 Where does the glare come? In the monitor? Oh, on you, directly on you. Is it okay? Is the recording okay? Will the recording be okay? Fine. So the first item is we wish all of you a happy new year. And the second item is we want to talk about the 3 basic components. Resistance, inductance and capacitance, energy considerations in them. Then we want to talk about sources, current and voltage and we want to work out a couple of problems. Then if time permits, we wish to enunciate 2 circuit laws. This would be the scope of today's lecture. Most of this material is known to you and therefore what we are doing essentially is a review of the knowledge that you already have. A resistance as you know is defined by a linear resistance, is defined by Ohm's law, that is the potential difference across its terminals, is proportional to the current that flows in it, V and I and the value is R. Experimental law is that V, the voltage drop is proportional to I, the more current goes the more is the drop and therefore the proportionality constant is given the name resistance and it is denoted by R. Now if it is a linear resistance then if you plot V versus I, note the polarity. Polarity is V is considered positive in this direction that is from where the current is originating, V is considered positive and the current going to the other terminal, that terminal is considered negative. And V, the relation between V and I naturally then shall be only in the first quadrant and it shall be a straight line whose slope is equal to R. Now if this relationship becomes non-linear then we say the resistance is non-linear. Most of the times we shall be concerned with linear resistances. In linear resistances the power that is the energy consumed per unit time, DWDT is given by the product VI and therefore this is P equals to I squared R and the energy that is dissipated in the resistance in time capital T shall be equal to I squared RT if I is a constant, if the current is a constant, if I is constant, if I is not constant then the relation shall be RT integral I squared DT from 0 to capital T. This would be the general form, is that okay? General form would be the integral of R I squared DT, no T, I am sorry, R I squared DT integral from 0 to capital T. If the current is a constant then it will be simply I squared R times capital T. This energy is dissipated in the resistor that is the dissipation means a transformation of electrical energy to heat energy alright and this is an irrecoverable process, irreversible process that is if a resistance dissipates heat, dissipates energy, this energy cannot be recovered alright. On the other, therefore in a resistance the energy is dissipated and not stored. On the other hand in an inductor for example, well first let us take a capacitor. Let us consider two parallel plates like this, two parallel plates which are separated by distance D and let us connect a battery of voltage V. There are two plates parallel plates like this alright separated by distance, separated by distance capital T, they are parallel to each other and we connect a battery across this. Then you know that the upper plate shall be charged positively and the lower plate shall be charged negatively. Now if I disconnect the battery, if I disconnect the battery the charges remain alright and therefore this device is capable of storing electrical charge, storing electrical charge. The charge does not disappear as soon as the battery is disconnected. Therefore it acts as a storage of charge and it is found that the charge stored in the device is proportional to the potential difference V that is applied. That is if you increase the battery voltage to twice its previous value then the charge doubles and therefore Q is proportional to V and the proportionality constant is given the name of capacitance. This is the capacitance for storing electrical charge and is given the symbol C. Now naturally if it is a linear capacitor then the relationship between Q and V is a straight line with a slope of C and this is called a linear capacitor. Not all capacitors are linear. For example there are devices called varactor diodes in which the capacitance varies as the square of the voltage and therefore there the capacitance is non-linear. The charge is not linearly related to the voltage. If it is linearly related then we say it is a linear capacitor. It is not a current voltage relationship. This is what I want to point out. The linearity does not exist between current and voltage. It is between charge and voltage. This is the basis of linearity. Now if Q is a constant if Q is a constant then naturally if V is a constant then Q is a constant alright and therefore no current flows. If there is a battery here the battery will not deliver any current. I shall be equal to 0 if V is a constant. On the other hand if V varies alright then obviously Q also varies. If V varies then Q varies and if Q varies then there shall be a flow of current which is proportional to the rate of change of charge. In other words if the charge in a capacitor varies then the current flowing through this shall be given by I equal to C, I equal to DQ DT which will be equal to C DV DT. Let us use a small symbol small V. I equal to C DV DT and therefore the current voltage relationship is no longer linear. What is linear is the current and DV DT relationship that is if I plot current versus DV DT not V mind you then this relationship shall be linear exactly like the charge is linearly related to the voltage alright. And the usual the symbol for a capacitance is this that is 2 lines parallel to each other and the value of the capacitance is written by its side. Similarly for well what about the energy? The power again in this case is given by VI power is the product of voltage and current and since I is C DV DT therefore power is C V DV DT alright. And therefore if we charge a capacitor from 0 to a voltage let us say capital V if we charge a capacitor from 0 voltage to a voltage V P is equal to C V DV DT if we charge a capacitor from 0 to a voltage V then the energy that is stored in the capacitor shall be given by integral P DT integral P DT from V equal to 0 to V equal to capital V and this therefore is equal to C integral 0 to capital V DV DT and DT cancel the rate of change with respect to time and therefore the energy stored in the capacitor is half C V square why is it stored and not dissipated like in a resistor because this energy can be extracted can be recovered from the capacitor if you leave the capacitor after charging to a voltage capital V if you leave it intact if you do not disturb it it shall maintain its charge for time immemorial time at infinity alright and therefore this energy can be recovered for example if you take a capacitor charge it to a voltage capital V then you short circuit the 2 terminals that is the 2 terminals you connect by means of a 0 resistance wire then a spark passes and the wire gets heated and therefore this energy can be recovered from the from the from the capacitor and therefore this energy is a stored energy. Now how does it store energy if there are 2 plates which are plus V and 0 then you know the lines of force the lines of force start from the positively charged plate and go towards the negatively charged plate this is the direction of the lines of force what is the direction of the electric field same as that of the lines of force and therefore if I put a positive charge here this positive charge shall go towards the lowest potential that is 0 potential alright the point that I was mentioning is that this energy is therefore stored in the electric field that exists between the 2 plates as I said if there exists a charge or a set of charges then an electric field is said to be created because another charge brought into this field fills a force of repulsion or attraction and therefore there is a field of force and this field of force is called the electric field therefore in a capacitor the energy is stored in the electric field on the other hand if I take an inductor an inductor physical manifestation is that of a coil if you take a 0 resistance wire and wind it let us say around this pen then it becomes a coil and it behaves as an inductor the property is that if a current passes through it I then a magnetic flux is generated around this coil and the flux experimentally it has been found that more the current the more the magnetic flux and therefore phi is proportional to I the flux is proportional to I and the proportionality constant is I linearity of an inductor implies that if I plot phi versus I it is a straight line with a slope of capital L if it is not linear if the relationship is not linear then you say the inductor is non-linear for example an iron cord inductor as you know if you increase the current sufficiently the core tends to tends to get saturated and that is why you get all those hysteresis phenomena and all that whenever saturation occurs it is in a display of non-linearity the flux current relationship no longer remains linear here also if the current remains constant then the flux remains constant on the other hand if current varies if current varies then the flux varies d phi phi varies and then a voltage a voltage is generated across the inductor and this voltage is given by d phi dt and this is equal to L di dt our convention would be that if this is the inductor and the voltage the potential difference is V the current is I then V is equal to L di dt this would be our convention whenever we draw a circuit element called an inductor this will be the sense of polarities that we shall adopt. Now an inductor stores energy it also stores energy but not in the electric field now it is in the magnetic field because an inductor is associated with moving charges unless the current flows it cannot generate a flux and therefore therefore the energy is stored in the magnetic field and this stored energy can now be utilized for example for inducing voltage in a nearby nearby inductor alright energy can be transferred from one inductor to another if you place a nearby inductor in the magnetic field of force then the second coil develops and emf across it alright so energy can be transferred and here also if you write the power expression it is V i and which is equal to L di dt L di dt and therefore the energy that is stored in the magnetic field is given by integral 0 to t in time capital T if the inductance is L then I di I beg your pardon 0 to capital I because t cancels out and this is what did I do I made a mistake I di okay dt cancels because W dw is equal to P dt and therefore dt and dt cancel and therefore what I get is half L i square half L capital I square therefore an amount of energy half L i square is stored in an inductor in the magnetic field and in that sense a capacitor and an inductor they are both energy storage elements and they are called dynamic elements dynamic whereas a resistance which does not store energy which only dissipates energy is called a static element alright a circuit which contains only resistors can be described by an algebraic equation a circuit which contains only resistors can be described by algebraic equations whereas a circuit which contains at least one energy storage element either an inductor or a capacitor at least one you cannot describe the circuit by means of an algebraic relationship you have to use a differential relationship and wherever differentiation with respect to time is involved the system or the circuit becomes dynamic because its property changes with time alright so capacitance and inductance are dynamic elements and a resistance is a static element you must understand I repeat what we mean by a linear resistor linear capacitor or linear inductor a linear inductor does not mean that the voltage current relationship is linear is not that right a linear inductor means the flux current relationship is linear alright a linear capacitor means that the charge voltage relationship q v relationship is linear a linear resistor means v i relationship is linear this must be kept in mind now as you know energy and under any circumstances cannot change instantaneously whether it is a mechanical system or an electrical system the total energy cannot change instantaneously for changing the total energy of a system you do require a certain non-zero amount of time because if energy can change instantaneously by definition d w d t which is the power becomes infinite and infinite power can neither be achieved nor can be conceived of and therefore one of the fundamental relationship is that if you fix any time let us say t 1 then the energy at time t 1 plus must be equal to the energy at time t 1 minus in other words just before t 1 whatever the energy is must be the same as the energy just after t 1 and therefore if that is so then you remember that energy in a capacitor is half c v squared and therefore since energy cannot change instantaneously the voltage across a capacitor cannot change instantaneously either is it not right in other words v c in general it shall be true that v c 0 minus would be equal to v c 0 plus where v c is the voltage across a capacitor in a similar manner energy in an inductor which is given by half l i squared since w cannot change instantaneously therefore i l current in an inductor at 0 minus must be equal to i l 0 plus I must point out that this 0 is an arbitrarily fixed reference it could be t 1 it could be t 2 or whatever it is we have we have taken this as 0 as the point of reference these 2 principles that the energy that the voltage across a capacitor cannot change instantaneously and that the current in an inductor cannot change instantaneously will be extremely useful in analysis of circuits electronic circuits or otherwise after introducing these 3 elements and their energy relationship let us look at some active elements active circuit elements the 3 elements are l and c are called passive because they cannot generate energy a resistance can only dissipate energy it cannot store it cannot generate an inductor in order that it stores magnetic energy it stores energy in the magnetic field it must be fed with a current it cannot generate on its own in a similar manner a capacitor cannot generate energy any device which cannot generate energy it can either dissipate or store is called a passive device on the other hand an active active device is one which can generate energy a battery for example is an active device a rotary converter or an electrical electromagnetic generator a rotating electrical machine which generates electrical energy is a generator. Now there are 2 kinds of active circuit elements that we shall basically be concerned with and these are these are voltage sources or current sources voltage sources and current sources now note these definitions carefully a voltage source is a generator which maintains its terminal voltage terminal voltage constant irrespective of what you connect across its terminals that is even if the load is changing if the load changes then what changes is the current but the voltage across the terminals remains v plus irrespective of what the load is. And therefore this is called a voltage source or sometimes the word the adjective ideal ideal is appended to it ideal ideal for an ideal voltage source naturally if I plot v versus i the terminal voltage versus current it remains a constant like this in an actual source however in an actual battery for example or an actual electromagnetic wave generator the voltage usually falls as the load current is increased in an ordinary power supply as you shall see in the laboratory if you take more current then the terminal voltage decreases and this non-idealness can be accounted for by introducing a small resistance here and this resistance is called the internal resistance of the source a non-ideal voltage source shall be modeled as an ideal voltage source v in series with a resistance r and therefore if you plot the terminal voltage now let us call it v t if you plot the terminal voltage v t versus i then what will happen the the curve shall no longer remain parallel but it shall droop because v t as you can see is v minus i r and therefore it is an equation to a straight line but it slopes it decreases as i increases and at any point at any point if this is the current then this difference is i times r is that clear the difference between an ideal voltage source and a non-ideal voltage source is this clear can will this line always be straight line yes if this resistance is linear then this line shall always be a straight line if you have a linear resistance because what you get is this the terminal relationship is given by v t equal to v minus i r which is an equation to a straight line provided small r is a linear resistance that is it does not depend on either the current or the voltage alright situation is slightly uncomfortable if we consider a current generator a current generator i capital i it maintains a constant current irrespective of what the load is if the load changes then what changes is the voltage v not the current the current capital i remains the same that is a constant a current generator maintains its output current the same irrespective of what load you connect so if we plot i in the load versus v then v changes but i does not change alright in a practical current source now you must follow this carefully in a practical current source usually the current drops as you increase the load resistance usually the current drops like this and this can be modeled this drop can be modeled by introducing a parallel resistance like this here small r under this condition you see the current in the load current in the load the green color I am using for non-idealness the current in the load shall be given by capital i minus minus what small v small v divided by small r which is a real linear relationship and therefore as v increases as v increases this term increases and therefore the current i current i decreases now if I had drawn we shall have such games in the in the tutorial class if I have drawn it the other way round suppose we draw v here and i here what kind of a characteristic I would have got for an ideal source it would be a vertical line for a non-ideal source tilted towards is it this way okay that is wonderful we will have all such orientations in the tutorial class and we shall consider consider tricky problems where it is likely it takes a bit of time to to figure out what kind of shape the voltage current voltage characteristics are 2 particular cases of termination shall be important termination that is you have a you have a let us say active active element it is either a current source or a voltage source it could be ideal or non-ideal I am representing this by means of a box with 2 terminals alright now 2 particular cases of termination are extremely important in electrical engineering and in electronics that is if there is no termination at all that is the load resistance is infinite then this condition is known as the open circuit OC open circuit means the circuit is open circuit and therefore no current can flow alright pardon me can I repeat that part one of the conditions of the load one extreme condition is that the load is infinity load is infinity means no resistance is connected across the 2 terminals if no resistance is connected then the circuit is open and therefore no current can flow and therefore this is called called the open circuit condition open circuit on the other hand you could also have a situation in which no voltage can drop across the terminals that is you connect them by a 0 resistance where that is RL is equal to 0 if RL equal to 0 then whatever current flows through RL it cannot drop a voltage and therefore V the terminal voltage is equal to 0 and this condition well open circuit is the condition for I equal to 0 and short circuit is the condition for V equal to 0 this is called a short circuited condition short circuit short circuit is usually accompanied by a spark because of the large amount of current that can flow virtually 0 resistance and if this is a this is an ideal voltage source if this active element is an ideal voltage source then how much current can flow infinite amount of current which also points out to the fallacy of conceptualizing an ideal voltage source an ideal voltage source naturally cannot exist if it if it can exist then it should be it should be capable of delivering infinite amount of power which is not possible agreed and therefore ideal voltage source is a conceptualization only it helps in in analyzing electrical and electronic circuits and therefore we have bit of an ideal voltage source or an ideal similarly an ideal current source cannot be cannot be realized in the laboratory what would happen if if you get an ideal current source you can generate almost infinite voltage isn't that right an ideal current source if it passes through an infinite resistance which is an open circuit you see the fallacy open circuit means current cannot flow but on the other hand if the source is ideal then obviously the total current should pass to an infinite resistance and the voltage should be infinity therefore the power that it is capable of delivering shall be infinity and therefore it is also a hypothetical element only. So well I have also introduced incidentally the concept of terminals which is very very easy to see that if you have a circuit if you have an electrical circuit which is only 2 terminals available then it is simply called a 2 terminal circuit okay it is also called a 1 port 1 port well the concept of a port is exactly like that of a ship docking at a port well since this is an electrical circuit an electrical ocean what you can dock in is either a voltage source or a current source or a load so what all that can be done is to connect from here to here you can connect a voltage source or a current source or a resistance so this is called 1 port it is only 1 port suppose you have a circuit in which there are 3 terminals like this well actually the both of these are the same 3 terminals like this then you can dock in 2 sources all right and therefore this is called a 3 terminal network 3 terminal circuit or a 2 port if a circuit has more than 2 ports it is called a multi port all right you can have a 3 port circuit the number of terminals shall depend on whether there are any common terminals or not for example in this circuit this terminal 3 let us say is common between port 1 and port 2 you could have a 2 port which has exactly 4 terminals like this this is a 2 port all right you can also have a 2 port which has 3 terminals in which these 2 are connected and usually they are connected to ground all right now enough of concepts let us work out a couple of examples the 1st example that you take is that of a fuse you know what is a fuse it is used in domestic power supply there is a fuse where which blows up if the current exceeds a certain limit all right a fuse is a non-linear resistor when the current goes high the voltage drop across it is such or the heat generated is such that it blasts it blows off so it disconnects the electricity and saves all the equipment in the house from being damaged from passing high current a fuse it melts when the current becomes excessive the resistance it is given that the resistance of a fuse is given by R equal to R R 1 plus C times T all right it is given that the resistance of a fuse is a function of temperature capital T is the temperature rise above the room temperature that is capital T is actual temperature minus room temperature okay capital T is the difference between the actual temperature and the room temperature or it is the rise above the room temperature C is a constant proportionality constant and R sub R is the resistance of the fuse at room temperature all right. It is also given that the temperature rise above the room temperature is given by is proportional to the power that is being fed to the fuse that is capital T the rise above room temperature is equal to some constant K times P what is wanted is determine an expression for R in terms of the current I and evaluate the current I such that the fuse blows off this is the problem all right you understand what you mean a fuse can be sufficiently accurately modeled by a relationship like this where the resistance is given by the resistance of the room temperature multiplied by 1 plus a constant times the rise of temperature of the fuse above the room temperature and this rise is understandably proportional to the power supply that is more power is supplied the more will be the heat generated the higher would be the would be the rise of temperature the question is to find out R as a function of I and to find out the current I at which the fuse blows off now you can see that capital R is equal to R R 1 plus C K P capital T is K P and P is I squared R and therefore capital R if I take this term to the left hand side becomes equal to R R divided by 1 minus C K I squared R R agreed this is therefore the expression now what is the relationship between R and I is it linear no it is a nonlinear relationship say capital R is a linear resistor should not depend on the current at all a linear resistor should not depend on the current that flows therefore it clearly shows that the fuse is a nonlinear resistor number 2 at what current does it blow off now when it blows off capital R becomes infinity blows off means what it becomes open circuit and therefore capital R becomes infinity capital R equal to infinity when I when this denominator term becomes equal to 0 and therefore capital I becomes equal to 1 over square root of C K R R that is the solution to the problem is that clear very simple problem any question on this no let us take another problem this problem is the connection of a resistance and the capacitance in parallel you know what is parallel connection and what is series connection in a parallel connection the potential difference across the elements are the same in a series connection the current in the elements are the same okay this is a parallel connection C and R and this circuit is used to smooth out to smooth out the fluctuations in a current I alright if I is not a constant if I is not a constant let us say I is something like this I is a constant on which is super imposed a small sinusoidal AC then this circuit smooths out this irregularities in other words the voltage V will approximately be a constant the voltage shall not display the ripples in the current and therefore this voltage shall be approximately constant and let this constant voltage be equal to capital V alright in that sense this circuit is said to be a filter it is as if it filters out the small ripples in the current alright now for good filtering it is said that the capacitance should be able to store W C should be able to store 10 times as much energy W C the criterion is that the energy stored in the capacitor should be 10 times the energy dissipated in the resistance R during one cycle during one cycle that is during if you take this sinusoidal then this is one cycle this is capital T alright during one cycle of the ripple the energy dissipated in the resistance 10 times that should be the energy stored in the capacitor this is the criterion that is specified it is also specified that capital R is 10 K capital R is 10 K you know what is a K K is 10 to the power 3 kilo is 10 to the 3 so 10 K is 10 to the 4 ohms R is 10 K and this ripples are at 60 cycles per second 60 cycles per second that is in one second there occurs 60 such cycles alright what you are required to do is to find out capital C alright now the solution do you understand the problem is a problem clear okay the solution to the problem lies in writing a relationship for W C and energy dissipated in the resistance W C is half C V square alright this we have already derived this should be 10 times the energies to energy dissipated in resistance what is the energy dissipated in the resistance I squared R but I squared is V squared divided by R squared times R times T capital T now what is capital T capital T is 1 over F alright so 1 over F F is the frequency okay cycles per second so it would be 1 over 60 and therefore I get C as equal to 20 alright V square and V square cancel C is 20 R and R squared 20 divided by R times 60 put down the value of R this is 20 divided by 10 K is 10 to the 4 times 60 and the unit will shall be Farad's okay after the name of Michael Faraday alright and this you can calculate this as 33.33 okay micro farad that is it these are 2 advanced problems given in the book and you can see that they are not not quite difficult as long as you understand what the problem is and it is extremely important that you understand the problem because it is said that once you understand the problem half of the problem is solved the other half is simply calculation I would like to conclude this class with a discussion on the 2 words which I have intentionally not used so far network we said that our course is introduction to electronic circuits alright and we have always said circuit element circuit elements could be passive or active and we have said energy may be energy shall be dissipated in a resistor energy shall be stored in an inductor or capacitor energy can be generated in an active device and so on however as you shall see as we go through the course in the textbook as also in my lectures I shall almost use these 2 words circuit and network interchangeably almost interchangeably but there is a difference between the 2 terms a circuit necessarily provides for at least one closed path for the flow of current a circuit necessarily should provide a path for flow of current so there and the path means a closed path that is if a current originates somewhere let us say in the battery then there is a resistance it cannot be left like this there must be a closed path for a current to flow the current if it is open circuit then the current shall not flow and therefore a circuit necessarily has at least one closed path now this path could be a short circuit or could be through another resistance alright so a circuit necessarily shall have at least one closed path on the other hand a network is a more general term a network is a more general term a network may contain circuits or may not contain circuits for example if you have a tree like this well this is a network you say it is a network of branches leaves but it does not contain a circuit that is if a chemical starts rising up like this well it cannot come back to the tree unless you provide another path like this which happens in the big benion trees for example you have those advantageous roots and therefore the chemical the juices flow in a circuit the point that I am mentioning is a network is not necessarily a circuit a network may not contain a circuit at all a network is a more general term than a circuit and when you say when you and say theorems for example we will use the term network theorems because they are applicable to more general situations alright whereas when you say about about laws which is circuit laws KVL Karchov voltage law and KCL Karchov current law they are both circuit laws because in a network there is no significance of this laws these laws apply to closed paths is that correct closed paths what about KCL KCL applies at a node the total current entering must be equal to total current leaving but there is a presumption that there exists a current one or more currents and therefore there must be a circuit right and therefore the KVL and KCL are circuit laws on the other hand when you enunciate theorems which apply to circuits as well as non-circuits or a combination of them we say network theorems and you shall study circuit laws and network theorems in tomorrow's lecture.