Control systems by N C Jagan

By N C Jagan

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38) Substituting for TI in eqn. 36), using eqn. 38), JI But, and '9 1 + BI 91 + KI ° =~O 2 N 2 I ° + ~. 1 N2 (J2 '0'2 + B2 92 + K2 O2 + T L) = TM ..... 39) 26 Control Systems Using these relations in eqn. 39), we get ..... 40) Replacing and We have, J I eq ·e·\ + BI 9\ + K leq el + T Leq = TM ..... 41) Thus the original system in Fig. 15 can be replaced by an equivalent system referred to primary side as shown in Fig. 16. Fig. 16 Equivalent system refered to primary side The moment of inertia J2, frictional coefficient B2 and torsional spring constant K2 of secondary side are represented by their equivalents referred to primary side by, J 12 ~ J, (~: r B12~B, (~:r 27 Mathematical Modelling of Physical Systems The load torque referred to primary side is given by, _ Nl TLeq-TL · - N2 These equations can be easily seen to be similar to the corresponding electrical quantities in the ideal transformer.

Mason and he has developed a formula to obtain the ratio of output to input, called as gain, for the given signal flow graph. 45 Mathematical Modelling of Physical Systems Mason's gain fonnula is given by, T = output variable input variable = 1: Mk ~k k ~ ..... 65) where Mk is the J(h forward path gain, ~ is the detenninant of the graph, given by, ~= 1 -1: Pml + 1: Pm2 •••••. 66) th where Pmr is the product of the gains of m possible combination of r non touching loops. or 1 - (sum of gains of individual loops) + sum of gain products of possible combinations of 2 non touching loops) - (sum of gain products of all possible combinations of 3 nontouching loops) + .....

Three non touching loops There is no combinations of 3 non touching loops or 4 nontouching loops etc. Pm3 =P m4 =·····O Let us now calculate the determinant of the flow graph. Ll = 1 - (L I + L2 + L3 + L4 + Ls + L6) + P I2 + P22 + P32 = 1 - (g23 h32 + g23 g34 h42 + g4s hS4 + gS6 h6S + g46 h6s hS4 + g23 g36 h6s hS4 h42 ) + (g23 g4S h32 hS4 + g23 gS6 h32 h6s + g23 g46 h32 h6s hs4) and LlI = value of Ll which is non touching with MI' Eliminate all terms in Ll which have any node in common with forward path MI' All loops LI to L6 have at least one node in common with MI and hence, LlI = 1 Similarly Ll2 = value of Ll which is non touching with M2.

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