CHAPTER VII. ADMITTANCE, CONDUCTANCE, SUSCEFTANCE. 38. If in a continuous-current circuit, a number of resistances, rj, rj, rg, . . . are connected in series, their joint resistance, Ry is the sum of the individual resistances ^ = ^1 + ^2 + 'a + • • • 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 : — rx n r^ Hence, in the latter case it is preferable to introduce, in- stead of the term resistance^ its reciprocal, or inverse value, the term conductance^ g =\ J r. If, then, a number of con- ductances, gxy g%i g^y . . . are connected in parallel, their joint conductance is the sum of the individual conductances, ox G = g^ -\- g^ -\- g^ -\- . . . When using the term con- ductance, the joint conductance of a number of series- connected conductances becomes similarly a complicated expression — 6^= . i + i- + i + ... g\ gi it Hence the term resistance is preferable in case of series connection, and the use of the reciprocal term conductance in parallel connections ; therefore, The joint resistance of a number of series -connected resis- tances is equal to the sum of the individual resistances ; the § 30] ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 53 joint conductance of a number of parallel-connected conduc- tances is equal to the sum of the individual conductances, 39. In alternating-current circuits, instead of the term resistance 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 energy component of the E.M.F., Ir\ the reactance, Xy 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 = lWr' + x\ Since E.M.Fs. are combined by adding their complex ex- pressions, we have : The joint impedance of a number of series-connected impe- dances is the sum of the individual impedances , when expressed in complex quantities. In graphical representation impedances have not to be added, but are combined in their proper phase by the law of parallelogram in the same manner as the E.M.Fs. corre- sponding to them. The term impedance becomes incon- venient, however, when dealing with parallel-connected circuits ; or, in other words, when several currents are pro- duced by the same E.M.F., such as in cases where Ohm's law is expressed in the form, -?• It is preferable, then, to introduce the reciprocal of impedance, which may be called the admittance of the circuit, or Z As the reciprocal of the complex quantity, Z =^ r — jxy the admittance is a complex quantity also, or 64 AL TERN A TING-CURRENT PHENOMENA . [ § 40 it consists of the component g^ which represents the co- efficient of current in phase with the E.M.F., or energy current, gE, in the equation of Ohm's law, — and the component ^, which represents the coefficient of current in quadrature with the K.M.F., or wattless com- ponent of current, bE, g may be called the conductance^ and b the susceptanccy of the circuit. Hence the conductance, g^ is the energy component, and the susceptance, by the wattless component, of the admittance, Y = g -\-jby while the numerical value of admittance is — the resistance, r, is the energy component, and the reactance^ Xy the wattless component, of the impedance, Z = r — jx\ the numerical value of impedance being — 40. As shown, the term admittance implies resolving the current into two components, in phase and in quadra- ture with the E.M.F., or the energy current and the watt- less current ; while the term impedance implies resolving the E.M.F. into two components, in phase and in quad- rature with the current, or the energy E.M.F. and the wattless E.M.F. It must be understood, however, that the conductance is not the reciprocal of the resistance, but depends upon the resistance as well as upon the reactance. Only when the reactance ^ = 0, or in continuous-current circuits, is the conductance the reciprocal of resistance. Again, only in circuits with zero resistance (r = 0) is the susceptance the reciprocal of reactance ; otherwise, the susceptance depends upon reactance and upon resistance. The conductance is zero for two values of the resistance : — 1.) If r = oc , or ^ = 00 , since in this case no current passes, and either component of the current = 0. §40] ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 55 2.) If r = 0, since in this case the current which passes through the circuit is in quadrature with the E.M.F., and thus has no energy component. Similarly, the susceptance, ^, is zero for two values of the reactance : — 1.) If ^ = Qo , or r = 00 . 2.) If.r=0. From the definition of admittance, Y ^ g ■\- jh^ as the reciprocal of the impedance, Z = r —jxy we have F = -^ , or, ^ +jb = r — jx or, multiplying numerator and denominator on the right side by {r +jx) hence, smce (r — jx) (r + Jx) = r^ + x^ = z\ g^ ' '' b== and conversely r» + A-^ z' X __ x_ r» + x' "" z^ g' -f l^' / ' x = —^^— = - g' + i^' y 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 : — ^g hence, zY = (^' + ^ (^' + ^^ = 1 ; and z = 1 = 1 I the absolute value of y -y/gi _|- ^i * ) impedance ; __ 1 __ 1 ) the absolute value of z -y/,.2 _|_ x'^ ' ) admittance. 6G AL TERNA TING-CURKENT PHENOMENA. [§41 41. If, in a circuit, the reactance, x, is constant, and the resistance, r, is varied from r = to r =■ ao , the susceptance, b, decreases from b = \ I x sX. r = 0, toi = at r=oo; while the conductance, ^ = at r=0, increases, reaches a maximum for r =x, where ^ = 1 / 2r is equal to the susceptance, or g = b, and then decreases again, reaching r ■= at r = 00 , Wll ""^N " \\ I %-'\ ■i" I \\ ". \^^ - \^ ^ u ^ %. a ^ s^ ^^ 3 ^^^ ^ 5"--^' ^ / ^^ '^^^^ " ' m ^5^^ ^ --^ '\ "s-^ .-%■' %, ~-^ = ;rl ^ i X ^- t ""-- 1 n BI8■^^ ii: '.Ol hs In Fig. 86, for constant reactance x = .5 ohm, the vari- ation of the conductance, g, and of the susceptance, b, are shown as functions of the varying resistance, r. As shown, the absolute value of admittance, susceptance, and conduc- tance are plotted in full lines, and in dotted line the abso- lute value of impedance. §41] ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 57 Obviously, if the resistance, r, is constant, and the reac- tance, Xy is varied, the values of conductance and susceptance are merely exchanged, the conductance decreasing steadily from ^ = 1 / r to 0, and the susceptance passing from at jr = to the maximum, ^ = l/2r = ^=l/2;r at-r=r, and to ^ = at ^ = 00 . The resistance, r, and the reactance, ;r, vary as functions of the conductance, g, and the susceptance, ^, varies, simi- larly to^ and by as functions of r and x. The sign in the complex expression of admittance is always opposite to that of impedance ; it follows that if the current lags behind the E.M.F., the E.M.F. leads the cur- rent, and conversely. We can thus express Ohm's law in the two forms — and therefore — The joint impedance of a number of series-connected im- pedances is equal to the sum of the individual impedances ; the joint admittance of a number of parallel-connected admit- tances, if expressed in complex quantities^ is equal to the sum of the individual admittances. In diagrammatic represen- tation, combination by the parallelogram law takes the place of addition of the complex quantities. / 58 ALTERNATING-CURRENT PHENOMENA. [§§42, 43