(v<4 HELL SYSTEM TECHXICAL JOVRSAL 



I'hc (liscriminanl of llu' mosl i;oiht;i1 two-mesh circuit is of tlu' form 



LnX + /eu + /)n\ '. /.,,X + i?,o + £),2X ' i 

 ^ = I , (44) 



I Ly{K + Rv, + Dv^ \ L,,X + i?.,., + /9«,X-' I 



where the three sets of coefficients, using Djic instead of 1 Cjk, are 

 suliject to the restriction thai the three determinants 



I L„ Ln I i ^11 ^12 i /)n D^i I 



i, !, and (4.")) 



■ L,o Loo /?,, i?,, /;,, /;,, 



are all positi\e or zero, as well as L\\, Rn, and Di\. This condition 

 retiuires La, R12, and D>2 also to be positive or zero. 



The most general dri\ing-point impedance of a two-mesh circuit 

 may be taken as the imjjedance in the first mesh of the circuit defined 

 by the discriminant (44). Set A An equal to the \alue of S given 

 by (lb). Kxpanding into polynomials in X, and etiualing coefficients 

 of the numerators and flonnniinators of ilie two ex|iressi()ns, the 

 lollowini; rclalinn> arr iihlainrd : 



LiiLoo-L;, = (/o/^-, (4()) 



LuR-2-2 + Li-2Ru-2Lv2Rii^aik-, (47) 



LnDy, + L.2-,Du + RnRr2-'2Li«Di.-Rl,=a2k-, (48) 



RnDyi + R22Dn-2Ri2Dn=a3k\ (49) 



DuDi2-D-,, = aik-, (50) 



L22 = bik-, (51) 



R22 = b2k-, (52) 



D22 = bik-, (53) 



where k has any real \alue other than zero. Intnuhue the notation 

 RiiR22-Ri.=dk-. (.")4) 



where /I is positi\e or zero. Then, using (46), (54), and (.")(•). eliminate 

 All, Ru. and J)\i from equations (47)r(49), obtaining 



(A ,../?,o - L,,/? ,.,)-' = k'{ - dLU+axL22Rii - chRh) . (55) 



{Di2L22-D22Lii)- = k-[-aoDl^+{a2-d)D22Lt2-aiLL]. (oli) 

 (V?,o/?oo- /?•.,/;, 0)- = ;t-'( -aiRi,+a3R22D22-dni). (57) 



Using (51) (5.'i), eliminate L..., R22, and P22 from the riv;lit-lian(l 

 members of (55) (57); extract the s(|uare rodi ; re.ur.miie ilie mder 

 of the equations, obtaining 



R,2lh2- R-i«.Dn= ±k\-atb2-+a3b2b,-db3-)' '', (58) 



/},oLo„-/)ooL,.,= ±P[-ra3'-|-(flo-rf)636,-a4/,,2]t'2, (.59) 



Ly.R,2 - /-■=/?,•• = ± kH -dbx'+d ibJ>2-ti,J)2') ''-'. «><» 



