By Galal Rabie

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**Example text**

Conventionally, the symbol G(s) is used for the transfer function, but if the element appears in the feedback path, the symbol H(s) is used. The linear, single-input single-output, system can be described by a single linear differential equation of zero initial conditions. This differential equation (DE), can be transformed into Laplace domain by replacing the term (d/dt) by (s). 2) For real systems, the order of P(s) should not exceed that of Q(s); n m . 1 Consider the linear system, of input x(t) and output y(t), described by the following differential equation of zero initial conditions.

17) 1 s(s 2) The inverse Laplace transform of this expression can be obtained after performing a partial fraction expansion as follows. e. Repeated real roots of the polynomial of the denominator). , n 1 s a 1 ds i! 10 Find the partial fractions of Y(s) Y( s ) s4 (s 1)(s 2) 2 (s 1)Y(s) 3 k 3 lim ( s 2 ) 2 Y ( s ) 2 s 2 (s 1)(s 2) 2 k3 k1 k 2 s 1 s 2 ( s 2) 2 k 1 lim s 1 s4 (s 1) (s 4) d s 4 d (s 2) 2 Y(s) lim lim k 2 lim 3 s 2 ds (s 1) 2 s 2 ds s 1 s 2 Y( s ) 3 3 2 s 1 s 2 ( s 2) 2 Or, calculate k1 and k3 as shown above then calculate k2 as follows.

Output node (Sink): is a node, which has only ingoing branches X5. Forward path: is a path from input node to output node along which no node is encountered more than once. Feedback loop: is a path, which originates and terminates at the same node, along which no node is encountered more than once. Path gain: is the product of all gains (transfer functions) along the forward path. 1. M1 t 12 t 23 t 34 t 45 M2 t12 t 24 t 45 M3 t12 t 25 Loop gain: is the product of the gains of branches forming the loop.