CHAPTER VII POLAR COORDINATES AND POLAR DIAGRAMS 42. The graphic representation of alternating waves in rec- tangular coordinates, with the time as abscissae and the instan- taneous values as ordinates, gives a picture of their wave structure, as shown in Figs. 1 to 5. It does not, however, show their periodic character as well as the representation in polar coordi- nates, with the time as the angle or the amplitude — one complete period being represented by one revolution — and the instan- taneous values as radius vectors; the polar coordinate system, in which the independent variable, the angle, is periodic, obvi- ously lends itself better to the representation of periodic functions, as alternating waves. Thus the two waves of Figs. 2 and 3 are represented in polar coordinates in Figs. 36 and 37 as closed characteristic curves, which, by their intersection with the radius vector, give the instantaneous value of the wave, corresponding to the time represented by the amplitude or angle of the radius vector. These instantaneous values are positive if in the direction of the radius vector, and negative if in opposition. Hence the two half-waves in Fig. 2 are represented by the same polar characteristic curve, which is traversed by the point of intersection of the radius vector twice per period — once in the direction of the vector, giving the positive half-wave, and once in opposition to the vector, giving the negative half-wave. In Figs. 3 and 37 where the two half-waves are different, they give different polar characteristics. 43. The sine wave. Fig. 1, is represented in polar coordinates by one circle, as shown in Fig. 38. The diameter of the char- acteristic curve of the sine wave, / = OC, represents the intensity of the wave; and the amplitude of the diameter OC, ^ 0o = AOC, is the phase of the wave, which, therefore, is represented analytic- ally by the function 1 = 1 cos {0 — ^o), 46 POLAR COORDINATES AND POLAR DIAGRAMS 47 where 0 = 2ir — is the instantaneous value of the ampHtude fo corresponding to the instantaneous value, i, of the wave. The instantaneous values are cut out on the movable radius vector by its intersection with the characteristic circle. Thus, for instance, at the amplitude, AOBi = 6i = 2w- (Fig- 38), the to instantaneous value is OB'; at the amphtude, AOB2 = 62 = 2 Try, the instantaneous value is OB", and negative, since in to opposition to the radius vector, OB^. The angle, 0, so represents the time, and increasing time is represented by an increase of angle B in counter-clockwise rota- FiG. 37 tion. That is, the positive direction, or increase of time, is chosen as counter-clockwise rotation, in conformity with general custom. The characteristic circle of the alternating sine wave is deter- mined by the length of its diameter — ^the intensity of the wave; and by the amplitude of the diameter — the phase of the wave. Hence wherever the integral value of the wave is considered alone, and not the instantaneous values, the characteristic circle may be omitted altogether, and the wave represented in intensity and in phase by the diameter of the characteristic circle. Thus, in polar coordinates, the alternating wave may be repre- sented in intensity and phase by the length and direction of- a vector, OC, Fig. 38, and its analytical expression would then be c = OC cos (0 - 0o). This leads to a second vector representation of alternating 48 ALTERNATING-CURRENT PHENOMENA waves, differing from the crank diagram discussed in Chapter IV. It may be called the time diagram or polar diagram, and is used to a considerable extent in the literature, thus must be familiar to the engineer, though in the following we shall in graphic representation and in the symbolic representation based thereon, use the crank diagram of Chapters IV and V. In the time diagram as well as in the crank diagram, instead of the maximum value of the wave, the effective value, or square root of mean square, may be used as the vector, which is more convenient ; and the maximum value is then \/2 times the vector OC, so that the instantaneous values, when taken from the dia- gram, have to be increased by the factor v^- Thus, the wave, b = Bcos2 Tfit - h) = B cos (d - 9i), is, in Fig. 39, represented by vector OB = B V2 of phase and the wave, AOB = di] = C cos 2 Tvfit + t2} = C cos {e + ^2) is, in Fig. 39, represented by vector OC = AOC = - C V2 of phase The former is said to lag by angle di, the latter to lead by angle 02, with regard to the zero position. The wave b lags by angle (^i + ^2) behind wave c, or c leads b by angle {di + ^2). 44. To combine different sine waves, their graphical repre- sentations, or vectors, are combined by the parallelogram law. From the foregoing considerations we have the conclusions: The sine wave is represented graphically in polar coordinates by a vector, which by its length OC, denotes the intensity, and by its amplitude, AOC, the phase, of the sine wave. Sine waves are combined or resolved graphically, in polar coordinates, by the law of the parallelogram or the polygon of sine waves. (Fig. 40.) POLAR COORDINATES AND POLAR DIAGRAMS 49 Kirchhoff's laws now assume, for alternating sine waves, the form : (o) The resultant of all the e.m.fs. in a closed circuit, as found by the parallelogram of sine waves, is zero if the counter e.m.fs. of resistance and of reactance are included. (6) The resultant of all the currents toward a distributing point, as found by the parallelo- gram of sine waves, is zero. The power equation expressed graphically is as follows: The power of an alternating- current circuit is represented in polar coordinates by the product of the current, I, into the projec- tion of the e.m.f., E, upon the current, or by the e.m.f., E, into the projection of the current, /, upon the e.m.f., or by IE cos d, where 9 = angle of time- phase displacement. 45. The instances represented by the vector representation of the crank diagram in Chapter IV as Figs. IG, 17, 18, 19, 20, ^i Fig. 41. Fig. 42. then appear in the vector representation of the time diagram or polar coordinate diagram, in the form of Figs. 41, 42, 43, 44, 45. These figures are the reverse, or mirror image of each other. That is, the crank diagrams, turned around the horizontal (or any other axis) , so as they would be seen in a mirror, are the time diagrams, and inversely. 4 50 ALTERNATING-CURRENT PHENOMENA The polar diagram, Fig. 46, of a current: i = I cos' (?9 - 0) represented by vector 01, E^-^ Fig. 43. Fig. 45. lagging behind the voltage: e = £' cos (?? — a) represented by vector OE, by angle e^ = ^ - a then means: Fig. 46. POLAR COORDINATES AND POLAR DIAGRAMS 51 The voltage e reaches its maxunum at the time ti, which is represented by angle a = 2 x— ' where to = period, and the cur- fo rent, i, reaches its maximum at the time (2, which is represented by U . angle /? = 2 wf, and since (3 > a, the current reaches its maximum to at a later time than the voltage, that is, lags behind the voltage, and the lag of the current behind the voltage is the difference between the times of their maxima, /? and a, in angular measure, that is, is fo At any moment of time t, represented by angle 9 = 2 tt— ' the in- to stantaneous values of current and voltage, i and e, are the projec- tions of 01 and OE on the time radius OX drawn under angle AOX = d. The crank diagram corresponding to the time diagram Fig. 46 is shown in Fig. 47. It means: The vectors 01 and OE, representing the current and the voltage respectively, rotate synchronously, and by their projections on the horizontal OA represent the instantaneous values of current and voltage. Angle lOA = /3 being larger than angle EOA = a, the current vector 01 passes its maximum, in position OA, later than the voltage vector OE, that is, the current lags behind the voltage, by the difference of time corresponding to the passage of the current and voltage vectors through their maxima, in the direc- tion OA, that is, by the time angle di = ^ — a. A polar diagram, Fig. 46, with the current, 01, lagging behind the voltage, OE, by the angle, di, thus considered as crank dia- gram would represent the current leading the voltage by the angle, 61, and a crank diagram. Fig. 47, with the current lagging behind the voltage by the angle, di, would as polar diagram represent a current leading the voltage by the angle, ^1. 46. The main difference in appearance between the crank dia- gram and the polar diagram therefore is that, with the same direction of rotation, lag in the one diagram is represented in the same manner as lead in the other diagram, and inversely. Or, a representation by the crank diagram looks like a representation by the polar diagram, with reversed direction of rotation, and vice versa. Or, the one diagram is the image of the other and can 62 ALTERNATING-CURRENT PHENOMENA be transformed into it by reversing right and left, or top and bottom. So the crank diagram, Fig. 47, is the image of the polar diagram, Fig. 46. In symbolic representation, based upon the crank diagram, the impedance was denoted by Z = r -\- jx, where x == inductive reactance. In the polar diagram, the impedance thus is denoted by: Z = r — jx since the latter is the mirror image of the crank diagram, that is, differs from it symbolically by the interchange of + j and — j. A treatise written in the symbolic repre- sentation by the polar diagram, thus can be translated to the representation by the crank diagram, and inversely, by simply reversing the signs of all imaginary quantities, that is, considering the signs of all terms with j Fig. 47. changed. A graphical representation in the polar dia- gram can be considered as a graphic representation in the crank diagram, with clockwise or right-handed rotation, and inversely. Thus, for the engineer familiar with one representation only, but less familiar with the other, the most convenient way when meet- ing with a treatise in the, to him, unfamiliar representation is to consider all the diagrams as clockwise and all the signs of j reversed. In conformity with the recommendation of the Turin Congress — however ill considered this may appear to many engineers — in the following the crank diagram will be used, and wherever conditions require the time diagram, the latter be translated to the crank diagram. It is not possible to entirely avoid the time diagram, since the crank diagram is more limited in its application. 47. The crank diagram offers the disadvantage, that it can be applied to sine waves only, while the polar diagram permits the construction of the curve of waves of any shapes, as those in Figs. 36 and 37. In most cases, this objection is not serious, and in the diagram- matic and symbolic representation, the alternating quantities can be assumed as sine waves, that is, the general wave repre- sented by the equivalent sine wave, that is, the sine wave of the same effective value as the general wave. POLAR COORDINATES AND POLAR DIAGRAMS 53 The transformation of the general wave into the equivalent sine wave, however, has to be carried out algebraically in the crank diagram, while the polar diagram permits a graphical transformation of the general wave into the equivalent sine wave. Let Fig. 48 represent a general alternating wave. An element BiOBo of this wave then has the area and the curve is total area 2 of the polar A = I '7,dd. The effective value of the wave is X2 ,r o 2' R = -s/mean square hence, ^£^■ 1 K \ w ^^9^ f, ^ ^^^^ ^ Fig. 48. R' If r'-d9 = A. The area of the polar curve of the general periodic wave, as measured by planimeter, therefore equals the area of a circle with the effective value of the wave as radius. The effective value of the equivalent sine wave therefore is the radius of a circle having the same area as the general wave, in polar coordinates: The diameter of the general polar circle, therefore, is RV2 = A \ TT And the phase of the equivalent sine wave, or the direction of the diameter of its polar circle, is the vector bisecting the area of the general wave, in polar coordinates. The transformation of the general alternating wave into the equivalent sine wave, therefore, is carried out by measuring the area of the general wave in polar coordinates, and drawing the sine wave circle of half this area. SECTION ir CIRCUITS