Chapter 10. Capacitors. Copyright 2011 by Pearson Education, Inc. publishing as Pearson [imprint] Introductory Circuit Analysis, 12/e Boylestad
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1 Chapter 10 Capacitors
2 OBJECTIVES Become familiar with the basic construction of a capacitor and the factors that affect its ability to store charge on its plates. Be able to determine the transient (time-varying) response of a capacitive network and plot the resulting voltages and currents. Understand the impact of combining capacitors in series or parallel and how to read the nameplate data. Develop some familiarity with the use of computer methods to analyze networks with capacitive elements.
3 INTRODUCTION The capacitor has a significant impact on the types of networks that you will be able to design and analyze. Like the resistor, it is a two-terminal device, but its characteristics are totally different from those of a resistor. In fact, the capacitor displays its true characteristics only when a change in the voltage or current is made in the network.
4 THE ELECTRIC FIELD This electric field is represented by electric flux lines, which are drawn to indicate the strength of the electric field at any point around the charged body. The denser the lines of flux, the stronger is the electric field.
5 THE ELECTRIC FIELD FIG Flux distribution from an isolated positive charge.
6 THE ELECTRIC FIELD
7 THE ELECTRIC FIELD
8 THE ELECTRIC FIELD FIG Determining the force on a unit charge r meters from a charge Q of similar polarity.
9 THE ELECTRIC FIELD FIG Electric flux distributions: (a) opposite charges; (b) like charges.
10 THE ELECTRIC FIELD In general, electric flux lines always extend from a positively charged body to a negatively charged body, always extend or terminate perpendicular to the charged surfaces, and never intersect.
11 CAPACITANCE FIG Fundamental charging circuit.
12 CAPACITANCE Capacitance is a measure of a capacitor s ability to store charge on its plates in other words, its storage capacity. In addition, the higher the capacitance of a capacitor, the greater is the amount of charge stored on the plates for the same applied voltage.
13 CAPACITANCE The relationship connecting the applied voltage, the charge on the plates, and the capacitance level is defined by the following equation:
14 CAPACITANCE Note the fringing that occurs at the edges as the flux lines originating from the points farthest away from the negative plate strive to complete the connection. FIG Electric flux distribution between the plates of a capacitor: (a) including fringing; (b) ideal.
15 CAPACITANCE The electric field strength between the plates is determined by the voltage across the plates and the distance between the plates as follows:
16 CAPACITANCE FIG Effect of a dielectric on the field distribution between the plates of a capacitor: (a) alignment of dipoles in the dielectric; (b) electric field components between the plates of a capacitor with a dielectric present.
17 CAPACITANCE FIG Demonstrating the effect of inserting a dielectric between the plates of a capacitor: (a) air capacitor; (b) dielectric being inserted.
18 CAPACITANCE TABLE 10.1 Relative permittivity (dielectric constant) Σr of various dielectrics.
19 CAPACITANCE TABLE 10.2 Dielectric strength of some dielectric materials.
20 CAPACITORS Capacitor Construction The basic components of a capacitor are: conductive plates, separation, and dielectric. Larger plates permit an increased area for the storage of charge, so the area of the plates should be in the numerator of the defining equation. The smaller the distance between the plates, the larger is the capacitance, so this factor should appear in the numerator of the equation. Finally, since higher levels of permittivity result in higher levels of capacitance, the factor should appear in the numerator of the defining equation.
21 CAPACITORS Capacitor Construction
22 CAPACITORS Capacitor Construction FIG Example 10.2.
23 CAPACITORS Capacitor Construction FIG Air capacitor for Example 10.3.
24 CAPACITORS Types of Capacitors Capacitors, like resistors, can be listed under two general headings: fixed and variable. The symbol for the fixed capacitor appears in Fig (a). FIG Symbols for the capacitor: (a) fixed; (b) variable.
25 CAPACITORS Types of Capacitors Fixed Capacitors Fixed-type capacitors come in all shapes and sizes. However, in general, for the same type of construction and dielectric, the larger the required capacitance, the larger is the physical size of the capacitor.
26 CAPACITORS Types of Capacitors FIG Demonstrating that, in general, for each type of construction, the size of a capacitor increases with the capacitance value: (a) electrolytic; (b) polyesterfilm; (c) tantalum.
27 CAPACITORS Types of Capacitors FIG Three ways to increase the area of a capacitor: (a) rolling; (b) stacking; (c) insertion.
28 CAPACITORS Types of Capacitors FIG Various types of electrolytic capacitors: (a) miniature radial leads; (b) axial leads; (c) flatpack; (d) surface-mount; (e) screw-in terminals.
29 CAPACITORS Types of Capacitors FIG (a) Film/foil polyester radial lead; (b) metalized polyester-film axial lead; (c) surface-mount polyester-film; (d) polypropylene-film, radial lead.
30 CAPACITORS Types of Capacitors FIG Ceramic (disc) capacitor: (a) construction; (b) appearance.
31 CAPACITORS Types of Capacitors FIG Mica capacitors: (a) and (b) surface-mount monolithic chips; (c) highvoltage/temperature mica paper capacitors. [(a) and (b) courtesy of Vishay Intertechnology, Inc.; (c) courtesy of Custom Electronics, Inc.]
32 CAPACITORS Types of Capacitors FIG Dipped capacitors: (a) polarized tantalum; (b) nonpolarized mica. FIG Oil-filled, metallic oval case snubber capacitor (the snubber removes unwanted voltage spikes).
33 CAPACITORS Types of Capacitors Variable Capacitors All the parameters in Eq. (10.11) can be changed to some degree to create a variable capacitor. For example, in Fig (a), the capacitance of the variable air capacitor is changed by turning the shaft at the end of the unit.
34 CAPACITORS Types of Capacitors FIG Variable capacitors: (a) air; (b) air trimmer; (c) ceramic dielectric compression trimmer. [(a) courtesy of James Millen Manufacturing Co.]
35 CAPACITORS Leakage Current and ESR Although we would like to think of capacitors as ideal elements, unfortunately, this is not the case. There is a dc resistance appearing as R s in the equivalent model of Fig due to the resistance introduced by the contacts, the leads, or the plate or foil materials.
36 CAPACITORS Leakage Current and ESR In addition, up to this point, we have assumed that the insulating characteristics of dielectrics prevent any flow of charge between the plates unless the breakdown voltage is exceeded. In reality, however, dielectrics are not perfect insulators, and they do carry a few free electrons in their atomic structure.
37 CAPACITORS Leakage Current and ESR FIG Leakage current: (a) including the leakage resistance in the equivalent model for a capacitor; (b) internal discharge of a capacitor due to the leakage current.
38 CAPACITORS Temperature Effects: ppm Every capacitor is temperature sensitive, with the nameplate capacitance level specified at room temperature. Depending on the type of dielectric, increasing or decreasing temperatures can cause either a drop or a rise in capacitance. If temperature is a concern for a particular application, the manufacturer will provide a temperature plot, such as shown in Fig , or a ppm/ C (parts per million per degree Celsius) rating for the capacitor.
39 CAPACITORS Temperature Effects: ppm FIG Variation of capacitor value with temperature.
40 CAPACITORS Capacitor Labeling Due to the small size of some capacitors, various marking schemes have been adopted to provide the capacitance level, tolerance, and, if possible, working voltage. In general, however, as pointed out above, the size of the capacitor is the first indicator of its value.
41 CAPACITORS Capacitor Labeling FIG Various marking schemes for small capacitors.
42 CAPACITORS Measurement and Testing of Capacitors The capacitance of a capacitor can be read directly using a meter such as the Universal LCR Meter. FIG Digital reading capacitance meter. (Courtesy of B+K Precision.)
43 CAPACITORS Measurement and Testing of Capacitors FIG Checking the dielectric of an electrolytic capacitor.
44 CAPACITORS Standard Capacitor Values The most common capacitors use the same numerical multipliers encountered for resistors. The vast majority are available with 5%, 10%, or 20% tolerances. There are capacitors available, however, with tolerances of 1%, 2%, or 3%, if you are willing to pay the price.
45 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE The placement of charge on the plates of a capacitor does not occur instantaneously. Instead, it occurs over a period of time determined by the components of the network. FIG Basic R-C charging network.
46 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG v C during the charging phase.
47 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG Universal time constant chart.
48 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE TABLE 10.3 Selected values of e -x.
49 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE The factor t, called the time constant of the network, has the units of time, as shown below using some of the basic equations introduced earlier in this text:
50 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG Plotting the equation y C = E(1 e -t/t ) versus time (t).
51 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE
52 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG Demonstrating that a capacitor has the characteristics of an open circuit after the charging phase has passed.
53 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG Revealing the short-circuit equivalent for the capacitor that occurs when the switch is first closed.
54 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE
55 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG Calculator key strokes to determine e -1.2.
56 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG Transient network for Example 10.6.
57 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG v C versus time for the charging network in Fig
58 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG Plotting the waveform in Fig versus time (t).
59 TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG i C and y R for the charging network in Fig
60 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE We now investigate how to discharge a capacitor while exerting some control on how long the discharge time will be. You can, of course, place a lead directly across a capacitor to discharge it very quickly and possibly cause a visible spark. For larger capacitors such those in TV sets, this procedure should not be attempted because of the high voltages involved unless, of course, you are trained in the maneuver.
61 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE FIG (a) Charging network; (b) discharging configuration.
62 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE For the voltage across the capacitor that is decreasing with time, the mathematical expression is:
63 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE FIG y C, i C, and y R for 5t switching between contacts in Fig (a).
64 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE FIG v C and i C for the network in Fig (a) with the values in Example 10.6.
65 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response
66 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG Effect of increasing values of C (with R constant) on the charging curve for v C.
67 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG Network to be analyzed in Example 10.8.
68 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG v C and i C for the network in Fig
69 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG Network to be analyzed in Example FIG The charging phase for the network in Fig
70 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG Network in Fig when the switch is moved to position 2 at t = 1t 1.
71 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG v C for the network in Fig
72 TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG i c for the network in Fig
73 INITIAL CONDITIONS The voltage across the capacitor at this instant is called the initial value, as shown for the general waveform in Fig FIG Defining the regions associated with a transient response.
74 INITIAL CONDITIONS FIG Example
75 INITIAL CONDITIONS FIG v C and i C for the network in Fig
76 INITIAL CONDITIONS FIG Defining the parameters in Eq. (10.21) for the discharge phase.
77 INSTANTANEOUS VALUES Occasionally, you may need to determine the voltage or current at a particular instant of time that is not an integral multiple of t. FIG Key strokes to determine (2 ms)(log e 2) using the TI-89 calculator.
78 THÉVENIN EQUIVALENT: t =R Th C You may encounter instances in which the network does not have the simple series form in Fig You then need to find the Thévenin equivalent circuit for the network external to the capacitive element.
79 THÉVENIN EQUIVALENT: t =R Th C FIG Example
80 THÉVENIN EQUIVALENT: t =R Th C FIG Applying Thévenin s theorem to the network in Fig
81 THÉVENIN EQUIVALENT: t =R Th C FIG Substituting the Thévenin equivalent for the network in Fig
82 THÉVENIN EQUIVALENT: t =R Th C FIG The resulting waveforms for the network in Fig
83 THÉVENIN EQUIVALENT: t =R Th C FIG Example FIG Network in Fig redrawn.
84 THÉVENIN EQUIVALENT: t =R Th C FIG yc for the network in Fig
85 THÉVENIN EQUIVALENT: t =R Th C FIG Example
86 THE CURRENT i C There is a very special relationship between the current of a capacitor and the voltage across it. For the resistor, it is defined by Ohm s law: i R = v R /R. The current through and the voltage across the resistor are related by a constant R a very simple direct linear relationship. For the capacitor, it is the more complex relationship defined by:
87 THE CURRENT i C FIG v C for Example
88 THE CURRENT i C FIG The resulting current i C for the applied voltage in Fig
89 CAPACITORS IN SERIES AND IN PARALLEL Capacitors, like resistors, can be placed in series and in parallel. Increasing levels of capacitance can be obtained by placing capacitors in parallel, while decreasing levels can be obtained by placing capacitors in series.
90 CAPACITORS IN SERIES AND IN PARALLEL FIG Series capacitors.
91 CAPACITORS IN SERIES AND IN PARALLEL FIG Parallel capacitors.
92 CAPACITORS IN SERIES AND IN PARALLEL FIG Example FIG Example
93 CAPACITORS IN SERIES AND IN PARALLEL FIG Example FIG Reduced equivalent for the network in Fig
94 CAPACITORS IN SERIES AND IN PARALLEL FIG Example FIG Determining the final (steady-state) value for yc.
95 CAPACITORS IN SERIES AND IN PARALLEL FIG Example
96 ENERGY STORED BY A CAPACITOR An ideal capacitor does not dissipate any of the energy supplied to it. It stores the energy in the form of an electric field between the conducting surfaces. A plot of the voltage, current, and power to a capacitor during the charging phase is shown in Fig The power curve can be obtained by finding the product of the voltage and current at selected instants of time and connecting the points obtained. The energy stored is represented by the shaded area under the power curve.
97 ENERGY STORED BY A CAPACITOR FIG Plotting the power to a capacitive element during the transient phase.
98 STRAY CAPACITANCES In addition to the capacitors discussed so far in this chapter, there are stray capacitances that exist not through design but simply because two conducting surfaces are relatively close to each other. Two conducting wires in the same network have a capacitive effect between them, as shown in Fig (a).
99 STRAY CAPACITANCES FIG Examples of stray capacitance.
100 APPLICATIONS Touch Pad FIG Laptop touch pad.
101 APPLICATIONS Touch Pad FIG Matrix approach to capacitive sensing in a touch pad.
102 APPLICATIONS Touch Pad FIG Capacitive shunt approach.
103 APPLICATIONS Touch Pad FIG Capacitive shunt sensors: (a) bottom, (b) slice.
104 APPLICATIONS Flash Lamp FIG Flash camera: general appearance.
105 APPLICATIONS Flash Lamp FIG Flash camera: basic circuitry.
106 APPLICATIONS Flash Lamp FIG Flash camera: internal construction.
107 APPLICATIONS Surge Protector (Line Conditioner) FIG Surge protector: general appearance.
108 APPLICATIONS Surge Protector (Line Conditioner) FIG Electrical schematic.
109 APPLICATIONS Surge Protector (Line Conditioner) FIG Internal construction of surge protector.
110 COMPUTER ANALYSIS PSpice FIG Circuit to be analyzed using PSpice.
111 COMPUTER ANALYSIS PSpice FIG Establishing a switching dc voltage level: (a) series dc voltageswitch combination; (b) PSpice pulse option.
112 COMPUTER ANALYSIS PSpice FIG The defining parameters of PSpice VPulse.
113 COMPUTER ANALYSIS PSpice FIG Using PSpice to investigate the transient response of the series R-C circuit in Fig
114 COMPUTER ANALYSIS PSpice Transient response for the voltage across the capacitor in Fig when VPulse is applied.
115 COMPUTER ANALYSIS PSpice FIG Using PSpice to verify the results in Example
116 COMPUTER ANALYSIS PSpice FIG The applied pulse and resulting current for the 2 mf capacitor in Fig
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