CHAPTER VIII ADMITTANCE, CONDUCTANCE, SUSCEPTANCE 48. If in a continuous-current circuit, a number of resistances, Ti, r2, ?'3, . . ., are connected in series, their joint resistance, R, is the sum of the individual resistances, K = ri + r2 + ra + . . . If, however, a number of resistances are connected in multiple or in parallel, their joint resistance, R, cannot be expressed in a simple form, but is represented by the expression 1 R = Ti n rz Hence, in the latter case it is preferable to introduce, instead of the term resistance, its reciprocal, or inverse value, the term conductance, g = ~- If, then, a number of conductances, 9iy Qij ds, • ' ' are connected in parallel, their joint conductance is the sum of the individual conductances, or G = gi -\- g2 -\- gs -\- . . . When using the term conductance, the joint con- ductance of a number of series-connected conductances becomes similarly a complicated expression gi gi gz Hence the term resistance is preferable in case of series con- nection, and the use of the reciprocal term conductance in parallel connections; therefore. The joint resistance of a number of series-connected resistances is equal to the sum of the individual resistances; the joint conduct- ance of a number of parallel-connected conductances is equal to the sum of the individual conductances. 64 ADMITTANCE, CONDUCTANCE, SUSCEPTANCE 55 49. In alternating-current circuits, instead of the term resist- ance we have the term impedance, Z = r -\- jx, with its two components, the resistance, r, and the reactance, x, in the formula of Ohm's law, E = IZ. The resistance, r, gives the component of e.m.f. in phase with the current, or the power component of the e.m.f., Ir; the reactance, x, gives the component of the e.m.f. in quadrature with the current, or the wattless component of e.m.f., Ix; both combined give the total e.m.f., Iz = iVr^ + x^. Since e.m.fs. are combined by adding their complex expressions, we have: The joint impedance of a number of series-connected impedances is the sum of the individual impedances, when expressed in com- plex quantities. In graphical representation impedances have not to be added, but are combined in their proper phase by the law of parallelo- gram in the same manner as the e.m.fs. corresponding to them. The term impedance becomes inconvenient, however, when dealing with parallel-connected circuits; or, in other words, when several currents are produced by the same e.m.f., such as in cases where Ohm's law is expressed in the form, / = I . Z It is preferable, then, to introduce the reciprocal of impe- dance, which may be called the admittance of the circuit, or Z As the reciprocal of the complex quantity, Z = r -{- jx, the admittance is a complex quantity also, or Y = g — jh; it con- sists of the component, g, which respresents the coefficient of current in phase with the e.m.f., or the power or active com- ponent, gE, of the current, in the equation of Ohm's law, I =YE ={g- jh)E, and the component, h, which represents the coefficient of current in quadrature with the e.m.f., or wattless or reactive component, hE, of the current. g is called the conductance, and h the susceptance, of the cir- cuit. Hence the conductance, g, is the power component, and 56 ALTERNATING-CURRENT PHENOMENA the susceptance, h, the wattless component, of the admittance, Y = g ~ jb, while the numerical value of admittance is y = Vg' + h^; the resistance, r, is the power component, and the reactance, X, the wattless component, of the impedance, Z = r -^ jx, the numerical value of impedance being z = Vr^ + x^. 50. As shown, the term admittance implies resolving the cur- rent into two components, in phase and in quadrature with the e.m.f., or the power or active component and the wattless or reactive component; while the term impedance implies resolving the e.m.f. into two components, in phase and in quadrature with the current, or the power component and the wattless or reactive component. It must be understood, however, that the conductance is not the reciprocal of the resistance, but depends upon the reactance as well as upon the resistance. Only when the reactance x = 0, or in continuous-current circuits, is the conductance the recip- rocal of resistance. Again, only in circuits with zero resistance (r = 0) is the susceptance the reciprocal of reactance; otherwise, the suscep- tance depends upon reactance and upon resistance. The conductance is zero for two values of the resistance: 1. If r = oo^ or a: = co ^ since in this case there is no current, and either component of the current = 0. 2. If r = 0, since in this case the current in the circuit is in quadrature with the e.m.f., and thus has no power component. Similarly, the susceptance, b, is zero for two values of the reactance: 1. If a; = 00, or r = oo . 2. Ux = 0. From the definition of admittance, Y = g — jb, as the recip- rocal of the impedance, Z = r -\- jx, we have \^ f or, multiplying numerator and denominator on the right side by (r - jx), h — r — jx ■ ^ ~ 3 - (r+jx) (r - jx)' ADMITTANCE, CONDUCTANCE, SUSCEPTANCE 57 hence, since (r + jx) (r - jx) = _ J.2 _j_ 2^2 = ^^ jb = r -^r X r 22 . aj J.2 _|_ -J.2 ■2 + a;2 •^2^ or 9 ^2 _^ 2-2 0 = o 1 o 2' and conversely r = J.2 g' + 62 6 2/' ^^ •" ~ ^2 _|_ 52 - ^2 By these equations, the conductance and susceptance can be calculated from resistance and reactance, and conversely. Multiplying the equations for g and r, we get fg . gr = -r^> hence, 22^2 = (^.2 _[_ 3.2) (-^2 -|- 52) = ;[. 11] the absolute value and ^ ~ 7. ~ y \/g2 _|_ 52 1 of impedance; 1 1 1 the absolute value y z -y/^.2 _(_ a;2 I of admittance. 51. If, in a circuit, the reactance, x, is constant, and the resistance, r, is varied from r = 0 to r = 00^ the susceptance, 6, decreases from 6 = - at r = 0, to 6 = 0 at r = 00 ; while the ' X conductance, g = 0 at r = 0, increases, reaches a maximum for r = X, where g = ^, is equal to the susceptance or g ^ h, and then decreases again, reaching gr = 0 at r = «» . In Fig. 49, for constant reactance x = 0.5 ohm, the variation of the conductance, g, and of the susceptance, h, are shown as functions of the varying resistance, r. As shown, the absolute value of admittance, susceptance, and conductance are plotted in full Unes, and in dotted line the absolute value of impedance, / 1 z = \/r2 + x2 = -• y 58 ALTERNATING-CURRENT PHENOMENA Obviously, if the resistance, r, is constant, and the reactance,* X, is varied, the values of conductance and susceptance are merely exchanged, the conductance decreasing steadily from ^ = - to 0, and the susceptance passing from 0 at x = 0 to the maximum, 6 = i7- = y = .^ata; = r, and to 6 = 0 at x = oo . The resistance, r, and the reactance, x, vary as functions of the conductance, g, and the susceptance, h, in the same manner as g and h vary as functions of r and x. OH It s a.o "^ i.y \ re; ct; NCI CO NS7 AN! = .! OH MS \ \ \ /• l.b \ \ \ / / l.b \ \ • ' 1.1 \ -P hs A •* l.J. \ 'y>! ^ [o" i.2 \ 4 k '^1 V 1.1 1.0 .9 .? .7 ,6 .5 .A ,3 ■X .1 \ A S" / / V ^' /N \ / \ y ' ^ ^ N / h^ ^> X \ ^ /> % V "" k f3 \ •^ /c i ? ^ ■ , ^ 8 —. RESISTAN de: •, OHMS 0 a _ 1 . i . a .-i 1 . & . 5 . 7 . s . 9 1 .0 i 1 1. i 1. i 1. 1 1. i 1. fi 1. 7 1.8 Fig. 49. The sign in the complex expression of admittance is always opposite to that of impedance; this is obvious, since if the cur- rent lags behind the e.m.f., the e.m.f. leads the current, and conversely. We can thus express Ohm's law in the two forms, E = /Z, / = EY, and therefore, ADMITTANCE, CONDUCTANCE, SUSCEPTANCE 59 The joint impedance of a number of series-connected impedances is equal to the sum of the individual impedances; the joint admit- tance of a number of parallel-connected admittances is equal to the sum of the individual admittances, if expressed in complex quantities. In diagrammatic representation, combination by the parallelogram law takes the place of addition of the complex quantities. 52. Experimentally, impedances and admittances are most conveniently determined by establishing an alternating current in the circuit, and measuring by voltmeter, ammeter and watt- meter, the volts, e, the amperes, i, and the watts, p. It is then, Impedance: z = -.' Resistance (effective) : r = Reactance: x = y/z^ — r^. 1 Admittance: y = — ^ e Conductance: o = -; ^ e'- Susceptance: b = \/y^ — g^. Regarding their calculation, see "Theoretical Elements of Electrical Engineering."