Kirchhoff’s Voltage Law

Kirchhoff’s Voltage Law: Kirchhoff’s Voltage Law (KVL) is Kirchhoff’s second law that deals with the conservation of energy around a closed circuit path. Gustav Kirchhoff’s Voltage Law is the second of his fundamental laws we can use for circuit analysis. His voltage law states that for a closed loop series path the algebraic sum of all the voltages around any closed loop in a circuit is equal to zero. This is because a circuit loop is a closed conducting path so no energy is lost. In other words the algebraic sum of ALL the potential differences around the loop must be equal to zero as: ΣV = 0. Note here that the term “algebraic sum” means to take into account the polarities and signs of the sources and voltage drops around the loop. This idea by Kirchhoff is commonly known as the Conservation of Energy, as moving around a closed loop, or circuit, you will end up back to where you started in the circuit and therefore back to the same initial potential with no loss of voltage around the loop. Hence any voltage drops around the loop must be equal to any voltage sources met along the way. So when applying Kirchhoff’s voltage law to a specific circuit element, it is important that we pay special attention to the algebraic signs, (+ and -) of the voltage drops across elements and the emf’s of sources otherwise our calculations may be wrong. But before we look more closely at Kirchhoff’s voltage law (KVL) lets first understand the voltage drop across a single element such as a resistor. A Single Circuit Element For this simple example we will assume that the current, I is in the same direction as the flow of positive charge, that is conventional current flow. Here the flow of current through the resistor is from point A to point B, that is from positive terminal to a negative terminal. Thus as we are travelling in the same direction as current flow, there will be a fall in potential across the resistive element giving rise to a -IR voltage drop across it. If the flow of current was in the opposite direction from point B to point A, then there would be a rise in potential across the resistive element as we are moving from a - potential to a + potential giving us a +I*R voltage drop. Thus to apply Kirchhoff’s voltage law correctly to a circuit, we must first understand the direction of the polarity and as we can see, the sign of the voltage drop across the resistive element will depend on the direction of the current flowing through it. As a general rule, you will loose potential in the same direction of current across an element and gain potential as you move in the direction of an emf source. The direction of current flow around a closed circuit can be assumed to be either clockwise or anticlockwise and either one can be chosen. If the direction chosen is different from the actual direction of current flow, the result will still be correct and valid but will result in the algebraic answer having a minus sign. To understand this idea a little more, lets look at a single circuit loop to see if Kirchhoff’s Voltage Law holds true. A Single Circuit Loop Kirchhoff’s voltage law states that the algebraic sum of the potential differences in any loop must be equal to zero as: ΣV = 0. Since the two resistors, R1 and R2 are wired together in a series connection, they are both part of the same loop so the same current must flow through each resistor. Thus the voltage drop across resistor, R1 = I*R1 and the voltage drop across resistor, R2 = I*R2 giving by KVL: We can see that applying Kirchhoff’s Voltage Law to this single closed loop produces the formula for the equivalent or total resistance in the series circuit and we can expand on this to find the values of the voltage drops around the loop. Kirchhoff’s Voltage Law Example No1 Three resistor of values: 10 ohms, 20 ohms and 30 ohms, respectively are connected in series across a 12 volt battery supply. Calculate: a) the total resistance, b) the circuit current, c) the current through each resistor, d) the voltage drop across each resistor, e) verify that Kirchhoff’s voltage law, KVL holds true. a) Total Resistance (RT) RT = R1 + R2 + R3 = 10Ω + 20Ω + 30Ω = 60Ω Then the total circuit resistance RT is equal to 60Ω b) Circuit Current (I) Thus the total circuit current I is equal to 0.2 amperes or 200mA c) Current Through Each Resistor The resistors are wired together in series, they are all part of the same loop and therefore each experience the same amount of current. Thus: IR1 = IR2 = IR3 = ISERIES = 0.2 amperes d) Voltage Drop Across Each Resistor VR1 = I x R1 = 0.2 x 10 = 2 volts VR2 = I x R2 = 0.2 x 20 = 4 volts VR3 = I x R3 = 0.2 x 30 = 6 volts e) Verify Kirchhoff’s Voltage Law Thus Kirchhoff’s voltage law holds true as the individual voltage drops around the closed loop add up to the total. Kirchhoff’s Circuit Loop: We have seen here that Kirchhoff’s voltage law, KVL is Kirchhoff’s second law and states that the algebraic sum of all the voltage drops, as you go around a closed circuit from some fixed point and return back to the same point, and taking polarity into account, is always zero. That is ΣV = 0 The theory behind Kirchhoff’s second law is also known as the law of conservation of voltage, and this is particularly useful for us when dealing with series circuits, as series circuits also act as voltage dividers and the voltage divider circuit is an important application of many series circuits. Kirchhoff’s Current Law: Kirchhoff’s Current Law (KCL) is Kirchhoff’s first law that deals with the conservation of charge entering and leaving a junction. To determine the amount or magnitude of the electrical current flowing around an electrical or electronic circuit, we need to use certain laws or rules that allows us to write down these currents in the form of an equation. The network equations used are those according to Kirchhoff’s laws, and as we are dealing with circuit currents, we will be looking at Kirchhoff’s current law, (KCL). Gustav Kirchhoff’s Current Law is one of the fundamental laws used for circuit analysis. His current law states that for a parallel path the total current entering a circuits junction is exactly equal to the total current leaving the same junction. This is because it has no other place to go as no charge is lost. In other words the algebraic sum of ALL the currents entering and leaving a junction must be equal to zero as: Σ IIN = Σ IOUT. This idea by Kirchhoff is commonly known as the Conservation of Charge, as the current is conserved around the junction with no loss of current. Lets look at a simple example of Kirchhoff’s current law (KCL) when applied to a single junction. A Single Junction Here in this simple single junction example, the current IT leaving the junction is the algebraic sum of the two currents, I1 and I2 entering the same junction. That is IT = I1 + I2. Note that we could also write this correctly as the algebraic sum of: IT – (I1 + I2) = 0. So if I1 equals 3 amperes and I2 is equal to 2 amperes, then the total current, IT leaving the junction will be 3 + 2 = 5 amperes, and we can use this basic law for any number of junctions or nodes as the sum of the currents both entering and leaving will be the same. Also, if we reversed the directions of the currents, the resulting equations would still hold true for I1 or I2. As I1 = IT – I2 = 5 – 2 = 3 amps, and I2 = IT – I1 = 5 – 3 = 2 amps. Thus we can think of the currents entering the junction as being positive (+), while the ones leaving the junction as being negative (-). Then we can see that the mathematical sum of the currents either entering or leaving the junction and in whatever direction will always be equal to zero, and this forms the basis of Kirchhoff’s Junction Rule, more commonly known as Kirchhoff’s Current Law, or (KCL). Resistors in Parallel Let’s look how we could apply Kirchhoff’s current law to resistors in parallel, whether the resistances in those branches are equal or unequal. Consider the following circuit diagram: In this simple parallel resistor example there are two distinct junctions for current. Junction one occurs at node B, and junction two occurs at node E. Thus we can use Kirchhoff’s Junction Rule for the electrical currents at both of these two distinct junctions, for those currents entering the junction and for those currents flowing leaving the junction. To start, all the current, IT leaves the 24 volt supply and arrives at point A and from there it enters node B. Node B is a junction as the current can now split into two distinct directions, with some of the current flowing downwards and through resistor R1 with the remainder continuing on through resistor R2 via node C. Note that the currents flowing into and out of a node point are commonly called branch currents. We can use Ohm’s Law to determine the individual branch currents through each resistor as: I = V/R, thus: For current branch B to E through resistor R1 For current branch C to D through resistor R2 From above we know that Kirchhoff’s current law states that the sum of the currents entering a junction must equal the sum of the currents leaving the junction, and in our simple example above, there is one current, IT going into the junction at node B and two currents leaving the junction, I1 and I2. Since we now know from calculation that the currents leaving the junction at node B is I1 equals 3 amps and I2 equals 2 amps, the sum of the currents entering the junction at node B must equal 3 + 2 = 5 amps. Thus ΣIN = IT = 5 amperes. In our example, we have two distinct junctions at node B and node E, thus we can confirm this value for IT as the two currents recombine again at node E. So, for Kirchhoff’s junction rule to hold true, the sum of the currents into point F must equal the sum of the currents flowing out of the junction at node E. As the two currents entering junction E are 3 amps and 2 amps respectively, the sum of the currents entering point F is therefore: 3 + 2 = 5 amperes. Thus ΣIN = IT = 5 amperes and therefore Kirchhoff’s current law holds true as this is the same value as the current leaving point A. Applying KCL to more complex circuits. We can use Kirchhoff’s current law to find the currents flowing around more complex circuits. We hopefully know by now that the algebraic sum of all the currents at a node (junction point) is equal to zero and with this idea in mind, it is a simple case of determining the currents entering a node and those leaving the node. Consider the circuit below. Kirchhoff’s Current Law Example No1 In this example there are four distinct junctions for current to either separate or merge together at nodes A, C, E and node F. The supply current IT separates at node A flowing through resistors R1 and R2, recombining at node C before separating again through resistors R3, R4 and R5 and finally recombining once again at node F. But before we can calculate the individual currents flowing through each resistor branch, we must first calculate the circuits total current, IT. Ohms law tells us that I = V/R and as we know the value of V, 132 volts, we need to calculate the circuit resistances as follows. Circuit Resistance RAC Thus the equivalent circuit resistance between nodes A and C is calculated as 1 Ohm. Circuit Resistance RCF Thus the equivalent circuit resistance between nodes C and F is calculated as 10 Ohms. Then the total circuit current, IT is given as: Giving us an equivalent circuit of: Kirchhoff’s Current Law Equivalent Circuit Therefore, V = 132V, RAC = 1Ω, RCF = 10Ω’s and IT = 12A. Having established the equivalent parallel resistances and supply current, we can now calculate the individual branch currents and confirm using Kirchhoff’s junction rule as follows. Thus, I1 = 5A, I2 = 7A, I3 = 2A, I4 = 6A, and I5 = 4A. We can confirm that Kirchoff’s current law holds true around the circuit by using node C as our reference point to calculate the currents entering and leaving the junction as: We can also double check to see if Kirchhoffs Current Law holds true as the currents entering the junction are positive, while the ones leaving the junction are negative, thus the algebraic sum is: I1 + I2 – I3 – I4 – I5 = 0 which equals 5 + 7 – 2 – 6 – 4 = 0. So we can confirm by analysis that Kirchhoff’s current law (KCL) which states that the algebraic sum of the currents at a junction point in a circuit network is always zero is true and correct in this example. Kirchhoff’s Current Law Example No2 Find the currents flowing around the following circuit using Kirchhoff’s Current Law only. IT is the total current flowing around the circuit driven by the 12V supply voltage. At point A, I1 is equal to IT, thus there will be an I1*R voltage drop across resistor R1. The circuit has 2 branches, 3 nodes (B, C and D) and 2 independent loops, thus the I*R voltage drops around the two loops will be: Loop ABC ⇒ 12 = 4I1 + 6I2 Loop ABD ⇒ 12 = 4I1 + 12I3 Since Kirchhoff’s current law states that at node B, I1 = I2 + I3, we can therefore substitute current I1 for (I2 + I3) in both of the following loop equations and then simplify. Kirchhoff’s Loop Equations We now have two simultaneous equations that relate to the currents flowing around the circuit. Eq. No 1 : 12 = 10I2 + 4I3 Eq. No 2 : 12 = 4I2 + 16I3 By multiplying the first equation (Loop ABC) by 4 and subtracting Loop ABD from Loop ABC, we can be reduced both equations to give us the values of I2 and I3 Eq. No 1 : 12 = 10I2 + 4I3 ( x4 ) ⇒ 48 = 40I2 + 16I3 Eq. No 2 : 12 = 4I2 + 16I3 ( x1 ) ⇒ 12 = 4I2 + 16I3 Eq. No 1 – Eq. No 2 ⇒ 36 = 36I2 + 0 Substitution of I2 in terms of I3 gives us the value of I2 as 1.0 Amps Now we can do the same procedure to find the value of I3 by multiplying the first equation (Loop ABC) by 4 and the second equation (Loop ABD) by 10. Again by subtracting Loop ABC from Loop ABD, we can be reduced both equations to give us the values of I2 and I3 Eq. No 1 : 12 = 10I2 + 4I3 ( x4 ) Eq. No 2 : 12 = 4I2 + 16I3 ( x10 ) ⇒ 48 = 40I2 + 16I3 ⇒ 120 = 40I2 + 160I3 Eq. No 2 – Eq. No 1 ⇒ 72 = 0 + 144I3 Thus substitution of I3 in terms of I2 gives us the value of I3 as 0.5 Amps As Kirchhoff’s junction rule states that : I1 = I2 + I3 The supply current flowing through resistor R1 is given as : 1.0 + 0.5 = 1.5 Amps Thus I1 = IT = 1.5 Amps, I2 = 1.0 Amps and I3 = 0.5 Amps and from that information we could calculate the I*R voltage drops across the devices and at the various points (nodes) around the circuit. We could have solved the circuit of example two simply and easily just using Ohm’s Law, but we have used Kirchhoff’s Current Law here to show how it is possible to solve more complex circuits when we can not just simply apply Ohm’s Law.