102 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1951 



Where 



P= Ro-\-Ri+ Xi(l - MlW +^2 (1 + M2) 

 Q = -Mii?i + Xi(^i - Ml) + X2O + fl2)k2 



5 = i?i + Xi(l + ^1) 

 i?o = i?p + i?2(l + M2) 

 ^1 = coupling factor = Mi/Xi 

 h = " " = M2/X2 



£3 and £4 = applied voltages 



While the derivation of most of the coefficients in these mesh equations 

 is obvious, the derivation of P and Q may require further explanation. Co- 

 efficient P can be considered as follows : 



1 — The term Rq equals Rp + -^2(1 + ^2)- The plate resistance Rp is in 

 the plate circuit and hence stands alone. The resistance R2 being between 

 cathode and ground. Fig. 6(a), produces negative feedback and must be 

 multiplied by (1 + /X2). 



2 — The resistance Ri is in the plate circuit, and here as in the case of Rp 

 the flow of current in Ri does not produce a voltage across the grid of the 

 same tube through which this current flows. 



3 — The reactance Xi is in the plate circuit of each tube, and by means 

 of the mutual reactance (Afi) is coupled to the respective grid of the 

 other tube. This coupling provides positive feedback for current flowing 

 through X\ which can be expressed as —fxiMi or — mi^i^i- Thus the term 

 Xi(l — Mi^i) is derived. 



4 — The reactance X2 being between cathode and ground, Fig. 6(a), 

 provides negative feedback. Hence X2 must be multiplied by (1 + ^2). 



Coefficient Q can be explained in similar fashion: 



1 — Although the flow of current through Ri does not produce a voltage 

 across the grid of the same tube through which this current flows, a voltage 

 drop is produced across the grid of the other tube because of the cross 

 coupling of these grids, Fig. 6(a). Thus voltage drop in one plate circuit 

 appears between grid and ground of the other tube in the direction to aid 

 the flow of current in this other mesh; hence the term — mi^i- 



2 — Likewise, the reactance Xi acts in the same manner as Ri in aiding 

 the flow of current in the other tube circuit. Furthermore, Xi is coupled by 

 the mutual reactance to this other mesh. These effects can be expressed as 

 Xi(^i — Ml). 



3 — The reactance X2 is coupled by mutual reactance to both tube circuits. 

 It appears in each circuit between cathode and ground in the polarity to 

 produce negative feedback equal to ^2(1 -|- ^2)^2. 



