FUNDAMENTALS OF ELECTRIC SYSTEMS CAPACITORS Figure 1.1 illustrates a capacitor. It consists of two insulated conductors a and b. They carry equal and opposite charges ϩq and Ϫq, respectively. All lines of force that originate on a terminate on b. The capacitor is characterized by the following parameters: ● q, the magnitude of the charge on each conductor ● V, the potential difference between the conductors The parameters q and V are proportional to each other in a capacitor, or q ϭ CV, where C is the constant of proportionality. It is called the capacitance of the capacitor. The capac- itance depends on the following parameters: ● Shape of the conductors ● Relative position of the conductors ● Medium that separates the conductors The unit of capacitance is the coulomb/volt (C/V) or farad (F). Thus 1 F ϭ 1 C/V It is important to note that ϭ C but since ϭ i Thus, i ϭ C This means that the current in a capacitor is proportional to the rate of change of the voltage with time. dV ᎏ dt dq ᎏ dt dV ᎏ dt dq ᎏ dt CHAPTER 1 1.1 Source: ELECTRICAL EQUIPMENT HANDBOOK Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. In industry, the following submultiples of farad are used: ● Microfarad (1 F ϭ 10 Ϫ6 F) ● Picofarad (1 pF ϭ 10 Ϫ12 F) Capacitors are very useful electric devices. They are used in the following applications: ● To store energy in an electric field. The energy is stored between the conductors, which are normally called plates. The electric energy stored in the capacitor is given by U E ϭ ● To reduce voltage fluctuations in electronic power supplies ● To transmit pulsed signals ● To generate or detect electromagnetic oscillations at radio frequencies ● To provide electronic time delays Figure 1.2 illustrates a parallel-plate capacitor in which the conductors are two parallel plates of area A separated by a distance d. If each plate is connected momentarily to the ter- minals of the battery, a charge ϩq will appear on one plate and a charge Ϫq on the other. If d is relatively small, the electric field E between the plates will be uniform. The capacitance of a capacitor increases when a dielectric (insulation) is placed between the plates. The dielectric constant of a material is the ratio of the capacitance with q 2 ᎏ C 1 ᎏ 2 1.2 CHAPTER ONE FIGURE 1.1 Two insulated conductors, totally isolated from their surroundings and carrying equal and opposite charges, form a capacitor. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS dielectric to that without dielectric. Table 1.1 illustrates the dielectric constant and dielec- tric strength of various materials. The high dielectric strength of vacuum (∞, infinity) should be noted. It indicates that if two plates are separated by vacuum, the voltage difference between them can reach infin- ity without having flashover (arcing) between the plates. This important characteristic of vacuum has led to the development of vacuum circuit breakers, which have proved to have excellent performance in modern industry. FUNDAMENTALS OF ELECTRIC SYSTEMS 1.3 FIGURE 1.2 A parallel-plate capacitor with conductors (plates) of area A. TABLE 1.1 Properties of Some Dielectrics* Dielectric Dielectric strength, † Material constant kV/mm Vacuum 1.00000 ∞ Air 1.00054 0.8 Water 78 — Paper 3.5 14 Ruby mica 5.4 160 Porcelain 6.5 4 Fused quartz 3.8 8 Pyrex glass 4.5 13 Bakelite 4.8 12 Polyethylene 2.3 50 Amber 2.7 90 Polystyrene 2.6 25 Teflon 2.1 60 Neoprene 6.9 12 Transformer oil 4.5 12 Titanium dioxide 100 6 *These properties are at approximately room temperature and for conditions such that the electric field E in the dielectric does not vary with time. † This is the maximum potential gradient that may exist in the dielectric without the occurrence of electrical break- down. Dielectrics are often placed between conducting plates to permit a higher potential difference to be applied between them than would be possible with air as the dielectric. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS CURRENT AND RESISTANCE The electric current i is established in a conductor when a net charge q passes through it in time t. Thus, the current is i ϭ The units for the parameters are ● i: amperes (A) ● q: coulombs (C) ● t: seconds (s) The electric field exerts a force on the electrons to move them through the conductor. A positive charge moving in one direction has the same effect as a negative charge moving in the opposite direction. Thus, for simplicity we assume that all charge carriers are positive. We draw the current arrows in the direction that positive charges flow (Fig. 1.3). A conductor is characterized by its resistance (symbol ). It is defined as the voltage difference between two points divided by the current flowing through the con- ductor. Thus, R ϭ where V is in volts, i is in amperes, and the resistance R is in ohms (abbreviated ⍀). The current, which is the flow of charge through a conductor, is often compared to the flow of water through a pipe. The water flow occurs due to the pressure difference between the inlet and outlet of a pipe. Similarly, the charge flows through the conductor due to the voltage difference. The resistivity is a characteristic of the conductor material. It is a measure of the resistance that the material has to the current. For example, the resistivity of copper is 1.7 ϫ 10 Ϫ8 ⍀иm; that of fused quartz is about 10 16 ⍀иm. Table 1.2 lists some electrical properties of common metals. The temperature coefficient of resistivity ␣ is given by ␣ϭ d ᎏ dT 1 ᎏ V ᎏ i q ᎏ t 1.4 CHAPTER ONE FIGURE 1.3 Electrons drift in a direction opposite to the electric field in a conductor. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS It represents the rate of variation of resistivity with temperature. Its units are 1/°C (or 1/°F). Conductivity (), is used more commonly than resistivity. It is the inverse of conductivity, given by ϭ The units for conductivity are (⍀иm) Ϫ1 . Across a resistor, the voltage and current have this relationship: V ϭ iR The power dissipated across the resistor (conversion of electric energy to heat) is given by P ϭ i 2 R or P ϭ where P is in watts, i in amperes, V in volts, and R in ohms. V 2 ᎏ R 1 ᎏ FUNDAMENTALS OF ELECTRIC SYSTEMS 1.5 TABLE 1.2 Properties of Metals as Conductors Temperature coefficient Resistivity of resistivity (at 20°C), ␣, per C° Metal 10 Ϫ8 ⍀иm(ϫ 10 Ϫ5 ) * Silver 1.6 380 Copper 1.7 390 Aluminum 2.8 390 Tungsten 5.6 450 Nickel 6.8 600 Iron 10 500 Steel 18 300 Manganin 44 1.0 Carbon † 3500 Ϫ50 * This quantity, defined from ␣ ϭ is the fractional change in resistivity d/ per unit change in temperature. It varies with temperature, the values here referring to 20°C. For copper (␣ϭ3.9 ϫ 10 Ϫ3 /°C) the resistivity increases by 0.39 percent for a temperature increase of 1°C near 20°C. Note that ␣ for carbon is negative, which means that the resistivity decreases with increasing temperature. † Carbon, not strictly a metal, is included for com- parison. d ᎏ dT 1 ᎏ Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS THE MAGNETIC FIELD A magnetic field is defined as the space around a magnet or a current-carrying conductor. The magnetic field B is represented by lines of induction. Figure 1.4 illustrates the lines of induction of a magnetic field B near a long current-carrying conductor. The vector of the magnetic field is related to its lines of induction in this way: 1. The direction of B at any point is given by the tangent to the line of induction. 2. The number of lines of induction per unit cross-sectional area (perpendicular to the lines) is proportional to the magnitude of B. Magnetic field B is large if the lines are close together, and it is small if they are far apart. The flux ⌽ B of magnetic field B is given by ⌽ B ϭ ͵B и dS The integral is taken over the surface for which ⌽ B is defined. The magnetic field exerts a force on any charge moving through it. If q 0 is a positive charge moving at a velocity v in a magnetic field B, the force F acting on the charge (Fig. 1.5) is given by F ϭ q 0 v ϫ B The magnitude of the force F is given by F ϭ q 0 vB sin where is the angle between v and B. 1.6 CHAPTER ONE FIGURE 1.4 Lines of B near a long, circular cylindrical wire. A current i, suggested by the central dot, emerges from the page. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS The force F will always be at a right angle to the plane formed by v and B. Thus, it will always be a sideways force. The force will disappear in these cases: 1. If the charge stops moving 2. If v is parallel or antiparallel to the direction of B The force F has a maximum value if v is at a right angle to B ( ϭ 90°). Figure 1.6 illustrates the force created on a positive and a negative electron mov- ing in a magnetic field B pointing out of the plane of the figure (symbol ᭪). The unit of B is the tesla (T) or weber per square meter (Wb/m 2 ). Thus 1 tesla (T) ϭ 1 weber/meter 2 ϭ The force acting on a current-carrying conductor placed at a right angle to a magnetic field B (Fig. 1.7) is given by F ϭ ilB where l is the length of conductor placed in the magnetic field. Ampère’s Law Figure 1.8 illustrates a current-carrying conductor surrounded by small magnets. If there is no current in the conductor, all the magnets will be aligned with the horizontal component 1 N ᎏ A и m FUNDAMENTALS OF ELECTRIC SYSTEMS 1.7 FIGURE 1.5 Illustration of F ϭ q 0 v ϫ B. Test charge q 0 is fired through the origin with velocity v. FIGURE 1.6 A bubble chamber is a device for rendering visible, by means of small bubbles, the tracks of charged par- ticles that pass through the chamber. The figure shows a photograph taken with such a chamber immersed in a magnetic field B and exposed to radiations from a large cyclotronlike accelerator. The curved υ at point P is formed by a positive and a neg- ative electron, which deflect in opposite directions in the magnetic field. The spirals S are tracks of three low-energy electrons. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS of the earth’s magnetic field. When a current flows through the conductor, the orientation of the magnets suggests that the lines of induction of the magnetic field form closed circles around the conductor. This observation is reinforced by the experiment shown in Fig. 1.9. It shows a current-carrying conductor passing through the center of a horizontal glass plate with iron filings on it. Ampère’s law states that ͶB и dl ϭ 0 i where B is the magnetic field, l is the length of the circumference around the wire, i is the current, 0 is the permeability constant ( 0 ϭ 4ϫ10 Ϫ7 T иm/A). The integration is car- ried around the circumference. If the current in the conductor shown in Fig. 1.8 is reverse direction, all the compass needles change their direction as well. Thus, the direction of B near a current-carrying conductor is given by the right-hand-rule: If the current is grasped by the right hand and the thumb points in the direction of the current, the fingers will curl around the wire in the direction B. 1.8 CHAPTER ONE FIGURE 1.7 A wire carrying a current i is placed at right angles to a magnetic field B. Only the drift velocity of the electrons, not their random motion, is suggested. FIGURE 1.8 An array of compass needles near a central wire carrying a strong current. The black ends of the compass needles are their north poles. The central dot shows the current emerging from the page. As usual, the direction of the current is taken as the direction of flow of positive charge. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS FUNDAMENTALS OF ELECTRIC SYSTEMS 1.9 FIGURE 1.9 Iron filings around a wire carrying a strong current. Magnetic Field in a Solenoid A solenoid (an inductor) is a long, current-carrying conductor wound in a close-packed helix. Figure 1.10 shows a “solenoid” having widely spaced turns. The fields cancel between the wires. Inside the solenoid, B is parallel to the solenoid axis. Figure 1.11 shows the lines of B for a real solenoid. By applying Ampere’s law to this solenoid, we have B ϭ 0 in where n is the number of turns per unit length. The flux ⌽ B for the magnetic field B will become ⌽ B ϭ B и A FARADAY’S LAW OF INDUCTION Faraday’s law of induction is one of the basic equations of electromagnetism. Figure 1.12 shows a coil connected to a galvanometer. If a bar magnet is pushed toward the coil, the galvanometer deflects. This indicates that a current has been induced in the coil. If the mag- net is held stationary with respect to the coil, the galvanometer does not deflect. If the magnet is moved away from the coil, the galvanometer deflects in the opposite direction. This indi- cates that the current induced in the coil is in the opposite direction. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS Figure 1.13 shows another experiment in which when the switch S is closed, thus estab- lishing a steady current in the right-hand coil, the galvanometer deflects momentarily. When the switch is opened, the galvanometer deflects again momentarily, but in the oppo- site direction. This experiment proves that a voltage known as an electromagnetic force (emf) is induced in the left coil when the current in the right coil changes. 1.10 CHAPTER ONE FIGURE 1.10 A loosely wound solenoid. FIGURE 1.11 A solenoid of finite length. The right end, from which lines of B emerge, behaves as the north pole of a compass needle does. The left end behaves as the south pole. Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS [...]... is subject to the Terms of Use as given at the website FUNDAMENTALS OF ELECTRIC SYSTEMS FUNDAMENTALS OF ELECTRIC SYSTEMS 1.13 B2l2v F1 ϭ ilB sin 90° ϭ ᎏ R The force pulling the loop must do a steady work given by B2l 2v2 P ϭ F1v ϭ ᎏ R Figure 1.16 illustrates a rectangular loop of resistance R, width l, and length a being pulled at constant speed υ through a magnetic field B of thickness d There is... Any use is subject to the Terms of Use as given at the website FUNDAMENTALS OF ELECTRIC SYSTEMS 1.19 FUNDAMENTALS OF ELECTRIC SYSTEMS and XL is called the inductive reactance As for the capacitive reactance, the unit for XL is the ohm Since Ᏹm is the maximum value of VL (ϭ VL,m), we can write iT iL iR R V L VL,m ϭ iL,mXL This indicated that when any alternating current of amplitude im and angular frequency... Companies All rights reserved Any use is subject to the Terms of Use as given at the website FUNDAMENTALS OF ELECTRIC SYSTEMS FUNDAMENTALS OF ELECTRIC SYSTEMS 1.21 The unit of this power is voltamperes (VA) or megavoltamperes (MVA) This power includes the combined effect of the real power and the reactive power All electrical equipment such as transformers, motors, and generators are rated by their apparent... Hoboken, N.J., 1978 2 A S Nasar, Handbook of Electric Machines, McGraw-Hill, New York, 1987 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website FUNDAMENTALS OF ELECTRIC SYSTEMS FUNDAMENTALS OF ELECTRIC SYSTEMS 1.25 FIGURE 1.32 Power in... following equations: Vc ϭ Ᏹm sin t and q Vc ϭ ᎏ C (definition of C) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website FUNDAMENTALS OF ELECTRIC SYSTEMS 1.17 FUNDAMENTALS OF ELECTRIC SYSTEMS VC From these relationships, we have C q ϭ.. .FUNDAMENTALS OF ELECTRIC SYSTEMS FUNDAMENTALS OF ELECTRIC SYSTEMS FIGURE 1.12 Galvanometer G deflects while the magnet is moving with respect to the coil Only their relative motion counts 1.11 FIGURE 1.13 Galvanometer G deflects momentarily when switch S is closed or opened No motion is involved Faraday’s law of induction is given by d⌽B Ᏹ ϭ ϪN ᎏ dt where Ᏹ ϭ emf for voltage N ϭ number of turns... subject to the Terms of Use as given at the website FUNDAMENTALS OF ELECTRIC SYSTEMS FUNDAMENTALS OF ELECTRIC SYSTEMS 1.15 ALTERNATING CURRENT An alternating current (ac) in a circuit establishes a voltage (emf) that varies with time as Ᏹ ϭ Ᏹm sin t where Ᏹm is the maximum emf and ϭ 2, where is the frequency measured in hertz (Hz) This type of emf is established by an ac generator in a power... subject to the Terms of Use as given at the website FUNDAMENTALS OF ELECTRIC SYSTEMS Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Source: ELECTRICAL EQUIPMENT HANDBOOK CHAPTER 2 INTRODUCTION TO MACHINERY PRINCIPLES ELECTRIC MACHINES... rate of variation of angular position with time (units: rad/s or deg/s) fm ϭ angular velocity, expressed in revolutions per second ϭ m/2 ␣ ϭ angular acceleration ϭ d/dt It is the rate of variation of angular velocity with time (units: rad/s2) ϭ torque ϭ (force applied) ϫ (perpendicular distance) Units are newton-meters (Nиm) Newton’s law of rotation: ϭ J␣ where J is the moment of inertia of the... 2.1 shows a rectangular core having a winding of N turns of wire wrapped on one leg of the core If the core is made of ferromagnetic material (such as iron), most of the magnetic field produced by the current will remain inside the core Ampere’s law becomes Hlc ϭ Ni where lc is the mean path length of the core The magnetic field intensity H is a measure of the “effort” that the current is putting out . the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS dielectric to that without dielectric. Table 1.1 illustrates the dielectric constant and dielec- tric strength of various. of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS Faraday’s law of induction is given by Ᏹ ϭϪN where Ᏹ ϭ emf for voltage N ϭ number of turns in coil d⌽ B /dt ϭ rate of change of. Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. FUNDAMENTALS OF ELECTRIC SYSTEMS FUNDAMENTALS OF ELECTRIC SYSTEMS 1.9 FIGURE 1.9 Iron filings around a wire