Second Order Process Electrical Engineering (EE) Notes | EduRev

Electrical Engineering (EE) : Second Order Process Electrical Engineering (EE) Notes | EduRev

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Second Order Process
A second order process is a process whose output is modeled by a second order differential equation.

Second Order Process Electrical Engineering (EE) Notes | EduRev      66

where, u(t) and y(t) are input and output of the process respectively. If  Second Order Process Electrical Engineering (EE) Notes | EduRev , then define the following:
Second Order Process Electrical Engineering (EE) Notes | EduRev

Hence, the second order differential equation takes the following form:
Second Order Process Electrical Engineering (EE) Notes | EduRev     67

At steady state condition Second Order Process Electrical Engineering (EE) Notes | EduRev, the equation can be re-written as

Second Order Process Electrical Engineering (EE) Notes | EduRev       68

Subtracting eq. (68) from eq. (67), we obtain
Second Order Process Electrical Engineering (EE) Notes | EduRev     69

Alternatively,
Second Order Process Electrical Engineering (EE) Notes | EduRev           70

Where,  Second Order Process Electrical Engineering (EE) Notes | EduRev  and  Second Order Process Electrical Engineering (EE) Notes | EduRev  are respectively the deviation forms of the output and input of the process around the steady state, whose initial conditions are assumed to be the following: 
Second Order Process Electrical Engineering (EE) Notes | EduRev

Taking Laplace Transform of the eq. (70) we obtain,
Second Order Process Electrical Engineering (EE) Notes | EduRev       71

Rearranging the above we obtain,
Second Order Process Electrical Engineering (EE) Notes | EduRev          72

Kp is called the gain of the process.

Example of a second order process

Second Order Process Electrical Engineering (EE) Notes | EduRev

Consider the U tube manometer as in Fig.6. The liquid inside the manometer has been shown in a pressurized state. Initially mercury levels at both the legs were at the same height. The present pressurized state is obtained upon exerting a pressure of  Second Order Process Electrical Engineering (EE) Notes | EduRev on Leg I.

Applying force balance on both the legs of the manometer across plane of initial pressurized state, we obtain: 
Second Order Process Electrical Engineering (EE) Notes | EduRev

Where,  Second Order Process Electrical Engineering (EE) Notes | EduRev  cross-sectional area of manometer leg(s), P= density of manometer liquid, f =Fanning' friction factor, v = velocity of manometer liquid, D = diameter of manometer leg(s), L =length of manometer liquid in the tube, m = mass of manometer liquid. Assuming laminar flow inside the manometer, the friction factor can be expressed as ,  Second Order Process Electrical Engineering (EE) Notes | EduRev  where  Second Order Process Electrical Engineering (EE) Notes | EduRev  is the Reynold's number. Hence the force balance equation takes the form:

Second Order Process Electrical Engineering (EE) Notes | EduRev      74

or

Second Order Process Electrical Engineering (EE) Notes | EduRev              75

The velocity of manometer liquid is rate of change of h . Hence,
Second Order Process Electrical Engineering (EE) Notes | EduRev                     76

or

Second Order Process Electrical Engineering (EE) Notes | EduRev                             77

Comparing eq.(77) with eq.(67), the following can be obtained: Second Order Process Electrical Engineering (EE) Notes | EduRev  and  Second Order Process Electrical Engineering (EE) Notes | EduRev  and  Second Order Process Electrical Engineering (EE) Notes | EduRev .

 Dynamic Response of a Second Order Process to a Step Change in the Input

For a step input of magnitude , the Laplace Transform of u(t) would be,

Second Order Process Electrical Engineering (EE) Notes | EduRev           78

Hence, second order process takes the following form,
Second Order Process Electrical Engineering (EE) Notes | EduRev

The process response will grossly depend upon the value of ξ and there can be three distinguished cases of ξ, i.e. ξ >1; ξ = 1 and ξ <1 .

Case A: ξ = 1

In this case the process response equation in the Laplace domain takes the following form:
Second Order Process Electrical Engineering (EE) Notes | EduRev         80

Using the following:
Second Order Process Electrical Engineering (EE) Notes | EduRev       81

in eq. (80), we obtain
Second Order Process Electrical Engineering (EE) Notes | EduRev               82

or

Second Order Process Electrical Engineering (EE) Notes | EduRev              83

For ξ ≠ 1, using the following:
Second Order Process Electrical Engineering (EE) Notes | EduRev           84

in eq. (79), we obtain

Second Order Process Electrical Engineering (EE) Notes | EduRev

Second Order Process Electrical Engineering (EE) Notes | EduRev

Second Order Process Electrical Engineering (EE) Notes | EduRev

Second Order Process Electrical Engineering (EE) Notes | EduRev

Case B: When ξ >1
Second Order Process Electrical Engineering (EE) Notes | EduRev

In the above equations, the following trigonometric identities have been used: Second Order Process Electrical Engineering (EE) Notes | EduRev  and  Second Order Process Electrical Engineering (EE) Notes | EduRev

Hence we get the final expression for process response when ξ >1,
Second Order Process Electrical Engineering (EE) Notes | EduRev         87

Case C: Second Order Process Electrical Engineering (EE) Notes | EduRev
Second Order Process Electrical Engineering (EE) Notes | EduRev            88

Second Order Process Electrical Engineering (EE) Notes | EduRev

In the above equations, the following trigonometric identities have been used: 
Second Order Process Electrical Engineering (EE) Notes | EduRev             89

Second Order Process Electrical Engineering (EE) Notes | EduRev          90

One can also use the following trigonometric identity for the above expression: 
Second Order Process Electrical Engineering (EE) Notes | EduRev                    91

Hencea

Second Order Process Electrical Engineering (EE) Notes | EduRev                       92

Second Order Process Electrical Engineering (EE) Notes | EduRev

Hence we get the final expression for process response for ξ < 1,
Second Order Process Electrical Engineering (EE) Notes | EduRev                        93

he frequency of oscillation is
Second Order Process Electrical Engineering (EE) Notes | EduRev                 94

whereas the phase lag is
Second Order Process Electrical Engineering (EE) Notes | EduRev       95

 

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