CHAPTER V. SYMBOLIC METHOD. 23. The graphical method of representing alternating, current phenomena by polar coordinates of time affords the best means for deriving a clear insight into the mutual rela- tion of the different alternating sine waves entering into the problem. For numerical calculation, however, the graphical method is generally not well suited, owing to the widely different magnitudes of the alternating sine waves repre- sented in the same diagram, which make an exact diagram- matic determination impossible. For instance, in the trans- former diagrams (cf. Figs. 18-20), the different magnitudes will have numerical values in practice, somewhat like El — 100 volts, and 1-^ = 75 amperes, for a non-inductive secon- dary load, as of incandescent lamps. Thus the only reac- tance of the secondary circuit is that of the secondary coil, or, x-^ = .08 ohms, giving a lag of ^ = 3.6°. We have also, n^ = 30 turns. n0 = 300 turns. CFi = 2250 ampere-turns. y = 100 ampere-turns. Er = 10 volts. JSX = 60 volts. E{ = 1000 volts. The corresponding diagram is shown in Fig. 21. Obvi- ously, no exact numerical values can be taken from a par- allelogram as flat as OF1FF0^ and from the combination of vectors of the relative magnitudes 1:6: 100. Hence the importance of the graphical method consists 34 ALTERNA TING-CURRENT PHENOMENA. not so much in its usefulness for practical calculation, as to aid in the simple understanding of the phenomena involved. 24. Sometimes we can calculate the numerical values trigonometrically by means of the diagram. Usually, how- ever, this becomes too complicated, as will be seen by trying Fig. 21. to calculate, from the above transformer diagram, the ratio of transformation. The primary M.M.F. is given by the equation : — ffo = Vfr2 + S^2 + 20^ sin Wi, an expression not well suited as a starting-point for further calculation. A method is therefore desirable which combines the exactness of analytical calculation with the clearness of the graphical representation. Fig. 22. 25. We have seen that the alternating sine wave is represented in intensity, as well as phase, by a vector, Of, which is determined analytically by two numerical quanti- ties — the length, Of, or intensity ; and the amplitude, AOf, or phase <3, of the wave, /. Instead of denoting the vector which represents the sine wave in the polar diagram by the polar coordinates, S YMB OL1C ME T11OD. 35 / and <3, we can represent it by its rectangular coordinates, a and b (Fig. 22), where — a = fcos u> is the horizontal component, b = I sin co is the vertical component of the sine wave. This representation of the sine wave by its rectangular components is very convenient, in so far as it avoids the use of trigonometric functions in the combination or reso- lution of sine waves. Since the rectangular components a and b are the hori- zontal and the vertical projections of the vector represent- ing the sine wave, and the projection of the diagonal of a parallelogram is equal to the sum of the projections of its sides, the combination of sine waves by the parallelogram law is reduced to the addition, or subtraction, of their rectangular components. That is, Sine waves are combined, or resolved, by adding, or subtracting, their rectangular components. For instance, if a and b are the rectangular components of a sine wave, /, and a' and b' the components of another sine wave, /' (Fig. 23), their resultant sine wave, I0, has the rectangular components a0 — (a -f- a!}, and b0 = (b -f- b'}. To get from the rectangular components, a and b, of a sine wave, its intensity, i, and phase, o>, we may combine a and b by the parallelogram, and derive, — tan 36 AL TERN A TING-CURRENT PHENOMENA . Hence we can analytically operate with sine waves, as with forces in mechanics, by resolving them into their rectangular components. 26. To distinguish, however, the horizontal and the ver- tical components of sine waves, so as not to be confused in lengthier calculation, we may mark, for instance, the vertical components, by a distinguishing index, or the addition of an otherwise meaningless symbol, as the letter /, and thus represent the sine wave by the expression, — I=a which now has the meaning, that a is the horizontal and b the vertical component of the sine wave /; and that both components are to be combined in the resultant wave of intensity, — _ / = V^ + //2, and of phase, tan <3 = b / a. Similarly, a —jb, means a sine wave with a as horizon- tal, and — b as vertical, components, etc. Obviously, the plus sign in the symbol, a -f- jb, does not imply simple addition, since it connects heterogeneous quan- tities — horizontal and vertical components — but implies combination by the parallelogram law. For the present,/ is nothing but a distinguishing index, and otherwise free for definition except that it is not an .ordinary number. 27. A wave of equal intensity, and differing in phase from the wave a + jb by 180°, or one-half period, is repre- sented in polar coordinates by a vector of opposite direction, and denoted by the symbolic expression, — a — jb. Or — Multiplying the symbolic expression, a + jb, of a sine wave by — 1 weans reversing' the wave, or rotating it through 180°, or one-half period. A wave of equal intensity, but lagging 90°, or one- quarter period, behind a -f jb, has (Fig. 24) the horizontal SYMBOLIC METHOD. 37 component, — b, and the vertical component, a, and is rep- resented symbolically by the expression, ja — b, Multiplying, however, a + jb by/, we get : — therefore, if we define the heretofore meaningless symbol, j, by the condition, — y2 = - i, we have — /(*+/*) =ja — 1>; hence : — Multiplying the symbolic expression, a -\- jb, of a sine wave by j means rotating the wave through 90°, or one-quarter pe- riod ; tJiat is, retarding the wave through one-quarter period. Fig. 24. Similarly, — Multiplying by — j means advancing the wave through one-quarter period. since y'2 = — 1, j = V— 1 ; that is, — j is the imaginary unit, and the sine wave is represented by a complex imaginary quantity, a -+- jb. As the imaginary unit j has no numerical meaning in the system of ordinary numbers, this definition of/ = V— 1 does not contradict its original introduction as a distinguish- ing index. For a more exact definition of this complex imaginary quantity, reference may be made to the text books of mathematics. 28. In the polar diagram of time, the sine wave is represented in intensity as well as phase by one complex quantity — 38 ALTERNATING-CURRENT PHENOMENA. where a is the horizontal and b the vertical component of the wave ; the intensity is given by — the phase by — tan -\-j sin <3), or, by substituting for cos w and sin w their exponential expressions, we obtain — id™. Since we have seen that sine waves may be combined or resolved by adding or subtracting their rectangular com- ponents, consequently : — Sine waves may be combined or resolved by adding or subtracting their complex algebraic expressions. For instance, the sine waves, — a +jb and combined give the sine wave — 7- (a + It will thus be seen that the combination of sine waves is reduced to the elementary algebra of complex quantities. 29. If /= i +/z' is a sine wave of alternating current, and r is the resistance, the E.M.F. consumed by the re- sistance is in phase with the current, and equal to the prod- uct of the current and resistance. Or — rl ' — ri -\- jri' . If L is the inductance, and x = 2 TT NL the reactance, the E.M.F. produced by the reactance, or the counter SYMBOLIC METHOD. 39 E.M.F. of self-induction, is the product of the current and reactance, and lags 90° behind the current ; it is, therefore, represented by the expression — The E.M.F. required to overcome the reactance is con- , sequently 90° ahead of the current (or, as usually expressed,-** the current lags 90° behind the E.M.F.), and represented by the expression — — jxl = — jxi -f- xi'. Hence, the E.M.F. required to overcome the resistance, r, and the reactance, x, is — that is — Z = r — jx is the expression of the impedance of the cir- cuit, in complex quantities. Hence, if / = i -\-ji' is the current, the E.M.F. required to overcome the impedance, Z = r — jx, is — hence, sincey"2 = — 1 or, if E = e -\- je' is the impressed E.M.F., and Z = r — jx the impedance, the current flowing through the circuit is : — or, multiplying numerator and denominator by (r+jx) to eliminate the imaginary from the denominator, we have — T _ or, if E = e -\-je' is the impressed E.M.F., and 7 = i ' -\- ji' the current flowing in the circuit, its impedance is — 0 +./>') O'-./*'') «'+^*'' . ' ~ ei' ' 40 ALTERNATING-CURRENT PHENOMENA. 30. If C is the capacity of a condenser in series in a circuit of current I = i + //', the E.M.F. impressed upon the terminals of the condenser is E = - - , 90° behind the current ; and may be represented by — - - , or jx^ /, where x^ = - is the capacity reactance or condensatice 2 TT NC of the condenser. Capacity reactance is of opposite sign to magnetic re- actance ; both may be combined in the name reactance. We therefore have the conclusion that If r = resistance and L = inductance, then x = 2 IT NL = magnetic reactance. If C = capacity, x^ = - = capacity reactance, or conden- sance ; Z = r — j (x — JCi), is the impedance of the circuit Ohm's law is then reestablished as follows : , -, . The more general form gives not only the intensity of the wave, but also its phase, as expressed in complex quantities. 31. Since the combination of sine waves takes place by the addition of their symbolic expressions, Kirchhoff's laws are now reestablished in their original form : — a.} The sum of all the E.M.Fs. acting in a closed cir- cuit equals zero, if they are expressed by complex quanti- ties, and if the resistance and reactance E.M.Fs. are also considered as counter E.M.Fs. b.) The sum of all the currents flowing towards a dis- tributing point is zero, if the currents are expressed as complex quantities. SYMBOLIC METHOD. 41 If a complex quantity equals zero, the real part as well as the imaginary part must be zero individually, thus if a +jb = 0, a = 0, b = 0. Resolving the E.M.Fs. and currents in the expression of Kirchhoff 's law, we find : — a.} The sum of the components, in any direction, of all the E.M.Fs. in a closed circuit, equals zero, 'if the resis- tance and reactance are considered as counter E.M.Fs. b.} The sum of the components, in any direction, of all the currents flowing to a distributing point, equals zero. Joule's Law and the energy equation do not give a simple expression in complex quantities, since the effect or power is a quantity of double the frequency of the current or E.M.F. wave, and therefore requires for its representa- tion as a vector, a transition from single to double fre- quency, as will be shown in chapter XII. In what follows, complex vector quantities will always be denoted by dotted capitals when not written out in full ; absolute quantities and real quantities by undotted letters. 32. Referring to the instance given in the fourth chapter, of a circuit supplied with an E.M.F., E, and a cur- rent, 7, over an inductive line, we can now represent the impedance of the line by Z = r — jx, where r = resistance, x = reactance of the line, and have thus as the E.M.F. at the beginning of the line, or at the generator, the expression — E0 = E + ZI. Assuming now again the current as the zero line, that is, / = /, we have in general — E0 = E -f ir —jix ; hence, with non-inductive load, or E = e, E0=(e + ir) -jix, + /r)2 + (/X)2, tan S>0 = 42 ALTERNATING-CURRENT PHENOMENA. In a circuit with lagging current, that is, with leading E.M.F., E = e -je', and *-*)2> tan In a circuit with leading current, that is, with lagging E.M.F., E = * +>', and — /V) , tan w0 = values which easily permit calculation. TOPOGRAPHIC METHOD. 43