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Mike Tooley Darren Ashby Robert Pease
1.1 Electrical Fundamentals
This chapter has been designed to provide you with the background knowledge required to help you understand the concepts introduced in the later chapters. If you have studied electrical science, electrical principles, or electronics then you will already be familiar with many of these concepts. If, on the other hand, you are returning to study or are a newcomer to electronics or electrical technology this chapter will help you get up to speed.
1.1.1 Fundamental Units
You will already know that the units that we now use to describe such things as length, mass and time are standardized within the International System of Units (SI). This SI system is based upon the seven fundamental units (see Table 1.1).
1.1.2 Derived Units
All other units are derived from these seven fundamental units. These derived units generally have their own names and those commonly encountered in electrical circuits are summarized in Table 1.2, together with the corresponding physical quantities.
(Note that 0K is equal to 273°C and an interval of 1K is the same as an interval of 1°C.)
If you find the exponent notation shown in Table 1.2 a little confusing, just remember that V ^{1} is simply 1/V, s ^{1} is 1/s, m ^{2} is 1/m ^{2}, and so on.
Example 1.1
The unit of flux density (the tesla) is defined as the magnetic flux per unit area. Express this in terms of the fundamental units.
Solution
The SI unit of flux is the weber (Wb). Area is directly proportional to length squared and, expressed in terms of the fundamental SI units, this is square meters (m^{2}). Dividing the flux (Wb) by the area (m^{2}) gives Wb/m^{2} or Wb m ^{2}. Hence, in terms of the fundamental SI units, the tesla is expressed in Wb m ^{2}.
Example 1.2
The unit of electrical potential, the volt (V), is defined as the difference in potential between two points in a conductor, which when carrying a current of one amp (A), dissipates a power of one watt (W). Express the volt (V) in terms of joules (J) and coulombs (C).
Solution
In terms of the derived units:
Volts = Watts/Amperes = Joules seconds/Amperes
= Joules/Amperes x seconds = Joules/Coulombs
Note that: Watts = Joules/seconds and also that Amperes x seconds = Coulombs. Alternatively, in terms of the symbols used to denote the units:
V = W/A = J s/A = J/As = J/C = JC ^{1}
One volt is equivalent to one joule per coulomb.
1.1.3 Measuring Angles
You might think it strange to be concerned with angles in electrical circuits. The reason is simply that, in analog and AC circuits, signals are based on repetitive waves (often sinusoidal in shape). We can refer to a point on such a wave in one of two basic ways, either in terms of the time from the start of the cycle or in terms of the angle (a cycle starts at 0° and finishes as 360°—see Figure 1.1). In practice, it is often more convenient to use angles rather than time; however, the two methods of measurement are interchangeable and it's important to be able to work in either of these units.
In electrical circuits, angles are measured in either degrees or radians (both of which are strictly dimensionless units). You will doubtless already be familiar with angular measure in degrees where one complete circular revolution is equivalent to an angular change of 360°. The alternative method of measuring angles, the radian, is defined somewhat differently. It is the angle subtended at the center of a circle by an arc having length that is equal to the radius of the circle (see Figure 1.2).
You may sometimes find that you need to convert from radians to degrees, and vice versa. A complete circular revolution is equivalent to a rotation of 360° or 2π radians (note that π is approximately equal to 3.142). Thus, one radian is equivalent to 360/2π degrees (or approximately 57.3°). Try to remember the following rules that will help you to convert angles expressed in degrees to radians and vice versa:
From degrees to radians, divide by 57.3.
From radians to degrees, multiply by 57.3.
Example 1.3
Express a quarter of a cycle revolution in terms of:
(a) degrees;
(b) radians.
Solution
(a) There are 360° in one complete cycle (i.e., one full revolution. Hence, there are (360/4)° or 90° in one quarter of a cycle).
(b) There are 2p radians in one complete cycle. Thus, there are 2p/4 or p/2 radians in one quarter of a cycle.
Example 1.4
Express an angle of 215° in radians.
Solution
To convert from degrees to radians, divide by 57.3. So, 215° is equivalent to 215/57.3 = 3.75 radians.
Example 1.5
Express an angle of 2.5 radians in degrees.
Solution
To convert from radians to degrees, multiply by 57.3. Hence, 2.5 radians is equivalent to 2.5 x 57.3 = 143.25°.
1.1.4 Electrical Units and Symbols
Table 1.3 shows the units and symbols that are commonly encountered in electrical circuits. It is important to get to know these units and also be able to recognize their abbreviations and symbols. You will meet all of these units later in this chapter.
1.1.5 Multiples and Sub-Multiples
Unfortunately, many of the derived units are either too large or too small for convenient everyday use, but we can make life a little easier by using a standard range of multiples and sub-multiples (see Table 1.4).
Example 1.6
An indicator lamp requires a current of 0.075A. Express this in mA.
Solution
You can express the current in mA (rather than in A) by simply moving the decimal point three places to the right. Hence, 0.075A is the same as 75 mA.
Example 1.7
A medium-wave radio transmitter operates on a frequency of 1,495 kHz. Express its frequency in MHz.
Solution
To express the frequency in MHz rather than kHz, we need to move the decimal point three places to the left. Hence, 1,495 kHz is equivalent to 1.495 MHz.
Example 1.8
Express the value of a 27,000 pF in µF.
Solution
To express the value in mF rather than pF we need to move the decimal point six places to the left. Hence, 27,000 pF is equivalent to 0.027 µF (note that we have had to introduce an extra zero before the 2 and after the decimal point).
1.1.6 Exponent Notation
Exponent notation (or scientific notation) is useful when dealing with either very small or very large quantities. It's well worth getting to grips with this notation as it will allow you to simplify quantities before using them in formulae.
Exponents are based on powers of ten. To express a number in exponent notation the number is split into two parts. The first part is usually a number in the range 0.1 to 100 while the second part is a multiplier expressed as a power of ten.
For example, 251.7 can be expressed as 2.517 x 100, i.e., 2.517 x 10^{2}. It can also be expressed as 0.2517 x 1,000, i.e., 0.2517 x 10^{3}. In both cases the exponent is the same as the number of noughts in the multiplier (i.e., 2 in the first case and 3 in the second case). To summarize:
251:7 = 2:517 x 10^{2} = 0:2517 x 10^{3}
As a further example, 0.01825 can be expressed as 1.825/100; that is, 1.825 x 10 ^{2}. It can also be expressed as 18.25/1,000, i.e., 18.25 x 10 ^{3}. Again, the exponent is the same as the number of zeros but the minus sign is used to denote a fractional multiplier. To summarize:
0:01825 = 1:825 x 10 ^{2} = 18:25 x 10 ^{3}
Example 1.9
A current of 7.25 mA flows in a circuit. Express this current in amperes using exponent notation.
Solution
1 mA = 1 x 10 ^{3} A; thus, 7.25 mA = 7:25 x 10 ^{3} A
Example 1.10
A voltage of 3.75 x 10 ^{6}V appears at the input of an amplifier. Express this voltage in (a) V, and (b) mV, using exponent notation.
Solution
(a) 1 x 10 ^{6} V = 1 µV so 3.75 x 10 ^{6} V = 3.75 µV
(b) There are 1,000 µV in 1 mV so we must divide the previous result by 1,000 in order to express the voltage in mV. So 3.75 µV = 0.00375 mV.
1.1.7 Multiplication and Division Using Exponents
Exponent notation really comes into its own when values have to be multiplied or divided. When multiplying two values expressed using exponents, you simply need to add the exponents. Here's an example:
(2 x 10^{2}) x (3 x 10^{6}) = (2 x 3) x 10^{(2+6)} = 6 x 10^{8}
Similarly, when dividing two values which are expressed using exponents, you only need to subtract the exponents. As an example:
(4 x 10^{6}) x (2 x 10^{4}) = 4 2 x 10^{(6 4)} = 2 x 10^{2}
In either case it's important to remember to specify the units, multiples and sub-multiples in which you are working (e.g., A, kσ, mV, µF, etc.).
Example 1.11
A current of 3 mA flows in a resistance of 33 kσ. Determine the voltage dropped across the resistor.
Solution
Voltage is equal to current multiplied by resistance. Thus:
V = I x R = 3 mA x 33 kΩ
(Continues...)
Excerpted from Circuit Design by Darren Ashby Bonnie Baker Stuart Ball J. Crowe Barrie Hayes-Gill Ian Hickman Walt Kester Ron Mancini Ian Grout Robert A. Pease Mike Tooley Tim Williams Peter Wilson Bob Zeidman Copyright © 2008 by Elsevier Inc.. Excerpted by permission of Newnes. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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