# Electrician'sMathand BasicElectricalFormulas

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1 Eletriian'sMathand BasiEletrialFormulas MikeHoltEnterprises,In NEC.CODE

2 Introdution Introdution This PDF is a free resoure from Mike Holt Enterprises, In. It s Unit 1 from the Eletrial NEC Exam Preparation textbook. It is always our pleasure to give bak to the industry as muh as we an. Enjoy! About the Author Mike Holt worked his way up through the eletrial tradefrom an apprentie eletriian to beome one of the most reognized experts in the world as it relates to eletrial power installation. He was a Journeyman Eletriian, Master Eletriian, and Eletrial Contrator. Mike ame from the real world, and his dediation to eletrial training is the result of his own struggles as an eletriian looking for a program that would help him sueed in this hallenging industry. It is for reasons like this that Mike ontinues to help the industry by providing free resoures suh as this doument. It is the goal of Mike Holt and everyone on the Mike Holt Team to do everything in our power to aid in your pursuit of exellene. About this Free PDF This Free Unit was extrated from Mike s Eletrial NEC Exam Preparation textbook whih is why you will notie that this pdf starts with page 3. To fully prepare for your National Eletrial Code exam you need to study the entire Eletrial NEC Exam Preparation textbook. Most eletrial exams ontain questions on eletrial theory, basi eletrial alulations, the Code, and important and diffiult National Eletrial Code Calulations. The Eletrial NEC Exam Preparation textbook ontains hundreds of illustrations, examples, almost 3,200 pratie questions overing all of these subjets and 36 pratie quizzes. This book is intended to be used with the 2008 National Eletrial Code. For more great Free resoures from Mike Holt visit Mike Holt Enterprises, In NEC.CODE iii

3 UNIT 1 Eletriian s Math and Basi Eletrial Formulas Introdution In order to onstrut a building that will last into the future, a strong foundation is a prerequisite. The foundation is a part of the building that is not visible in the finished struture, but is extremely essential in ereting a building that will have the neessary strength to endure. The math and basi eletrial onepts of this unit are very similar to the foundation of a building. The onepts in this unit are the essential basis that you must understand, beause you will build upon them as you study eletrial iruits and systems. As your studies ontinue, you ll find that a good foundation in eletrial theory and math will help you understand why the NEC ontains ertain provisions. This unit inludes math, eletrial fundamentals, and an explanation of the operation of eletrial meters to help you visualize some pratial appliations. You ll be amazed at how often your eletrial studies return to the basis of this unit. Ohm s law and the eletrial formulas related to it, are the foundation of all eletrial iruits. Every student begins at a different level of understanding, and you may find this unit an easy review, or you may find it requires a high level of onentration. In any ase, be ertain that you fully understand the onepts of this unit and are able to suessfully omplete the questions at the end of the unit before going on. A solid foundation will help in your suessful study of the rest of this book. Part A Eletriian s Math 1.0 Introdution Numbers an take different forms: Whole numbers: 1, 20, 300, 4,000, 5,000 Deimals: 0.80, 1.25, 0.75, 1.15 Frations: 1/2, 1/4, 5/8, 4/3 Perentages: 80%, 125%, 250%, 500% You ll need to be able to onvert these numbers from one form to another and bak again, beause all of these number forms are part of eletrial work and eletrial alulations. You ll also need to be able to do some basi algebra. Many people have a fear of algebra, but as you work through the material here you will see there is nothing to fear but fear itself. 1.1 Whole Numbers Whole numbers are exatly what the term implies. These numbers do not ontain any frations, deimals, or perentages. Another name for whole numbers is integers. 1.2 Deimals The deimal method is used to display numbers other than whole numbers, frations or perentages, suh as, 0.80, 1.25, 1.732, et. 1.3 Frations A fration represents part of a whole number. If you use a alulator for adding, dividing, subtrating, or multiplying, you need to onvert the fration to a deimal or whole number. Mike Holt Enterprises, In NEC.CODE

4 Chapter 1 Eletrial Theory To hange a fration to a deimal or whole number, divide the numerator (top number) by the denominator (bottom number). Examples: 1/6 = one divided by six = /5 = two divided by five = /6 = three divided by six = /4 = five divided by four = /2 = seven divided by two = Perentages A perentage is another method used to display a value. One hundred perent (100%) means all of the value; fifty perent (50%) means one-half of a value, and twenty-five perent (25%) means one-fourth of a value. For onveniene in multiplying or dividing by a perentage, onvert the perentage value to a whole number or deimal, and then use this value for the alulation. When hanging a perent value to a deimal or whole number, drop the perentage symbol and move the deimal point two plaes to the left. Figure Multiplier When a number needs to be hanged by multiplying it by a perentage, this perentage is alled a multiplier. The first step is to onvert the perentage to a deimal, then multiply the original number by the deimal value. Example A An overurrent protetion devie (iruit breaker or fuse) must be sized no less than 125 perent of the ontinuous load. If the load is 80A, the overurrent protetion devie will have to be sized no smaller than. Figure 1 2 (a) 80A (b) 100A () 125A (d) 75A Answer: (b) 100A Step 1: Convert 125 perent to a deimal: 1.25 Step 2: Multiply the value 80 by 1.25 = 100A Figure 1 2 Examples Figure 1 1 Perentage Number 32.50% % % % 2.50 Example B The maximum ontinuous load on an overurrent protetion devie is limited to 80 perent of the devie rating. If the protetive devie is rated 50A, what is the maximum ontinuous load permitted on the protetive devie? Figure 1 3 (a) 40A (b) 50A () 75A (d) 100A Answer: (a) 40A Step 1: Convert 80 perent to a deimal: 0.80 Step 2: Multiply the value 50A by 0.80 = 40A Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

5 Eletriian s Math and Basi Eletrial Formulas Unit 1 Step 2: Add one to the deimal: = 1.15 Step 3: Multiply 8 by the multiplier 1.15: 8 kw x 1.15 = 9.20 kw Figure Perent Inrease The following steps aomplish inreasing a number by a speifi perentage: Step 1: Step 2: Step 3: Convert the perent to a deimal value. Add one to the deimal value to reate the multiplier. Multiply the original number by the multiplier found in Step 2. Example A Inrease the whole number 45 by 35 perent. Step 1: Convert 35 perent to deimal form: 0.35 Step 2: Add one to the deimal value: = 1.35 Step 3: Multiply 45 by the multiplier: 1.35: 45 x 1.35 = Example B If the feeder demand load for a range is 8 kw and it is required to be inreased by 15 perent, the total alulated load will be. Figure 1 4 (a) 8 kw (b) 15 kw () 6.80 kw (d) 9.20 kw Answer: (d) 9.20 kw Step 1: Convert the perentage inrease required to deimal form: 15 perent = Reiproals To obtain the reiproal of a number, onvert the number into a fration with the number one as the numerator (top number). It is also possible to alulate the reiproal of a deimal number. Determine the reiproal of a deimal number by following these steps: Step 1: Step 2: Convert the number to a deimal value. Divide the value into the number one. Example A What is the reiproal of 80 perent? (a) 0.80 (b) 100% () 125% (d) 150% Answer: () 125% Step 1: Convert 80 perent into a deimal (move the deimal two plaes to the left): 80 perent = 0.80 Step 2: Divide 0.80 into the number one: 1/0.80 = 1.25 or 125 perent Example B Figure 1 4 What is the reiproal of 125 perent? Mike Holt Enterprises, In NEC.CODE

6 Chapter 1 Eletrial Theory (a) 0.80 (b) 100% () 125% (d) 75% Answer: (a) 0.80 Step 1: Convert 125 perent into a deimal: 125 perent = 1.25 Step 2: Divide 1.25 into the number one: 1/1.25 = 0.80 or 80 perent 1.8 Squaring a Number Squaring a number means multiplying the number by itself = 10 x 10 = = 23 x 23 = 529 Example A What is the power onsumed in watts by a 12 AWG ondutor that is 200 ft long, and has a total resistane of 0.40 ohms, if the urrent (I) in the iruit ondutors is 16A? Formula: Power = I 2 x R (Answers are rounded to the nearest 50). (a) 50 (b) 150 () 100 (d) 200 Answer: () 100 P = I 2 x R I = 16A R = 0.40 ohms P = 16A 2 x 0.40 ohms P = 16A x 16A x 0.40 ohms P = W Example C What is the sq in. area of an 8 in. pizza? Figure 1 5A (a) 50 (b) 75 () 25 (d) 64 Answer: (a) 50 Area = π x r 2 Area = 3.14 x (0.50 x 8) 2 Area = 3.14 x 4 2 Area = 3.14 x 4 x 4 Area = 3.14 x 16 Area = 50 sq in. Example D What is the sq in. area of a 16 in. pizza? Figure 1 5B (a) 100 (b) 200 () 150 (d) 256 Answer: (b) 200 Example B What is the area in square inhes (sq in.) of a trade size 1 raeway with an inside diameter of in.? Formula: Area = ¹ x r 2 ¹ = 3.14 r = radius (is equal to 0.50 of the diameter) (a) 1 (b) 0.86 () 0.34 (d) 0.50 Answer: (b) 0.86 Area = π x r 2 Area = 3.14 x (0.50 x 1.049) 2 Area = 3.14 x Area = 3.14 x ( x ) Area = 3.14 x Area = 0.86 sq in. Figure 1 5 Area = π x r 2 Area = 3.14 x (0.50 x 16) 2 Area = 3.14 x 8 2 Area = 3.14 x 8 x 8 Area = 3.14 x 64 Area = 200 sq in. Author s Comment: As you see in Examples C and D, if you double the diameter of the irle, the area ontained in the irle is inreased by a fator of four! By the way, a large pizza is always heaper per sq in. than a small pizza. Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

7 Eletriian s Math and Basi Eletrial Formulas Unit Parentheses Whenever numbers are in parentheses, omplete the mathematial funtion within the parentheses before proeeding with the rest of the problem. What is the urrent of a 36,000W, 208V, three-phase load? Figure 1 6 Formula: Ampere (I) = Watts/(E x 1.732) (a) 50A (b) 100A () 150A (d) 360A Answer: (b) 100A Step 1: Perform the operation inside the parentheses first determine the produt of: 208V x = 360 Step 2: Divide 36,000W by 360 = 100A squaring a number. The square root of 36 is a number that, when multiplied by itself, gives the produt 36. The 36 equals six (6), beause six, multiplied by itself (6 2 ) equals the number 36. Beause it s diffiult to do this manually, just use the square root key of your alulator. Example 3: Following your alulator s instrutions, enter the number 3, then press the square root key = ,000: enter the number 1,000, then press the square root key = If your alulator does not have a square root key, don t worry about it. For all pratial purposes of this textbook, the only number you need to know the square root of is 3. The square root of 3 equals approximately To multiply, divide, add, or subtrat a number by a square root value, determine the deimal value, then perform the math funtion. Example A 36,000W/(208V x 3) is equal to. (a) 120A (b) 208A () 360A (d) 100A Answer: (d) 100A Step 1: Determine the deimal value for the 3 = Step 2: Divide 36,000W by (208V x 1.732) = 100A Example Parenthesis are used to group steps of a proess in the orret order. For instane, the sum of 3 and 15 added to the produt of 4 and 2. (3 + 15) + (4 x 2) = = Square Root Figure 1 6 Deriving the square root ( n) of a number is the opposite of Example B The phase voltage of a 120/208V system is equal to 208V/ 3 whih is. (a) 120V (b) 208V () 360V (d)480v Answer: (a) 120V Step 1: Determine the deimal value for the 3 = Step 2: Divide 208V by = 120V 1.11 Volume The volume of an enlosure is expressed in ubi inhes (u in.). It is determined by multiplying the length, by the width, by the depth of the enlosure. Mike Holt Enterprises, In NEC.CODE

8 Chapter 1 Eletrial Theory Example What is the volume of a box that has the dimensions of 4 x 4 x 1½ in.? Figure 1 7 (a) 20 u in. (b) 24 u in. () 30 u in. (d) 12 u in. Answer: (b) 24 u in. 1½ = x 4 x 1.50 = 24 u in. To onvert a unit value to a k value, divide the number by 1,000 and add the k suffix. Example B A 300W load will have a kw rating. Figure 1 8 (a) 300 kw (b) 3,000 kw () 30 kw (d) 0.30 kw Answer: (d) 0.30 kw kw = Watts/1,000 kw = 300W/1,000 = 0.30 kw Figure 1 7 Figure 1 8 Author s Comment: The atual volume of a 4 in. square eletrial box is less than 24 u in. beause the interior dimensions may be less than the nominal size and often orners are rounded, so the allowable volume is given in NEC. Table (A). Author s Comment: The use of the letter k is not limited to kw. It is also used for kva (1,000 volt-amps), and kmil (1,000 irular mils) and other units suh as kft (1,000 feet) Kilo The letter k is used in the eletrial trade to abbreviate the metri prefix kilo whih represents a value of 1,000. To onvert a number whih inludes the k prefix to units, multiply the number preeding the k by 1, Rounding Off There is no speifi rule for rounding off, but rounding to two or three signifiant figures should be suffiient for most eletrial alulations. Numbers below five are rounded down, while numbers five and above are rounded up. Example A What is the wattage value for an 8 kw rated range? (a) 8W (b) 8,000W () 4,000W (d) 800W Answer: (b) 8,000W Examples fourth number is five or above = rounded up fourth number is below five = 1.67 rounded down Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

10 Chapter 1 Eletrial Theory 1.15 Eletrial Ciruit An eletrial iruit onsists of the power soure, the ondutors, and the load. A swith an be plaed in series with the iruit ondutors to ontrol the operation of the load (on or off). Figure 1 10 Diret urrent is used for eletroplating, street trolley and railway systems, or where a smooth and wide range of speed ontrol is required for a motor-driven appliation. Diret urrent is also used for ontrol iruits and eletroni instruments. Figure 1 11 Figure 1 10 Alternating Current Author s Comment: Aording to the eletron urrent flow theory, urrent always flows from the negative terminal of the soure, through the iruit and load, to the positive terminal of the soure. Alternating-urrent power soures produe a voltage that hanges polarity and magnitude. Alternating urrent is produed by an a power soure suh as an a generator. The major advantage of a over d is that voltage an be hanged through the use of a transformer. Figure Power Soure The power neessary to move eletrons out of their orbit around the nuleus of an atom an be produed by hemial, magneti, photovoltai, and other means. The two ategories of power soures are diret urrent (d) and alternating urrent (a). Diret Current The polarity and the output voltage from a d power soure never hange diretion. One terminal is negative and the other is positive, relative to eah other. Diret-urrent power is often produed by batteries, d generators, and eletroni power supplies. Figure 1 11 Figure Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

11 Eletriian s Math and Basi Eletrial Formulas Unit 1 Author s Comment: Alternating urrent aounts for more than 90 perent of all eletri power used throughout the world Condutane Condutane or ondutivity is the property of a metal that permits urrent to flow. The best ondutors in order of their ondutivity are: silver, opper, gold, and aluminum. Copper is most often used for eletrial appliations. Figure 1 13 Figure Ohm s Law Ohm s Law expresses the relationship between a d iruit s urrent intensity (I), eletromotive fore (E), and its resistane (R). This is expressed by the formula: I = E/R. Author s Comment: The German physiist Georg Simon Ohm ( ) stated that urrent is diretly proportional to voltage, and inversely proportional to resistane. Figure 1 15 Diret proportion means that hanging one fator results in a diret hange to another fator in the same diretion and by the same magnitude. Figure 1 15A 1.18 Ciruit Resistane Figure 1 13 The total resistane of a iruit inludes the resistane of the power supply, the iruit wiring, and the load. Applianes suh as heaters and toasters use high-resistane ondutors to produe the heat needed for the appliation. Beause the resistane of the power soure and ondutor are so muh smaller than that of the load, they are generally ignored in iruit alulations. Figure 1 14 If the voltage inreases 25 perent, the urrent inreases 25 perent in diret proportion (for a given resistane). If the voltage dereases 25 perent, the urrent dereases 25 perent in diret proportion (for a given resistane). Inverse proportion means that inreasing one fator results in a derease in another fator by the same magnitude, or a derease in one fator will result in an inrease of the same magnitude in another fator. Figure 1 15B If the resistane inreases by 25 perent, the urrent dereases by 25 perent in inverse proportion (for a given voltage), or if the resistane dereases by 25 perent, the urrent inreases by 25 perent in inverse proportion (for a given voltage). Mike Holt Enterprises, In NEC.CODE 11

12 Chapter 1 Eletrial Theory Author s Comment: Plae your thumb on the unknown value in Figure 1 16, and the two remaining variables will show you the orret formula. Figure 1 16 Figure Ohm s Law and Alternating Current Diret Current In a d iruit, the only opposition to urrent flow is the physial resistane of the material that the urrent flows through. This opposition is alled resistane and is measured in ohms. Alternating Current In an a iruit, there are three fators that oppose urrent flow: the resistane of the material, the indutive reatane of the iruit, and the apaitive reatane of the iruit. Current Example 120V supplies a lamp that has a resistane of 192 ohms. What is the urrent flow in the iruit? Figure 1 17 (a) 0.60A (b) 0.50A () 2.50A (d) 1.30A Answer: (a) 0.60A Step 1: What is the question? What is I? Step 2: What do you know? E = 120V, R = 192 ohms Step 3: The formula is I = E/R Step 4: The answer is I = 120V/192 ohms Step 5: The answer is I = 0.625A Author s Comment: For now, we will assume that the effets of indutane and apaitane on the iruit are insignifiant and they will be ignored Ohm s Law Formula Cirle Ohm s Law, the relationship between urrent, voltage, and resistane expressed in the formula, E = I x R, an be transposed to I = E/R or R = E/I. In order to use these formulas, two of the values must be known. 12 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

13 Eletriian s Math and Basi Eletrial Formulas Unit 1 Step 1: What is the question? What is E? Step 2: What do you know about the ondutors? I = 16A, R = 0.20 ohms. The NEC lists the a resistane of 1,000 ft of 12 AWG as 2 ohms [Chapter 9, Table 8]. The resistane of 100 ft is equal to 0.20 ohms. Figure 1 19 Step 3: The formula is E = I x R Step 4: The answer is E = 16A x 0.20 ohms Step 5: The answer is E = 3.20V Figure 1 17 Voltage-Drop Example What is the voltage drop over two 12 AWG ondutors (resistane of 0.20 ohms per 100 ft) supplying a 16A load loated 50 ft from the power supply? Figure 1 18 (a) 16V (b) 32V () 1.60V (d) 3.20V Answer: (d) 3.20V Figure 1 19 Figure 1 18 Resistane Example What is the resistane of the iruit ondutors when the ondutor voltage drop is 3V and the urrent flowing in the iruit is 100A? Figure 1 20 (a) 0.03 ohms () 30 ohms (b) 2 ohms (d) 300 ohms Answer: (a) 0.03 ohms Step 1: What is the question? What is R? Step 2: What do you know about the ondutors? E = 3V dropped, I = 100A Step 3: The formula is R = E/I Step 4: The answer is R = 3V/100A Step 5: The answer is R = 0.03 ohms Mike Holt Enterprises, In NEC.CODE 13

14 Chapter 1 Eletrial Theory Power Loss Example What is the power loss in watts for two ondutors that arry 12A and have a voltage drop of 3.6V? Figure 1 22 (a) 4.3W (b) 43.2W () 432W (d) 24W Answer: (b) 43.2W Step 1: What is the question? What is P? Step 2: What do you know? I = 12A, E = 3.60 VD Step 3: The formula is P = I x E Step 4: The answer is P = 12A x 3.60V Step 5: The answer is 43.2W Figure PIE Formula Cirle The PIE formula irle demonstrates the relationship between power, urrent, and voltage, and it is expressed in the formula P = I x E. This formula an be transposed to I = P/E or E = P/I. In order to use these formulas, two of the values must be known. Author s Comment: Plae your thumb on the unknown value in Figure 1 21 and the two remaining variables will show you the orret formula. Figure 1 22 Figure 1 21 Current Example What is the urrent flow in amperes through a 7.50 kw heat strip rated 230V when onneted to a 230V power supply? Figure 1 23 (a) 25A (b) 33A Answer: (b) 33A () 39A (d) 230A Step 1: What is the question? What is I? Step 2: What do you know? P = 7,500W, E = 230V Step 3: The formula is I = P/E 14 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

15 Eletriian s Math and Basi Eletrial Formulas Unit 1 Step 4: The answer is I = 7,500/230V Step 5: The answer is 32.6A Author s Comment: Unity power fator is explained in Unit 3. For the purpose of this Unit, we will assume a power fator of 1.0 for all a iruits Using the Formula Wheel The formula wheel is divided into four setions with three formulas in eah setion. Figure When working the formula wheel, the key to getting the orret answer is to follow these steps: Step 1: Know what the question is asking for: I, E, R, or P. Step 2: Determine the knowns: I, E, R, or P. Figure 1 23 Step 3: Step 4: Determine whih setion of the formula wheel applies: I, E, R, or P and selet the formula from that setion based on what you know. Work out the alulation Formula Wheel The formula wheel is a ombination of the Ohm s Law and the PIE formula wheels. The formulas in the formula wheel an be used for d iruits or a iruits with unity power fator. Figure 1 24 Figure 1 25 Figure 1 24 Example The total resistane of two 12 AWG ondutors, 75 ft long is 0.30 ohms, and the urrent through the iruit is 16A. What is the power loss of the ondutors? Figure 1 26 (a) 20W (b) 75W () 150W (d) 300W Answer: (b) 75W Step 1: What is the question? What is the power loss of the ondutors P? Mike Holt Enterprises, In NEC.CODE 15

16 Chapter 1 Eletrial Theory Step 2: What do you know about the ondutors? I = 16A, R = 0.30 ohms Step 3: What is the formula? P = I 2 x R Step 4: Calulate the answer: P = 16A 2 x 0.30 ohms = 76.8W The answer is 76.80W Step 1: What is the problem asking you to find? What is wasted P? Step 2: What do you know about the ondutors? I = 24A E = 240V x 3% E = 240V x 0.03 E = 7.20 VD Step 3: The formula is P = I x E Step 4: Calulate the answer: P = 24A x 7.20V = W The answer is W Figure 1 26 Figure Power Losses of Condutors Power in a iruit an be either useful or wasted. Most of the power used by loads suh as fluoresent lighting, motors, or stove elements is onsumed in useful work. However, the heating of ondutors, transformers, and motor windings is wasted work. Wasted work is still energy used; therefore it must be paid for, so we all these power losses. Example What is the ondutor power loss in watts for a 10 AWG ondutor that has a voltage drop of 3 perent and arries a urrent flow of 24A? Figure 1 27 (a) 17W () 350W Answer: (b) 173W (b) 173W (d) none of these 1.26 Cost of Power Sine eletri bills are based on power onsumed in watts, we should understand how to determine the ost of power. Example What does it ost per year (at 8.60 ents per kwh) for the power loss of two 10 AWG iruit ondutors that have a total resistane of 0.30 ohms with a urrent flow of 24A? Figure 1 28 (a) \$1.30 (b) \$13.00 () \$130 (d) \$1,300 Answer: () \$130 Step 1: Determine the power onsumed: P = I 2 x R P = 24A 2 x 0.30 ohms P = W 16 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

17 Eletriian s Math and Basi Eletrial Formulas Unit 1 Step 2: Convert answer in Step 1 to kw: P = W/1,000W P = kw Step 3: Determine ost per hour: (0.086 dollars per kwh) x kw = dollars per hr Step 4: Determine dollars per day: dollars per hr x (24 hrs per day) = dollars per day Step 5: Determine dollars per year: dollars per day x (365 days per year) = \$ per year Figure 1 29 Power Example at 230V What is the power onsumed by a 9.60 kw heat strip rated 230V onneted to a 230V iruit? Figure 1 30 (a) 7.85 kw (b) 9.60 kw () kw (d) 9.60W Answer: (b) 9.60 kw Figure 1 28 Author s Comment: That s a lot of money just to heat up two 10 AWG ondutors for one iruit. Imagine how muh it osts to heat up the ondutors for an entire building! 1.27 Power Changes with the Square of the Voltage The voltage applied to a resistor dramatially affets the power onsumed by that resistor. Power is determined by the square of the voltage. This means that if the voltage is doubled, the power will inrease four times. If the voltage is dereased 50 perent, the power will derease to 25 perent of its original value. Figure 1 29 Figure 1 30 Mike Holt Enterprises, In NEC.CODE 17

18 Chapter 1 Eletrial Theory Step 1: What is the problem asking you to find? Power onsumed by the resistane. Step 2: What do you know about the heat strip? You were given P = 9.60 kw in the statement of the problem. Power Example at 208V What is the power onsumed by a 9.60 kw heat strip rated 230V onneted to a 208V iruit? Figure 1 31 (a) 7.85 kw () kw Answer: (a) 7.85 kw (b) 9.60 kw (d) 208 kw Step 1: What is the problem asking you to find? Power onsumed by the resistane. Step 2: What do you know about the heat strip? E = 208V, R = E2/P R = 230V x 230V/9,600W R = 5.51 ohms Step 3: The formula to determine power is: P = E 2 /R Step 4: The answer is: P = 208V 2 /5.51 ohms P = 7,851W or 7.85 kw Thus, for a small hange in voltage, there is a onsiderable hange in power onsumption by this heater. Author s Comment: The urrent flow for this heat strip is I = P/E. P = 7,851W E = 208V I = 7,851W/208V I = 38A Power Example at 240V What is the power onsumed by a 9.60 kw heat strip rated 230V onneted to a 240V iruit? Figure 1 32 (a) 7.85 kw () kw Answer: () kw (b) 9.60 kw (d) kw Figure 1 32 Step 1: What is the problem asking you to find? Power onsumed by the resistane. Figure 1 31 Author s Comment: It is important to realize that the resistane of the heater unit does not hange it is a property of the material that the urrent flows through and is not dependent on the voltage applied. Step 2: What do you know about the resistane? R = 5.51 ohms* *The resistane of the heat strip is determined by the formula R = E 2 /P. E = Nameplate voltage rating of the resistane, 230V P = Nameplate power rating of the resistane, 9,600W 18 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

19 Eletriian s Math and Basi Eletrial Formulas Unit 1 R = E 2 /P R = 230V 2 /9,600W R = 5.51 ohms Step 3: The formula to determine power is: P = E 2 /R Step 4: The answer is: P = 240V x 240V/5.51 ohms P = 10,454W P = kw Author s Comment: The urrent flow for this heat strip is I = P/E. P = 10,454W E = 240V I = 10,454W/240V I = 44A As you an see, when the voltage hanges, the power hanges by the square of the hange in the voltage, but the urrent hanges in diret proportion. UNIT 1 Conlusion You ve gained skill in working with Ohm s Law and the power equation, and an use the power wheel to solve a wide variety of eletrial problems. You also know how to alulate voltage drop and power loss, and an relate the osts in real dollars. As you work through the pratie questions, you ll see how well you have mastered the mathematial onepts and how ready you are to put them to use in eletrial formulas. Always remember to hek your answer when you are done then you ll know you have a right answer every time. As useful as these skills are, there is still more to learn. But, your mastery of the basi eletrial formulas means you are well prepared. Work through the questions that follow, and go bak over the instrutional material if you have any diffiulty. When you believe you know the material in Unit 1, you are ready to takle the eletrial iruits of Unit 2. Mike Holt Enterprises, In NEC.CODE 19

20 Chapter 1 Eletrial Theory UNIT 1 Calulation Pratie Questions PART A eletriian s MATH 1.3 Frations 1. The deimal equivalent for the fration 1/2 is. (a) 0.50 (b) 5 () 2 (d) The approximate deimal equivalent for the fration 4/18 is. (a) 4.50 (b) 3.50 () 2.50 (d) Perentages 3. To hange a perent value to a deimal or whole number, drop the perentage sign and move the deimal point two plaes to the. (a) right (b) left () depends (d) none of these 4. The deimal equivalent for 75 perent is. (a) (b) 0.75 () 7.50 (d) The deimal equivalent for 225 perent is. (a) 225 (b) () 2.25 (d) The deimal equivalent for 300 perent is. (a) 0.03 (b) 0.30 () 3 (d) Multiplier 7. The method of inreasing a number by another number is done by using a. (a) perentage (b) deimal () fration (d) multiplier 8. An overurrent protetion devie (iruit breaker or fuse) must be sized no less than 125 perent of the ontinuous load. If the load is 16A, the overurrent protetion devie will have to be sized at no less than. (a) 20A (b) 23A () 17A (d) 30A 9. The maximum ontinuous load on an overurrent protetion devie is limited to 80 perent of the devie rating. If the overurrent devie is rated 100A, the maximum ontinuous load is. (a) 72A (b) 80A () 90A (d) 125A 20 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

21 Eletriian s Math and Basi Eletrial Formulas Unit Perent Inrease 10. The feeder alulated load for an 8 kw load, inreased by 20 perent is. (a) 8 kw (b) 9.60 kw () 6.40 kw (d) 10 kw 1.7 Reiproals 11. What is the reiproal of 1.25? (a) 0.80 (b) 1.10 () 1.25 (d) A ontinuous load requires an overurrent protetion devie sized no smaller than 125 perent of the load. What is the maximum ontinuous load permitted on a 100A overurrent protetion devie? (a) 100A (b) 125A () 80A (d) 75A 1.8 Squaring a Number 13. Squaring a number means multiplying the number by itself. (a) True (b) False 14. What is the power onsumed in watts by a 12 AWG ondutor that is 100 ft long and has a resistane (R) of 0.20 ohms, when the urrent (I) in the iruit is 16A? Formula: Power = I 2 x R. (a) 75W (b) 50W () 100W (d) 200W 15. What is the area in sq in. of a trade size 2 raeway? Formula: Area = Pi x r 2, Pi = 3.14, r = radius (1/2 of the diameter) (a) 1 sq in. (b) 2 sq in. () 3 sq in. (d) 4 sq in. 16. The numeri equivalent of 4 2 is. (a) 2 (b) 8 () 16 (d) The numeri equivalent of 12 2 is. (a) 3.46 (b) 24 () 144 (d) 1, Parentheses 18. What is the maximum distane that two 14 AWG ondutors an be run if they arry 16A and the maximum allowable voltage drop is 10V? D = (Cmil x VD)/(2 x K x I) D = (4,110 mil x 10V)/(2 x ohms x 16A) (a) 50 ft (b) 75 ft () 100 ft (d) 150 ft Mike Holt Enterprises, In NEC.CODE 21

22 Chapter 1 Eletrial Theory 19. What is the urrent in amperes of an 18 kw, 208V, three-phase load? Current: I = VA/(E x 3) Current: I = 18,000W/(208V x 1.732) (a) 25A (b) 50A () 100A (d) 150A 1.10 Square Root 20. Deriving the square root of a number is almost the same as squaring a number. (a) True (b) False 21. What is the approximate square root of 1,000 ( 1,000)? (a) 3 (b) 32 () 100 (d) The square root of 3 ( 3) is. (a) (b) 9 () 729 (d) Volume 23. The volume of an enlosure is expressed in, and it is alulated by multiplying the length, by the width, by the depth of the enlosure. (a) ubi inhes (b) weight () inh-pounds (d) none of these 24. What is the volume (in ubi inhes) of a 4 x 4 x 1.50 in. box? (a) 20 u in. (b) 24 u in. () 30 u in. (d) 33 u in Kilo 25. What is the kw of a 75W load? (a) 75 kw (b) 7.50 kw () 0.75 kw (d) kw 1.13 Rounding Off 26. The approximate sum of 2, 7, 8, and 9 is equal to. (a) 20 (b) 25 () 30 (d) Testing Your Answer for Reasonableness 27. The output power of a transformer is 100W and the transformer effiieny is 90 perent. What is the transformer input if the output is lower than the input? Formula: Input = Output/Effiieny (a) 90W (b) 110W () 100W (d) 125W 22 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

23 Eletriian s Math and Basi Eletrial Formulas Unit 1 PART B basi ELECTRICAL FORMULA 1.15 Eletrial Ciruit 28. An eletrial iruit onsists of the. (a) power soure (b) ondutors () load (d) all of these 29. Aording to the eletron flow theory, eletrons leave the terminal of the soure, flow through the ondutors and load(s), and return to the terminal of the soure. (a) positive, negative (b) negative, positive () negative, negative (d) positive, positive 1.16 Power Soure 30. The polarity and the output voltage from a d power soure hanges diretion. One terminal will be negative and the other will be positive. (a) True (b) False 31. Diret urrent is used for eletroplating, street trolley and railway systems, or where a smooth and wide range of speed ontrol is required for a motor-driven appliation. (a) True (b) False 32. The polarity and the output voltage from an a power soure never hange diretion. (a) True (b) False 33. The major advantage of a over d is the ease of voltage regulation by the use of a transformer. (a) True (b) False 1.17 Condutane 34. Condutane is the property that permits urrent to flow. (a) True (b) False 35. The best ondutors, in order of their ondutivity, are gold, silver, opper, and aluminum. (a) True (b) False 36. Condutane or ondutivity is the property of metal that permits urrent to flow. The best ondutors in order of their ondutivity are:. (a) gold, silver, opper, aluminum () silver, gold, opper, aluminum (b) gold, opper, silver, aluminum (d) silver, opper, gold, aluminum 1.18 Ciruit Resistane 37. The iruit resistane inludes the resistane of the. (a) power soure (b) ondutors () load (d) all of these Mike Holt Enterprises, In NEC.CODE 23

24 Chapter 1 Eletrial Theory 38. Often the resistane of the power soure and ondutor are ignored in iruit alulations. (a) True (b) False 1.19 Ohm s Law 39. The Ohm s Law formula, I = E/R, states that urrent is proportional to the voltage, and proportional to the resistane. (a) indiretly, inversely (b) inversely, diretly () inversely, indiretly (d) diretly, inversely 40. Ohm s Law demonstrates the relationship between iruit. (a) intensity (b) EMF () resistane (d) all of these 1.20 Ohm s Law and Alternating Current 41. In a d iruit, the only opposition to urrent flow is the physial resistane of the material. This opposition is alled reatane and is measured in ohms. (a) True (b) False 42. In an a iruit, the fators that oppose urrent flow are. (a) resistane (b) indutive reatane () apaitive reatane (d) all of these 1.21 Ohm s Law Formula Cirle 43. What is the voltage drop of two 12 AWG ondutors (0.40 ohms) supplying a 16A load, loated 100 ft from the power supply? Formula: E VD = I x R (a) 6.40 ohms (b) ohms () 1.60 ohms (d) 3.20 ohms 44. What is the resistane of the iruit ondutors when the ondutor voltage drop is 7.20V and the urrent flow is 50A? Formula: R = E/I (a) 0.14 ohms (b) 0.30 ohms () 3 ohms (d) 14 ohms 1.22 PIE Formula Cirle 45. What is the power loss in watts of a ondutor that arries 24A and has a voltage drop of 7.20V? Formula: P = I x E (a) 175W (b) 350W () 700W (d) 2,400W 46. What is the approximate power onsumed by a 10 kw heat strip rated 230V, when onneted to a 208V iruit? Formula: P = E 2 /R (a) 8.20 kw (b) 9.3 kw () kw (d) kw 24 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

25 Eletriian s Math and Basi Eletrial Formulas Unit Formula Wheel 47. The formulas in the power wheel apply to. (a) d (b) a with unity power fator () d or a iruits (d) a and b 1.24 Using the Formula Wheel 48. When working any formula, the key to getting the orret answer is following these four steps: Step 1: Know what the question is asking you to find. Step 2: Determine the knowns of the iruit. Step 3: Selet the formula. Step 4: Work out the formula alulation. (a) True (b) False 1.25 Power Losses of Condutors 49. Power in a iruit an be either useful or wasted. Wasted work is still energy used; therefore it must be paid for, so we all this. (a) resistane (b) indutive reatane () apaitive reatane (d) power loss 50. The total iruit resistane of two 12 AWG ondutors (eah 100 ft long) is 0.40 ohms. If the urrent of the iruit is 16A, what is the power loss of both ondutors? Formula: P = I 2 x R (a) 75W (b) 100W () 300W (d) 600W 51. What is the ondutor power loss for a 120V iruit that has a 3 perent voltage drop and arries a urrent flow of 12A? Formula: P = I x E (a) 43W (b) 86W () 172W (d) 1,440W 1.26 Cost of Power 52. What does it ost per year (at 8 ents per kwh) for the power loss of a 12 AWG ondutor (100 ft long) that has a total resistane of 0.40 ohms and a urrent flow of 16A? Formula: Cost per Year = Power for the Year in kwh x \$0.08 (a) \$33 (b) \$54 () \$72 (d) \$ Power Changes with the Square of the Voltage 53. The voltage applied to a resistor dramatially affets the power onsumed by that resistor beause power is affeted in diret proportion to the voltage. (a) True (b) False 54. What is the power onsumed by a 10 kw heat strip that s rated 230V, if it s onneted to a 115V iruit? Formula: P = E 2 /R (a) 2.50 kw (b) 5 kw () 7.50 kw (d) 15 kw Mike Holt Enterprises, In NEC.CODE 25

26 Chapter 1 Eletrial Theory UNIT 1 Calulation Challenge Questions ( Indiates that 75% or fewer of those who took this exam answered the question orretly.) PART A eletriian s MATH 1.12 Kilo 1. One kva is equal to. (a) 100 VA (b) 1,000V () 1,000W (d) 1,000 VA PART B basi ELECTRICAL FORMULAS 1.17 Condutane 2. Whih of the following is the most ondutive? (a) Bakelite (b) Oil () Air (d) Salt water 1.19 Ohm s Law 3. If the ontat resistane of a onnetion inreases and the urrent of the iruit (load) remains the same, the voltage dropped aross the onnetion will. (a) inrease (b) derease () remain the same (d) annot be determined 4. To double the urrent of a iruit when the voltage remains onstant, the R (resistane) must be. (a) doubled (b) redued by half () inreased (d) none of these 5. An ohmmeter is being used to test a relay oil. The equipment instrutions indiate that the resistane of the oil should be between 30 and 33 ohms. The ohmmeter indiates that the atual resistane is less than 22 ohms. This reading would most likely indiate. (a) the oil is okay (b) an open oil () a shorted oil (d) a meter problem 1.23 Formula Wheel 6. To alulate the energy onsumed in watts by a resistive appliane, you need to know of the iruit. (a) the voltage and urrent (b) the urrent and resistane () the voltage and resistane (d) any of these pairs of variables 26 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

27 Eletriian s Math and Basi Eletrial Formulas Unit 1 7. The power onsumed of a resistor an be expressed by the formula I 2 x R. If 120V is applied to a 10 ohm resistor, the power onsumed will be. (a) 510W (b) 1,050W () 1,230W (d) 1,440W 8. Power loss in a iruit beause of heat an be determined by the formula. (a) P = R x I (b) P = I x R () P = I 2 x R (d) none of these 9. The energy onsumed by a 5 ohm resistor is than the energy onsumed by a 10 ohm resistor, assuming the urrent in both ases remains the same. (a) more (b) less 10. When a load is rated 500W at 115V is onneted to a 120V power supply, the urrent of the iruit will be. Tip: At 120V, the load onsumed more than 500 ohms, but the resistane of the load remains onstant. (a) 3.80A (b) 4.50A () 2.70A (d) 5.50A 1.27 Power Changes with the Square of the Voltage 11. A 120V-rated toaster will produe heat when supplied by 115V. (a) more (b) less () the same (d) none of these 12. When a 100W, 115V lamp operates at 230V, the lamp will onsume approximately. (a) 150W (b) 300W () 400W (d) 450W 13. A 1,500W resistive heater is rated 230V and it is onneted to a 208V supply. The power onsumed for this load at 208V will be approximately. (a) 1,625W (b) 1,750W () 1,850W (d) 1,225W 14. The total resistane of a iruit is ohms. The load has a resistane of 10 ohms and the wire has a resistane of 0.20 ohms. If the urrent of the iruit is 12A, then the power onsumed by the iruit ondutors (0.20 ohms) is approximately. (a) 8W (b) 29W () 39W (d) 45W Mike Holt Enterprises, In NEC.CODE 27

28 Chapter 1 Eletrial Theory UNIT 1 Notes 28 Mike Holt s Illustrated Guide to Eletrial NEC Exam Preparation

29 UNIT 1 CALCULATION PRACTICE QUESTIONS ANSWERS 1. (a) (d) (b) left 4. (b) () () 3 7. (d) multiplier 8. (a) 20A The overurrent protetion devie must be sized 1.25 times larger than the load. 16A x 1.25 = 20A 9. (b) 80A The ontinuous load must be limited to 80 perent of the rating of the protetion devie. 100 x 0.8 = 80A 10. (b) 9.6 kw Step 1: -Change the % to its deimal multiplier 20% inrease = 1.20 Step 2: -Multiply the number by the multiplier 8 kw x 1.2 = 9.6 kw 11. (a) 0.8 Reiproal of 1.25 = 1/1.25 Reiproal of 1.25 = () 80A The ontinuous load must be limited to 80 perent of the rating of the protetion devie. 100A x 0.8 = 80A 13. (a) True 14. (b) 50W P = I 2 x R P = 16A 2 x 0.2 ohms P = (16A x 16a) x 0.2 ohms P = 51.2W 15. () 3 sq in. Area = -Pie x r 2 Pie = 3.14 r = radius (1/2 of the diameter) Area = 3.14 x (1/2 x 2) 2 Area = 3.14 sq in. 16. () = 4 x 4 = () = 12 x 12 = () 100 ft D = (Cmil x VD)/(2 x K x I) D = -(4,110 Cmil x 10V)/(2 wires x 12.9 ohms x 16A) D = 41,100/4,128 D = 99 ft 19. (b) 50A

30 I = VA/(E x 3) I = 18,000W/(208V x 1.732) Current = 18,000W/360 Current = 50A 20. (b) False 21. (b) 32 Enter the number on your alulator, then push the square root key ( ). 22. (a) Enter the number on your alulator, then push the square root key ( ). 23. (a) ubi inhes 24. (b) 24 u in. Volume = 4 in. x 4 in. x 1.5 in. Volume = 24 u in. 25. (d) kw kw = W/1000 kw = 0.75W/1000 kw = kw 26. (b) = 26, the multiple hoie seletions are rounded to the nearest fives. 27. (b) 110W The input must be greater than the output. Input = Output/Effiieny Input = 100W/0.9 Input = 111W 28. (d) all of these 29. (b) negative, positive 30. (b) False 31. (a) True 32. (b) False 33. (a) True 34. (a) True 35. (b) False 36. (d) silver, opper, gold, aluminum 37. (d) all of these 38. (a) True 39. (d) diretly, inversely 40. (d) all of these 41. (b) False 42. (d) all of these 43. (a) 6.4 ohms E VD = I x R E VD = 16A x 0.4 ohms E VD = 6.4 ohms 44. (a) 0.14 ohms

31 R = E/I R = 7.2V/50A R = 0.14 ohms 45. (a) 175W P= I x E P = 24A x 7.2V P = 172.8W 46. (a) 8.2 kw The power onsumed by this resistor will 10,000W if onneted to a 230V soure. But, beause the applied voltage (208V) is less than the equipment voltage rating (230V), the atual power onsumed will be less. Step 1: -Determine the resistane rating of a 10 kw, 230V load. R = E 2 /P R = 230V 2 /10,000W R = 5.29 ohms Step 2: --Determine the power onsumed for a 5.29 ohm load onneted to a 208V soure. P = E 2 /R P = 208V 2 /5.29 ohms P = (208V x 208V)/5.29 ohms P = 43,264/5.29 P = 8,178W or 8.2 kw 47. (d) a and b 48. (a) True 49. (d) power loss 50. (b) 100W P = I 2 x R P = 16A 2 x 0.4 ohms P = (16A x 16a) x 0.4 ohms P = 102.4W 51. (a) 43W P = I x E P = 12A x (120V x 3%) P = 12A x 3.6V P = 43.2W 52. () \$70 Formula: Cost per Year = Power for the Year in kwh x \$0.08 Step 1: Determine the power loss per hour. P = I 2 x R P = 16A 2 x 0.4 ohms P = (16A x 16a) x 0.4 ohms P = 102.4W per hour Step 2: -Determine the power loss in kwh for the year. Power for the Year in kwh = (Power per hour x 24 hours per day x 365 days)/1,000 Power for the Year in kwh = (102.4W x 24 hours x 365 days)/1,000 Power for the Year in kwh = 897 kwh (ontinued) Step 3: -Determine the ost per year for the ondutor power losses. Formula: Cost per Year = kwh per year x Cost per kwh

32 Cost per Year = 897 kwh x \$0.08 Cost per Year = \$ (b) False 54. (a) 2.5 kw The power onsumed by this resistor will be 10,000W if onneted to a 230V soure. But, beause the applied voltage (115V) is less than the equipment voltage rating (230V), the atual power onsumed will be less. Step 1: -Determine the resistane rating of a 10 kw, 230V load. R = E 2 /P R = 230V 2 /10,000W R = (230V x 230V)/10,000W R = 52,900/10,000 R = 5.29 ohms Step 2: -Determine the power onsumed for a 5.29 ohm load onneted to a 115V soure. P = E 2 /R P = 115V 2 /5.29 ohms P = (115V x 115V)/5.29 ohms P = 13,225/5.29 ohms P = 2,500W or 2.5 kw Note: Power hanges with the square of the voltage. If the voltage is redued to 50%, then the power onsumed will be equal to the new voltage perent 2 or 50% 2, or 10,000 x (0.50 x 0.50 = 0.25 = 25%) = 2,500W. UNIT 1 CALCULATION CHALLENGE QUESTIONS ANSWERS 1. (d) 1,000 VA 2. (d) Salt water 3. (a) inrease 4. (b) redued by half Aording to Ohm s Law, urrent is inversely proportional to resistane. This means that if the resistane goes down, assuming voltage remains the same, the urrent will inrease. It also works in the opposite diretion; if the resistane inreases, again assuming the voltage remains the same, the urrent will derease. Example: What is the urrent of a 120V iruit if the resistane is 5 ohms, 10 ohms or 20 ohms? Formula: I =E/R Answer: At 5 ohmss the urrent is equal to 24A, at 10 ohms the urrent is equal to 12A, and at 20 ohms, the urrent is only equal to 6A I = 120V/5 ohms = 24A I = 120V/10 ohms = 12A I = 120V/20 ohms = 6A 5. () a shorted oil If the reading is less than 30V, this indiates that the length of the oil s ondutor must be shorted. 6. (d) any of these pairs of variables 7. (d) 1,440W The formula I 2 x R in the question has nothing to do with the atual alulation. If we know the voltage of the iruit and the resistane in ohms of the resistor, the formula we need to use is: P = E 2 /R P = 120V2/10 ohms

33 P = (120V x 120V)/10 ohms P = 1,440W 8. () P = I 2 x R 9. (b) Less If urrent remains the same and resistane inreases, then energy onsumed will inrease. Example: P = I 2 x R P = 10A 2 x 5 ohms P = 500W P = 10A2 x 10 ohms P = 1,000W 10. (b) 4.5A The power onsumed by this resistor will be 500W if onneted to a 115V soure. But, beause the applied voltage (120V) is greater than the equipment voltage rating (115V), the atual power onsumed will be greater than 500W. Step 1: -Determine the resistane rating of a 500W, 115V load. R = E 2 /P R = 115V 2 /500W R = 13,225/500 R = ohms Step 2: -Determine the urrent of a ohm load onneted to a 120V soure. I = E/R P = 120/26.45 ohms P = 4.54A 11. (b) less When the resistane is not hanged, the power will derease with dereasing voltage. For example a 144 ohm resistor will onsume 144W of power at 120V, but only 132W of power at 115V. P = E 2 /R P = 120V 2 /100 ohms P = 144W P = 115V 2 /100 ohms P = 132W 12. () 400W The power onsumed by this resistor will be 100W if onneted to a 115V soure. But, beause the applied voltage (120V) is greater than the equipment voltage rating (115V), the atual power onsumed will be greater than 100W. Step 1: -Determine the resistane rating of a 100W, 115V lamp. R = E 2 /P R = 115V 2 /100W R = (115V x 115V)/100W R = 13,225/100 R = ohms Step 2: -Determine the power onsumed for a ohm load onneted to a 120V soure. P = E 2 /R P = 230V 2 / ohms P = (230V x 230V)/ ohms P = 52,900/ ohms P = 400W

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