Fundamental Concepts of Electricity

Good day everyone and welcome. Today we'll be starting the course off with some basic concepts and laws of electricity. Most if not all of the elements of this presentation will be familiar to you all as you would have already covered it in O-level physics. The contents of this lecture is as follows. We'll take a brief look at conduction in metals as it relates to electrical charges. We'll explore the definitions of and laws associated with voltage and current and then power and energy. We'll then review resistivity and resistance, typical electrical symbols, circuit connections and most importantly, culture sports. So these points are very basic and should already be familiar to you. So I'll just read them. The most basic. quantity in an electric circuit is the electric charge. Each atom consists of electrons, protons, and neutrons. Charge E on an electron is negative and equal in magnitude to 1.602 by 10 to the minus 19th power coulombs. The coulomb is a large unit for charges. In one coulomb of charge, there are 1 over 1.602 by 10 to the minus 19, which gives us 6.24 by 10 to the 18th power electrons. Because the electric charge in a single electron is very small, there are a lot of electrons in one coulomb of charge. Thus, realistic laboratory values of charges are in the order of microcoulombs, nanocoulombs, or picocoulombs. We are all familiar with Bohr's model of the atom, even though you may not remember, it is called that, where the protons and neutrons are packed in the center of the atom, its nucleus, and the electrons orbit it. Metal atoms tend to have conductive properties because of the solitary electron in their valence shell. These electrons have loose bonds and are free to move randomly on ductile show. To further our understanding of this, we can recall metallic bonds from O-level chemistry. Metals bond in a lattice structure and the valence electrons are not bound to any single atom but are shared in a sea of electrons around the positive ions on nuclei. The motion of these free electrons mentioned is what we call an electric current. When an electric potential is applied across the conductor, the electrons flow toward the high positive potential. This is called drift direction, shown in the right direction here. However, the convention... for the electric field is from the high potential to the low potential. So the electric field will be from the right to the left. The law of conservation of charge states that charge can neither be created nor destroyed, only transferred. Thus the algebraic sum of the electric charges in a system does not change. The total amount of energy gained per unit charge must be equal to the amount of energy lost per unit charge, as energy and charge are both consumed. So what actually happens in the circuit is charge is created and injected from the negative terminal of the source, and the same amount of charge is neutralized at the positive end of the source. This is what constitutes a current flow in the circuit. I'll just read these points going forward. In an atom, each orbiting electron is at a certain energy level. In order for an electron to move from its orbit, it must be acted on by some force. In insulators, the orbiting electrons are tightly held by the attracting force of the nucleus and within the atomic matrix. In conductors, however, the outer electrons respond to stimuli and can leave their orbit. Some examples of those stimuli are heat, which causes random emission of these valence electrons, and an electromotive force. a potential difference which causes the unidirectional emission of electrons. Some examples of devices that create an electromotive force is a battery or a generator. The electromotive forces an electrical potential and is able to impart energy to outer electrons. The unit of emf is the volt. It is a measure of the amount of energy transferred to the electron. Current is the measure of the amount of charge flow in unit time. So therefore, the flow rate of charges in a conductor is current. Yeah, it is given by the equation dq dt, charge per unit time. You look at the unit analysis of this equation, it would be coulomb. per second and this is called the ampere. To move the electron in a conductor in a particular direction requires some work or energy transfer. So the voltage VAB between two points A and B in an electric circuit is the energy or work needed to move a unit charge from the point A to the point B. The voltage between the points A and B is the work exerted per charge, DW, DQ, where W is energy in joules and Q, as mentioned before, is charge in coulomb. The unit analysis here is joule per coulomb on newton meter per coulomb. The unit measure for work is joule on newton meter. this is called is the volt power is the time rate of expending or absorbing energy and it is measured in watts is given by the equation a DW DT a power expended per unit time and we can split DW DT into dw dq multiplied by dq dt. And we know these two things are v by i and this is how we derive the equation for power. Energy on the other hand is the capacity to do work measured in joules. So power is really how much energy given off or taken in per unit time and energy is the rate at which this is done and it's time dependent. it's time dependent and it's just uh power multiplied by the time in which that power is dissipated which is v by i multiplied by the time t is measured in joules so i decided to include a quick example to illustrate this how much energy it does on a 100 watt electric bulb consuming two hours so 100 watts the power rating of the bulb. It represents the power dissipated per unit time at any given time. So in two hours the energy dissipated would simply be 100 watts multiplied by a number of seconds in two hours which is 2 multiplied by 60 multiplied by 60 which gives us 0.72 mega joules of energy. Another example, a bit more complicated one this time, an electric heater consumes 1.8 mega joules when connected to a 250 volt supply for 30 minutes. Find the power rating of the heater and the current taken from the supply. So we have the energy dissipated by the electric heater or consumed by it for 30 minutes. So we can use the equation for energy to rearrange take the energy divided by the number of seconds in 30 minutes 60 by 30 and we'll get 1000 watts which is the power rating of the electric heater electric heater dissipates 1000 watts per unit time at any given now that we have the power we can use the power equation rearranged to find the current drawn by the supply. 1000 watts divided by the voltage source, 250 volts, will give us 4 amps. 4 amps is drawn from the source to the heater to produce 100 watts for 30 minutes, which dissipates at 1.8 megajoules of energy. Resistivity. Resistivity is a measure of a material's ability to oppose the flow of an electric current. It depends on the type of material and its temperature. So all materials have resistivity. It's a physical property of it. The value of this resistivity depends on the type of material and the temperature it is at. Metals tend to have a low resistivity. Therefore, gold and silver are used for high quality contacts in computers and instruments because of their low resistivity. So I have a table here with some examples of resistivity. I have four metals, silver, copper, aluminum, and gold. You can see how low their resistivity is. Then I have one insulator, glass, which has a very high range for resistivity. So what is resistance and how does it relate to resistivity? Resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area measured in ohms. This equation gives the relationship between resistance, resistivity and its physical properties. It is directly proportional to resistivity but is also dependent on the physical parameters length and area of the conductor. The reciprocal of resistance R is called conductance G and is measured in Siemens. So we expect the greater the resistance of a conductor the lesser its conductance or ability. to conduct electricity. So the definition of the ohm can be defined mathematically. When a constant potential of one volt is applied between two points, it produces a current flow of one ampere. This table on the left shows the resistivity values of different parts of the human body. Yep, I have in there for reference the first value and we have for all the muscles and bones and other tissue the respective resistivity values in ohm centimeters. A diagram on the right gives us ranges of resistance paths of the body. Considering the dimensions of a typical human, of course, we know it would vary from human to human. There are ranges based on a typical human. So we have 100 ohms from ear to ear, for example, and from hand to foot, 400 to 600 ohms. You can see for dry skin it's very high 100 kilo ohms to 600 kilo ohms and significantly lower for wet skin 1000 ohms. So why are these resistances important? Well we'll come to that soon. First we need to define Ohm's law. This law states that the voltage across a resistor is directly proportional to the current flowing through the resistor. provided that its temperature remains constant. So this is the relationship mathematically V is equal to IR. Rearrange the resistance is equal to V on I. So I have a diagram here that shows the relationship as well. Voltage is proportional to current and resistance serves to impede the current. So it's inversely proportional to it. If a voltage was plotted against a current on a graph, it would show a positive gradient. As you can see, the graph on the right and this gradient is equal to the resistance V on I. So if we apply Ohm's law to the body, given the resistances, we just... that was just given we can calculate the typical currents that would flow through the body when subjected to a given vote for example if i applied 120 volts by touching a live wire and the current travels from my hand to my foot i can use ohm's law to calculate how much current would flow through me or why would I need to do this? Why should I consider this beforehand if I'm handling you know live conductors? Well different magnitudes of current can cause different effects on my body as shown in the table. If one milliamp only one milliamp flows through my body I would just feel a slight tingling sensation. However if five milliamps flow I will feel a slight shock. it will be disturbing but not too painful. From 6 milliamps to 16 milliamps, I'll feel a painful shock. I'll lose some muscle control. Above that, from 17 to 19, sorry 99 milliamps, I would feel extreme pain. I may exhibit respiratory problems and I will experience muscle contraction. I may not be able to let go of the conductors. from 100 milliamps to 2000 i will exhibit a ventricular fibrillation my heartbeat will start to uh follow an irregular pattern yep i will also experience muscle contraction and some nerve damage and obviously i death is likely from these things over 2000 milliamps death is highly probable i would go into cardiac arrest i would experience organ damage and severe burn so let's look at resistors used in electric circuitry these are actual resistors manufactured to put in circuits to fulfill certain functions yeah they use a color code to denote resistance the resistance value and tolerance Each color represents a number and there are four bands of colors on the resistor as shown in the diagram. The first band represents the first figure of the resistance value. The second band represents the second figure. The third band tells us the multiplier of the resistance value. And the fourth gives us the tolerance. Just to recap, tolerance is simply an error margin or X. expected deviation from the ideal value of the resistance since practically no value would be exact yeah let's do an example uh we have the illustrated resistor up here let's do that one what is its resistance value well the first band is black and if we look at the table black is zero so the first figure is zero the second band is red red is two you The third band is green. Green is 5, so it's 10 to the 5th power. 2 by 10 to the 5th power is 200,000 ohms. Yeah, so 200 kilo ohms is the resistance of this resistor. The fourth band is silver, so it has a 10% tolerance as we could see in the table. What does this mean? It is All the resistances like these are manufactured with a 10% tolerance. It could be plus. or minus 10% of that value which is 180k that is minus 10% of 200k and 220k which is plus 10% of 200k. So if I for example buy 10 of these resistors all 10 will not be 200 kilo ohms. They will lie within the tolerance band which is from plus or minus 10% of that ideal value which is 180 kilo ohms to 220 kilo ohms. So now I want to look at the two extreme cases of resistance in a circuit, the open circuit and the short circuit. Theoretically, an open circuit refers to an infinite resistance. which would cause no current to flow. So between the terminals here we see an infinite resistance between r is equal to infinity and the current due to Ohm's law that would flow through here would have to be zero. Theoretically a short circuit refers to as zero resistance between the terminals which would cause an infinite current to flow. So what about this practically? Practical open circuits and short circuits, will they behave in this manner when we see these values? Practically there is no such thing as infinite and zero resistance. resistances in the real world may be very very high or very low but they will still be finite values so therefore an open circuit practically is simply a very large resistance which can be achieved by just open circuiting the terminal just breaking the conductor yeah and a short circuit is a very small resistance which can be achieved by connecting a conductor or a piece of wire between the terminals So this slide gives us the effect of temperature on resistance of a metal conductor. I'll just read the points. When temperature increases in a conductor, the number of free electrons per unit volume in the metal remains unchanged. However, the metal atoms in the crystal lattice vibrate with greater amplitude. The number of collisions between the free electrons and metal atoms increase because of this increase in vibration. As such, the electron flow slows down. And if we have a slowing of electron flow, that means we have a smaller current. And if we have a smaller current, we have a resistance increase in the metal. Yep, and vice versa happens when the temperature decreases. You will have less vibrations, less collisions, more electron flow and hence less resistance. The temperature coefficient of resistance is a measure of how much the resistance of a conductor. increases with the increase in temperature. So it is given by the formula shown. I don't usually ask you all to use this formula in your calculations. It's just there for completeness. From the graph, we could see resistance as a positive gradient for temperature, meaning as temperature increases, the resistance increases. This positive coefficient is the reason for keeping electrical equipment cool. the resistances will not increase as as it heats up under use if you keep it cool so this is why when we have a lot of electronic circuitry we try to keep it cool using uh air conditioners and so on so that it would not modify the value of resistances in the circuit this increase in temperature due to the increase in resistance sorry due to the increase in temperature may cause malfunctions in the electrical equipment this is the purpose of keeping it cool so i've listed a few important points i'd like you to note before we go further when you're doing your circuit analysis and you expressing a variable of a particular electrical parameter, be it voltage, current, or power. A common letter signifies time varying values. Yep. So you remember we have V of T, I of T, and so on. A capital letter, however, signifies average or constant values. Current is the flow of electrons. as i mentioned that is from negative to the positive terminals of a circuit however conventional current flow is from the positive terminal to the negative terminal so you'll see a lot of circuits or all these circuits i will be analyzing i will usually choose a conventional from the positive terminal of the source to the negative terminal of the source through the circuit this is not actually how the electrons flow but it is the convention we take when we're doing our circuit analysis current leaves active components and thus they supply power so active components supply power they produce currents current enters passive components like resistors and thus they absorb electrical compost electrical power so the passive components don't supply any power they actually absorb power that is supplied from active components so we look at active and passive components uh in more detail So there are two general classifications of electrical elements or symbols, active and passive. Active elements include independent and dependent sources. Dependent sources are not considered in this course. It won't be tested on it. So we will just focus on independent voltage sources and independent current. So passive elements, on the other hand, include. those that do not initially possess some stored electrical energy. And we have some, for example here, some resistors and the symbols for them. Let's take a closer look at some active independent elements. We have both DC sources, which include voltage sources and current sources. We have different options for symbols for them. It also includes AC sources which only include voltage supplies and you have two options for those. And now onto some passive elements. These are some common symbols for passive elements. We have resistors, general loads, unpolarized and polarized capacitors, and inductors which have air cores or iron cores. They also include some devices, filament lamps or fuses. You also have transformers. which is a topic by itself in this course and motors there are other symbols that form the connections between the aforementioned elements as well as those for measurement apparatus so pay particular attention here at how connections are drawn these are connecting in leader as well as how they are drawn when they intersect to show if they're electrically connected or not if they connect you'll see that duck yeah If not, you will see it has a loop over one of the conductors. Yeah. So you all should be familiar with switches. A switch is a device that could switch between or alternate between open circuit and a short circuit. Yeah. The earth symbol indicates a voltage potential of zero at that point in a circuit. So where about that? symbol is that touch it symbolizes that that point is at zero volts some measuring apparatus here volt meters meters and oh meters there's also watt meters which is a similar symbol with just with a w within the circle okay so now on to the more important part of circuit analysis given that we understand the laws the basic laws we introduced the basic concepts yeah the parameters voltage current power energy and we are able to form circuits we can go ahead and analyze them. So before we analyze them, we have to be able to identify the following. A node. A node is the point of connection between two or more branches. So I'll show you what our branch is next. But the node is symbolized by a dot as shown in the diagram. You have two nodes there. And we mostly consider nodes between more than two branches for our analysis. So what is a branch? A branch represents any element or elements between two nodes. So if you have any elements between two identified nodes, that is called a branch. How about a loop? A loop is simply any closed path in a circuit that is made by multiple branches. So these three important elements are highlighted in the circuit given and we need to be able to identify these things or elements of the circuit in order to be able to analyze the circuit. So let's apply what we know so far to identify the node branches and loops of this given circuit. So we could see the dots there. I made it easy for you. So we could identify the nodes. Firstly, what we actually see here in terms of just counting dots is eight dots. But what actually is happening here is we only have three nodes. Why is that? Well, remember, a node is on either side of our branch. Two nodes are on either side of our branch. so those three points on top d circuit which is b is actually one node simply because there are no elements between them they are actually one common point it's only because it is drawn in a kind of a structured fashion that it looks like three nodes same thing for Node C, these four points here are not separated by any element. So they are actually at the same voltage potential and therefore is the same node. So if we draw it in a different way, an unorthodox way, we could see that B is actually just one point and C is one point. So we have three nodes in the circuit. Now that we know the nodes, how do we know how much branches there are? We could just count the elements between these nodes. We have one, two, three, four, five branches. Now on to loops. How many loops are there in this circuit? Now this is a bit more complicated because loops can be any... combination of branches. So let's look at the combinations. we have the voltage source the 5 ohm and the 2 ohm making a complete path that's one the voltage source the 5 ohm and the 3 ohm making another complete path that's two loops the voltage source the 5 ohm and the 2 amp current source that's three loops Then we have a little loop in the middle here with just the resistance, the 2 ohm resistor and the 3 ohm resistor. So that's 4. Then we have the 2 ohm resistor and the current source. That's 5. The 3 ohm resistor and the current source finally, which is 6. So the different combinations there of branches. gives us six different loops. So the answers are three nodes, five branches, and six loops. All right, so now on to Kirchhoff's laws. Kirchhoff has two laws. His first law is based on the conservation of charge. Recall the law of conservation of charge states that the charge entering a system is equal to the charge leaving. So following this law, Kirchhoff's current law states that the algebraic sum of currents entering a node or a closed boundary is equal to zero. This is expressed mathematically by the summation given the sum of currents. where k is equal to 1 from 1 to n, that's the number of branches, is equal to 0. So for this given node in the diagram, for example, the currents entering the node, I1, I2, and I3, would be positive. But the currents leaving the node... would be negative so this is the convention when interacting with currents in and out of a node yep applying and expanding the summation for this example would give us i1 plus i2 plus i3 plus minus i4 plus minus i5 because i4 and i5 are in opposite directions to i1 2 and 3. So the sum of them is equal to zero. And if you rearrange, we could get I1 plus I2 plus I3 is equal to I4 plus I5. And this means that the sum of the currents entering the node is equal to the sum of the currents leaving the same node, which supports the law of conservation. This is Kirchhoff's current law. Kuchoff's second law is based on the conservation, the law of conservation of energy. It states that the algebraic sum of all voltages around a closed path or loop is equal to zero. So this is expressed mathematically by the summation shows the sum of voltages in a loop. When a number of voltages in that loop is k is equal to one. n is equal to zero. Considering this circuit, the simple circuit for example, for a clockwise current flowing out from the positive of the source through the resistors and back through the negative of the source, conventional current flow, we will have these voltage directions. So how did I get these voltage directions? Well, a voltage source direction is already defined for us remember our voltage source has a positive terminal and a negative terminal so the direction will point in the positive terminal yeah but for inactive components like resistors the voltage drop across them is not predetermined the voltage drop direction across a passive element is opposite to that of the current flow defined. So if I define conventional current like I did clockwise, then the voltage drop across V2, V3, V4 and V5 would be in the opposite direction to the current flow. So when we expand the summation for this example, we have V1 plus. minus V2 minus V3 plus minus V4 plus minus V5. Remember if we take the clockwise direction this source voltage is in the opposite direction to all the voltage drops. Yeah so this is why all the voltage drops are negative. If we rearrange this formula We have V1, the voltages in this positive direction, is equal to V2 plus V3 plus V4 plus V5. The voltage is in the opposite direction to that source voltage, and hence the source current. So, hence, in a loop, the sum of voltages in one direction is equal to the sum of voltages in the opposite direction. Yeah? this is Kirchhoff's voltage. Now this makes sense because it is based on the law of conservation of energy where energy cannot be created or destroyed so if energy is produced by the source it is dissipated in the resistors the passive elements of that particular loop yeah remember energy used to move charges in a loop is called voltage. All right, so let's apply what we learned so far to our question. Given this circuit, whose nodes and loops are already identified for you, use Kirchhoff's current law to obtain equations of the nodes A and B, and use Kirchhoff's voltage law. top-down equations for loops one two and three so starting with the nodes a and b analyzing node a we could see that i1 goes into the node i2 goes into the node as well and i3 leaves the node so we have i1 plus i2 is equal to i3 this is what we end up with here yeah at node b we have the same thing because it is actually the return path of this node i3 will be flowing into this node while i1 and i2 will be leaving and rearranging the equation we will get the same formula so these nodes connect the same branches and this is why they have the same equations yeah so let's use kuchoff's voltage law for the loops the three loops loop one is just this loop here on the left The voltage source V0 is in the upward direction. Since I1 is in the clockwise direction, the voltage drops across R1. R3 will be opposing that current direction. So we have V0 minus V1 minus V3 is equal to 0. We could rearrange and we get V0 is equal to V1 plus V3. Now we could substitute. Using Ohm's law for the voltage drops, V1 and V3, which is just the current flowing through that resistive element multiplied by the resistance of that element. And we get I1R1 for V1 and I3R3 for V3. Here we just use Ohm's law and substitute it for voltage. so looking at loop two remember loop two the current defined for us in that loop which is the right loop is counterclockwise so we have v4 which is the positive direction is upward and the current is flowing counterclockwise so the voltage drops across r2 and r3 would be in the opposite direction of the current yep so we will end up with a v4 minus v2 minus v3 could rearrange this time we get v4 is equal to v2 plus v3 and we could substitute using ohm's law to get them in terms of current and resistance the voltage drops Let's apply Kirchhoff's voltage law to the outer loop. This outer loop has two voltage sources. So let's see how that works. Yeah. The current direction is clockwise. It was chosen for us clockwise. V0 is in the positive direction. If the current is clockwise, then the voltage drop across R1 would be in the opposite direction to that current direction. R2 as well. So you will have minus V1 minus V2. And also V4 will be minus. Why? Because the positive is upward. There's a positive terminal on top, a negative terminal below. So the voltage across this is upward. And this is in the opposite direction to V0. So what we have here is V0. minus v1 minus v2 minus v4 remember for the supplies the directions are already predetermined based on the terminals of the supply so rearranging we have v naught minus v4 is equal to v1 plus v2 we put all the voltage drops on one side of the equation so we could substitute it using all yeah so let's move on you'll have a lot more examples like this to look at before you are able to do one of these on you yeah you have to get a lot more practice and i will give you a lot more examples before you could attempt them on your own so what about equivalent resistance of a circuit If you have resistances in series, what does that mean? A series connection is when any two elements share the same node on one side of each other. If they share the same node on one side, they are in series with each other. Yep, so R1 is in series with R2, which is in series with R3, etc. Yeah. So how do we find the equivalent resistance of this circuit with n resistors in series? Well, the equivalent resistors resistance can be derived by using Kirchhoff's voltage law to the circuit. So how do we do that? Remember, Kirchhoff's voltage law is the voltages in. a loop yeah for a specific current direction so you're assuming the current direction is clockwise yeah from the positive terminal to the negative terminal conventional current and so you'll have a positive upward direction for voltage for the from the source and an opposing voltage drop from all the series resistances yeah so therefore true volt for Kirchhoff's voltage though we have v is equal the voltage of the supply is equal to all the drops across the resistances v1 plus v2 plus v3 and so on yeah using Ohm's law we could substitute v for ir as shown so you'll have ir1 is v1 ir2 is v2 and so on remember i flows through all of the resistance so it's have a common current flowing through all of the resistances yeah but we have different values r1 r2 r3 and so on so we could factor we could factor the current out because it's common the current is flowing through all the resistors and we'll have this expression here and if we take the current and we rearrange the formula we put it under the v we will actually find an expression for Req, the equivalent resistance. Now you'll see Req is the source voltage divided by the source current and is also derived to be the sum of all the series resistances. So this is why resistances in series can be added to find its equivalent resistance, the circuit's equivalent resistance. This is how it's derived. yeah what about parallel connections a parallel connection is when two or more elements share the same nodes on either side of each element so on both sides there must be the same node recall this entire area is one node in this parallel circuit here and below this entire area is one node and all these resistances share these nodes on either side. So by definition, they are in parallel. Yeah? R1 is in parallel with R2, is in parallel with R3, and so on. So how do we find the equivalent resistance of this circuit? How do we resolve resistors in parallel this time? Well, it can be derived using Kirchhoff's current law. applied to the node of this circuit. How do we do that? Well, we consider the node, and we can see at the node, I, the total source current entering the node, is equal to I1 plus I2 plus I3, which are the branch currents exiting those nodes, or that node, sorry. So this is what we have here due to Kirchhoff's Current law applied to this node, we have I, the source current, is equal to the sum of all the branch currents. Using Ohm's law to substitute for I, you have V on R. Remember, V, the voltage. at the source is across all the parallel resistance of this circuit yeah this voltage is across all the resistances which are in parallel in the circuit simply because they are in parallel yeah so we have the same voltage over each resistance the sum of those yeah that is ohm's law substituted Since V is common, we could factor that out. And we have the sum of the reciprocals of the resistances in each branch. Yeah. If we rearrange this, put the V in the denominator on the left hand side of the formula. We have I on V is equal to the sum of the reciprocals of the parallel resistances. Yeah. So I on V, we know V on I is Req. So I on V would be 1 on Req. So what we have derived here when simplifying is the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of all the parallel resistances. And this is how we derive. the equation for equivalent resistances in parallel. The reciprocal of the equivalent resistance is the sum of the reciprocals of all the branch resistances in parallel to each other. So I'm sure you've seen these formulae before, but you may not have derived them yet. So we have done that and I hope it's understandable. So now we're going to look at how voltage supplied in a circuit divides among elements in a loop. If we have to find the voltage across a particular passive element in a loop, say a resistor, it will be a fraction. of the source voltage applied to the loop. This fraction is the resistance of that resistor over the total resistance in the loop. So, for example, we have a source, a voltage source supplying R1 and R2 in series with each other. We want to find the voltage across either of these resistances. We know that the voltage across these resistances is a fraction of the source voltage. Now that fraction is defined by that resistance value divided by the total resistance in the loop. So the voltage across R1 is R1 on R1 plus. R2 multiplied by the total voltage. So this is the fraction of the total voltage that is dropped across R1. Similarly, the voltage drop across R2 is R2 over R1 plus R2, the total, multiplied by the supply voltage. So this is the fraction. the voltage that drops across R2. Yeah and according to Kirchhoff's voltage law this total voltage here is equal to the sum of these two voltage drops but their specific value depends on their resistance value. Yeah it depends on a fraction which is the resistance of that value sorry the resistance value of that resistor divided by the total resistance in the circuit so what about the current divider rule we are now going to look at how current supplied it divides among different parallel branches in our circuit yeah if we have to find the current through a particular element or branch say a resistor it will be a fraction of the total current applied from the source yeah this fraction we are talking about is the equivalent resistance of the resistors in the other branches over the total resistances in the circuit let's see what i'm talking about it must be reduced to two branches before analysis so let's see what we are talking about here if we have a circuit with a set of parallel branches and we want to know the current through these branches we could use the current divider rule. The current divider rule tells us that the currents through these branches are a fraction of the total current entering the node and that makes sense because it follows the law of conservation of charge and Kuchov's current law. So Ix here is a fraction of the total current i t. What is that fraction? That fraction is given by the resistance of all the other branches over the sum of all the branches. So the current i x is given by r t which is the parallel combination of all the other branches divided by That value plus rx. And that fraction of the total current is the current flowing in that branch. So it's a bit more complicated than the voltage divider rule. Because it's not that resistance over the sum. It's the equivalent resistance of the other branches. divided by the sum of the parallel branches, all the parallel branches. Yeah. So RT is resolved in parallel. Remember, you are not summing these because they are not in series. Yeah. They have to be resolved in parallel. And we just really derived how to do that to get RT. And then to get the current through. our x, it is our t. divided by the sum of the two branches we're analyzing, which is Rx plus Rt, right? Through examples, you will solidify this rule, yeah? So let's look at a practical example for some appreciation. So we have a transmission line. from A to B. It is divided into three segments of resistance R1, R2 and R3 with these values 1 ohm, 2 ohm and 3 ohms respectively. A generating station acts as the source and supplies 1000 volts. through the transmission line to the load which is residential customers their houses they receive a voltage of 940 volts so we have a circuit they are practical circuits and we could apply the laws we've learned in this presentation in this lesson to calculate the voltage across the transmission. So what is the voltage VAB? VAB is simply the difference in voltages between A and B. In this case we have the voltages at A and B. What we have to do is find the potential difference between the two. which is a 1000 volts minus 940 volts that means that 60 volts is dropped across the transmission line yeah next what is the current through the transmission line we have the voltage across the transmission line and we have the resistance of the transmission line so we could use ohm's law and calculate the current I line through the transmission line. It will be 60 volts, we got before, divided by 1 R1 plus R2 plus R3 which is 1 plus 2 plus 3. That's 60 divided by 6, 10 amps. Next, what is the voltage across R3? The segment R3. the transmission now to do that we need to use the voltage divider rule yeah because we want the voltage across one component we have the total voltage the total voltage across a and B is 60 we calculated that and we want the voltage now across one component of out of the three so we could use the voltage divider rule to find V3. That would be the total voltage, 60, multiplied by the resistance. We're looking to find the voltage across, divided by the total, 1 plus 2 plus 3, which is 6. So the voltage across R3 would be 30 volts. So let's switch things up a little bit. A bird flies and lands on the transmission line. It lands across the R3 section of the transmission line as shown. Its body resistance is 3000 ohms. So if we look closely this bird introduces a new element to the circuit in parallel with R3. So the new current must factor in this resistance. What is the new current to the transmission line? We have to rework this using Ohm's law. But now we have 3R3 in parallel with R bird, which is 3 Ohms in parallel with 3000 Ohms. So it's no longer just 60 divided by 1 plus 2 plus 3. It's 60 volts divided by 1 plus 2 plus 3 in parallel with the resistance of the bullet, 3,000 ohms. This gives us not 10 amps, but 10.005 amps. So we have a 5 milliamp difference from the last case. Yeah. So using the current divider rule, we could actually calculate how much current is flowing through the bird. We'd like to know if the bird is going to be electrocuted or if it's going to be fine. So we have to calculate the current flowing through this bird. Yeah. How do we do that? We use the current divider rule because we know the total current flowing through the transmission line, 10.005 amps. and we know the two branches it splits up at this node. Where the bird makes contact with the transmission line is actually an introduced node where current splits. Yeah, so current flowing through the bird, iBird, is equal to the total current flowing through the line multiplied by the resistance in the other branch which is the transmission line divided by the total of the two branches three plus three thousand which is three thousand and three and we get the current flowing through 0.01 amp which is 10 milliamps so this bird is definitely dead because it's 10 milliamps is a sizable current for a bird yeah so in reality even less current if any at all it passed through the bird We see birds land on wires all the time and nothing happens to them. Yeah, this is because the resistance of a real bird is much higher and since the bird is small, the segment of wire between the bird's leg when it lands on the transmission line is very small. So our wire shown here is actually smaller. than my example and our bird is actually larger than my example so therefore the resistance of the bird is much greater than the resistance of the wire causing a much smaller current draw if any current at all flowing through the bird yeah so the bird is usually fine when it lands on a transmission line So what happens though when we see birds die of electrocution? What happened there? Well, if the bird flaps its wings, for example, or it steps on two separate conductors of different voltage potentials, then the potential difference across the bird will be significant. And then a significant current would flow. and the bird will be electrocuted. So the potential on one line like our example is smaller than if the bird steps on one line and touches another. So the larger the potential difference between the legs of the bird or the wings of the bird means a larger current will flow and if this current is large enough the bird will die. So high voltage line workers use this principle to do live line work on transmission lines. They and their equipment are at the same potential on the line so there's no difference in potential and no current flows through them to electrocute them. There are some illustrations of them. in this So a much more interesting and applicable situation is how we get shocked. So let's consider if we have some sort of electrical device where the insulation of a live wire is exposed and it comes into contact with the metal case. So you have some wire inside this case here and the insulation is degraded and the live part of the wire touches the metal case. If someone touches that case, their body creates a parallel circuit with the device as represented by the equivalent circuit shown. So touching the live metal case creates a parallel circuit with the device where current flows through you once the person is grounded. So once that person, you know, makes contact with ground, current will flow through their body and they will experience a shock. So that's usually what happens. We create a parallel path for current to flow. If we come into contact with a positive voltage potential and our foot makes contact with a zero potential, we have a potential difference now across us and therefore our current will flow. based on that resistance path we make for the voltage potential the value of current a specific value of current will flow that value of current is high enough we will experience uh the different symptoms as shown in in the previous slide in this presentation for different uh current fluid yeah if it's large enough we will uh experience some more severe symptoms like cardiac arrest and all that and it may be lethal so that's it for today um i hope you all understood the presentation i want you all to go through it thoroughly and highlight the parts you do not understand. And feel free to ask in the next class or post your questions in the audience.

The contents of this lecture is as follows. We'll take a brief look at conduction in metals as it relates to electrical charges. We'll explore the definitions of and laws associated with voltage and current and then power and energy. We'll then review resistivity and resistance, typical electrical symbols, circuit connections and most importantly, culture sports.

So these points are very basic and should already be familiar to you. So I'll just read them. The most basic.

quantity in an electric circuit is the electric charge. Each atom consists of electrons, protons, and neutrons. Charge E on an electron is negative and equal in magnitude to 1.602 by 10 to the minus 19th power coulombs.

The coulomb is a large unit for charges. In one coulomb of charge, there are 1 over 1.602 by 10 to the minus 19, which gives us 6.24 by 10 to the 18th power electrons. Because the electric charge in a single electron is very small, there are a lot of electrons in one coulomb of charge. Thus, realistic laboratory values of charges are in the order of microcoulombs, nanocoulombs, or picocoulombs.

We are all familiar with Bohr's model of the atom, even though you may not remember, it is called that, where the protons and neutrons are packed in the center of the atom, its nucleus, and the electrons orbit it. Metal atoms tend to have conductive properties because of the solitary electron in their valence shell. These electrons have loose bonds and are free to move randomly on ductile show. To further our understanding of this, we can recall metallic bonds from O-level chemistry. Metals bond in a lattice structure and the valence electrons are not bound to any single atom but are shared in a sea of electrons around the positive ions on nuclei.

The motion of these free electrons mentioned is what we call an electric current. When an electric potential is applied across the conductor, the electrons flow toward the high positive potential. This is called drift direction, shown in the right direction here. However, the convention...

for the electric field is from the high potential to the low potential. So the electric field will be from the right to the left. The law of conservation of charge states that charge can neither be created nor destroyed, only transferred.

Thus the algebraic sum of the electric charges in a system does not change. The total amount of energy gained per unit charge must be equal to the amount of energy lost per unit charge, as energy and charge are both consumed. So what actually happens in the circuit is charge is created and injected from the negative terminal of the source, and the same amount of charge is neutralized at the positive end of the source.

This is what constitutes a current flow in the circuit. I'll just read these points going forward. In an atom, each orbiting electron is at a certain energy level. In order for an electron to move from its orbit, it must be acted on by some force. In insulators, the orbiting electrons are tightly held by the attracting force of the nucleus and within the atomic matrix.

In conductors, however, the outer electrons respond to stimuli and can leave their orbit. Some examples of those stimuli are heat, which causes random emission of these valence electrons, and an electromotive force. a potential difference which causes the unidirectional emission of electrons. Some examples of devices that create an electromotive force is a battery or a generator. The electromotive forces an electrical potential and is able to impart energy to outer electrons.

The unit of emf is the volt. It is a measure of the amount of energy transferred to the electron. Current is the measure of the amount of charge flow in unit time.

So therefore, the flow rate of charges in a conductor is current. Yeah, it is given by the equation dq dt, charge per unit time. You look at the unit analysis of this equation, it would be coulomb.

per second and this is called the ampere. To move the electron in a conductor in a particular direction requires some work or energy transfer. So the voltage VAB between two points A and B in an electric circuit is the energy or work needed to move a unit charge from the point A to the point B. The voltage between the points A and B is the work exerted per charge, DW, DQ, where W is energy in joules and Q, as mentioned before, is charge in coulomb.

The unit analysis here is joule per coulomb on newton meter per coulomb. The unit measure for work is joule on newton meter. this is called is the volt power is the time rate of expending or absorbing energy and it is measured in watts is given by the equation a DW DT a power expended per unit time and we can split DW DT into dw dq multiplied by dq dt.

And we know these two things are v by i and this is how we derive the equation for power. Energy on the other hand is the capacity to do work measured in joules. So power is really how much energy given off or taken in per unit time and energy is the rate at which this is done and it's time dependent.

it's time dependent and it's just uh power multiplied by the time in which that power is dissipated which is v by i multiplied by the time t is measured in joules so i decided to include a quick example to illustrate this how much energy it does on a 100 watt electric bulb consuming two hours so 100 watts the power rating of the bulb. It represents the power dissipated per unit time at any given time. So in two hours the energy dissipated would simply be 100 watts multiplied by a number of seconds in two hours which is 2 multiplied by 60 multiplied by 60 which gives us 0.72 mega joules of energy. Another example, a bit more complicated one this time, an electric heater consumes 1.8 mega joules when connected to a 250 volt supply for 30 minutes.

Find the power rating of the heater and the current taken from the supply. So we have the energy dissipated by the electric heater or consumed by it for 30 minutes. So we can use the equation for energy to rearrange take the energy divided by the number of seconds in 30 minutes 60 by 30 and we'll get 1000 watts which is the power rating of the electric heater electric heater dissipates 1000 watts per unit time at any given now that we have the power we can use the power equation rearranged to find the current drawn by the supply. 1000 watts divided by the voltage source, 250 volts, will give us 4 amps. 4 amps is drawn from the source to the heater to produce 100 watts for 30 minutes, which dissipates at 1.8 megajoules of energy.

Resistivity. Resistivity is a measure of a material's ability to oppose the flow of an electric current. It depends on the type of material and its temperature.

So all materials have resistivity. It's a physical property of it. The value of this resistivity depends on the type of material and the temperature it is at.

Metals tend to have a low resistivity. Therefore, gold and silver are used for high quality contacts in computers and instruments because of their low resistivity. So I have a table here with some examples of resistivity. I have four metals, silver, copper, aluminum, and gold.

You can see how low their resistivity is. Then I have one insulator, glass, which has a very high range for resistivity. So what is resistance and how does it relate to resistivity? Resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area measured in ohms.

This equation gives the relationship between resistance, resistivity and its physical properties. It is directly proportional to resistivity but is also dependent on the physical parameters length and area of the conductor. The reciprocal of resistance R is called conductance G and is measured in Siemens. So we expect the greater the resistance of a conductor the lesser its conductance or ability. to conduct electricity.

So the definition of the ohm can be defined mathematically. When a constant potential of one volt is applied between two points, it produces a current flow of one ampere. This table on the left shows the resistivity values of different parts of the human body.

Yep, I have in there for reference the first value and we have for all the muscles and bones and other tissue the respective resistivity values in ohm centimeters. A diagram on the right gives us ranges of resistance paths of the body. Considering the dimensions of a typical human, of course, we know it would vary from human to human.

There are ranges based on a typical human. So we have 100 ohms from ear to ear, for example, and from hand to foot, 400 to 600 ohms. You can see for dry skin it's very high 100 kilo ohms to 600 kilo ohms and significantly lower for wet skin 1000 ohms. So why are these resistances important? Well we'll come to that soon.

First we need to define Ohm's law. This law states that the voltage across a resistor is directly proportional to the current flowing through the resistor. provided that its temperature remains constant. So this is the relationship mathematically V is equal to IR.

Rearrange the resistance is equal to V on I. So I have a diagram here that shows the relationship as well. Voltage is proportional to current and resistance serves to impede the current. So it's inversely proportional to it.

If a voltage was plotted against a current on a graph, it would show a positive gradient. As you can see, the graph on the right and this gradient is equal to the resistance V on I. So if we apply Ohm's law to the body, given the resistances, we just... that was just given we can calculate the typical currents that would flow through the body when subjected to a given vote for example if i applied 120 volts by touching a live wire and the current travels from my hand to my foot i can use ohm's law to calculate how much current would flow through me or why would I need to do this? Why should I consider this beforehand if I'm handling you know live conductors?

Well different magnitudes of current can cause different effects on my body as shown in the table. If one milliamp only one milliamp flows through my body I would just feel a slight tingling sensation. However if five milliamps flow I will feel a slight shock.

it will be disturbing but not too painful. From 6 milliamps to 16 milliamps, I'll feel a painful shock. I'll lose some muscle control. Above that, from 17 to 19, sorry 99 milliamps, I would feel extreme pain. I may exhibit respiratory problems and I will experience muscle contraction.

I may not be able to let go of the conductors. from 100 milliamps to 2000 i will exhibit a ventricular fibrillation my heartbeat will start to uh follow an irregular pattern yep i will also experience muscle contraction and some nerve damage and obviously i death is likely from these things over 2000 milliamps death is highly probable i would go into cardiac arrest i would experience organ damage and severe burn so let's look at resistors used in electric circuitry these are actual resistors manufactured to put in circuits to fulfill certain functions yeah they use a color code to denote resistance the resistance value and tolerance Each color represents a number and there are four bands of colors on the resistor as shown in the diagram. The first band represents the first figure of the resistance value.

The second band represents the second figure. The third band tells us the multiplier of the resistance value. And the fourth gives us the tolerance. Just to recap, tolerance is simply an error margin or X.

expected deviation from the ideal value of the resistance since practically no value would be exact yeah let's do an example uh we have the illustrated resistor up here let's do that one what is its resistance value well the first band is black and if we look at the table black is zero so the first figure is zero the second band is red red is two you The third band is green. Green is 5, so it's 10 to the 5th power. 2 by 10 to the 5th power is 200,000 ohms.

Yeah, so 200 kilo ohms is the resistance of this resistor. The fourth band is silver, so it has a 10% tolerance as we could see in the table. What does this mean? It is All the resistances like these are manufactured with a 10% tolerance.

It could be plus. or minus 10% of that value which is 180k that is minus 10% of 200k and 220k which is plus 10% of 200k. So if I for example buy 10 of these resistors all 10 will not be 200 kilo ohms. They will lie within the tolerance band which is from plus or minus 10% of that ideal value which is 180 kilo ohms to 220 kilo ohms. So now I want to look at the two extreme cases of resistance in a circuit, the open circuit and the short circuit.

Theoretically, an open circuit refers to an infinite resistance. which would cause no current to flow. So between the terminals here we see an infinite resistance between r is equal to infinity and the current due to Ohm's law that would flow through here would have to be zero.

Theoretically a short circuit refers to as zero resistance between the terminals which would cause an infinite current to flow. So what about this practically? Practical open circuits and short circuits, will they behave in this manner when we see these values?

Practically there is no such thing as infinite and zero resistance. resistances in the real world may be very very high or very low but they will still be finite values so therefore an open circuit practically is simply a very large resistance which can be achieved by just open circuiting the terminal just breaking the conductor yeah and a short circuit is a very small resistance which can be achieved by connecting a conductor or a piece of wire between the terminals So this slide gives us the effect of temperature on resistance of a metal conductor. I'll just read the points. When temperature increases in a conductor, the number of free electrons per unit volume in the metal remains unchanged. However, the metal atoms in the crystal lattice vibrate with greater amplitude.

The number of collisions between the free electrons and metal atoms increase because of this increase in vibration. As such, the electron flow slows down. And if we have a slowing of electron flow, that means we have a smaller current. And if we have a smaller current, we have a resistance increase in the metal.

Yep, and vice versa happens when the temperature decreases. You will have less vibrations, less collisions, more electron flow and hence less resistance. The temperature coefficient of resistance is a measure of how much the resistance of a conductor.

increases with the increase in temperature. So it is given by the formula shown. I don't usually ask you all to use this formula in your calculations.

It's just there for completeness. From the graph, we could see resistance as a positive gradient for temperature, meaning as temperature increases, the resistance increases. This positive coefficient is the reason for keeping electrical equipment cool. the resistances will not increase as as it heats up under use if you keep it cool so this is why when we have a lot of electronic circuitry we try to keep it cool using uh air conditioners and so on so that it would not modify the value of resistances in the circuit this increase in temperature due to the increase in resistance sorry due to the increase in temperature may cause malfunctions in the electrical equipment this is the purpose of keeping it cool so i've listed a few important points i'd like you to note before we go further when you're doing your circuit analysis and you expressing a variable of a particular electrical parameter, be it voltage, current, or power.

A common letter signifies time varying values. Yep. So you remember we have V of T, I of T, and so on. A capital letter, however, signifies average or constant values.

Current is the flow of electrons. as i mentioned that is from negative to the positive terminals of a circuit however conventional current flow is from the positive terminal to the negative terminal so you'll see a lot of circuits or all these circuits i will be analyzing i will usually choose a conventional from the positive terminal of the source to the negative terminal of the source through the circuit this is not actually how the electrons flow but it is the convention we take when we're doing our circuit analysis current leaves active components and thus they supply power so active components supply power they produce currents current enters passive components like resistors and thus they absorb electrical compost electrical power so the passive components don't supply any power they actually absorb power that is supplied from active components so we look at active and passive components uh in more detail So there are two general classifications of electrical elements or symbols, active and passive. Active elements include independent and dependent sources. Dependent sources are not considered in this course.

It won't be tested on it. So we will just focus on independent voltage sources and independent current. So passive elements, on the other hand, include. those that do not initially possess some stored electrical energy. And we have some, for example here, some resistors and the symbols for them.

Let's take a closer look at some active independent elements. We have both DC sources, which include voltage sources and current sources. We have different options for symbols for them. It also includes AC sources which only include voltage supplies and you have two options for those. And now onto some passive elements.

These are some common symbols for passive elements. We have resistors, general loads, unpolarized and polarized capacitors, and inductors which have air cores or iron cores. They also include some devices, filament lamps or fuses. You also have transformers. which is a topic by itself in this course and motors there are other symbols that form the connections between the aforementioned elements as well as those for measurement apparatus so pay particular attention here at how connections are drawn these are connecting in leader as well as how they are drawn when they intersect to show if they're electrically connected or not if they connect you'll see that duck yeah If not, you will see it has a loop over one of the conductors.

Yeah. So you all should be familiar with switches. A switch is a device that could switch between or alternate between open circuit and a short circuit. Yeah.

The earth symbol indicates a voltage potential of zero at that point in a circuit. So where about that? symbol is that touch it symbolizes that that point is at zero volts some measuring apparatus here volt meters meters and oh meters there's also watt meters which is a similar symbol with just with a w within the circle okay so now on to the more important part of circuit analysis given that we understand the laws the basic laws we introduced the basic concepts yeah the parameters voltage current power energy and we are able to form circuits we can go ahead and analyze them. So before we analyze them, we have to be able to identify the following.

A node. A node is the point of connection between two or more branches. So I'll show you what our branch is next. But the node is symbolized by a dot as shown in the diagram.

You have two nodes there. And we mostly consider nodes between more than two branches for our analysis. So what is a branch?

A branch represents any element or elements between two nodes. So if you have any elements between two identified nodes, that is called a branch. How about a loop? A loop is simply any closed path in a circuit that is made by multiple branches.

So these three important elements are highlighted in the circuit given and we need to be able to identify these things or elements of the circuit in order to be able to analyze the circuit. So let's apply what we know so far to identify the node branches and loops of this given circuit. So we could see the dots there.

I made it easy for you. So we could identify the nodes. Firstly, what we actually see here in terms of just counting dots is eight dots.

But what actually is happening here is we only have three nodes. Why is that? Well, remember, a node is on either side of our branch. Two nodes are on either side of our branch. so those three points on top d circuit which is b is actually one node simply because there are no elements between them they are actually one common point it's only because it is drawn in a kind of a structured fashion that it looks like three nodes same thing for Node C, these four points here are not separated by any element.

So they are actually at the same voltage potential and therefore is the same node. So if we draw it in a different way, an unorthodox way, we could see that B is actually just one point and C is one point. So we have three nodes in the circuit. Now that we know the nodes, how do we know how much branches there are?

We could just count the elements between these nodes. We have one, two, three, four, five branches. Now on to loops.

How many loops are there in this circuit? Now this is a bit more complicated because loops can be any... combination of branches. So let's look at the combinations.

we have the voltage source the 5 ohm and the 2 ohm making a complete path that's one the voltage source the 5 ohm and the 3 ohm making another complete path that's two loops the voltage source the 5 ohm and the 2 amp current source that's three loops Then we have a little loop in the middle here with just the resistance, the 2 ohm resistor and the 3 ohm resistor. So that's 4. Then we have the 2 ohm resistor and the current source. That's 5. The 3 ohm resistor and the current source finally, which is 6. So the different combinations there of branches.

gives us six different loops. So the answers are three nodes, five branches, and six loops. All right, so now on to Kirchhoff's laws. Kirchhoff has two laws. His first law is based on the conservation of charge.

Recall the law of conservation of charge states that the charge entering a system is equal to the charge leaving. So following this law, Kirchhoff's current law states that the algebraic sum of currents entering a node or a closed boundary is equal to zero. This is expressed mathematically by the summation given the sum of currents. where k is equal to 1 from 1 to n, that's the number of branches, is equal to 0. So for this given node in the diagram, for example, the currents entering the node, I1, I2, and I3, would be positive. But the currents leaving the node...

would be negative so this is the convention when interacting with currents in and out of a node yep applying and expanding the summation for this example would give us i1 plus i2 plus i3 plus minus i4 plus minus i5 because i4 and i5 are in opposite directions to i1 2 and 3. So the sum of them is equal to zero. And if you rearrange, we could get I1 plus I2 plus I3 is equal to I4 plus I5. And this means that the sum of the currents entering the node is equal to the sum of the currents leaving the same node, which supports the law of conservation.

This is Kirchhoff's current law. Kuchoff's second law is based on the conservation, the law of conservation of energy. It states that the algebraic sum of all voltages around a closed path or loop is equal to zero. So this is expressed mathematically by the summation shows the sum of voltages in a loop. When a number of voltages in that loop is k is equal to one.

n is equal to zero. Considering this circuit, the simple circuit for example, for a clockwise current flowing out from the positive of the source through the resistors and back through the negative of the source, conventional current flow, we will have these voltage directions. So how did I get these voltage directions?

Well, a voltage source direction is already defined for us remember our voltage source has a positive terminal and a negative terminal so the direction will point in the positive terminal yeah but for inactive components like resistors the voltage drop across them is not predetermined the voltage drop direction across a passive element is opposite to that of the current flow defined. So if I define conventional current like I did clockwise, then the voltage drop across V2, V3, V4 and V5 would be in the opposite direction to the current flow. So when we expand the summation for this example, we have V1 plus.

minus V2 minus V3 plus minus V4 plus minus V5. Remember if we take the clockwise direction this source voltage is in the opposite direction to all the voltage drops. Yeah so this is why all the voltage drops are negative. If we rearrange this formula We have V1, the voltages in this positive direction, is equal to V2 plus V3 plus V4 plus V5. The voltage is in the opposite direction to that source voltage, and hence the source current.

So, hence, in a loop, the sum of voltages in one direction is equal to the sum of voltages in the opposite direction. Yeah? this is Kirchhoff's voltage. Now this makes sense because it is based on the law of conservation of energy where energy cannot be created or destroyed so if energy is produced by the source it is dissipated in the resistors the passive elements of that particular loop yeah remember energy used to move charges in a loop is called voltage.

All right, so let's apply what we learned so far to our question. Given this circuit, whose nodes and loops are already identified for you, use Kirchhoff's current law to obtain equations of the nodes A and B, and use Kirchhoff's voltage law. top-down equations for loops one two and three so starting with the nodes a and b analyzing node a we could see that i1 goes into the node i2 goes into the node as well and i3 leaves the node so we have i1 plus i2 is equal to i3 this is what we end up with here yeah at node b we have the same thing because it is actually the return path of this node i3 will be flowing into this node while i1 and i2 will be leaving and rearranging the equation we will get the same formula so these nodes connect the same branches and this is why they have the same equations yeah so let's use kuchoff's voltage law for the loops the three loops loop one is just this loop here on the left The voltage source V0 is in the upward direction. Since I1 is in the clockwise direction, the voltage drops across R1.

R3 will be opposing that current direction. So we have V0 minus V1 minus V3 is equal to 0. We could rearrange and we get V0 is equal to V1 plus V3. Now we could substitute. Using Ohm's law for the voltage drops, V1 and V3, which is just the current flowing through that resistive element multiplied by the resistance of that element.

And we get I1R1 for V1 and I3R3 for V3. Here we just use Ohm's law and substitute it for voltage. so looking at loop two remember loop two the current defined for us in that loop which is the right loop is counterclockwise so we have v4 which is the positive direction is upward and the current is flowing counterclockwise so the voltage drops across r2 and r3 would be in the opposite direction of the current yep so we will end up with a v4 minus v2 minus v3 could rearrange this time we get v4 is equal to v2 plus v3 and we could substitute using ohm's law to get them in terms of current and resistance the voltage drops Let's apply Kirchhoff's voltage law to the outer loop.

This outer loop has two voltage sources. So let's see how that works. Yeah.

The current direction is clockwise. It was chosen for us clockwise. V0 is in the positive direction.

If the current is clockwise, then the voltage drop across R1 would be in the opposite direction to that current direction. R2 as well. So you will have minus V1 minus V2.

And also V4 will be minus. Why? Because the positive is upward. There's a positive terminal on top, a negative terminal below.

So the voltage across this is upward. And this is in the opposite direction to V0. So what we have here is V0. minus v1 minus v2 minus v4 remember for the supplies the directions are already predetermined based on the terminals of the supply so rearranging we have v naught minus v4 is equal to v1 plus v2 we put all the voltage drops on one side of the equation so we could substitute it using all yeah so let's move on you'll have a lot more examples like this to look at before you are able to do one of these on you yeah you have to get a lot more practice and i will give you a lot more examples before you could attempt them on your own so what about equivalent resistance of a circuit If you have resistances in series, what does that mean?

A series connection is when any two elements share the same node on one side of each other. If they share the same node on one side, they are in series with each other. Yep, so R1 is in series with R2, which is in series with R3, etc. Yeah.

So how do we find the equivalent resistance of this circuit with n resistors in series? Well, the equivalent resistors resistance can be derived by using Kirchhoff's voltage law to the circuit. So how do we do that?

Remember, Kirchhoff's voltage law is the voltages in. a loop yeah for a specific current direction so you're assuming the current direction is clockwise yeah from the positive terminal to the negative terminal conventional current and so you'll have a positive upward direction for voltage for the from the source and an opposing voltage drop from all the series resistances yeah so therefore true volt for Kirchhoff's voltage though we have v is equal the voltage of the supply is equal to all the drops across the resistances v1 plus v2 plus v3 and so on yeah using Ohm's law we could substitute v for ir as shown so you'll have ir1 is v1 ir2 is v2 and so on remember i flows through all of the resistance so it's have a common current flowing through all of the resistances yeah but we have different values r1 r2 r3 and so on so we could factor we could factor the current out because it's common the current is flowing through all the resistors and we'll have this expression here and if we take the current and we rearrange the formula we put it under the v we will actually find an expression for Req, the equivalent resistance. Now you'll see Req is the source voltage divided by the source current and is also derived to be the sum of all the series resistances. So this is why resistances in series can be added to find its equivalent resistance, the circuit's equivalent resistance. This is how it's derived.

yeah what about parallel connections a parallel connection is when two or more elements share the same nodes on either side of each element so on both sides there must be the same node recall this entire area is one node in this parallel circuit here and below this entire area is one node and all these resistances share these nodes on either side. So by definition, they are in parallel. Yeah?

R1 is in parallel with R2, is in parallel with R3, and so on. So how do we find the equivalent resistance of this circuit? How do we resolve resistors in parallel this time?

Well, it can be derived using Kirchhoff's current law. applied to the node of this circuit. How do we do that?

Well, we consider the node, and we can see at the node, I, the total source current entering the node, is equal to I1 plus I2 plus I3, which are the branch currents exiting those nodes, or that node, sorry. So this is what we have here due to Kirchhoff's Current law applied to this node, we have I, the source current, is equal to the sum of all the branch currents. Using Ohm's law to substitute for I, you have V on R. Remember, V, the voltage. at the source is across all the parallel resistance of this circuit yeah this voltage is across all the resistances which are in parallel in the circuit simply because they are in parallel yeah so we have the same voltage over each resistance the sum of those yeah that is ohm's law substituted Since V is common, we could factor that out.

And we have the sum of the reciprocals of the resistances in each branch. Yeah. If we rearrange this, put the V in the denominator on the left hand side of the formula. We have I on V is equal to the sum of the reciprocals of the parallel resistances. Yeah.

So I on V, we know V on I is Req. So I on V would be 1 on Req. So what we have derived here when simplifying is the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of all the parallel resistances.

And this is how we derive. the equation for equivalent resistances in parallel. The reciprocal of the equivalent resistance is the sum of the reciprocals of all the branch resistances in parallel to each other.

So I'm sure you've seen these formulae before, but you may not have derived them yet. So we have done that and I hope it's understandable. So now we're going to look at how voltage supplied in a circuit divides among elements in a loop.

If we have to find the voltage across a particular passive element in a loop, say a resistor, it will be a fraction. of the source voltage applied to the loop. This fraction is the resistance of that resistor over the total resistance in the loop. So, for example, we have a source, a voltage source supplying R1 and R2 in series with each other. We want to find the voltage across either of these resistances.

We know that the voltage across these resistances is a fraction of the source voltage. Now that fraction is defined by that resistance value divided by the total resistance in the loop. So the voltage across R1 is R1 on R1 plus. R2 multiplied by the total voltage.

So this is the fraction of the total voltage that is dropped across R1. Similarly, the voltage drop across R2 is R2 over R1 plus R2, the total, multiplied by the supply voltage. So this is the fraction.

the voltage that drops across R2. Yeah and according to Kirchhoff's voltage law this total voltage here is equal to the sum of these two voltage drops but their specific value depends on their resistance value. Yeah it depends on a fraction which is the resistance of that value sorry the resistance value of that resistor divided by the total resistance in the circuit so what about the current divider rule we are now going to look at how current supplied it divides among different parallel branches in our circuit yeah if we have to find the current through a particular element or branch say a resistor it will be a fraction of the total current applied from the source yeah this fraction we are talking about is the equivalent resistance of the resistors in the other branches over the total resistances in the circuit let's see what i'm talking about it must be reduced to two branches before analysis so let's see what we are talking about here if we have a circuit with a set of parallel branches and we want to know the current through these branches we could use the current divider rule.

The current divider rule tells us that the currents through these branches are a fraction of the total current entering the node and that makes sense because it follows the law of conservation of charge and Kuchov's current law. So Ix here is a fraction of the total current i t. What is that fraction?

That fraction is given by the resistance of all the other branches over the sum of all the branches. So the current i x is given by r t which is the parallel combination of all the other branches divided by That value plus rx. And that fraction of the total current is the current flowing in that branch. So it's a bit more complicated than the voltage divider rule.

Because it's not that resistance over the sum. It's the equivalent resistance of the other branches. divided by the sum of the parallel branches, all the parallel branches. Yeah.

So RT is resolved in parallel. Remember, you are not summing these because they are not in series. Yeah. They have to be resolved in parallel. And we just really derived how to do that to get RT.

And then to get the current through. our x, it is our t. divided by the sum of the two branches we're analyzing, which is Rx plus Rt, right? Through examples, you will solidify this rule, yeah?

So let's look at a practical example for some appreciation. So we have a transmission line. from A to B.

It is divided into three segments of resistance R1, R2 and R3 with these values 1 ohm, 2 ohm and 3 ohms respectively. A generating station acts as the source and supplies 1000 volts. through the transmission line to the load which is residential customers their houses they receive a voltage of 940 volts so we have a circuit they are practical circuits and we could apply the laws we've learned in this presentation in this lesson to calculate the voltage across the transmission.

So what is the voltage VAB? VAB is simply the difference in voltages between A and B. In this case we have the voltages at A and B.

What we have to do is find the potential difference between the two. which is a 1000 volts minus 940 volts that means that 60 volts is dropped across the transmission line yeah next what is the current through the transmission line we have the voltage across the transmission line and we have the resistance of the transmission line so we could use ohm's law and calculate the current I line through the transmission line. It will be 60 volts, we got before, divided by 1 R1 plus R2 plus R3 which is 1 plus 2 plus 3. That's 60 divided by 6, 10 amps. Next, what is the voltage across R3? The segment R3.

the transmission now to do that we need to use the voltage divider rule yeah because we want the voltage across one component we have the total voltage the total voltage across a and B is 60 we calculated that and we want the voltage now across one component of out of the three so we could use the voltage divider rule to find V3. That would be the total voltage, 60, multiplied by the resistance. We're looking to find the voltage across, divided by the total, 1 plus 2 plus 3, which is 6. So the voltage across R3 would be 30 volts. So let's switch things up a little bit. A bird flies and lands on the transmission line.

It lands across the R3 section of the transmission line as shown. Its body resistance is 3000 ohms. So if we look closely this bird introduces a new element to the circuit in parallel with R3. So the new current must factor in this resistance.

What is the new current to the transmission line? We have to rework this using Ohm's law. But now we have 3R3 in parallel with R bird, which is 3 Ohms in parallel with 3000 Ohms.

So it's no longer just 60 divided by 1 plus 2 plus 3. It's 60 volts divided by 1 plus 2 plus 3 in parallel with the resistance of the bullet, 3,000 ohms. This gives us not 10 amps, but 10.005 amps. So we have a 5 milliamp difference from the last case.

Yeah. So using the current divider rule, we could actually calculate how much current is flowing through the bird. We'd like to know if the bird is going to be electrocuted or if it's going to be fine.

So we have to calculate the current flowing through this bird. Yeah. How do we do that? We use the current divider rule because we know the total current flowing through the transmission line, 10.005 amps. and we know the two branches it splits up at this node.

Where the bird makes contact with the transmission line is actually an introduced node where current splits. Yeah, so current flowing through the bird, iBird, is equal to the total current flowing through the line multiplied by the resistance in the other branch which is the transmission line divided by the total of the two branches three plus three thousand which is three thousand and three and we get the current flowing through 0.01 amp which is 10 milliamps so this bird is definitely dead because it's 10 milliamps is a sizable current for a bird yeah so in reality even less current if any at all it passed through the bird We see birds land on wires all the time and nothing happens to them. Yeah, this is because the resistance of a real bird is much higher and since the bird is small, the segment of wire between the bird's leg when it lands on the transmission line is very small. So our wire shown here is actually smaller. than my example and our bird is actually larger than my example so therefore the resistance of the bird is much greater than the resistance of the wire causing a much smaller current draw if any current at all flowing through the bird yeah so the bird is usually fine when it lands on a transmission line So what happens though when we see birds die of electrocution?

What happened there? Well, if the bird flaps its wings, for example, or it steps on two separate conductors of different voltage potentials, then the potential difference across the bird will be significant. And then a significant current would flow. and the bird will be electrocuted. So the potential on one line like our example is smaller than if the bird steps on one line and touches another.

So the larger the potential difference between the legs of the bird or the wings of the bird means a larger current will flow and if this current is large enough the bird will die. So high voltage line workers use this principle to do live line work on transmission lines. They and their equipment are at the same potential on the line so there's no difference in potential and no current flows through them to electrocute them.

There are some illustrations of them. in this So a much more interesting and applicable situation is how we get shocked. So let's consider if we have some sort of electrical device where the insulation of a live wire is exposed and it comes into contact with the metal case.

So you have some wire inside this case here and the insulation is degraded and the live part of the wire touches the metal case. If someone touches that case, their body creates a parallel circuit with the device as represented by the equivalent circuit shown. So touching the live metal case creates a parallel circuit with the device where current flows through you once the person is grounded. So once that person, you know, makes contact with ground, current will flow through their body and they will experience a shock.

So that's usually what happens. We create a parallel path for current to flow. If we come into contact with a positive voltage potential and our foot makes contact with a zero potential, we have a potential difference now across us and therefore our current will flow. based on that resistance path we make for the voltage potential the value of current a specific value of current will flow that value of current is high enough we will experience uh the different symptoms as shown in in the previous slide in this presentation for different uh current fluid yeah if it's large enough we will uh experience some more severe symptoms like cardiac arrest and all that and it may be lethal so that's it for today um i hope you all understood the presentation i want you all to go through it thoroughly and highlight the parts you do not understand.

And feel free to ask in the next class or post your questions in the audience.

Transcript for:Fundamental Concepts of Electricity

Transcript for:
Fundamental Concepts of Electricity