The two series-connected lines along with their constants as shown in Fig.For line No. 1
VS = A1V + B1I ; IS = C1V + D1I ...(i)
For line No. 2
V = A2VR + B2IR ; I = C2VR + D2I2 ...(ii)
Substituting the values of V and I from Eq. (ii) into Eq. (i), we get
VS = A1 (A2VR + B2IR) + B1(C2VR + D2IR) = (A1A2 + B1C2 )VR + (A1B2 + B1D2 )IR
and IS = C1(A2VR + B2IR) + D1(C2VR + D2IR) = (C1A2 + D1C2 )VR + (C1B2 + D1D2 )IR
Hence, the two lines connected in series have equivalent auxiliary constants of
A = A1A2 + B1C2, B = A1B2 + B1D2
C = C1A2 + D1C2 and D = C1B2 + D1D2