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A 3-phase transmission line consists of two lines 1 and 2 connected in series, line 1 being at the sending end and 2 at the receiving end. The respective auxiliary constants of the two lines are : A1, B1, C1, D1 and A2, B2, C2, D2. Find the A, B, C, D constants of the whole line which is equivalent to two series-connected lines.

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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

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