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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Modeling g of Electrical Elements D Bi Dr. Bishakh h kh Bh Bhattacharya tt h Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD Module 1- Lecture 7 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System This Lecture Contains Modeling of electrical elements Kirchoff’s Laws and modeling electrical circuit A A multi‐loop system lil Modeling of a DC servomotor Joint Initiative of IITs and IISc - Funded by MHRD Module 1- Lecture 7 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 7 Modeling g of Electrical Elements In the last lecture we have discussed about modeling of dynamic mechanical systems. However, today you will hardly find dynamic systems which are purely mechanical in nature. There will be invariably electrical systems coupled with the mechanical elements. Similar to mechanical systems electrical systems consist of three basic electrical elements elements. These are: Resistors (R) (R), Capacitors (C) and Inductors (L) Resistors are electric elements for which voltage g across it is proportional p p to the current passing through it. The constant of proportionality is known as Resistance. For a wire of length ‘l’, cross-sectional area –’A’ and resistivity – ‘ρ’: R Joint Initiative of IITs and IISc ‐ Funded by MHRD l A NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 7 Electrical Elements – Resistors Resistors are dissipative in nature. Energy (E) dissipated from a resistor (R) could be modeled using ‘Joule Effect’. Accordingly: 2 V E i2R t t R where ‘i’ is the current p passed through g the resistor for a time ‘t’ and ‘V’ is the applied Voltage. Figure below shows different types of resistors. Joint Initiative of IITs and IISc ‐ Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 7 Electrical Elements: Capacitors p Capacitors are electrical elements used to store the electrostatic energy. Voltage (V) across a capacitor of capacitance (C) changes according to the following equation: t 1 V (t ) i dt C0 Th energy stored The t d in i a capacitor it could ld be b expressed d as: 1 E CV 2 2 Capacitance of commonly used configuration of a capacitor and the corresponding capacitance is shown below: oint Initiative of IITs and IISc ‐ Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 7 Electrical Elements: Inductors Inductors refer to the coiled conductors where a variable current generates voltage, which, for a linear inductor is proportional to the current rate. Accordingly, the voltage (V) is given by: V (t ) L d i (t ) dt Where, L refers to the inductance of the coil. For a cylindrical coil of diameter d length g l,, no. of turns n,, and magnetic g permeability p y µ, the inductance is given g by: y L n2 d 2 4l The magnetic energy stored by an inductor is: 1 E L i 2 (t ) 2 Joint Initiative of IITs and IISc ‐ Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 7 Modeling of Electrical Circuits When the electrical elements get connected into a circuit, the dynamic behavior of the electrical system could be modeled by using the following governing laws: • Kirchoff’s Current Law: The algebraic sum of currents leaving a junction or node equals the algebraic sum of currents entering the node. • Kirchoff s Voltage Law: The algebraic sum of all voltages taken around a closed Kirchoff’s path in a circuit is zero. Parameters Mechanical Electrical Static storage of Static storage of Energy Spring Capacitor Dynamic Storage of Inertia gy Energy Inductor Energy Dissipation Dashpot/Damper Resistor Excitation Force Voltage R Response Di l Displacement t Ch Charge Joint Initiative of IITs and IISc ‐ Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Modeling g of a LRC Circuit The governing equation of the electrical circuit could be obtained using Kirchoff ‘s Laws and is given by: d q (t ) dt d 2 q (t ) d q (t ) 1 V (t ) L R q d t2 dt C i (t ) Joint Initiative of IITs and IISc ‐ Funded by MHRD Module 1- Lecture 7 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Impedance p based representation p Joint Initiative of IITs and IISc ‐ Funded by MHRD Module 1- Lecture 7 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System A multi-loop presentation Similar to multiple DOF mechanical systems, one can have multi‐loop electrical circuits. Figure below shows a multi‐loop electrical circuit in time and frequency domain and the corresponding transfer function for the system Joint Initiative of IITs and IISc ‐ Funded by MHRD Module 1- Lecture 7 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Assignment: How to Model a DC Servomotor? The example below shows an electromechanical system. Find out the governing equations of the system (electrical and mechanical). Joint Initiative of IITs and IISc ‐ Funded by MHRD Module 1- Lecture 7 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Hints: Development p of state-space p representation p • Torque developed by the motor Tm = Kt im • Back emf em = Kb dΘ/dt • Kirchoff’s Law: ea = Lm dim/dt + Rm im + em • Force Balance: Tm – Tl = J d2Θ/dt2 + B dΘ/dt • Take states as x1 = Θ, x2 = dΘ/dt, x3=im, Obtain EOM in state-space form Joint Initiative of IITs and IISc ‐ Funded by MHRD Module 1- Lecture 7 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 7 Special p References for this lecture: System Dynamics for Engineering Students: Nicolae Lobontiu, Lobontiu Academic Publisher Feedback Control of Dynamic Systems Systems, Frankline, Frankline Powell and Emami Emami-Naeini, Naeini Pearson Publisher Control C lS Systems Engineering, E i i N Norman S Ni Nise, J John h Wil Wiley & Sons S Systems Dynamics and Response, S. Graham Kelly, Thomson Publisher Joint Initiative of IITs and IISc ‐ Funded by MHRD