POWERS OF TEN 3

POWERS OF TEN
It should be apparent from the relative magnitude of the various units of measurement that very large and very small numbers will frequently be encountered in the sciences. To ease the difficulty of mathematical operations with numbers of such varying size, powers of ten are usually employed. This notation takes full advantage of the mathematical properties of powers of ten. The notation used to represent numbers that are integer powers of ten is as follows: 1 100 1/10 0.1 10 1 10 101 1/100 0.01 10 2 100 102 1/1000 0.001 10 3 1000 103 1/10,000 0.0001 10 4 In particular, note that 100 1, and, in fact, any quantity to the zero power is 1 (x0 1, 10000 1, and so on). Also, note that the numbers in the list that are greater than 1 are associated with positive powers of ten, and numbers in the list that are less than 1 are associated with negative powers of ten.
S   I
POWERS OF TEN  13
A quick method of determining the proper power of ten is to place a caret mark to the right of the numeral 1 wherever it may occur; then count from this point to the number of places to the right or left before arriving at the decimal point. Moving to the right indicates a positive power of ten, whereas moving to the left indicates a negative power. For example,
Some important mathematical equations and relationships pertaining to powers of ten are listed below, along with a few examples. In each case, n and m can be any positive or negative real number.
(1.2)
Equation (1.2) clearly reveals that shifting a power of ten from the denominator to the numerator, or the reverse, requires simply changing the sign of the power.
EXAMPLE 1.2
a. 10 3
b. 10 5
The product of powers of ten:
(1.3)
EXAMPLE 1.3 a. (1000)(10,000) (103)(104) 10(3 4) 107 b. (0.00001)(100) (10 5)(102) 10( 5 2) 10 3
The division of powers of ten:
(1.4)
EXAMPLE 1.4
a. 10(5 2) 103
b. 10(3 ( 4)) 10(3 4) 107
Note the use of parentheses in part (b) to ensure that the proper sign is established between operators.
103 10 4
1000 0.0001
105 102
100,000 100
1 1 0 0 m n 10(n m)
(10n)(10m) 10(n m)
1 10 5
1 0.00001
1 10 3
1 1000
1 1 0n 10 n 10 1 n 10n
10,000.0 1 0 , 0 0 0 . 10 4
0.00001 0 . 0 0 0 0 1 10 5
1 2 3 4
1234 45
S   I
14  INTRODUCTION
The power of powers of ten:
(1.5)
EXAMPLE 1.5 a. (100)4 (102)4 10(2)(4) 108 b. (1000) 2 (103) 2 10(3)( 2) 10 6 c. (0.01) 3 (10 2) 3 10( 2)( 3) 106
Basic Arithmetic Operations
Let us now examine the use of powers of ten to perform some basic arithmetic operations using numbers that are not just powers of ten. The number 5000 can be written as 5 1000 5 103, and the number 0.0004 can be written as 4 0.0001 4 10 4.Ofcourse, 105 can also be written as 1 105 if it clarifies the operation to be performed.
Addition and Subtraction To perform addition or subtraction using powers of ten, the power of ten must be the same for each term; that is,
(1.6)
Equation (1.6) covers all possibilities, but students often prefer to remember a verbal description of how to perform the operation. Equation (1.6) states
when adding or subtracting numbers in a powers-of-ten format, be sure that the power of ten is the same for each number. Then separate the multipliers, perform the required operation, and apply the same power of ten to the result.
EXAMPLE 1.6
a. 6300 75,000 (6.3)(1000) (75)(1000) 6.3 103 75 103 (6.3 75) 103 81.3 103 b. 0.00096 0.000086 (96)(0.00001) (8.6)(0.00001) 96 10 5 8.6 10 5 (96 8.6) 10 5 87.4 10 5
Multiplication In general,
(1.7)( A 10n)(B 10m) (A)(B) 10n m
A 10n B 10n (A B) 10n
(10n)m 10(nm)
S   I
POWERS OF TEN  15
revealing that the operations with the powers of ten can be separated from the operation with the multipliers. Equation (1.7) states
when multiplying numbers in the powers-of-ten format, first find the product of the multipliers and then determine the power of ten for the result by adding the power-of-ten exponents.
EXAMPLE 1.7
a. (0.0002)(0.000007) [(2)(0.0001)][(7)(0.000001)] (2 10 4)(7 10 6) (2)(7) (10 4)(10 6) 14 10 10 b. (340,000)(0.00061) (3.4 105)(61 10 5) (3.4)(61) (105)(10 5) 207.4 100 207.4
Division In general,
(1.8)
revealing again that the operations with the powers of ten can be separated from the same operation with the multipliers. Equation (1.8) states
when dividing numbers in the powers-of-ten format, first find the result of dividing the multipliers. Then determine the associated power for the result by subtracting the power of ten of the denominator from the power of ten of the numerator.
EXAMPLE 1.8
a. 23.5 10 2
b. 5.31 1012
Powers In general,
(1.9)
which again permits the separation of the operation with the powers of ten from the multipliers. Equation (1.9) states
when finding the power of a number in the power-of-ten format, first separate the multiplier from the power of ten and determine each separately. Determine the power-of-ten component by multiplying the power of ten by the power to be determined.
(A 10n)m Am 10nm
104 10 8
69 13
69 104 13 10 8
690,000 0.00000013
10 5 10 3
47 2
47 10 5 2 10 3
0.00047 0.002
B A
1 1 0 0 m n A B 10n m
S   I
16  INTRODUCTION
EXAMPLE 1.9 a. (0.00003)3 (3 10 5)3 (3)3 (10 5)3 27 10 15 b. (90,800,000)2 (9.08 107)2 (9.08)2 (107)2 82.4464 1014
In particular, remember that the following operations are not the same. One is the product of two numbers in the powers-of-ten format, while the other is a number in the powers-of-ten format taken to a power. As noted below, the results of each are quite different: (103)(103) (103)3 (103)(103) 106 1,000,000 (103)3 (103)(103)(103) 109 1,000,000,000
Fixed-Point, Floating-Point, Scientific, and Engineering Notation
There are, in general, four ways in which numbers appear when using a computer or calculator. If powers of ten are not employed, they are written in the fixed-point or floating-point notation. The fixed-point format requires that the decimal point appear in the same place each time. In the floating-point format, the decimal point will appear in a location defined by the number to be displayed. Most computers and calculators permit a choice of fixed- or floating-point notation. In the fixed format, the user can choose the level of precision for the output as tenths place, hundredths place, thousandths place, and so on. Every output will then fix the decimal point to one location, such as the following examples using thousandths place accuracy:
0.333 0.063 1150.000
If left in the floating-point format, the results will appear as follows for the above operations:
0.333333333333 0.0625 1150
Powers of ten will creep into the fixed- or floating-point notation if the number is too small or too large to be displayed properly. Scientific (also called standard) notation and engineering notation make use of powers of ten with restrictions on the mantissa (multiplier) or scale factor (power of the power of ten). Scientific notation requires that the decimal point appear directly after the first digit greater than or equal to 1 but less than 10. A power of ten will then appear with the number (usually following the power notation E), even if it has to be to the zero power. A few examples:
3.33333333333E 1 6.25E 2 1.15E3
Within the scientific notation, the fixed- or floating-point format can be chosen. In the above examples, floating was employed. If fixed is chosen and set at the thousandths-point accuracy, the following will result for the above operations:
2300 2
1 16
1 3
2300 2
1 16
1 3
2300 2
1 16
1 3
S   I
POWERS OF TEN  17
3.333E 1 6.250E 2 1.150E3
The last format to be introduced is engineering notation, which specifies that all powers of ten must be multiples of 3, and the mantissa must be greater than or equal to 1 but less than 1000. This restriction on the powers of ten is due to the fact that specific powers of ten have been assigned prefixes that will be introduced in the next few paragraphs. Using engineering notation in the floating-point mode will result in the following for the above operations:
333.333333333E 3 62.5E 3 1.15E3
Using engineering notation with three-place accuracy will result in the following:
333.333E 3 62.500E 3 1.150E3
Prefixes
Specific powers of ten in engineering notation have been assigned prefixes and symbols, as appearing in Table 1.2. They permit easy recognition of the power of ten and an improved channel of communication between technologists.
2300 2
1 16
1 3
2300 2
1 16
1 3
2300 2
1 16
1 3
TABLE 1.2
Multiplication Factors SI Prefix SI Symbol
1 000 000 000 000 1012 tera T 1 000 000 000 109 giga G 1 000 000 106 mega M 1 000 103 kilo k 0.001 10 3 milli m 0.000 001 10 6 micro m 0.000 000 001 10 9 nano n 0.000 000 000 001 10 12 pico p
EXAMPLE 1.10 a. 1,000,000 ohms 1 106 ohms 1 megohm (M ) b. 100,000 meters 100 103 meters 100 kilometers (km) c. 0.0001 second 0.1 10 3 second 0.1 millisecond (ms) d. 0.000001 farad 1 10 6 farad 1 microfarad (mF)
Here are a few examples with numbers that are not strictly powers of ten.
S   I
18  INTRODUCTION
EXAMPLE 1.11 a. 41,200 m is equivalent to 41.2 103 m 41.2 kilometers 41.2 km. b. 0.00956Jisequivalentto9.56 10 3 J 9.56millijoules 9.56mJ. c. 0.000768 s is equivalent to 768 10 6 s 768 microseconds 768 ms.
d. m 103 10 2 8.4 6 8.4 103 m 6 10 2 8400 m 0.06 1.4 105 m 140 103 m 140 kilometers 140 km e. (0.0003)4 s (3 10 4)4 s 81 10 16 s 0.0081 10 12 s 0.008 picosecond 0.0081 ps
1.7 CONVERSION BETWEEN LEVELS OF POWERS OF TEN
It is often necessary to convert from one power of ten to another. For instance, if a meter measures kilohertz (kHz), it may be necessary to find the corresponding level in megahertz (MHz), or if time is measured in milliseconds (ms), it may be necessary to find the corresponding time in microseconds (ms) for a graphical plot. The process is not a difficult one if we simply keep in mind that an increase or a decrease in the power of ten must be associated with the opposite effect on the multiplying factor. The procedure is best described by a few examples.
EXAMPLE 1.12
a. Convert 20 kHz to megahertz. b. Convert 0.01 ms to microseconds. c. Convert 0.002 km to millimeters.
Solutions:
a. In the power-of-ten format: 20 kHz 20 103 Hz
The conversion requires that we find the multiplying factor to appear in the space below:
Since the power of ten will be increased by a factor of three, the multiplying factor must be decreased by moving the decimal point three places to the left, as shown below:
and 20 103 Hz 0.02 106 Hz 0.02 MHz
b. In the power-of-ten format: 0.01 ms 0.01 10 3 s
and 0.01 10 3 s 10 6 s
Reduce by 3
Increase by 3
020. 0.02 3
20 103 Hz
7
106 Hz
Increase by 3
Decrease by 3
S   I
CONVERSION WITHIN AND BETWEEN SYSTEMS OF UNITS  19
Since the power of ten will be reduced by a factor of three, the multiplying factor must be increased by moving the decimal point three places to the right, as follows:
and 0.01 10 3 s 10 10 6 s 10 ms
There is a tendency when comparing 3 to 6 to think that the power of ten has increased, but keep in mind when making your judgment about increasing or decreasing the magnitude of the multiplier that 10 6 is a great deal smaller than 10 3.
c.
In this example we have to be very careful because the difference between 3 and 3 is a factor of 6, requiring that the multiplying factor be modified as follows:
and 0.002 103 m 2000 10 3 m 2000 mm
1.8 CONVERSION WITHIN AND BETWEEN SYSTEMS OF UNITS
The conversion within and between systems of units is a process that cannot be avoided in the study of any technical field. It is an operation, however, that is performed incorrectly so often that this section was included to provide one approach that, if applied properly, will lead to the correct result. There is more than one method of performing the conversion process. In fact, some people prefer to determine mentally whether the conversion factor is multiplied or divided. This approach is acceptable for some elementary conversions, but it is risky with more complex operations. The procedure to be described here is best introduced by examining a relatively simple problem such as converting inches to meters. Specifically, let us convert 48 in. (4 ft) to meters. If we multiply the 48 in. by a factor of 1, the magnitude of the quantity remains the same:
48 in. 48 in.(1) (1.10)
Let us now look at the conversion factor, which is the following for this example:
1 m 39.37 in.
Dividing both sides of the conversion factor by 39.37 in. will result in the following format:
(1)
1 m 39.37 in.
0.002000 2000 6
0.002 103 m
7
10 3 m
Reduce by 6
Increase by 6
0.010 10 3
S   I
20  INTRODUCTION
Note that the end result is that the ratio 1 m/39.37 in. equals 1, as it should since they are equal quantities. If we now substitute this factor (1) into Eq. (1.10), we obtain 48 in.(1) 48 in.
which results in the cancellation of inches as a unit of measure and leaves meters as the unit of measure. In addition, since the 39.37 is in the denominator, it must be divided into the 48 to complete the operation:
m 1.219 m
Let us now review the method, which has the following steps:
1. Set up the conversion factor to form a numerical value of (1) with the unit of measurement to be removed from the original quantity in the denominator. 2. Perform the required mathematics to obtain the proper magnitude for the remaining unit of measurement.
EXAMPLE 1.13
a. Convert 6.8 min to seconds. b. Convert 0.24 m to centimeters.
Solutions:
a. The conversion factor is
1 min 60 s
Since the minute is to be removed as the unit of measurement, it must appear in the denominator of the (1) factor, as follows: Step 1: (1) Step 2: 6.8 min(1) 6.8 min (6.8)(60) s 408 s
b. The conversion factor is
1 m 100 cm
Since the meter is to be removed as the unit of measurement, it must appear in the denominator of the (1) factor as follows: Step 1: 1 Step 2: 0.24 m(1) 0.24 m (0.24)(100) cm 24 cm
The products (1)(1) and (1)(1)(1) are still 1. Using this fact, we can perform a series of conversions in the same operation.
100 cm 1 m
100 cm 1 m
60 s 1 min
60 s 1 min
48 39.37
1 m 39.37 in.
S   I
SYMBOLS  21
EXAMPLE 1.14
a. Determine the number of minutes in half a day. b. Convert 2.2 yards to meters.
Solutions:
a. Working our way through from days to hours to minutes, always ensuring that the unit of measurement to be removed is in the denominator, will result in the following sequence: 0.5 day (0.5)(24)(60) min 720 min
b. Working our way through from yards to feet to inches to meters will result in the following: 2.2 yards m 2.012 m
The following examples are variations of the above in practical situations.
EXAMPLE 1.15
a. In Europe and Canada, and many other locations throughout the world, the speed limit is posted in kilometers per hour. How fast in miles per hour is 100 km/h? b. Determine the speed in miles per hour of a competitor who can run a 4-min mile.
Solutions: a. (1)(1)(1)(1)
62.14 mi/h
Many travelers use 0.6 as a conversion factor to simplify the math involved; that is, (100 km/h)(0.6) 60 mi/h and (60 km/h)(0.6) 36 mi/h
b. Invertingthefactor4min/1mito1mi/4min,wecanproceedasfollows: mi/h 15 mi/h
1.9SYMBOLS
Throughout the text, various symbols will be employed that the reader may not have had occasion to use. Some are defined in Table 1.3, and others will be defined in the text as the need arises.
60 4
60 min h
1 mi 4 min
mi h
(100)(1000)(39.37) (12)(5280)
1 mi 5280 ft
1 ft 12 in.
39.37 in. 1 m
1000 m 1 km
100 km h
100 km h
(2.2)(3)(12) 39.37
1 m 39.37 in.
12 in. 1 ft
3 ft 1 yard
60 min 1 h
24 h 1 day
TABLE 1.3
Symbol Meaning
Not equal to 6.12 6.13 > Greater than 4.78 > 4.20 k Much greater than 840 k 16 < Less than 430 < 540 K Much less than 0.002 K 46 ≥ Greater than or equal to x ≥ y is satisfied for y 3 and x > 3 or x 3 ≤ Less than or equal to x ≤ y is satisfied for y 3 and x < 3 or x 3 Approximately equal to 3.14159 3.14 Σ Sum of Σ (4 6 8) 18 || Absolute magnitude of |a| 4, where a 4 or 4 ∴ Therefore x 4 ∴ x 2 By definition Establishes a relationship between two or more quantities
S   I
22  INTRODUCTION
1.10 CONVERSION TABLES
Conversion tables such as those appearing in Appendix B can be very useful when time does not permit the application of methods described in this chapter. However, even though such tables appear easy to use, frequent errors occur because the operations appearing at the head of the table are not performed properly. In any case, when using such tables, try to establish mentally some order of magnitude for the quantity to be determined compared to the magnitude of the quantity in its original set of units. This simple operation should prevent several impossible results that may occur if the conversion operation is improperly applied. For example, consider the following from such a conversion table:
A conversion of 2.5 mi to meters would require that we multiply 2.5 by the conversion factor; that is, 2.5 mi(1.609 103) 4.0225 103 m
A conversion from 4000 m to miles would require a division process:
2486.02 10 3 2.48602 mi
In each of the above, there should have been little difficulty realizing that 2.5 mi would convert to a few thousand meters and 4000 m would be only a few miles. As indicated above, this kind of anticipatory thinking will eliminate the possibility of ridiculous conversion results.
1.11 CALCULATORS
In some texts, the calculator is not discussed in detail. Instead, students are left with the general exercise of choosing an appropriate calculator and learning to use it properly on their own. However, some discussion about the use of the calculator must be included to eliminate some of the impossible results obtained (and often strongly defended by the user—because the calculator says so) through a correct understanding of the process by which a calculator performs the various tasks. Time and space do not permit a detailed explanation of all the possible operations, but it is assumed that the following discussion will enlighten the user to the fact that it is important to understand the manner in which a calculator proceeds with a calculation and not to expect the unit to accept data in any form and always generate the correct answer. When choosing a calculator (scientific for our use), be absolutely sure that it has the ability to operate on complex numbers (polar and rectangular) which will be described in detail in Chapter 13. For now simply look up the terms in the index of the operator’s manual, and be sure that the terms appear and that the basic operations with them are discussed. Next, be aware that some calculators perform the operations with a minimum number of steps while others can require a downright lengthy or complex series of steps. Speak to your instructor if unsure about your purchase. For this text, the TI-86 of Fig. 1.5 was chosen because of its treatment of complex numbers.
4000 m 1.609 103
Multiply by 1.609 103
To Meters
To convert from Miles
FIG. 1.5 Texas Instruments TI-86 calculator. (Courtesy of Texas Instruments, Inc.)
S   I
CALCULATORS  23
Initial Settings
Format and accuracy are the first two settings that must be made on any scientific calculator. For most calculators the choices of formats are Normal, Scientific, and Engineering. For the TI-86 calculator, pressing the 2nd function (yellow) key followed by the key will provide a list of options for the initial settings of the calculator. For calculators without a choice, consult the operator’s manual for the manner in which the format and accuracy level are set. Examples of each are shown below:
Normal: 1/3 0.33 Scientific: 1/3 3.33E 1 Engineering: 1/3 333.33E 3
Note that the Normal format simply places the decimal point in the most logical location. The Scientific ensures that the number preceding the decimal point is a single digit followed by the required power of ten. The Engineering format will always ensure that the power of ten is a multiple of 3 (whether it be positive, negative, or zero). In the above examples the accuracy was hundredths place. To set this accuracy for the TI-86 calculator, return to the selection and choose 2 to represent two-place accuracy or hundredths place. Initially you will probably be most comfortable with the Normal mode with hundredths-place accuracy. However, as you begin to analyze networks, you may find the Engineering mode more appropriate since you will be working with component levels and results that have powers of ten that have been assigned abbreviations and names. Then again, the Scientific mode may the best choice for a particular analysis. In any event, take the time now to become familiar with the differences between the various modes, and learn how to set them on your calculator.
Order of Operations
Although being able to set the format and accuracy is important, these features are not the source of the impossible results that often arise because of improper use of the calculator. Improper results occur primarily because users fail to realize that no matter how simple or complex an equation, the calculator will perform the required operations in a specific order. For instance, the operation
3 8
1
is often entered as
8 3 1 2.67 1 3.67
which is totally incorrect (2 is the answer). The user must be aware that the calculator will not perform the addition first and then the division. In fact, addition and subtraction are the last operations to be performed in any equation. It is therefore very important that the reader carefully study and thoroughly understand the next few paragraphs in order to use the calculator properly.
1. The first operations to be performed by a calculator can be set using parentheses ( ). It does not matter which operations are within
8 3
1
MODE
MODE
MODE
the parentheses. The parentheses simply dictate that this part of the equation is to be determined first. There is no limit to the number of parentheses in each equation—all operations within parentheses will be performed first. For instance, for the example above, if parentheses are added as shown below, the addition will be performed first and the correct answer obtained:
(3 8
1) 8 4 2 2. Next, powers and roots are performed, such as x2, x , and so on. 3. Negation (applying a negative sign to a quantity) and single-key operations such as sin, tan 1, and so on, are performed. 4. Multiplication and division are then performed. 5. Addition and subtraction are performed last.
It may take a few moments and some repetition to remember the order, but at least you are now aware that there is an order to the operations and are aware that ignoring them can result in meaningless results.
EXAMPLE 1.16 a. Determine
9 3
b. Find
3 4
9
c. Determine
1 4 1 6 2 3
Solutions:
a. The following calculator operations will result in an incorrect answer of 1 because the square-root operation will be performed before the division.
3
9 3 3 1
However, recognizing that we must first divide 9 by 3, we can use parentheses as follows to define this operation as the first to be performed, and the correct answer will be obtained: 9 3 3 1.67
b. If the problem is entered as it appears, the incorrect answer of 5.25 will result.
3 9 4 5.25
Using brackets to ensure that the addition takes place before the division will result in the correct answer as shown below:
(3 4
9) 1 4 2 343
( )9
493

93
√ ()
9
√ 3
3
( ) 18
S   I
24  INTRODUCTION
c. Since the division will occur first, the correct result will be obtained by simply performing the operations as indicated. That is,
1 4 1 6 2 3 1.08
1.12 COMPUTER ANALYSIS
The use of computers in the educational process has grown exponentially in the past decade. Very few texts at this introductory level fail to include some discussion of current popular computer techniques. In fact, the very accreditation of a technology program may be a function of the depth to which computer methods are incorporated in the program. There is no question that a basic knowledge of computer methods is something that the graduating student should carry away from a twoyear or four-year program. Industry is now expecting students to have a basic knowledge of computer jargon and some hands-on experience. For some students, the thought of having to become proficient in the use of a computer may result in an insecure, uncomfortable feeling. Be assured, however, that through the proper learning experience and exposure, the computer can become a very “friendly,” useful, and supportive tool in the development and application of your technical skills in a professional environment. For the new student of computers, two general directions can be taken to develop the necessary computer skills: the study of computer languages or the use of software packages.
Languages
There are several languages that provide a direct line of communication with the computer and the operations it can perform. A language is a set of symbols, letters, words, or statements that the user can enter into the computer. The computer system will “understand” these entries and will perform them in the order established by a series of commands called a program. The program tells the computer what to do on a sequential, line-by-line basis in the same order a student would perform the calculations in longhand. The computer can respond only to the commands entered by the user. This requires that the programmer understand fully the sequence of operations and calculations required to obtain a particular solution. In other words, the computer can only respond to the user’s input—it does not have some mysterious way of providing solutions unless told how to obtain those solutions. A lengthy analysis can result in a program having hundreds or thousands of lines. Once written, the program has to be checked carefully to be sure the results have meaning and are valid for an expected range of input variables. Writing a program can, therefore, be a long, tedious process, but keep in mind that once the program has been tested and proven true, it can be stored in memory for future use. The user can be assured that any future results obtained have a high degree of accuracy but require a minimum expenditure of energy and time. Some of the popular languages applied in the electrical/electronics field today include C , QBASIC, Pascal, and FORTRAN. Each has its own set of commands and statements to communicate with the computer, but each can be used to perform the same type of analysis.
4
6
1 3
21
COMPUTER ANALYSIS  25 S   I
S   I
26  INTRODUCTION
This text includes C in its development because of its growing popularity in the educational community. The C language was first developed at Bell Laboratories to establish an efficient communication link between the user and the machine language of the central processing unit (CPU) of a computer. The language has grown in popularity throughout industry and education because it has the characteristics of a high-level language (easily understood by the user) with an efficient link to the computer’s operating system. The C language was introduced as an extension of the C language to assist in the writing of complex programs using an enhanced, modular, top-down approach. In any event, it is not assumed that the coverage of C in this text is sufficient to permit the writing of additional programs. The inclusion is meant as an introduction only: to reveal the appearance and characteristics of the language, and to follow the development of some simple programs. A proper exposure to C would require a course in itself, or at least a comprehensive supplemental program to fill in the many gaps of this text’s presentation.
Software Packages
The second approach to computer analysis—software packages— avoids the need to know a particular language; in fact, the user may not be aware of which language was used to write the programs within the package. All that is required is a knowledge of how to input the network parameters, define the operations to be performed, and extract the results; the package will do the rest. The individual steps toward a solution are beyond the needs of the user—all the user needs is an idea of how to get the network parameters into the computer and how to extract the results. Herein lie two of the concerns of the author with packaged programs—obtaining a solution without the faintest idea of either how the solution was obtained or whether the results are valid or way off base. It is imperative that the student realize that the computer should be used as a tool to assist the user—it must not be allowed to control the scope and potential of the user! Therefore, as we progress through the chapters of the text, be sure that concepts are clearly understood before turning to the computer for support and efficiency. Each software package has a menu, which defines the range of application of the package. Once the software is entered into the computer, the system will perform all the functions appearing in the menu, as it was preprogrammed to do. Be aware, however, that if a particular type of analysis is requested that is not on the menu, the software package cannot provide the desired results. The package is limited solely to those maneuvers developed by the team of programmers who developed the software package. In such situations the user must turn to another software package or write a program using one of the languages listed above. In broad terms, if a software package is available to perform a particular analysis, then it should be used rather than developing routines. Most popular software packages are the result of many hours of effort by teams of programmers with years of experience. However, if the results are not in the desired format, or if the software package does not provide all the desired results, then the user’s innovative talents should be put to use to develop a software package. As noted above, any program the user writes that passes the tests of range and accuracy can be considered a software package of his or her authorship for future use.
26  INTRODUCTION S   I
Three software packages will be used throughout this text: Cadence’s OrCAD PSpice 9.2, Electronics Workbench’s Multisim, and MathSoft’s Mathcad 2000, all of which appear in Fig. 1.6. Although PSpice and Electronics Workbench are both designed to analyze electric circuits, there are sufficient differences between the two to warrant covering each approach separately. The growing use of some form of mathematical support in the educational and industrial environment justifies the introduction and use of Mathcad throughout the text. There is no requirement that the student obtain all three to proceed with the content of this text. The primary reason for their inclusion was simply to introduce each and demonstrate how they can support the learning process. In most cases, sufficient detail has been provided to actually use the software package to perform the examples provided, although it would certainly be helpful to have someone to turn to if questions arise. In addition, the literature supporting all three packages has improved dramatically in recent years and should be available through your bookstore or a major publisher. Appendix A lists all the system requirements, including how to get in touch with each company.
PROBLEMS  27 S   I
FIG. 1.6 Software packages: (a) Cadence’s OrCAD (PSpice) release 9.2; (b) Electronics Workbench’s Multisim; (c) MathSoft’s Mathcad 2000.
PROBLEMS
Note: More difficult problems are denoted by an asterisk (*) throughout the text.
SECTION 1.2 A Brief History
1. Visit your local library (at school or home) and describe the extent to which it provides literature and computer support for the technologies—in particular, electricity, electronics, electromagnetics, and computers.
2. Choose an area of particular interest in this field and write a very brief report on the history of the subject.
3. Choose an individual of particular importance in this field and write a very brief review of his or her life and important contributions.
SECTION 1.3 Units of Measurement
4. Determine the distance in feet traveled by a car moving at 50 mi/h for 1 min.
5. How many hours would it take a person to walk 12 mi if the average pace is 15 min/mile?
SECTION 1.4 Systems of Units
6. Are there any relative advantages associated with the metric system compared to the English system with
(a)
(b)
(c)
18. Perform the following operations and express your answer as a power of ten:
a. b.
c. d.
19. Perform the following operations and express your answer as a power of ten: a. (100)3 b. (0.0001)1/2 c. (10,000)8 d. (0.00000010)9
20. Perform the following operations and express your answer as a power of ten: a. (2.2 103)3 b. (0.0006 102)4 c. (0.004)(6 102)2 d. ((2 10 3)(0.8 104)(0.003 105))3
21. Perform the following operations and express your answer in scientific notation: a. ( 0.001)2 b.
c. d.
e. *f.
*22. Perform the following operations and express your answer in engineering notation: a. b. [(40,000)2][(20) 3]
c. d.
e.
f. [(0.000016)1/2][(100,000)5][0.02]
g. (a challenge)
SECTION 1.7 Conversion between Levels of Powers of Ten
23. Fill in the blanks of the following conversions: a. 6 103 ___ 106 b. 4 10 4 ___ 10 6 c. 50 105 ___ 103 ___ 106 ___ 109 d. 30 10 8 ___ 10 3 ___ 10 6 ___ 10 9
24. Perform the following conversions: a. 2000 ms to milliseconds b. 0.04 ms to microseconds c. 0.06 mF to nanofarads d. 8400 ps to microseconds e. 0.006 km to millimeters f. 260 103 mm to kilometers
[(0.003)3][(0.00007)2][(800)2] [(100)(0.0009)]1/2
[(4000)2][300] 0.02
(0.000027)1/3 210,000
(60,000)2 (0.02)2
(300)2(100) 104
[(100)(0.01)] 3 [(100)2][0.001]
(0.0001)3(100) 1,000,000
(102)(10,000) 0.001
(0.001)2(100) 10,000
(100)(10 4) 10
78 109 4 10 6
0.000215 0.00005
0.00408 60,000
2000 0.00008
28  INTRODUCTION
respect to length, mass, force, and temperature? If so, explain.
7. Which of the four systems of units appearing in Table 1.1 has the smallest units for length, mass, and force? When would this system be used most effectively?
*8. Which system of Table 1.1 is closest in definition to the SI system? How are the two systems different? Why do you think the units of measurement for the SI system were chosen as listed in Table 1.1? Give the best reasons you can without referencing additional literature.
9. What is room temperature (68°F) in the MKS, CGS, and SI systems?
10. How many foot-pounds of energy are associated with 1000 J? 11. How many centimeters are there in 1⁄2 yd?
SECTION 1.6 Powers of Ten
12. Express the following numbers as powers of ten: a. 10,000 b. 0.0001 c. 1000 d. 1,000,000 e. 0.0000001 f. 0.00001
13. Using only those powers of ten listed in Table 1.2, express the following numbers in what seems to you the most logical form for future calculations: a. 15,000 b. 0.03000 c. 7,400,000 d. 0.0000068 e. 0.00040200 f. 0.0000000002
14. Perform the following operations and express your answer as a power of ten: a. 4200 6,800,000 b. 9 104 3.6 103 c. 0.5 10 3 6 10 5 d. 1.2 103 50,000 10 3 0.006 105
15. Perform the following operations and express your answer as a power of ten: a. (100)(100) b. (0.01)(1000) c. (103)(106) d. (1000)(0.00001) e. (10 6)(10,000,000) f. (10,000)(10 8)(1035)
16. Perform the following operations and express your answer as a power of ten: a. (50,000)(0.0003) b. 2200 0.08 c. (0.000082)(0.00007) d. (30 10 4)(0.0002)(7 108)
17. Perform the following operations and express your answer as a power of ten:
a. b.
c. d.
e. f.
(100)1/2 0.01
1038 0.000100
0.0000001 100
10,000 0.00001
0.01 100
100 1000
S   I
GLOSSARY  29
SECTION 1.8 Conversion within and between Systems of Units
For Problems 25 to 27, convert the following:
25. a. 1.5 min to seconds b. 0.04 h to seconds c. 0.05 s to microseconds d. 0.16 m to millimeters e. 0.00000012 s to nanoseconds f. 3,620,000 s to days g. 1020 mm to meters 26. a. 0.1 mF (microfarad) to picofarads b. 0.467 km to meters c. 63.9 mm to centimeters d. 69 cm to kilometers e. 3.2 h to milliseconds f. 0.016 mm to micrometers g. 60 sq cm (cm2) to square meters (m2)
*27. a. 100 in. to meters b. 4 ft to meters c. 6 lb to newtons d. 60,000 dyn to pounds e. 150,000 cm to feet f. 0.002 mi to meters (5280 ft 1 mi) g. 7800 m to yards
28. What is a mile in feet, yards, meters, and kilometers?
29. Calculate the speed of light in miles per hour using the defined speed of Section 1.4.
30. Find the velocity in miles per hour of a mass that travels 50 ft in 20 s.
31. How long in seconds will it take a car traveling at 100 mi/h to travel the length of a football field (100 yd)?
32. Convert 6 mi/h to meters per second.
33. If an athlete can row at a rate of 50 m/min, how many days would it take to cross the Atlantic ( 3000 mi)? 34. How long would it take a runner to complete a 10-km race if a pace of 6.5 min/mi were maintained?
35. Quarters are about 1 in. in diameter. How many would be required to stretch from one end of a football field to the other (100 yd)?
36. Compare the total time in hours to cross the United States ( 3000 mi) at an average speed of 55 mi/h versus an average speed of 65 mi/h.What is your reaction to the total time required versus the safety factor?
*37. Find the distance in meters that a mass traveling at 600 cm/s will cover in 0.016 h.
*38. Each spring there is a race up 86 floors of the 102-story Empire State Building in New York City. If you were able to climb 2 steps/second, how long would it take you to reach the 86th floor if each floor is 14 ft. high and each step is about 9 in.?
*39. The record for the race in Problem 38 is 10 minutes, 47 seconds. What was the racer’s speed in min/mi for the race?
*40. If the race of Problem 38 were a horizontal distance, how long would it take a runner who can run 5-minute miles to cover the distance? Compare this with the record speed of Problem 39. Gravity is certainly a factor to be reckoned with!
SECTION 1.10 Conversion Tables
41. Using Appendix B, determine the number of a. Btu in 5 J of energy. b. cubic meters in 24 oz of a liquid. c. seconds in 1.4 days. d. pints in 1 m3 of a liquid.
SECTION 1.11 Calculators
Perform the following operations using a calculator: 42. 6(4 8) 43. 3 2 4 2 44. tan 1 4 3 45. 6 24 0 01 0
SECTION 1.12 Computer Analysis
46. Investigate the availability of computer courses and computer time in your curriculum. Which languages are commonly used, and which software packages are popular?
47. Develop a list of five popular computer languages with a few characteristics of each. Why do you think some languages are better for the analysis of electric circuits than others?
GLOSSARY
C A computer language having an efficient communication link between the user and the machine language of the central processing unit (CPU) of a computer. CGS system The system of units employing the Centimeter, Gram, and Second as its fundamental units of measure. Difference engine One of the first mechanical calculators. Edison effect Establishing a flow of charge between two elements in an evacuated tube.
Electromagnetism The relationship between magnetic and electrical effects. Engineering notation A method of notation that specifies that all powers of ten used to define a number be multiples of 3 with a mantissa greater than or equal to 1 but less than 1000. ENIAC The first totally electronic computer.
S   I
MKS system The system of units employing the Meter, Kilogram, and Second as its fundamental units of measure. Newton (N) A unit of measurement for force in the SI and MKS systems. Equal to 100,000 dynes in the CGS system. Pound (lb) A unit of measurement for force in the English system. Equal to 4.45 newtons in the SI or MKS system. Program A sequential list of commands, instructions, etc., to perform a specified task using a computer. PSpice A software package designed to analyze various dc, ac, and transient electrical and electronic systems. Scientific notation A method for describing very large and very small numbers through the use of powers of ten, which requires that the multiplier be a number between 1 and 10. Second (s) A unit of measurement for time in the SI, MKS, English, and CGS systems. SI system The system of units adopted by the IEEE in 1965 and the USASI in 1967 as the International System of Units (Système International d’Unités). Slug A unit of measure for mass in the English system. Equal to 14.6 kilograms in the SI or MKS system. Software package A computer program designed to perform specific analysis and design operations or generate results in a particular format. Static electricity Stationary charge in a state of equilibrium. Transistor The first semiconductor amplifier. Voltaic cell A storage device that converts chemical to electrical energy.
30  INTRODUCTION
Fixed-point notation Notation using a decimal point in a particular location to define the magnitude of a number. Fleming’s valve The first of the electronic devices, the diode. Floating-point notation Notation that allows the magnitude of a number to define where the decimal point should be placed. Integrated circuit (IC) A subminiature structure containing a vast number of electronic devices designed to perform a particular set of functions. Joule (J) A unit of measurement for energy in the SI or MKS system. Equal to 0.7378 foot-pound in the English system and 107 ergs in the CGS system. Kelvin (K) A unit of measurement for temperature in the SI system. Equal to 273.15 °C in the MKS and CGS systems. Kilogram (kg) A unit of measure for mass in the SI and MKS systems. Equal to 1000 grams in the CGS system. Language A communication link between user and computer to define the operations to be performed and the results to be displayed or printed. Leyden jar One of the first charge-storage devices. Menu A computer-generated list of choices for the user to determine the next operation to be performed. Meter (m) A unit of measure for length in the SI and MKS systems. Equal to 1.094 yards in the English system and 100 centimeters in the CGS system.