Modeling of Electrical Elements Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD
This Lecture Contains Modeling of electrical elements Kirchoff s Laws and modeling electrical circuit A multi loop lil system Modeling of a DC servomotor Joint Initiative of IITs and IISc - Funded by MHRD
Modeling 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. These are: Resistors (R), Capacitors (C) and Inductors (L) Resistors are electric elements for which voltage across it is proportional 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 l A
Electrical Elements Resistors Resistors are dissipative in nature. Energy (E) dissipated from a resistor (R) could be modeled using Joule Effect. Accordingly: E i V Rt R where i is the current passed through the resistor for a time t and V is the applied Voltage. Figure below shows different types of resistors. t
Electrical Elements: Capacitors 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 C The energy stored in a capacitor could be expressed as: 0 1 E CV Capacitance of commonly used configuration of a capacitor and the corresponding capacitance is shown below: oint Initiative of IITs and IISc Funded by MHRD
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: d i( t) V ( t ) L dt Where, L refers to the inductance of the coil. For a cylindrical coil of diameter d length l,, no. of turns n, and magnetic permeability µ, the inductance is given by: n d L 4l The magnetic energy stored by an inductor is: E 1 L i ( ) t
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 path in a circuit is zero. Parameters Mechanical Electrical Staticstorage of Spring Capacitor Energy Dynamic Storage of Energy Inertia Inductor Energy Dissipation Dashpot/Damper Resistor Excitation Force Voltage Response Displacement Charge
Modeling 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) i( t) dt d q( t) d q( t) 1 V ( t ) L R d t dt C q
Impedance based representation
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
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).
Hints: Development of state-space representation Torque developed by the motor T m = K t i m Back emf e m =K b dθ/dt Kirchoff s Law: e a = L m di m /dt + R m i m + e m Force Balance: T m T l = J d Θ/dt + B dθ/dt Take states as x 1 = Θ, x = dθ/dt, x 3 =i m, Obtain EOM in state-space form
Special References for this lecture: System Dynamics for Engineering Students: Nicolae Lobontiu, Academic Publisher Feedback Control of Dynamic Systems, Frankline, Powell and Emami-Naeini, Pearson Publisher Control Systems Engineering, i Norman S Nise, John Wiley & Sons Systems Dynamics and Response, S. Graham Kelly, Thomson Publisher