186 BELL SYSTEM TECHNICAL JOURNAL 



voltage in the disturbed circuit is 



e = — jo}{miix + m2t'2 + m^iz + • • • mJn)- 

 The mutual impedance, Zm, may therefore be written 



„ _ e _ jcx}(miii + W2«2 + fii'ds + • • • niJn) 



Zm ~ ~ y ~ f ' 



where I = ii -{- ii -\- is -\- • • • in- This is a general expression and 

 holds for any type of current distribution in the disturbing conductor. 

 In the case of symmetrical current distribution (Case III) all 

 filamentary currents having the value ii lie in a ring concentric with 

 the center of the wire. The voltage induced in a disturbed filamentary 

 wire due to all the currents in one such ring is the same as if their 

 total value, /i, was concentrated in the center of the ring. This 

 voltage is equal to — jwMiIi where Mi is the coefficient of mutual 

 inductance between the center filament of the disturbing wire and the 

 disturbed filamentary wire. The same reasoning holds for currents 

 having values i^, iz, • • • in and the mutual impedance may be written 



^ _ e _ jo^iMJi + M2/2 + M3/3 4- • • • MJn) 

 Zm - -j- ~ J 



But Ml = M2 = Ms = ■ • • Mn = M since all are computed from the 

 center filament in the disturbing wire to the disturbed filamentary 

 wire. Then 



Zm = jooM J 



= jcoM 



since /i + /2 + /a + • • • In = I- This is the same expression for Zm 

 as given in the discussion on symmetrical current distribution. 



However, when the current distribution in the disturbing wire is 

 unsymmetrical in phase and magnitude it is impossible to make the 

 above simplifications. In the general expression 



e jooimiii + ^2^*2 + msis + • • • ninin) 

 Zm - -J- J 



there is no correspondingly simple way to separate the w's from the 

 i's in the complex expression in brackets and the phase angle of the 

 expression may be quite different from that of /. Therefore, e cannot 

 be in phase quadrature with respect to /. In order to put the equation 



