CHAPTER V SYMBOLIC METHOD 25. The graphical method of representing alternating-current phenomena affords the best means for deriving a clear insight into the mutual relation 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 rep- resented in the same diagram, which make an exact diagram- matic determination impossible. For instance, in the trans- former diagrams (c/. Figs. 18-20), the different magnitudes have numerical values in practice somewhat like the following: Ei = 100 volts, and 7i = 75 amp. For a non-inductive second- ary load, as of incandescent lamps, the only reactance of the secondaiy circuit thus is that of the secondary coil, or Xi = 0.08 ohms, giving a lag of ^i = 3.6°. We have also, rii = 30 turns. rio = 300 turns. Fi = 2250 ampere-turns. F =100 ampere-turns. Er = 10 volts. E:, = 60 volts. Ei = 1000 volts. Fig. 21. — Vector diagram of transformer. The corresponding diagram is shown in Fig. 21. Obviously, no exact numerical values can be taken from a parallelogram as flat as OFiFFo, and from the combination of vectors of the relative magnitudes 1 :6 :100. Hence the importance of the graphical method consists not 30 SYMBOLIC METHOD 31 so much in its usefulness for practical calculation as to aid in the simple understanding of the phenomena involved. 26. Sometimes we can calculate the numerical values trigo- nometrically by means of the diagram. Usually, however, this becomes too complicated, as will be seen by trying to calculate, from the above transformer diagram, the ratio of transformation. The primary m.m.f. is given by the equation Fo = VF2 + Fi2 _^ 2i^i^isin Bi, 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. 27. We have seen that the alternating sine wave is repre- sented in intensity, as well as phase, by a vector, 01, which is determined analytically by two numerical quantities — the length, 01, or intensity; and the amplitude, AOI, or phase, 6, of the wave, /. Instead of denoting the vector which repre- Fig. 22. sents the sine wave in the polar diagram by the polar coordinates, I and B, we can represent it by its rec- tangular coordinates, a and h (Fig. 22), where a = I cos B is the horizontal component, h = 7 sin ^ is the vertical component of the sine wave. This representation of the sine wave by its rectangular com- ponents is very convenient, in so far as it avoids the use of trigonometric functions in the combination or solution of sine waves. Since the rectangular components, a and h, are the horizontal and the vertical projections of the vector representing 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, hij adding, or subtracting, their rectangular corwponents. For instance, if a and b are the rectangular components of a sine wave, /, and a' and b' the components of another sine wave, 32 ALTERNA TING-C URREN T PHENOMENA I' (Fig. 23), their resultant sine wave, 7o, has the rectangular components a^ = (a + a'), and 60 = (& + h'). To get from the rectangular components, a and 6, of a sine wave its intensity, i, and phase, d, we may combine a and h by the parallelogram, and derive i = Va^ + 62. 6 tan 6 ^ — a Hence we can analytically operate with sine waves, as with forces in mechanics, by resolving them into their rectangular components. 28. To distinguish, however, the horizontal and the vertical com- ponents 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 mean- ingless symbol, as the letter j, and thus represent the sine wave by the expression 7 = a + jh, which now has the meaning that a is the horizontal and 6 the vertical component of the sine wave I, and that both components are to be combined in the resultant wave of intensity, and of phase. tan 6 i = Va2 + 62, h Similarly, a — jh means a sine wave with a as horizontal, and — 6 as vertical, components, etc. Obviously, the plus sign in the symbol, a -f jh, does not imply simple addition, since it connects heterogeneous quan- tities— horizontal and vertical components — but implies com- bination by the parallelogram law. For the present, j is nothing but a distinguishing index, and otherwise free for definition except that it is not an ordinary number. 29. A wave of equal intensity, and differing in phase from the wave, a -\- jh, by 180°, or one-half period, is represented in SYMBOLIC METHOD 33 Fig. 24. 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 means reversing the wave, or rotating it through 180°, or one- half period. A wave of equal intensity, but leading a + jb by 90°, or one-quarter period, has (Fig. 24) the horizontal component, — b, and the vertical component, a, and is represented symbolically by the expres- sion, ja — b. Multiplying, however, a -f jb by j, we get ja -h j-b; therefore, if we define the heretofore meaningless symbol, j, by the condition, j' = - 1, we have j{a + jb) ^ ja - b; hence, Multiplying the symbolic expression, a + jb, of a siyie wave by j means rotating the wave through 90°, or one-quarter period; that is, leading the wave by one-quarter period. Similarly — Multiplying by — j jneans lagging the wave by one-quarter period. Since j^ = - 1, it is j = v^^=n:; and j is the imaginary unit, and the sine wave is represented by a complex imaginary quantity or general number, a ^- jb. As the imaginary unit, j, has no numerical meaning in the system of ordinary numbers, this definition of j = V — 1 does not contradict its original introduction as a distinguishing index. For the Algebra of Complex Quantities see Appendix I. For a more complete discussion thereof see " Engineering Mathematics." 30. In the vector diagram, the sine wave is represented in intensity as well as phase by one complex quantity, a + jb, 3 34 ALTERNATING-CURRENT PHENOMENA where a is the horizontal and h the vertical component of the wave; the intensity is given by i = Va2 + 62, the phase by tan 6 = — a and a = i cos 6, b = i sin 6] hence the wave, a + jh, can also be expressed by ?'(cos d -\- j sin d), or, by substituting for cos 6 and sin 6 their exponential expres- sions, we obtain Since we have seen that sine waves may be combined or resolved by adding or subtracting their rectangular components, consequently, Sine waves may be combined or resolved by adding or subtracting their complex algebraic expressions. For instance, the sine waves, a + jb and a' + jb', combined give the sine wave, I = {a + a') +j(6 + 6'). It will thus be seen that the combination of sine waves is reduced to the elementary algebra of complex quantities. 31. If / = I + ji' is a sine wave of alternating current, and r is the resistance, the voltage consumed by the resistance is in phase with the current, and equal to the product of the current and resistance. Or rl = ri -\- jri'. If L is the inductance, and x = 2x/L the inductive react- ance, the e.m.f. produced by the reactance, or the counter e.m.f. 1 In this representation of the sine wave by the exponential expression of the complex quantity, the angle 0 necessarily must be expressed in radians, and not in degrees, that is, with one complete revolution or cycle as 2 tt. or 180 with — = 57.3° as unit. SYMBOLIC METHOD 35 of self-induction, is the product of the current and reactance, and lags in phase 90° behind the current; it is, therefore, repre- sented by the expression — jxl = — jxi -\- xi'. The voltage required to overcome the reactance is consequently 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 — xi' . Hence, the voltage required to overcome the resistance, r, and the reactance, x, is {r -\- jx)I; that is, Z = r •\- jx is the expression of the impedance of t he circuit in complex quantities. Hence, if / = ^ + ji' is the current, the voltage required to overcome the impedance, Z = r -\- jx, is E ^ ZI = {r+ jx) {i + ji') = {ri + j^xi') -\- j{ri' + xi) ; hence, since j^ = — 1 E = (ri — xi') + j(ri' + xi) ; or, ii E = e -\- je' is the impressed voltage and Z = r -\- jx the impedance, the current through the circuit is I _^ _e^je'_ Z r + jx' or, multiplying numerator and denominator by (r — jx) to eliminate the imaginary from the denominator, we have Y _ (e -\- je') (r — jx) _er -\- e'x . e'r — ex ^ or, if £" = e + je' is the impressed voltage and 7 = t + ji' the current in the circuit, its impedance is jE ^ e + je' ^ (e + je') (i - ji') ^ ei + e'i' . e'i - ei' I i + ji' i^ -F i'^ i^ -\- i"- "^ ^ i- + i'^ ' 32. If C is the capacity of a condenser in series in a circuit in which exists a current I = i + ji' , the voltage impressed upon the terminals of the condenser is E = ^ .^, 90° behind the cur- 36 ALTERNATING-CURRENT PHENOMENA ji rent; and may be represented by — o'— 779 or — jxj, where Ztt/u ^1 ~ o — Tr* i^ ^^6 condensive reactance or condensance of the Z irjL condenser. Condensive reactance is of opposite sign to inductive reactance; both may be combined in the name reactance. We therefore have the conclusion that If r = resistance and L — inductance, thus X = 2 TcJL = inductive reactance. If C = capacity, Xi = ^ — 77-, = condensive reactance, Z — r -{- j(x — Xi) is the impedance of the circuit. Ohm's law is then re-established as follows: E = ZI, I = y, Z = -J- The more general form gives not only the intensity of the wave but also its phase, as expressed in complex quantities. 33. Since the combination of sine waves takes place by the addition of their symbolic expressions, Kirchhoff's laws are now re-established in their original form: (a) The sum of all the e.m.fs. acting in a closed circuit equals zero, if they are expressed by complex quantities, and if the resistance and reactance e.m.fs. are also considered as counter e.m.fs. {h) The sum of all the currents directed toward a distributing point is zero, if the currents are expressed as complex quantities. If a complex quantity equals zero, the real part as well as the imaginary part must be zero individually; thus, if a + i6 = 0, a = 0, 6 = 0. Resolving the e.m.fs. and currents in the expression of Kirch- hoff'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 resistance and reactance are represented as counter e.m.fs. (6) The sum of the components, in any direction, of all the currents at a distributing point equals zero. Joule's law and the power equation do not give a simple expression in complex quantities, since the effect or power is SYMBOLIC METHOD 37 a quantity of double the frequency of the current or e.m.f. wave, and therefore requires for its representation as a vector a transition from single to double frequency, as will be shown in Chapter XVI. In what follows, complex vector quantities will always be denoted by dotted capitals when not written out in full; abso- lute quantities and real quantities by undotted letters. 34. Referring to the example given in the fourth chapter, of a circuit supplied with a voltage, E, and a current, I, 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 voltage at the beginning of the line, or at the generator, the expression Eo = E -\- ZI. Assuming now again the current as the zero line, that is, I = i, we have in general Eo = E -i- ir + jix] hence, with non-inductive load, or E = e, Eo =- (e ■{■ ir) + jix, or e.o = \/(e + ir)- + {ix^, tan ^o = — 77^- In a circuit with lagging current, that is, with leading e.m.f., E = e -{- je', and Eo = e -{- je' + (r + jx)i = (e 4- ir) + j(e' + ix), 6 ~t~ ix or Co = V (e + ir)^ + (e' + ix)^ tan do = . • In a circuit with leading current, that is, with lagging e.m.f., E = e — je', and Eo = (e - je') + (r 4- jx)i = (e + ir) — j{e' - ix), or eo = V{e + ir)^ -{- {e' — ixY, tan Qo = T^"' values which easily permit calculation. 35. When transferring from complex quantities to absolute values, it must be kept in mind that: The absolute value of a product or a ratio of complex quanti- ties is the product or ratio of their absolute values. 38 ALTERNATING-CURRENT PHENOMENA The phase angle of a product or a ratio of complex quantities is the sum or difference of their phase angles. That is, if A = a' -\- ja" = a (cos a -\- j sin a) B = 6' + jh" = 6(cos 13 + j sin /3) 0 = 0'+ jc" = c(cos T + j sin 7) AB . ah the absolute value of ^ is given by — ' and its phase angle by O c a + /3 — 7, that is, it is AB ab ^ = —[cos (a + |3 - 7) + j sin (a + |8 - 7)], where are the absolute values of A, 5 and C. This rule frequently simplifies greatly the derivation of the absolute value and phase angle, from a comphcated complex expression.