', . . . ae, and the square root can be developed by the binomial (14), thus, u= yyj ; n= --, and gives h(r)T*=-i(7)"-s(r)'-f.(")'-- <■:) In this series (17), if x = 0.1r or less; that is, the reactance is not more than 10 per cent of the resistance, the third term, Q ( - ) , is less than 0.01 per cent; hence, negligible, and the series is approximated with sufficient exactness by the fii-st two terms, and equation (16) of the current then gives -tI'-I©! « This expression is simpler for numerical calculations than the expression (15), as it contains no square root. 47. Development into a series may become necessary, if further operations have to be carried out with an expression for which the expression is not suited, or at least not well suited. This is often the case where the expression has to be integrated, since very few expressions can be integrated. { Expressions under an integral sign therefore very commonly have to be developed into an infinite series to carry out the integration. \ POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 61 EXAMPLE 1. Of the equilateral hyperbola (Fig. 26), xy = a^, (20) the length L of the arc between Xi=2a and 0:2 = 10a is to be calculated. An element dl of the arc is the hypothenuse of a right triangle with dx and dy as cathetes. It, therefore, is, dl = Vdx^+dy'^ -MH)'" «» \ \ \ \ \ \ V V d> % xy= -.a- iCi -£ r. and from (20), Fig. 26. Equilateral Hyperbola. dy ^=7 ^^^ dx Substituting (22) in (21) gives, hence, the length L of the arc, from xi to X2 is. (22) (23) (24) 62 ENGINEERING MATHEMATICS. Substituting - = v; that is, dx = adv, also substituting Vi=-=2 and 2;, = - =10, .... (25) gives r^' i r ,4^^' (26) 4SyF^^ The expression under the integral is inconvenient for integra- tion; it is preferably developed into an infinite series, by the binomial theorem (14). Write u = -2 and n = -^, then V 1^^ ±_J_ _i ^ and L 1 1 1 [ 2X3XV* 7X8X2;8 11X16X2;^^ + I + 1 3X128X1-16 J and substituting the numerical values, L = a\ (10-2) +i(0.125-0.001) -^(0.0078-0) +-47^(0.0001 -0) ■ 5b 17d = a{8 +0.0207-0.0001} = 8.0206a. As seen, in this series, only the first two terms are appreciable in value, the third term less than 0.01 per cent of the total, and hence negligible, therefore the series converges very rapidly, and numerical values can easily be calculated by it. POTENTIAL SERIES AND EXPONENTIAL FUNCTION, 63 For xi<2 a; that is, Vi <2, the series converges less rapidly, and becomes divergent for xil, but near this limit of convergency it is of no use for engineering calculation, as it does not converge with sufficient rapidity, and it becomes suitable for engineering calculation only when V} approaches 2. EXAMPLE 2. 48. (log 1=0, and, therefore log (1-hx) is a small quantity if X is small. / log (l+x) shall therefore be developed in such a series of powers of x, which permits its rapid calculation without using logarithm tables. It is log«=J -; then, substituting (1+x) for u gives, ^log(l+.r)=Jj^^ (18) From equation (4) ■z = l — X-hX^ — X^ + . . . , 1+x ' • hence, substituted into (18), I0g(l^.)^=/(1-X..-X3....).. ( = i dx- I xdx + j x^dx — I x^dx + . . . x^ x^ x/^ ,^^, = :c— 2+3--4-+ ••• ; (19) V x hence, if x is very small, — is negligible, and, therefore, all terms beyond the first are negligible, thus, log(l4-a;j=x; (20) while, if the second term is still appreciable in value, the more complete, but still fairly simple expression can be used. ( ^ log(l+a:) = x-|' = x(l-|) (21) 64 ENGINEERING MATHEMATICS. If instead of the natural logarithm, as used above^ the decimal logarithm is required, the following relation may be applied : logio a = logio£ logs a = 0.4343 logs a, . . (22) logio ci is expressed by logs cl, and thus (19), (20) (21) assume the form. logio(l+:r)=0.4343(^x--+-— j+... j; . (23) or, approximately, logio(l+a:)=0.4343x; • (24) or, more accurately, logio(l+.T)=0.4343x(l-|). . ; . (25) B. DIFFERENTIAL EQUATIONS. 49. The representation by an infinite series is of special value in those cases, in which no finite expression of the func- tion is known, as for instance, if the relation between x and y is given by a differential equation. Differential equations are solved by separating the variables, that is, bringing the terms containing the one variable, 2/, on one side of the equation, the terms with the other variable x on the other side of the equation, and then separately integrat- ing both sides of the equation. Very rarely, however, is it possible to separate the variables in this manner, and where it cannot be done, usually no systematic method of solving the differential equation exists, but this has to be done by trying different functions, until one is found which satisfies the equation. In electrical engineering, currents and voltages are dealt with as functions of time. The current and c.m.f. giving the power lost in resistance are related to each other by Ohm's law. Current also produces a magnetic field, and this magnetic field by its changes generates an e.m.f. — the e.m.f. of self- inductance. In this case, e.m.f. is related to the change of current; that is, the differential coefficient of the current, and thus also to the differential coefficient of e.m.f., since the e.m.f. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 65 is related to the current by Ohm's law. In a condenser, the current and therefore, b}^ Ohm's law, the e.m.f., depends upon and is proportional to the rate of change of the e.m.f. impressed upon the condenser; that is, it is proportional to the differential coefficient of e.m.f. Therefore, in circuits having resistance and inductance, or resistance and capacity, a relation exists between currents and e.m.f s., and their differential coefficients, and in circuits having resistance, inductance and capacity, a double relation of this kind exists; that is, a relation between current or e.m.f. and their first and second differential coefficients. The most common differential equations of electrical engineer- ing thus are the relations betw^een the function and its differential coefficient, which in its simplest form is, i=»^ ««) or '!""■ ' (»' and where the circuit has capacity as well as inductance, the second differential coefficient also enters, and the relation in its simplest form is, §'-r. (=») or S-^' (29) and the most general form of this most common differential equation of electrical engineering then is, g+2c|+«2/+6=0 (30) The differential equations (26) and (27) can be integrated by separating the variables, but not so with equations (28), (29) and (30); the latter require' solution by trial. 50. The general method of solution may be illustrated with the equation (26) : i-» « 66 ENGINEERING MATHEMATICS. To determine whether this equation can be integrated by an infinite series, choose such an infinite series, and then, by sub- stituting it into equation (26), ascertain whether it satisfies the equation (26) ; that is, makes the left side equal to the right side for every value of x. Let, y^ao-^aix±a2x'^ + a'6X^-\-a^x^-\- (31) be an infinite series, of which the coefficients (Iq, a\, ao, as. . . are still unknown, and by substituting (31) into the differential equation (26), determine whether such values of these coefficients can be found, which make the series (31) satisfy the equation (26). Differentiating (31) gives, -^--ai+2a2X+Sa3X^+4:a4X^ + (32) ^ The differential equation (26) transposed gives, J-^^=0 • • (33). Substituting (31) and (32) into (33), and arranging the terms in the order of x, gives, (ai — ao) + (2a2 — ai)x + {Sas — a2)x^ + {4:a4'^3)x^ + (5a5-ai)x^ + . . .=0. . (34) If then the above series (31) is a solution of the differential equation (26), the expression (34) must be an identity; that is, must hold for every value of x. If, however, it holds for every value of x, it does so also for x = 0, and in this case, all the terms except the first vanish, and (34) becomes, ai — ao = 0; or, ai=ao (35) To make (31) a solution of the differential equation (ai — ao) must therefore equal 0. This being the case, the term (ai — ao) can be dropped in (34), which then becomes, (2a2~ai)x + {Sa3-a2)x^ + {4:a4—a3)x^ + (^a5-a4)x'^ + . . .=0; or, a'{(2a2-ai) + (3a3-a2)x- + (4a4-a3\r2 + . . .{=0. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 67 Since this equation must hold for every value of x, the second term of the equation must be zero, since the first term, x, is not necessarily zero. This gives, fe— «i)+(3a3 — a2)a: + (4a4- 03)^2 + . . .=0. As this equation holds for every value of x, it holds also for x = 0. In this case, however, all terms except the first vanish, and, 2a2-ai=0; (36) hence. 02=2' from (35), Continuing the same reasoning, 3a3-a2 = 0, 4a4— 03=0, etc. Therefore, if an expression of successive powers of x, such as (34), is an identity, that is, holds for every value of x, then all the coefficients of all the powers of x must separately he zero."^ Hence, ai — aQ=0] or ai = ao; (37) 2a2-ai=0; or GL2—^=-^] 3a3-;^2 = 0; a2 «o or a3=^3=-3; 4a4-3a3=0; «,3 «0 ov a^=j=j; etc. etc., * The reader must realize the difference between an expression (34), as equation in x, and as substitution product of a function; that is, an as identity. Regardless of the values of the coefficients, an expression (34) as equation gives a number of separate values of a:, the roots of the equation, which make the left side of (34) equal zero, that is, solve the equation. If, however, the infinite series (31) is a solution of the differential equation (26), then the expression (34), which is the result of substituting (31) into (26), must be correct not only for a limited number of values of x, which are the roots of the equation, but for all values of x, that is, no matter what value is chosen for x, the left side of (34) must always give the same result, 0, that is, it must not be changed by a change of x, or in other words, it must not contain x, hence all the cpefficients of the powers of x must be zero. 68 ENGINEERING MATHEMATICS. (39) Therefore, if the coefficients of the series (31) are chosen by equation (37), this series satisfies the differential equation (is); that is, r ^2 ^3 ^4 ] 2/ = ao{l+x+-2+T3+p- + ...j. . . . (38) is the solution of the differential equation, 51. In the same manner, the differential equation (27), dz Tx-""'' ■ ■ • is solved by an infinite series, z = ao^-aiX-\-a2x'^-\-azX^-\-. . ., .... (40) and the coefficients of this series determined by substituting (40) into (39), in the same manner as done above. This gives, (ai — aao) + {2a2 — aai)x + {Za^ — aa2)x'^ + (4a4-aa3)a:3 + . . .=0, . (41) and, as this equation must be an identity, all its coefficients must be zero; that is, ai — aao = 0; or ai = aao; a a2 2a2—aai=0; or a2 = ai-^ = ao-:^ Sas — aa2 = 0 ; or as = 02 ^ = ao 75- 4a4— aa3 = 0; or 04 = as ao^ (42) 3 a 4 etc., etc. and the solution of differential equation (39) is, ( a%2 a^x^ a^x^ ] ,^^^ 2: = ao|l+ax+-2-+-|3-+-n- + ...j. . . (43) 52. These solutions, (38) and (43), of the differential equa- tions (26) and (39), are not single solutions, but each contains an infinite number of solutions, as it contains an arbitrary POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 69 constant ao] that is, a constant which may have any desired numerical vahie. This can easily be seen^ since, if ^ is a solution of the dif- ferential equation, dz dx then, any multiple, or fraction of z, hz, also is a solution of the differential equation; d{hz) .. . since the 6 cancels. Such a constant, ao, which is not determined by the coeffi- cients of the mathematical problem, but is left arbitrary, and requires for its determinations some further condition in addition to the differential equation, is called an integration constant. It usually is determined by some additional require- ments of the physical problem, which the differential equation represents; that is, by a so-called terminal condition, as, for instance, by having the value of y given for some particular value of X, usually for x = 0, or x=qc. The differential equation, i-r. («> thus, is> solved by the function, y = aoyo, (45) where, 7>2 'Y'3 '1^4 i/o = l+x+-+i3+jj + ..., .... (46) and the differential equation, -r = a^, (47) dx ' ' is solved by the function, z = aoZo, . (48) where, a^x^ aV a4.r4 Zo = l+ax-\-—^+-r^+-r^+ (49) 70 ENGINEERING MATHEMATICS. yo and zq thus are the simplest forms of the solutions y and z of the differential equations (26) and (39). 53. It is interesting now to determine the value of ?/". To raise the infinite series (46), which represents 2/0 ; to the nth power, would obviously be a very complicated operation. However, -i^'^y-'d' ^50) dv and since from (44) w~""2/; • • • (51) by substituting (51) into (50), dv"' -h-^r-' (52) hence, the same equation as (47), but with ?/" instead of z. Hence, if y is the solution of the differential equation, dy then z = y'^ is the solution of the differential equation (52), dz -y- = nz. dx However, the solution of this differential equation from (47), (48), and (49), is z = aQZo'y ZQ = l+nx-\- that is, if 2o = l+na:+-Y--f-j^-f-. then, ZQ = yo'^ = ^+rix^-Y+-w-^'") • • • (53) therefore the series y is raised to the nth power by multiply- ing the variable x by n. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 71 Substituting now in equation (53) for n the value - gives 2^^T = i + i+i+-+_ + ... : .... (54) that is, a constant numerical value. This numerical value equals 2.7182828. . ., and is usually represented by the symbol e. Therefore, j_ 2/0^ = ^' hence, ^2 j3 j4 2/o = c- = l+x+-+jj+p- + ..., (55) and n~X^ Tl^X^ 71'^X'^ 2;o = 2/o'' = (-'^)"=-'"'' = l+^-c+-^+-|^+-j^ + . . . ; (56) therefore, the infinite series, which integrates above differential equation, is an exponential function with the base £ = 2.7182818 (57) The solution of the differential equation, t- '=«' thus is, 2/ = «o^^ (5^) and the solution of the differential equation, %=^y^ ^'''^ is, 2/ = «o-'^^ (^>1) where aQ is an integration constant. The exponential function thus is one of the most common functions met in electrical engineering problems. The above described method of solving a problem, by assum- ing a solution in a form containing a number of unknown coefficients, Oq, ai, a2 . . ., substituting the solution in the problem and thereby determining the coefficients, is called the method of indeterminate coefficients. It is one of the most convenient 72 ENGINEERING MATHEMATICS. and most frequently used methods of solving engineering problems. EXAMPLE 1. 54. In a 4-pole 500-volt 50-kw. direct-current shunt motor, the resistance of the field circuit, inclusive of field rheostat, is 250 ohms. Each field pole contains 4000 turns, and produces at 500 volts impressed upon the field circuit, 8 megalines of magnetic flux per pole. What is the equation of the field current, and how much time after closing the field switch is required for the field cur- rent to reach 90 per cent of its final value? Let r be the resistance of the field circuit, L the inductance of the field circuit, and i the field current, then the voltage consumed in resistance is, In general, in an electric circuit, the current produces a magnetic field; that is, lines of magnetic flux surrounding the conductor of the current ; or, it is usually expressed, interlinked with the current. This magnetic field changes with a change of the current, and usually is proportional thereto. A change of the magnetic field surrounding a conductor, however, gen- erates an e.m.f. in the conductor, and this e.m.f. is proportional to the rate of change of the magnetic field; hence, is pro- portional to the rate of change of the current, or to di . -T., with a proportionality factor L, which is called the induct- ance of the circuit. This counter-generated e.m.f. is in oppo- di sition to the current, —L-r, and thus consumes an e.m.f., ' dt' di . , + L-Z, which is called the e.m.f, consumed by self -inductances at or inductance e.m.f. Therefore, by the inductance, L, of the field circuit, a voltage is consumed which is proportional to the rate of change of the field current, thus, ^di '^ = ^di' POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 73 Since the supply voltage, and thus the total voltage consumed in the field circuit, is e = 500 volts, or, rearranged, di e—ri JC~L~' Substituting herein, u = e—ri] (63) hence, dii di gives. dt ^ dV du r ,^,, *=-L« (64) T This is the same differential equation as (39), with a=— y, Ij and therefore is integrated by the function, therefore, resubstituting from (63), and --rt e—ri = ao£ ^ , e , ao -L^ t=-i-s ^ (65) r r ^ ^ This solution (65), still contains the unknown quantity ao; or, the integration constant, and this is determined by know- ing the current i for some particular value of the time t. Before closing the field switch and thereby impressing the voltage on the field, the field current obviously is zero. In the moment of closing the field switch, the current thus is still zero; that is, i = 0 for ; = 0 (66) 74 ENGINEERING MATHEMATICS. Substituting these values in (65) gives, hence, (67) is the final solution of the differential equation (62); that is, it is the value of the field current, i, as function of the time, i, after closing the field switch. After infinite time, ^ = oo , the current i assumes the final value io, which is given by substituting t = oo into equation (67), thus, V to = -=77^ = 2 amperes; .... (o8) hence, by substituting (68) into (67), this equation can also be written, = 2[i-b-~l')^ .... . (69) where io = 2 is the final value assumed by the field current. The time h, after which the field current i has reached 90 per cent of its final value i^, is given by substituting i = 0.9fo into (69), thus, 0.9to=io(l-^"~^''), and Taking the logarithm of both sides, r -^^ilog£=-l; and h=-^ (70) r log £ POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 75 55. The inductance L is calculated from the data given in the problem. Inductance is measured by the number of interlinkages of the electric circuit, with the magnetic flux produced by one absolute unit of current in the circuit; that is, it equals the product of magnetic flux and number of turns divided by the absolute current. A current of ^0=2 amperes represents 0.2 absolute units, since the absolute unit of current is 10 amperes. The number of field turns per pole is 4000; hence, the total number of turns n = 4x4000 = 16,000. The magnetic flux at full excitation, or i'o =0.2 absolute units of current, is given as (^=8x10® lines of magnetic force. The inductance of the field thus is: ^ n0 16000X8X106 ^^^ ^^^ ^ , ^ .^ ^,^, L = -^ = jr^ = 640 X 109 absolute units = 640.^ , Iq 0.2 ' the practical unit of inductance, or the henry (h) being 10^ absolute units. Substituting L = 640 r = 250 and e = oOO, into equation (67) and (70) gives i = 2(l-£-«-9f), and 640 ^^^250X0.4343^'^-^-'^^^ ^^^^ Therefore it takes about 6 sec. before the motor field has reached 90 per cent of its final value. The reader is advised to calculate and plot the numerical values of i from equation (71), for ^ = 0, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 3, 4, 5, 6, 8, 10 sec. This calculation is best made in the form of a table, thus; ,-o.39< = jV_o.39nog., and, log £ =0.4343; hence. 0.39 logs =0.1694i; and. .-0-39' = iV_o.l694<. 76 ENGINEERING MATHEMATICS, The values of £-o-39< ^^^ ^jgQ ^g taken directly from the tables of the exponential function, at the end of the book. t 0.1694« -0.169^4 ,-0.39^ l_^-0.39i 2(1- c- -390 = -/V-U.1694< ; 0.0 0.1 0.2 0.4 0.6 0.8 etc. 0 0.0170 0.0339 0.0678 0.1016 0.1355 0 0.9830-1 0.9661-1 0.9322-1 0.8984-1 0.8645-1 1 0.962 0.925 0.855 0.791 0.732 0 0.038 0.075 0.145 0.209 0.268 0 0.076 0.150 0.290 0.418 0.536 EXAMPLE 2. 56. A condenser of 20 mf. capacity, is charged to a potential of 60 = 10,000 volts, and then discharges through a resistance of 2 megohms. What is the equation of the discharge current, and after how long a time has the voltage at the condenser dropped to 0.1 its initial value? A condenser acts as a reser- voir of electric energy, similar to a tank as water reservoir. If in a water tank, Fig. 27, A is the sectional area of the tank, e, the height of water, or water pressure, and water flows out of the tank, then the height e decreases by the flow of water; that is the tank empties, and the current of water, i, is proportional to the change of the de water level or height of water, — , and to the area A of the Fig. 27. Water Reservoir. tank; that is, it is. (U' de dt' (72) The minus sign stands on the right-hand side, as for positive i; that is, out-flow, the height of the water decreases; that is, de is negative. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 77 In an electric reservoir, the electric pressure or voltage e corresponds to the water pressure or height of the water, and to the storage capacity or sectional area A of the water tank corresponds the electric storage capacity of the condenser, called capacity C. The current, or, flow out of an electric condenser, thus is, --4I (-3) The capacity of condenser is, C = 20 mf = 20x10-6 farads. The resistance of the discharge path is, r = 2X106 ohms; hence, the current taken by the resistance, r, is and thus . e ^ dt r and dt ~ Cr ^' Therefore, from (60) (61), t_ and for ^ = 0, 6 = ^0 = 10,000 volts; hence and 0.1 of the initial value: Ls reached at : 10,000 = ao, (74) t_ e = €Q£ Cr = 10,000£-^^25< volts; 6 = 0.1^0, =Rl = 92sec (75) 78 ENGINEERING MATHEMATICS. The reader is advised to calculate and plot the numerical values of e, from equation (74), for / = 0; 2; 4; 6; 8; 10; 15; 20; 30; 40; 60; 80; 100; 150; 200 sec. 57. Wherever in an electric circuit, in addition to resistance, inductance and capacity both occur, the relations between currents and voltages lead to an equation containing the second differential coefficient, as discussed above. The simplest form of such equation is: ih'^y (7«) To integrate this by the method of indeterminate coefficients, we assume as solution of the equation (76) the infinite series, y=-^ao+aiX+a2X^+fi'iX^+a^x'^ + {77)- in which the coefficients ao, ai, a2, as, a^. . . are indeterminate. Differentiating -(67) twice, gives ^==2a2 + 2X3a3X + 3X4a4a:2 + 4x5a5a-3 + ..., . (78) and substituting (77) and (78) into (76) gives the identity, 2a2+2x3a3X+3x4a4x2+4x5a5x3 + . . . =a{ao-\-aiX-\-a2X^+a^x^-\-. . .); or, arranged in order of x, {2a2-aao) +x{2xSa^-aai) +x^{3Xia4-aa2) + x3(4x5a5-aa3)+. . .=0 (79) Since this equation (79) is an identity, the coefficients of all powers of x must individually equal zero. This gives for the determination of these hitherto indeterminate coefficients the equations, 2a2 — aao==0: 2x3a3-aai=0: 3x4a^-aa2 = 0; 4Xoao — aa3 = 0, etc. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 79 Therefore aao afi] 03- 3 ; aa2 ^^"3X4 = aoa2 aar, "^-4X5 = aia2 5 aa4 "^^-5X6 = aoa^ 6 ' "^"6X7 = 7 aae ^«-7x8 = = 8 ' aa7 ^^-8X9 = 9 etc., etc. Substituting these values in (77), 2/ = a„|l+-2-+^+^+.. o^x^ aH^ a'^x'' \ ax+ ,„ I ,. I |3 li (80) In tWs case, two coefficients ao and ai thus remain inde- terminate, as was to be expected, as a differential equation of second order must have two integration constants in its most general form of solution. Substituting into this equation, 62 = a; that is, h = \/a, (81) S=+^2/, (82) and 2/ = aoi 62^ 6%4 66^ h^x^ ¥x^ 67^7 +a]6 ''^+ir+^+-i7-+--- •• (83) 80 ENGINEERING MATHEMATICS. In this case, instead of the integration constants ao and a\y the two new integration constants A and B can be introduced by the equations aQ = A-\-B and aih = A — B', hence, A = — ^ — and B= — ; and, substituting these into equation (83), gives, , r , &2^2 53^3 54^4 ^ ^ f ^ ^ h^^ h^x^ ¥x^ 1 +B^l-hx+-^- 13- +-|4- "+•••}• • ^^4) The first series, however, from (56), for n = h is e"^^^, and the second series from (56), for n=—b is £~^*. Therefore, the infinite series (83) is, 2/ = A£+&^+5£-^*; (85) that is, it is the sum of two exponential functions, the one with a positive, the other with a negative exponent. Hence, the difTerential equation, S=a2/, ........ (76) is integrated by the function, 2/-^£+^^+5s-^^ (86) where, b = Va. (87) However, if a is a negative quantity, h = Va is imaginary, and can be represented by b = jc, (88) where c^=-a (89) In this case, equation (86) assumes the form. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 81 that is, if in the differential equation (76) a is a positive quantity, = +6^^ this differential equation is integrated by the sum of the two exponential functions (86) ; if, however, a is a negative quantity, = — c^, the solution (86) appears in the form of exponen- tial functions with imaginary exponents (90). 58. In the latter case, a form of the solution of differential equation (76) can be derived which does not contain the imaginary appearance, by turning back to equation (80), and substituting therein a=—c^, which gives, = -r2 c^y y-% {'- cH^ c^x^ c^x^ -^r- + -n 177- + , 11 li aic ex 77r-+-T^ 1-. |3 |5 (91) ■h or, writing* A = 00 and B= — aic. y=A\l-^^ + c^x^ c'^x^ c^x^ li I + +B\ ex +-r^ +, I li (92) The solution then is given by the sum of two infinite series, thus. , , , c^x^ c%4 c^x^ and as It |6 v(cx)=cx— r^T^ c^x^ - +, (93) y = Au(cx)-\-Bv(cx) (94) In the i^-series, a change of the sign of x does not change the value olu. u{ — cx)=u{+cx). . Such a function is called an even function. (95) 82 ENGINEERING MATHEMATICS. In the v-scries, a change of the sign of x reverses the sign of V, as seen from (93) : v{-cx) = -v{+cx) (96) Such a function is called an odd function. It can be shown that u{cx)=Qo^cx and v{cx)=^mcx; . . . (97) hence, 2/ = -A cos ex +5 sin ex, ..... (98) where A and B are the integration constants, which are to be determined by the terminal conditions of the physical problem. Therefore, the solution of the differential equation d^^=^y^ • (9^) has two different forms, an exponential and a trigonometric. If 4t is positive, ^^ dhi ^^-+^y' (100) it is: y = As + ^'^+BB-^^, (101) If a negative, ■'O" d-y it is: ^=-cY. (102) 2/ = A cos ex +5 sin ex (103) In the latter case, the solution (101) would appear as ex- ponential function with imaginary exponents; y^Ae+^'^+Be-^''^ (104) As (104) obviously must be the same function as (103), it follows that exponential functions with imaginary exponents must be expressible by trigonometric functions. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 83 59. The exponential functions and the trigonometric func- tions, according to the preceding discussion, are expressed by the infinite series, x^ x^ r^ x^ j>2 -j«4 /p6 COS X=l—-r+'r-r—-r^ + —. . . 2 |4 |D /v»3 'Y'O -vT sin^ = x-i3+j^-j^ + -... (105) Therefore, substituting ju for x, ■' 2 -^ |3 |4 •'5 6 ■' |/ /, u^ u^ u^ \ ./ u^ u^ u^ \ However, the first part of this series is cos u, the latter part sin u, by (105); that is, £^" = cos 16+/ sin \i. Substituting —u for +u gives, £ - 2" = cos u — j sin u. Combining (106) and (107) gives, (106) (107) ^+jM_|_ g-iu COS u = and c+iu_ f—ju sm w = 2/ rio8) Substituting in (106) to (108), jv for u, gives, £~'' = cos /iJ+ysin p, 1 and. "^ "^ = cos p — y sin jv. . (109) 84 ENGINEERING MATHEMATICS. Adding and subtracting gives respectively, cosp = , and sm ]v = (110) 2j -J By these equations, (106) to (110), exponential functions with imaginary exponents can be transformed into trigono- metric functions with real angles, and exponential functions with real exponents into trignometric functions with imaginary angles, and inversely. Mathematically, the trigonometric functions thus do not constitute a separate class of functions, but may be considered as exponential functions with imaginary angles, and it can be said broadly that the solution of the above differential equa- tions is given by the exponential function, but that in this function the exponent may be real, or may be imaginary, and in the latter case, the expression is put into real form by intro- ducing the trigonometric functions. EXAMPLE 1. 6o. A condenser (as an underground high-potential cable) of 20 mf. capacity, and of a voltage of eo = 10,000, discharges through an inductance of 50 mh. and of negligible resistance, What is the equation of the discharge current? The current consumed by a condenser of capacity C and potential difference e is proportional to the rate of change of the potential difference, and to the capacity; hence, it is de C—^, and the current from the condenser; or, its discharge CLl current, is --^l-- • (1") The voltage consumed by an inductance L is proportional to the rate of change of the current in the inductance, and to the inductance; hence, ^-4 • • • (112) POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 85 ■* Differentiating (112) gives, de dH and substituting this into (111) gives, dH dH 1 . '^-^^^ ^'' W^^'cV^ ' ' ' (1^^) as the differential equation of the problem. This equation (113) is the same as (102), for c^^ttTj thus is solved by the expression, and the potential difference at the condenser or at the inductance is, by substituting (114) into (112), These equations (114) and (115) still contain two unknown constants, A and B, which have to be determined by the terminal conditions, that is, by the known conditions of current and voltage at some particular time. At the moment of starting the discharge; or, at the time ^ = 0, the current is zero, and the voltage is that to which the condenser is charged, that is, -^ = 0, and e = eo. Substituting these values in equations (114) and (115) gives, fL 0 = A and eQ=^\-7^B] hence B = e^ \L and, substituting for A and B the values in (114) and (115), gives \C . t and t P=en COS — =^ ^ ° VCL (116) 86 ENGINEERING MATHEMATICS. Substituting the numerical values, ^0 = 10,000 volts, (7 = 20 mf.= 20X10- 6 farads, L = 50 mh.=0.05h. gives, J ^ = 0.02 and VC'L^IO-^; hence, i = 200 sin 1000 t and 6 = 10,000 cos 1000 t. 6 1. The discharge thus is alternating. ' In reality, due to the unavoidable resistance in the discharge path, the alterna- tions gradually die out, that is, the discharge is oscillating. The time of one complete period is given by, 1000^0=2;:; or, to=^. Hence the frenquency, /= — = —^ — = 159 cycles per second. As the circuit in addition to the inductance necessarily contains resistance r, besides the voltage consumed by the inductance by equation (112), voltage is consumed by the resistance, thus er = ri, . (117) and the total voltage consumed by resistance r and inductance L, thus is e = n + Lj^ (118) Differentiating (118) gives, dc di dH jr'dt'^^'Tt^^ -=r-+L—^ , ni9) and, substituting this into equation (111), gives, dt^^^dt + Cr^.+CL~=0, (120) as the differential equation of the problem. This differential equation is of the more general form, ("2^ 62. The more general differential equation {22% ^^ g+2^+«, + 6 = 0, (121) POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 87 can, by substituting, y+l-^, . ■ (122) which gives dy dz dx dx be transformed into the somewhat simpler form, S-^4>«-^ (123) It may also be solved by the method of indeterminate coefficients, by substituting for z an infinite series of powers of Xj and determining thereby the coefficients of the series. As, however, the simpler forms of this equation were solved by exponential functions, the applicability of the exponential functions to this equation (123) may be directly tried, by the method of indeterminate coefficients. That is, assume as solu- tion an exponential function, z = Ae-^', (124) where A and h are unls:nown constants. Substituting (124) into (123), if such values of A and h can be found, which make the substitution product an identity, (124) is a solution of the differential equation (123). From (124) it follows that, dz d^'^ y-=-bAE-^^] and ^ = hUe-^^, . . (125) dx ' d^x ' and substituting (124) and (125) into (123), gives, AB-^^{b^ + 2ch+a}=0 (126) As seen, this equation is satisfied for every value of x, that is, it is an identity, if the parenthesis is zero, thus, b^+2cb+a = 0, (127) and the value of b, calculated by the quadratic equation (127), thus makes (124) a solution of (123), and leaves A still undeter- mined, as integration constant. 88 ENGINEERING MATHEMATICS. From (127), h=-c±Vc^-a; (128) or, substituting, Vc^-a = p, (129) into (128), the equation becomes, b^-c±p (130) Hence, two values of h exist, hi=—c-\-p and 62=— c— p, . . . (131) and, therefore, the differential equation, g+2c|+««=0, (132) is solved by Ae^^^; or, by Ae^^*, or, by any combination of these two solutions. The most general solution is, that is, ^ . . . (131) h '■ = £-^^{^l£+P^+^2£~^'=! . As roots of a quadratic equation, 61 and 62 may both be real quantities, or may be complex imaginary, and in the latter case, the solution (131) appears in imaginary form, and has to be reduced or modified for use, so as to eliminate the imaginary appearance, by the relations (106) and (107). EXAMPLE. A 63. Assume, in the example in paragraph 9, the discharge circuit of the condenser of (7 = 20 mf. capacity, to contain, besides the inductance, L = 0.05 h, the resistance, r = 125 ohms. The general equation of the problem, (120), dividing by C L, becomes, dH r di i ^ ,,^^^ POTENTIAL SERIES' AND EXPONENTIAL FUNCTION, 89 This is the equation (123), for: x = t, 2c = 7=2500: z = i, a iCL) = 106 . . (133) If p = Vc^ — a, then and, writing P = 7: and since 2L' 2^ = 10 and ^ = 2500, s = 75 and p = 750. The equation of the current from (131) then is. i=.Aie 2l' +^^,. 2L (134) (135) (136) (137) This equation still contains two unknown quantities, the inte- gration constants Ai and A2, which are determined by the terminal condition : The values of current and of voltage at the beginning of the discharge, or ^ = 0. This requires the determination of the equation of the voltage at the condenser terminals . This obviously is the voltage consumed by resistance and inductance, and is expressed by equation (118), e = ri + Lj^; . (118) 90 ENGINEERING MATHEMATICS. di hence, substituting herein the value of i and -jt, from equation (137), gives (r-8 _L±f/l f r—s ^~^t r-\-s ^-^-y) = c 2L . LtlA,,^^U'^A.,s-^'\, (138) and, substituting the numerical values (133) and (136) into equations (137) and (138), gives (139) and, e = 100Ai£-^oof+25A2£-2ooo' J At the moment of the beginning of the discharge, ^ = 0, the current is zero and the voltage is 10,000; that is, t = 0; i = 0; c= 10,000 ..... .(140) Substituting (140) into (139) gives, 0=^1+^2, 10,000 = 100^1+25^2; hence, A2=-Ai] Ai=: 133.3; .42= -133.3. Therefore, the current and voltage are, t= 133.3 js-^^^^-s-^'^^^M, e=13,333c-soof-3333£-2ooo^ (142) The reader is advised to calculate and plot the numerical values of i and e, and of their two components, for, ^ = 0, 0.2, 0.4, 0.6, 1, 1.2, 1.5, 2, 2.5, 3, 4, 5, 6x10-^ sec. POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 91 64. Assuming, however, that the resistance of the discharge circuit is only r = 80 ohms (instead of 125 ohms, as assumed above : r^—-p in equation (134) then becomes —3600, and there- fore : s = \/-3600 = 60V-1 = 60/, and P=^=6ooy. The equation of the current (137) thus appears in imaginary form, i = £-8ooq^^£ + 6oo;< + ^2£-6ooyn^ . . . (143) The same is also true of the equation of voltage. As it is obvious, however, physically, that a real current must be coexistent with a real e.m.f., it follows that this imaginary form of the expression of current and voltage is only apparent, and that in reality, by substituting for the exponential functions with imaginary exponents their trigononetric expres- sions, the imaginary terms must ehminate, and the equation (ii6)- appear in real form. ''^According to equations (106) and (107), e+6ooyf_cos 600f +/ sin 600^; 1 (144) £-600?^ = cos 600^-/ sin 600^. J Substituting (144) into (143) gives, 1 = £-800^1 Bi cos 600^+^2 sin 600^!, . . (145) where Bi and B2 are combinations of the previous integration constants Ai and A2 thus, Bi = Ai+A2, and B2=j(Ai-A2). . . (146) By substituting the numerical values, the condenser e.m.f., given by equation (138), then becomes, e = £-800^1 (40 +30j)Ai(cos 600^ +/ sin 6000 + (40-30y)A2(cos 600i-/sin 6000 ! = £ - 800/ 1 (40Bi + 30B2)cos 600^ + (40^2 - 30Bi) sin 600^ \ . (147) 92 ENGINEERING MATHEMATICS. Since for i=0, i = 0 and e = 10,000 volts (140), substituting into (145) and (147), 0 = Bi and 10,000 = 40 Bi+SO B2. Therefore, Bi = 0 and 52 = 333 and, by (145) and (147), ?: = 333£-8oo^sin600i; 1 • . (148) e = 10,000£-8oo^ (cos 600 ^ + 1.33 sin 600 t. J As seen, in this case the current i is larger, and current and e.m.f. are the product of an exponential term (gradually decreasing value) and a trigonometric term (alternating value) ; that is, they consist of successive alternations of gradually decreasing amplitude. Such functions are called oscillating functions. Practically all disturbances in electric circuits consist of such oscillating currents and voltages. 600^ = 2;: gives, as the time of one complete period, and the frequency is ^ = ^ = 0.0105 sec; 600 ' /=-^ = 95.3 cycles per sec. In this particular case, as the resistance is relatively high, the oscillations die out rather rapidly. The reader is advised to calculate and plot the numerical values of i and e, and of their exponential terms, for every 30 T T T degrees, that is, for ^ = 0, -rx, 2 j^, 3 t^, etc., for the first two periods, and also to derive the equations, and calculate and plot the numerical values, for the same capacity, C = 20 mf., and same inductance, L = 0.05/i, but for the much lower resistance, r = 20 ohms. 65. Tables of e"*"^ and £~^, for 5 decimals, and tables of log e"^^ and log £~^, for 6 decimals, are given at the end of the book, and also a table of e~^ for 3 decimals. For most engineering purposes the latter is sufficient; where a higher accuracy is required, the 5 decimal table may be used, and for POTENTIAL SERIES AND EXPONENTIAL FUNCTION. 93 highest accuracy interpolation by the logarithmic table may be employed. For instance, ♦ g — 13.6847^9 From the logarithmic table, log £-10 =5.657055, log £-3 =8.697117, log £-0-6 =9.739423, * log £-0.08 = 9 965256, log £-0.0047 = 9.998133 added interpolated, between log £-0004 = 9993263, . and log £-0 005 = 9.997829), log £-13-6847 = 4.056984 = 0.056984 - 6. From common logarithmic tables, ^-13.6847 = 1.14021x10-6.