Recovering a Differential Equation From the Transfer Function of a Cascaded System

1. How do you write a differential equation from a transfer function?
2. How do you find the transfer function of a second order system?
3. Why Laplace transform is used in transfer function?

How do you write a differential equation from a transfer function?

Transfer Function to Single Differential Equation

To find the transfer function, first write an equation for X(s) and Y(s), and then take the inverse Laplace Transform. Recall that multiplication by "s" in the Laplace domain is equivalent to differentiation in the time domain.

How do you find the transfer function of a second order system?

Here, an open loop transfer function, ω2ns(s+2δωn) is connected with a unity negative feedback. Substitute, G(s)=ω2ns(s+2δωn) in the above equation. The power of 's' is two in the denominator term. Hence, the above transfer function is of the second order and the system is said to be the second order system.

Why Laplace transform is used in transfer function?

Because the Laplace transform is a linear operator, each term can be transformed separately. With a zero initial condition the value of y is zero at the initial time or y(0)=0. Putting these terms together gives the first-order differential equation in the Laplace domain.