 Namaste, welcome to this session Mathematical Modeling of Electrical Elements. At the end of this session, students will be able to describe the mathematical equation of basic electrical elements. Now in this session we are going to discuss mathematical modeling, electrical system and basic elements and we will see mathematical model of electrical system. Mathematical Modeling The analysis of any control system requires mathematical models of the processes in the system which is obtained through equations and formulae that can predict how the various devices will behave. The set of mathematical equations describing the dynamic characteristics of a system is called as mathematical model of the system. A set of differential equations are used to represent dynamic behavior of a system and these equations are based on the fundamental laws of physical system. For example, motion of a mechanical system to find out the mathematical model we use Newton's law of motion. And for electrical system we use Kirchhoff's laws to find out differential equations. Now let us see mathematical modeling of an electrical system. Now just we have seen that for electrical systems to find out mathematical modeling or to write differential equations we have to use Kirchhoff's laws. So take a pause here and recall what are the Kirchhoff's laws. So I think you are recall Kirchhoff's laws. So there are two Kirchhoff's laws. One is Kirchhoff's voltage law. It says that the algebraic sum of potential drop and emf around a closed path of a circuit must be equal to 0. Now let us have example here where you can see there is a closed path AB, BC, CD and DA. In each of these path we have resistors. So when you find out the potential drop at each path like here between AB we have voltage AB then VBC between B and C then voltage drop between C and D is represented as VCD and voltage drop between D and A denoted by VDE. So therefore the summation of these voltage drops in a closed path is always equal to 0 and it must be 0. Now Kirchhoff's second law is Kirchhoff's current law which says that the algebraic sum of the current at a junction must be equal to 0. Now for example in this figure we have different currents current I1 and I2 are incoming to this node along with I3 is also incoming whereas current I4 and I5 are outgoing currents. So the algebraic sum of all these currents is given as I1 plus I2 plus I3 minus I4 minus I5 that must be equal to 0. So these are the Kirchhoff's laws which we are going to use to find out mathematical modeling of electrical systems. Now let us see electrical system and basic elements. So in the analysis of electrical systems there are three essential basic elements which are resistors, capacitors and inductors which may occur in various ways. In an electrical system or a circuit. So very first and basic element is resistor. So it is an energy dissipative element. For a resistor let us see the relation between current and voltage for a circuit. So here you can see there are various types of resistors with different values. And here is a circuit where you can see the voltage V of t is applied across the resistor R and then current I flows through the resistor. Then the relationship between current and voltage is given by V of t is equal to R into I of t. So the same equation can be written in terms of current as I of t is equal to V of t by R where R is the resistance in Ohm, V of t is voltage across the resistor and I of t is current through the resistor. Now let us consider the second circuit where you can see there are two currents flowing through the resistor in opposite direction. Then let us find out the equation for the circuit. So here the equation V of t is equal to R into I t becomes V of t is equal to R into I 1 minus I 2. As current I 1 is flowing from positive terminal to negative. So it is positive and I 2 current is flowing from negative terminal to positive. So the equation becomes V of t is equal to R into I 1 minus I 2. The second basic element is capacitor which stores energy in an electrical form. So for a capacitor let us see the relation between current and voltage. Now the capacitors are of different types and shapes where you can see here. Now let us consider this circuit where voltage V of t is applied across the capacitor and then current flows through the circuit. Now the relation between current and voltage for capacitor is given by V of t is equal to 1 upon C integration minus infinity t of I of t into d t. Or the same equation can be written in terms of current as I of t is equal to C d by d t of V of t. Let us consider this circuit where two currents are flowing in opposite direction. So the equation for this circuit will be V of t is equal to 1 upon C integration minus infinity to t in the bracket I 1 minus I 2 into d t. So again I 1 is flowing from positive terminal and I 2 is flowing from negative terminal. So the equation is like this. Now the third basic element is inductor which stores energy in a magnetic field for inductor. Let us see the relation between current and voltage. So again inductors are of different types and shapes and this is the circuit where we have connected inductor with a source V of t and then current I flows. Then the equation between current and voltage is given as V of t is equal to L into d by d t of I t. The same equation can be written in terms of current. Now let us consider the second circuit where two currents are flowing in opposite direction through the inductor L. Then the equation for this circuit is V of t is equal to L into d by d t of I 1 minus I 2 as two currents are flowing in opposite direction. Now let us see mathematical model of electrical elements through this table where we have relation between voltage and current. Current and voltage and voltage and charge. So for resistor the symbol is like this where the relation between voltage and current is given as V is equal to I into R. For resistor current and voltage relationship is given as I is equal to V by R and voltage charge relation is V is equal to R into d cube by d t. For inductor the symbol is as shown here. The relation between voltage and current is given by V is equal to L d I by d t. Then current voltage relation is I is equal to 1 upon L integration V d t. Then voltage charge relation is given as V is equal to L d square cube by d t square. Then for capacitor the symbol is two parallel plates. Then voltage current relationship is V is equal to 1 upon C integration of I d t. Then current voltage relation for capacitor is I is equal to C d V by d t. And voltage charge relation for capacitor is V is equal to cube by C. So for resistor the resistance is calculated in Ohm. For inductor the inductance is calculated in Henry. And for capacitor the capacitance is calculated in Farad. Now let us see mathematical model of an electrical circuit. So let us have an example here. Obtain mathematical model of the given electrical system. So this is the electrical circuit where you can see resistor, inductor and capacitor are connected in series. So let us consider the current I is flowing through the circuit. So when voltage is applied then current flows through the circuit. Then we can find out the voltage across R, L and C. So applying Kirchhoff's voltage law we can write the equation as V is equal to V R plus V L plus V C. Whereas V R is voltage across R, V L is voltage obtained across L and V C voltage obtained across capacitor. So the equation becomes V is equal to R into I for resistor plus L d i by d t for inductor and plus 1 upon C integration of I d t for the capacitor. So this is the mathematical model of given electrical circuit. So in this way you can apply Kirchhoff's laws and find out mathematical modeling of an electrical circuit or system. These are references. Thank you.