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## Capacitors

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**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.**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.**FIG. 10.1 Flux distribution from an isolated positive**charge. THE ELECTRIC FIELD • Electric field (E) ⇨electric flux lines ⇨ to indicate the strength of E at any point around the charged body. • Denser flux lines ⇨ stronger E.**FIG. 10.2 Determining the force on a unit charge r meters**from a charge Q of similar polarity. THE ELECTRIC FIELD**FIG. 10.3 Electric flux distributions: (a) opposite charges;**(b) like charges. THE ELECTRIC FIELD • Electric flux lines always extend from a +ve charged body to a -ve charged body, ⊥ to the charged surfaces, and never intersect.**FIG. 10.4 Fundamental charging circuit.**CAPACITANCE ⇨V=IR**FIG. 10.7 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. CAPACITANCE**TABLE 10.1 Relative permittivity (dielectric constant) Σr**of various dielectrics. CAPACITANCE**FIG. 10.9 Example 10.2.**CAPACITOR Construction ⇨ R =ρL/A**FIG. 10.11 Symbols for the capacitor: (a) fixed; (b)**variable. CAPACITORSTypes of Capacitors • Capacitors, like resistors, can be listed under two general headings: fixed and variable.**FIG. 10.12 Demonstrating that, in general, for each type of**construction, the size of a capacitor increases with the capacitance value: (a) electrolytic; (b) polyester-film; (c) tantalum. CAPACITORSTypes of Capacitors**FIG. 10.20 Variable capacitors: (a) air; (b) air trimmer;**(c) ceramic dielectric compression trimmer. [(a) courtesy of James Millen Manufacturing Co.] CAPACITORSTypes of Capacitors • Variable Capacitors • All the parameters can be changed to create a variable capacitor. • For example; the capacitance of the variable air capacitor is changed by turning the shaft at the end of the unit.**FIG. 10.21 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. CAPACITORSLeakage Current and ESR**FIG. 10.23 Various marking schemes for small capacitors.**CAPACITORSCapacitor Labeling**FIG. 10.24 Digital reading capacitance meter. (Courtesy of**B+K Precision.) CAPACITORSMeasurement and Testing of Capacitors • The capacitance of a capacitor can be read directly using a meter such as the Universal LCR Meter.**FIG. 10.26 Basic R-C charging network.**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. 10.27 vC during the charging phase.**TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE The current ( ic ) through a capacitive network is essentially zero after five time constants of the capacitor charging phase.**FIG. 10.28 Universal time constant chart.**TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE**TABLE 10.3 Selected values of e-x.**TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE**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:**FIG. 10.29 Plotting the equation yC =E(1 –e-t/t) versus**time (t). TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE**FIG. 10.31 Demonstrating that a capacitor has the**characteristics of an open circuit after the charging phase has passed. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE**FIG. 10.32 Revealing the short-circuit equivalent for the**capacitor that occurs when the switch is first closed. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE**FIG. 10.35 Transient network for Example 10.6.**TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASEUsing the Calculator to Solve Exponential Functions**FIG. 10.36 vC versus time for the charging network in Fig.**10.35. TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASEUsing the Calculator to Solve Exponential Functions**FIG. 10.37 Plotting the waveform in Fig. 10.36 versus time**(t). TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASEUsing the Calculator to Solve Exponential Functions**FIG. 10.38 iC and yR for the charging network in Fig. 10.36.**TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASEUsing the Calculator to Solve Exponential Functions**TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE**• How to discharge a capacitor and 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.**FIG. 10.39 (a) Charging network; (b) discharging**configuration. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE • For the voltage across the capacitor that is decreasing with time, the mathematical expression is:**FIG. 10.40 yC, iC, and yR for 5t switching between contacts**in Fig. 10.39(a). TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE**FIG. 10.41 vC and iC for the network in Fig. 10.39(a) with**the values in Example 10.6. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE**TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASEThe**Effect of on the Response**FIG. 10.43 Effect of increasing values of C (with R**constant) on the charging curve for vC. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASEThe Effect of on the Response**FIG. 10.44 Network to be analyzed in Example 10.8.**TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASEThe Effect of on the Response**FIG. 10.45 vC and iC for the network in Fig. 10.44.**TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASEThe Effect of on the Response**FIG. 10.46 Network to be analyzed in Example 10.9.**FIG. 10.47 The charging phase for the network in Fig. 10.46. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASEThe Effect of on the Response**FIG. 10.48 Network in Fig. 10.47 when the switch is moved to**position 2 at t =1t1. TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASEThe Effect of on the Response**FIG. 10.49 vC for the network in Fig. 10.47.**TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASEThe Effect of on the Response**FIG. 10.50 ic for the network in Fig. 10.47.**TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASEThe Effect of on the Response**FIG. 10.51 Defining the regions associated with a transient**response. INITIAL CONDITIONS • The voltage across the capacitor at this instant is called the initial value, as shown for the general waveform in Fig. 10.51.**FIG. 10.52 Example 10.10.**INITIAL CONDITIONS**FIG. 10.53 vC and iC for the network in Fig. 10.52.**INITIAL CONDITIONS**FIG. 10.54 Defining the parameters in Eq. (10.21) for the**discharge phase. INITIAL CONDITIONS**THÉVENIN EQUIVALENT: t =RThC**• You may encounter instances in which the network does not have the simple series form in Fig. 10.26. • You then need to find the Thévenin equivalent circuit for the network external to the capacitive element.**FIG. 10.56 Example 10.11.**THÉVENIN EQUIVALENT: t =RThC**FIG. 10.57 Applying Thévenin’s theorem to the network in**Fig. 10.56. THÉVENIN EQUIVALENT: t =RThC**FIG. 10.58 Substituting the Thévenin equivalent for the**network in Fig. 10.56. THÉVENIN EQUIVALENT: t =RThC