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A Differential Equation that Says 'Second Order of Time Response and Stability in Physical Systems'

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University Of Melbourne

   

Electrical Engineering (ET4047)

   

Added on  2020-03-02

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In this document, we will discuss Physical System Modelling for timely response and stability. We cover the following topics which are a differential equation that relates two variables, the time response of the system that the input is a unit impulse input, the response of the system to a unit-step input and the root locus of the given system.

A Differential Equation that Says 'Second Order of Time Response and Stability in Physical Systems'

   

University Of Melbourne

   

Electrical Engineering (ET4047)

   Added on 2020-03-02

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Time Response and Stability1PHYSICAL SYSTEM MODELING, TIME RESPONSE AND STABILITYStudent’s NameCourseProfessor’s NameUniversityCity (State)Date
A Differential Equation that Says 'Second Order of Time Response and Stability in Physical Systems'_1
Time Response and Stability2Physical System Modeling, Time Response and Stability1.A differential equation that relates ea(t) and θL(t)We know that ea(t)=Raia(t)+Kbdθm(t)dt and θm(t)=N2N1θL(t)dθm(t)dt =ddt(θm(t))=N2N1dθL(t)dtSubstituting the value of dθm(t)dt in ea(t) gives:ea(t)=Kb(N2N1dθL(t)dt)+Raia(t)Therefore, ea(t)=Raia(t)+(KbN2N1dθL(t)dt)2.G(S)=θL(S)Ea(S) of the systemea(t)=Raia(t)+Kbdθm(t)dt in the s-Domain becomesEa(s)=RaIa(s)+KbSθm(s)Also, Tm(S)=KtIa(s), Ia(S)=Tm(S)KtEa(S)=RaTm(S)Kt+Kbsθm(s)...........(i)We also know that Tm(S)=(J¿¿ms2+Dm(s))θm(s)...........(ii)¿Substituting equation (ii) in equation (i)
A Differential Equation that Says 'Second Order of Time Response and Stability in Physical Systems'_2
Time Response and Stability3Ea(S)=RaKt(J¿¿ms2+Dm(s))θm(s)+Kbsθm(s)¿Ea(S)=sθm(s)¿After simplification, θm(s)Ea(S) is found to be:θm(s)Ea(S)=Kt(R¿¿aJm)[s+1Jm(Dm+KtKbRa)]s¿But from θm(t)=N2N1θL(t) we find θm(s)=N2N1θL(s) , Hence θL(s)Ea(S)=KtRaN1N2Jm[s+1Jm(Dm+KtKbRa)]s3.Time response of the system given that the input is a unit impulse input Jm=Ja+JL(N1N2)2=5+700(1001000)2=12Dm=Da+DL(N1N2)2=2+800(1001000)2=10To get the electrical constant KtRa we use the torque-speed curve.
A Differential Equation that Says 'Second Order of Time Response and Stability in Physical Systems'_3

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