Transfer function and Characteristic Equation

Transfer Function:

Mathematically Transfer Function  is defined as the ratio of Laplace transform of output of the system to the Laplace transform of input under the assumption that all initial conditions are zero.
Symbolically system can be given as below and its transfer function of system can be shown as given below,

TF1

G(s) = Laplace transform of output/ Laplace transform of input

         = C(s) / R(s) = L[ c(t)] /  L[r(t)]

By taking Laplace transform of the differential equations for nth order system,

TF2

Characteristic Equation of a transfer function:

Characteristic Equation of a linear system is obtained by equating the denominator polynomial of the transfer function to zero. Thus the Characteristic Equation is,

\\a_{ n }s^{ n }+a_{ n-1 }s^{ n-1 }+…..a_{ 1 }s+a_{ 0 } = {0} \
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