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Second Order Process Notes - Electrical Engineering (EE)

Document Description: Second Order Process for Electrical Engineering (EE) 2022 is part of Electrical Engineering (EE) preparation. The notes and questions for Second Order Process have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Second Order Process covers topics like and Second Order Process Example, for Electrical Engineering (EE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Second Order Process.

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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 Notes - Electrical Engineering (EE)      66

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

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

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

Second Order Process Notes - Electrical Engineering (EE)       68

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

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

Where,  Second Order Process Notes - Electrical Engineering (EE)  and  Second Order Process Notes - Electrical Engineering (EE)  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 Notes - Electrical Engineering (EE)

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

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

Kp is called the gain of the process.

Example of a second order process

Second Order Process Notes - Electrical Engineering (EE)

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 Notes - Electrical Engineering (EE) on Leg I.

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

Where,  Second Order Process Notes - Electrical Engineering (EE)  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 Notes - Electrical Engineering (EE)  where  Second Order Process Notes - Electrical Engineering (EE)  is the Reynold's number. Hence the force balance equation takes the form:

Second Order Process Notes - Electrical Engineering (EE)      74

or

Second Order Process Notes - Electrical Engineering (EE)              75

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

or

Second Order Process Notes - Electrical Engineering (EE)                             77

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

 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 Notes - Electrical Engineering (EE)           78

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

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 Notes - Electrical Engineering (EE)         80

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

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

or

Second Order Process Notes - Electrical Engineering (EE)              83

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

in eq. (79), we obtain

Second Order Process Notes - Electrical Engineering (EE)

Second Order Process Notes - Electrical Engineering (EE)

Second Order Process Notes - Electrical Engineering (EE)

Second Order Process Notes - Electrical Engineering (EE)

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

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

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

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

Second Order Process Notes - Electrical Engineering (EE)

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

Second Order Process Notes - Electrical Engineering (EE)          90

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

Hencea

Second Order Process Notes - Electrical Engineering (EE)                       92

Second Order Process Notes - Electrical Engineering (EE)

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

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

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

 

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