Characteristics of transfer functions of second-order lag systems

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■Calculation of the transfer function of a second-order delay system

We explained the motor and disk model here. We calculate the transfer function when the output is the rotational speed w(t) for the input V(t).

■Laplace transform

Laplace transform the above formula. From equation (1)



And from equation (2)

■Expression in a block diagram

If equations (3) and (4) above are expressed in a block diagram, they will look like this.



This can be expressed as a single transfer function as follows. For information on how to synthesize transfer functions, see here.




This is called a second-order delay transfer function because the denominator is in the form of s squared.

■Consider the characteristics of the system

As explained here, you can find out the characteristics of this transfer function by finding the solution (pole) of the characteristic equation (denominator polynomial) of equation (5). First, consider the generalization of equation (5) as shown below.



The characteristic equation above is:



From the formula for solving quadratic equations,



The poles of this equation change significantly depending on whether they are real numbers or include imaginary numbers, depending on the value of ζ in the square root. This is because, as explained here, real numbers represent the damping components, and imaginary numbers represent the oscillating components.

Now let's actually check the operation with a simulation. Design it as follows in Scilab. Here, ω is set to 2.





The results are summarized below. You can see that the characteristics change depending on the value of ζ (the solution to the quadratic equation).

■ Expressing the poles on the complex plane

The characteristics can be seen by where the poles are placed on the complex plane.



The positive side of the real axis is unstable, the negative side is convergent, and if an imaginary axis component is included, it will oscillate. (As explained earlier)

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