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# ELEMENTARY ELECTRONICS LAB PHYS 3036

GPA 3.58

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This 26 page Class Notes was uploaded by Carmen Jerde on Thursday October 15, 2015. The Class Notes belongs to PHYS 3036 at Montana Tech of the University of Montana taught by Xiaobing Zhou in Fall. Since its upload, it has received 86 views. For similar materials see /class/223559/phys-3036-montana-tech-of-the-university-of-montana in Physics 2 at Montana Tech of the University of Montana.

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Date Created: 10/15/15

Chapter 6 DIVIDER CIRCUITS AND KIRCHHOFF S LAWS Contents 61 Voltage divider circuits 165 62 Kirchhoff s Voltage Law KVL 173 63 Current divider circuits 184 64 Kirchhoff s Current Law KCL 187 65 Contributors 189 61 Voltage divider Circuits Let s analyze a simple series circuit determining the voltage drops across individual resistors 45V 75 k9 R1 SkQ 10 k9 ltR2 ltgt R3 165 166 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS R1 R2 R3 Total E 45 Volts l Amps R 5k 10k 7 5k Oh ms From the given values of individual resistances we can determine a total circuit resistance knowing that resistances add in series R1 R2 R3 Total E 45 Volts l Amps R 5k 10k 75k 225k Ohms From here we can use Ohm s Law lER to determine the total current which we know will be the same as each resistor current currents being equal in all parts of a series circuit R1 R2 R3 Total E 45 Volts 2m 2m 2m 2m Amps R 5k 10k 75k 225k Ohms Now knowing that the circuit current is 2 mA we can use Ohm s Law EIR to calculate voltage across each resistor R1 R2 R3 Total E 10 20 15 45 Volts 2m 2m 2m 2m Amps R 5k 10k 75k 225k Ohms It should be apparent that the voltage drop across each resistor is proportional to its resistance given that the current is the same through all resistors Notice how the voltage across R2 is double that of the voltage across R1 just as the resistance of R2 is double that of R1 If we were to change the total voltage we would nd this proportionality of voltage drops remains constant R1 R2 R3 Total E 40 80 60 180 Volts 8m 8m 8m 8m Amps R 5k 10k 75k 225k Ohms 61 VOLTAGE DIVIDER CIRCUITS 167 The voltage across R2 is still exactly twice that of Rl s drop despite the fact that the source voltage has changed The proportionality of voltage drops ratio of one to another is strictly a function of resistance values With a little more observation it becomes apparent that the voltage drop across each resistor is also a xed proportion of the supply voltage The voltage across R1 for example was 10 volts when the battery supply was 45 volts When the battery voltage was increased to 180 volts 4 times as much the voltage drop across R1 also increased by a factor of 4 from 10 to 40 volts The ratio between R1 s voltage drop and total voltage however did not change E R1 10 V 022222 Emml 45 V 180 V Likewise none of the other voltage drop ratios changed with the increased supply voltage either E R2 20V 80V 044444 Emml 45 V 180 V E R3 15V 60V 033333 Emml 45 V 180 V For this reason a series circuit is often called a voltage divider for its ability to proportion 7 or divide 7 the total voltage into fractional portions of constant ratio With a little bit of algebra we can derive a formula for determining series resistor voltage drop given nothing more than total voltage individual resistance and total resistance Voltage drop across any resistor Equot 1quot Rquot Etotal Current In a serIes crrcurt Itotal Rmtal Etota Substltutlng for In In the rst equation total 7 Etotal Voltage drop across any series resIstor Equot 7 Rn Rmtal or En Etotal Rtotal The ratio of individual resistance to total resistance is the same as the ratio of individual voltage 168 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS drop to total supply voltage in a voltage divider circuit This is known as the voltage divider formula and it is a shortcut method for determining voltage drop in a series circuit without going through the current calculations of Ohm s Law Using this formula we can reanalyze the example circuit s voltage drops in fewer steps R1 5 k9 45 v 10 k9 ltR2 ltgt 75 kg R3 ERI 45v 10 225 kg ER2 45v amp 20v 225 kg 75 kg ER3 45v 15v 225 kg Voltage dividers nd wide application in electric meter circuits where speci c combinations of se ries resistors are used to 77divide77 a voltage into precise proportions as part of a voltage measurement dev1ce lt lt In ut vol age AAA One device frequently used as a 1 d39 39d39 A is the 39 which is a resistor with a movable element positioned by a manual knob or lever The movable element typically called a wiper makes contact with a resistive strip of material commonly called the slidewi39re if made of resistive metal wire at any point selected by the manual control 61 VOLTAGE DIVIDER CIRCUITS 169 A Potentiometer wiper contact 2 The wiper contact is the leftfacing arrow symbol drawn in the middle of the vertical resistor elementi As it is moved up it contacts the resistive strip closer to terminal 1 and further away from terminal 2 lowering resistance to terminal 1 and raising resistance to terminal 2 As it is moved down the opposite effect results The resistance as measured between terminals 1 and 2 is constant for any wiper position 1 V I less resistance more resistance l it more resistance less resistance 4 2 2 Shown here are internal illustrations of two potentiometer types rotary and linear Terminals l 0 0 0 Rotary potentiometer construe Ion V per Resistive strip 170 CHAPTER 6 DIVIDER CIRCUITS AND KIRCHHOFF S LAWS Linear potentiometer construction Wiper Resistive strip Terminals Some linear potentiometers are actuated by straightrline motion of a lever or slide button Others like the one depicted in the previous illustration are actuated by a turnrscrew for fine adjustment ability The latter units are sometimes referred to as trimpots because they work well for applications requiring a variable resistance to be 77trimmed77 to some precise value It should be noted that not all linear potentiometers have the same terminal assignments as shown in this illustration With some the wiper terminal is in the middle between the two end terminals The following photograph shows a real rotary potentiometer with exposed wiper and slidewire for easy viewing The shaft which moves the wiper has been turned almost fully clockwise so that the wiper is nearly touching the left terminal end of the slidewire Here is the same potentiometer with the wiper shaft moved almost to the fullrcounterclockwise position so that the wiper is near the other extreme end of travel 61 VOLTAGE DIVIDER CIRCUITS 171 If a constant voltage is applied between the outer terminals across the length of the slidewire the wiper position will tap off a fraction of the applied voltage measurable between the wiper contact and either of the other two terminals The fractional value depends entirely on the physical position of the wiper Using a potentiometer as a variable voltage divider ltgt gt AA less voltage Just like the fixed voltage divider the potentiometer s voltage division ratio is strictly a function of resistance and not of the magnitude of applied voltage In other words if the potentiometer ob or lever is moved to the 50 percent exact center position the voltage dropped between wiper and either outside terminal would be exactly 12 of the applied voltage no matter what that voltage happens to be or what the endrtorend resistance of the potentiometer is In other words a potentiometer functions as a variable voltage divider where the voltage division ratio is set by wiper position This application of the potentiometer is a very useful means of obtaining a variable voltage from a fixedevoltage source such as a batter If a circuit you re building requires a certain amount of voltage that is less than the value of an available battery s volta e you may connect the outer terminals of a potentiometer across that battery and 77dial up whatever voltage you need between the potentiometer wiper and one of the outer terminals for use in your circuit 172 CHAPTER 6 DIVIDER CIRCUITS AND KIRCHHOFF S LAWS Adjust potentiometer to obtain desired lt Battery j V V V VV AAA vvv Circuit requiring ess voltage than what the battery rovides When used in this manner the name potentiometer makes perfect sense they meter control the potential Voltage applied across them by creating a Variable Voltageedivider ratio This use of the threerterminal potentiometer as a Variable Voltage divider is Very popular in circuit design Shown here are several small potentiometers of the kind commonly used in consumer electronic equipment and by hobbyists and students in constructing circuits The smaller units on the Very left and Very right are designed to plug into a solderless breadboard or be soldered into a printed circuit board The middle units are designed to be mounted on a at panel With Wires soldered to each of the three terminals Here are three more potentiometers more specialized than the set just shown 62 KIRCHHOFF S VOLTAGE LAW KVL 173 The large 7 Helipot unit is a laboratory potentiometer designed for quick and easy connection to a circuit The unit in the lowerrleft corner of the photograph is the same type of potentiometer just Without a case or 107turn counting dial Both of these potentiometers are precision units using multiiturn helicaletrack resistance strips and Wi er mechanisms for making small adj ustments The unit on the lowerrright is a panelrmount potentiometer designed for rough service in industrial applications 0 REVIEW 0 Series circuits proportion or divide the total supply voltage among individual voltage drops the proportions being strictly dependent upon resistances ER ETOW Rn RTOW o A potentiometer is a variablerresistance component with three connection points frequently used as an adjustable voltage divider 62 Kirchhoff s Voltage Law KVL Let s take another look at our example series circuit this time numbering the points in the circuit for voltage reference R 2 1 3 SkQ ltgt 45v IOkQ R2 39 75kQ 39 1 R3 4 If We Were to connect a voltmeter between points 2 and 1 red test lead to point 2 and black test lead to point 1 the meter would register 45 volts Typically the 77 sign is not shown but rather 174 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS implied for positive readings in digital meter displays However for this lesson the polarity of the voltage reading is very important and so I will show positive numbers explicitly EH 45 v When a voltage is speci ed with a double subscript the characters 772177 in the notation 77 22177 it means the voltage at the rst point 2 as measured in reference to the second point A voltage speci ed as 77Ecg77 would mean the voltage as indicated by a digital meter with the red test lead on point 77c77 and the black test lead on point 77g the voltage at 77c77 in reference to 77g The meaning of Ecd d If we were to take that same voltmeter and measure the voltage drop across each resistor stepping around the circuit in a clockwise direction with the red test lead of our meter on the point ahead and the black test lead on the point behind we would obtain the following readings E34 10 v 1343 20 v EM 15v 62 KIRCHHOFF S VOLTAGE LAW KVL 175 E32 E174 We should already be familiar with the general principle for series circuits stating that individual voltage drops add up to the total applied voltage but measuring voltage drops in this manner and paying attention to the polarity mathematical sign of the readings reveals another facet of this principle that the voltages measured as such all add up to zero E21 45 V voltage from point2to point 1 10 V voltage from point 3to point 2 E43 20 V voltage from point4to point 3 15 V voltage from point 1 to point 4 0V This principle is known as Kirchho us Voltage Law discovered in 1847 by Gustav R Kirchhoff a German physicist and it can be stated as such 7 The algebraic sum of all voltages in a loop must equal zero By algebraic I mean accounting for signs polarities as well as magnitudes By loop I mean any path traced from one point in a circuit around to other points in that circuit and nally back to the initial point In the above example the loop was formed by following points in this order 12341 It doesn t matter which point we start at or which direction we proceed in tracing the loop the voltage sum will still equal zero To demonstrate we can tally up the voltages in loop 3214 3 of the same circuit 176 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS E23 10 V voltage from point2to point 3 E12 45 V voltage from point 1 to point 2 E41 15 V voltage from point4to point 1 E34 20 V voltage from point3to point 4 0 V This may make more sense if we redraw our example series circuit so that all components are represented in a straight line current SkQ 39 1015239 7st39 39 45 V current It s still the same series circuit just with the components arranged in a di erent form Notice the polarities of the resistor voltage drops with respect to the battery the battery s voltage is negative on the left and positive on the right whereas all the resistor voltage drops are oriented the other way positive on the left and negative on the right This is because the resistors are resisting the ow of electrons being pushed by the battery In other words the 77push77 exerted by the resistors against the ow of electrons must be in a direction opposite the source of electromotive force Here we see what a digital voltmeter would indicate across each component in this circuit black lead on the left and red lead on the right as laid out in horizontal fashion current 10V 20V 15V 45V 1332 1343 E14 E21 If we were to take that same voltmeter and read voltage across combinations of components starting with only R1 on the left and progressing across the whole string of components we will see how the voltages add algebraically to zero 62 KIRCHHOFF S VOLTAGE LAW KVL 177 current R2 4 R3 1 2 10kg 75kQ 3945V J The fact that series voltages add up should be no mystery but we notice that the polarity of these voltages makes a lot of difference in how the gures addl While reading voltage across R1 RliiRQ and RliiRgiiRg I m using a 77doubledash77 symbol 777 to represent the series connection between resistors R1 R2 and R3 we see how the voltages measure successively larger albeit negative magnitudes because the polarities of the individual voltage drops are in the same orientation positive left negative right The sum of the voltage drops across R1 R2 and R3 equals 45 volts which is the same as the battery s output except that the battery s polarity is opposite that of the resistor voltage drops negative left positive right so we end up with 0 volts measured across the whole string of components That we should end up with exactly 0 volts across the whole string should be no mystery eitherl Looking at the circuit we can see that the far left of the string left side of R1 point number 2 is directly connected to the far right of the string right side of battery point number 2 as necessary to complete the circuit Since these two points are directly connected they are electrically common to each other And as such the voltage between those two electrically common points must be zero KirchhoH s Voltage Law sometimes denoted as KVL for short will work for my circuit con g uration at all not just simple seriesl Note how it works for this parallel circuit 178 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS 1 2 3 4 lt lt lt 6 V ERI ERZ ER3 8 7 6 5 Being a parallel circuit the voltage across every resistor is the same as the supply voltage 6 volts Tallying up voltages around loop 234 5672 we get E32 0 V voltage from point 3to point 2 E43 0 V voltage from point4to point 3 E54 6 V voltage from point 5to point 4 E55 0 V voltage from point 6to point 5 E76 0 V voltage from point 7to point 6 E27 6 V voltage from point 2to point 7 Note how I label the nal sum voltage as E272 Since we began our loopstepping sequence at point 2 and ended at point 2 the algebraic sum of those voltages will be the same as the voltage measured between the same point E22 which of course must be zero The fact that this circuit is parallel instead of series has nothing to do with the validity of KirchhoH s Voltage Law For that matter the circuit could be a 77black box77 7 its component con guration completely hidden from our view with only a set of exposed terminals for us to measure voltage between 7 and KVL would still hold true 62 KIRCHHOFF S VOLTAGE LAW KVL 179 39 I 8V 11V Try any order of steps from any terminal in the above diagram stepping around back to the original terminal and you ll nd that the algebraic sum of the voltages always equals zero Furthermore the 77loop77 we trace for KVL doesn t even have to be a real current path in the closedcircuit sense of the word All we have to do to comply with KVL is to begin and end at the same point in the circuit tallying voltage drops and polarities as we go between the next and the last point Consider this absurd example tracing 77loop77 23632 in the same parallel resistor circuit 1 2 3 4 i lt lt lt 6V iR1 jiRz iR3 8 7 6 5 180 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS E32 0 V voltage from point 3to point 2 E53 6 V voltage from point 6to point 3 E36 6 V voltage from point 3to point 6 E23 0 V voltage from point 2to point 3 E2 2 0 V KVL can be used to determine an unknown voltage in a complex circuit where all other voltages around a particular 77loop77 are known Take the following complex circuit actually two series circuits joined by a single wire at the bottom as an examp e 1 2 5 6 lt lt E15v 13v E 35 V E 3 4 E 25 V 39 ltgt ltgt jgt20V 12v jgt gt gt 7 8 9 10 To make the problem simpler l ve omitted resistance values and simply given voltage drops across each resistori The two series circuits share a common wire between them wire 78910 making voltage measurements between the two circuits possible If we wanted to determine the voltage between points 4 and 3 we could set up a KVL equation with the voltage between those points as the unknown 1343 1394 E89 E38 0 E43120200 E43 32 0 1343 32 v 62 KIRCHHOFF S VOLTAGE LAW KVL 181 1 2 5 6 35 V 25 V 7 8 9 10 Measuring voltage from point 4 to point 3 unknown amount 1343 1 2 5 6 35 V T 25 V 7 8 9 10 Measuring voltage from point 9 to point 4 12 volts E43 12 182 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS 1 2 5 6 35 V E 25 V 7 10 Measuring voltage from point 8 to point 9 0 volts E43 12 0 1 2 5 6 35 V 25 V 391 7 8 9 10 Measuring voltage from point 3 to point 8 20 volts E43120200 Stepping around the loop 34983 we write the voltage drop gures as a digital voltmeter would register them measuring with the red test lead on the point ahead and black test lead on the point behind as we progress around the loop Therefore the voltage from point 9 to point 4 is a positive 12 volts because the red lead is on point 9 and the blacllt lead is on point 4 The voltage from point 3 to point 8 is a positive 20 volts because the red lead is on point 3 and the black lead is on point 8 The voltage from point 8 to point 9 is zero of course because those two points are electrically common Our nal answer for the voltage from point 4 to point 3 is a negative 32 volts telling us that point 3 is actually positive with respect to point 4 precisely what a digital voltmeter would indicate 62 KIRCHHOFF S VOLTAGE LAW KVL 183 with the red lead on point 4 and the black lead on point 3 1 2 5 6 TZSV 7 8 9 1o 1343 32 In other words the initial placement of our 77 meter leads77 in this KVL problem was 77backwards77 Had we generated our KVL equation starting with E34 instead of E473 stepping around the same loop with the opposite meter lead orientation the nal answer would have been E34 32 volts 1 2 5 6 7 8 9 1 0 E34 32 It is important to realize that neither approach is 77wrong77 In both cases we arrive at the correct assessment of voltage between the two points 3 and 4 point 3 is positive with respect to point 4 and the voltage between them is 32 volts 0 REVIEW 0 KirchhoH s Voltage Law KVL 7 The algebraic sum of all voltages in a loop must equal zeroquot 184 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS 63 Current divider Circuits Let s analyze a simple parallel circuit determining the branch currents through individual resistors AAA Rz gtR3 1ko 3ko ZkQ Knowing that voltages across all components in a parallel circuit are the same we can ll in our voltagecurrentresistance table with 6 volts across the top row R1 R2 R3 Total E 6 6 6 6 Volts l Amps R 1k 3k 2k Oh ms Using Ohm s Law lER we can calculate each branch current R1 R2 R3 Total E 6 6 6 6 Volts 6m 2m 3m Amps R 1k 3k 2k Oh ms Knowing that branch currents add up in parallel circuits to equal the total current we can arrive at total current by summing 6 mA 2 mA and 3 mA R1 R2 R3 Total E 6 6 6 6 Volts 6m 2m 3m 11m Amps R 1k 3k 2k Oh ms The nal step of course is to gure total resistance This can be done with Ohm s Law REl in the 77total77 column or with the parallel resistance formula from individual resistances Either way we ll get the same answer 63 CURRENT DIVIDER CIRCUITS 185 R1 R2 R3 Total E 6 6 6 6 Volts 6m 2m 3m 1 1m Amps R 1k 3k 2k 54545 Oh ms Once again it should be apparent that the current through each resistor is related to its resistance given that the voltage across all resistors is the same Rather than being directly proportional the relationship here is one of inverse proportion For example the current through R1 is half as much as the current through R3 which has twice the resistance of R1 If we were to change the supply voltage of this circuit we nd that surprise these proportional ratios do not change R1 R2 R3 Total E 24 24 24 24 Volts 24m 8111 12m 44m Amps R 1k 3k 2k 54545 Ohms The current through R1 is still exactly twice that of R2 despite the fact that the source volt age has changed The proportionality between different branch currents is strictly a function of resistance Also reminiscent of voltage dividers is the fact that branch currents are xed proportions of the total current Despite the fourfold increase in supply voltage the ratio between any branch current and the total current remains unchanged 1R1 6 mA 24 mA 054545 Imtal 11 mA 44 mA I R2 2mA SmA 018182 Imtal 11 mA 44 mA I R3 3 mA 12 mA 027273 Imtal 11 mA 44 mA For this reason a parallel circuit is often called a current divider for its ability to proportion 7 or divide 7 the total current into fractional parts With a little bit of algebra we can derive a formula for determining parallel resistor current given nothing more than total current individual resistance and total resistance 186 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS E Current through any resistor In Rquot n Voltage in a parallel circuit Etotal En Itotal Rm11 Substituting tom Rtoml for Equot in the rst equation Itotal Rtotal R 11 Current through any parallel resistor In Ol39 I 71 Rtotal n 7 total The ratio of total resistance to individual resistance is the same ratio as individual branch current to total current This is known as the current divider formula and it is a shortcut method for determining branch currents in a parallel circuit when the total current is known Using the original parallel circuit as an example we can re calculate the branch currents using this formula if we start by knowing the total current and total resistance 1R111mA 6mA lk 1R211mA 2mA 3kg 1R311mA 3mA ZkQ If you take the time to compare the two divider formulae you ll see that they are remarkably similar Notice however that the ratio in the voltage divider formula is R7 individual resistance divided by RTotal and how the ratio in the current divider formula is RTWZ divided by Rn 64 KIRCHHOFF S CURRENT LAW KCL 187 Voltage divider Current divider formula formula R Rt t 1 En Etotal n In Itotal 0 a Rtotal It is quite easy to confuse these two equations getting the resistance ratios backwards One way to help remember the proper form is to keep in mind that both ratios in the voltage and current divider equations must equal less than one After all these are divider equations not multiplier equations If the fraction is upsidedown it will provide a ratio greater than one which is incorrect Knowing that total resistance in a series voltage divider circuit is always greater than any of the individual resistances we know that the fraction for that formula must be Rn over RToml Conversely knowing that total resistance in a parallel current divider circuit is always less then any of the individual resistances we know that the fraction for that formula must be RTota over 7 Current divider circuits also nd application in electric meter circuits where a fraction of a measured current is desired to be routed through a sensitive detection device Using the current divider formula the proper shunt resistor can be sized to proportion just the right amount of current for the device in any given instance Itotal 4 REM Itoml fraction of total current ea sensitive device 0 REVIEW 0 Parallel circuits proportion or 77divide77 the total circuit current among individual branch currents the proportions being strictly dependent upon resistances In ITO RTWZ R7 64 Kirchhoff s Current Law KCL Let s take a closer look at that last parallel example circuit 1 2 3 4 L Item lt lt 6 V IRil R1 IRZT R2 IR3 l R3 1 k9 3 k9 2 k9 Itotal gt 188 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS Solving for all values of voltage and current in this circuit R1 R2 R3 Total E 6 6 6 6 Volts 6m 2m 3m 1 1m Amps R 1k 3k 2k 54545 Oh ms At this point we know the value of each branch current and of the total current in the circuit We know that the total current in a parallel circuit must equal the sum of the branch currents but there s more going on in this circuit than just that Taking a look at the currents at each wire junction point node in the circuit we should be able to see something e se IR1 IR2 IR3 IR2 IR3 IR3 1 lt lt 3 lt 4 L Item lt 6 V IRIT R1 IRZT R2 IR3 T R3 gt1k 2 gt3kg gt214 Itotal gt gt gt 8 IR1 IR2 IR3 7 IR2 IR3 6 IR3 5 At each node on the negative 77rail77 wire 8765 we have current splitting off the main ow to each successive branch resistor At each node on the positive 77rail77 wire 1234 we have current merging together to form the main ow from each successive branch resistor This fact should be fairly obvious if you think of the water pipe circuit analogy with every branch node acting as a 77tee77 tting the water ow splitting or merging with the main piping as it travels from the output of the water pump toward the return reservoir or sump If we were to take a closer look at one particular 77tee77 node such as node 3 we see that the current entering the node is equal in magnitude to the current exiting the node 1R2 1R3 1R3 lt 3 lt IR2 T R2 39 3 k9 From the right and from the bottom we have two currents entering the wire connection labeled as node 3 To the left we have a single current exiting the node equal in magnitude to the sum of the two currents entering To refer to the plumbing analogy so long as there are no leaks in the piping what ow enters the tting must also exit the tting This holds true for any node 77 tting77 no matter how many ows are entering or exiting Mathematically we can express this 65 CONTRIBUTORS 189 general relationship as such Iexiu39ng Ientering Mr Kirchho decided to express it in a slightly di erent form though mathematically equiva lent calling it Kirchho f s Current Law K Ientering 39Iexiting 0 Summarized in a phrase KirchhoH s Current Law reads as such 7 The algebraic sum of all currents entering and exiting a node must equal zero That is if we assign a mathematical sign polarity to each current denoting whether they enter or exit a node we can add them together to arrive at a total of zero guaranteed Taking our example node number 3 we can determine the magnitude of the current exiting from the left by setting up a KCL equation with that current as the unknown value 12I3I0 2mA3mAI0 soving for l2mA3mA I 5mA The negative sign on the value of 5 milliamps tells us that the current is exiting the node as opposed to the 2 milliamp and 3 milliamp currents which must were both positive and therefore entering the node Whether negative or positive denotes current entering or exiting is entirely arbitrary so long as they are opposite signs for opposite directions and we stay consistent in our notation KCL will work Together KirchhoH s Voltage and Current Laws are a formidable pair of tools useful in analyzing electric circuits Their usefulness will become all the more apparent in a later chapter Network Analysis but suf ce it to say that these Laws deserve to be memorized by the electronics student every bit as much as Ohm s Law 0 REVIEW 0 KirchhoH s Current Law KCL 7The algebraic sum of all currents entering and exiting a node must equal zeroquot 65 Contributors Contributors to this chapter are listed in chronological order of their contributions from most recent to rst See Appendix 2 Contributor List for dates and contact information 190 CHAPTER 6 DI VIDER CIRCUITS AND KIRCHHOFF S LAWS Jason 5133er June 2000 HTML document formatting which led to a much betterlooking second edition Ron LaPlante October 1998 helped create 77table77 method of series and parallel circuit analysis

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