Capacitors

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.

THE ELECTRIC FIELD Electric field (E) ⇨ electric flux lines ⇨ to indicate the strength of E at any point around the charged body. FIG. 10.1 Flux distribution from an isolated positive charge. Denser flux lines ⇨ stronger E.

THE ELECTRIC FIELD FIG. 10.2 Determining the force on a unit charge r meters from a charge Q of similar polarity.

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. 10.3 Electric flux distributions: (a) opposite charges; (b) like charges.

CAPACITANCE ⇨V=IR FIG. 10.4 Fundamental charging circuit.

CAPACITANCE 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.

CAPACITOR Construction ⇨ R =ρL/A FIG. 10.9 Example 10.2.

CAPACITORS Types of Capacitors Capacitors, like resistors, can be listed under two general headings: fixed and variable. FIG. 10.11 Symbols for the capacitor: (a) fixed; (b) variable.

CAPACITORS Types of Capacitors 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.

CAPACITORS Types 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. 10.20 Variable capacitors: (a) air; (b) air trimmer; (c) ceramic dielectric compression trimmer. [(a) courtesy of James Millen Manufacturing Co.]

CAPACITORS Leakage Current and ESR 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.

CAPACITORS Capacitor Labeling FIG. 10.23 Various marking schemes for small capacitors.

CAPACITORS Measurement and Testing of Capacitors FIG. 10.24 Digital reading capacitance meter. (Courtesy of B+K Precision.) The capacitance of a capacitor can be read directly using a meter such as the Universal LCR Meter.

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. 10.26 Basic R-C charging network.

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. 10.27 vC during the charging phase.

TRANSIENTS IN CAPACITIVE NETWORKS: THE 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 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:

TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE FIG. 10.29 Plotting the equation yC = E(1 – e-t/t) versus time (t).

TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE

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

TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using the Calculator to Solve Exponential Functions FIG. 10.35 Transient network for Example 10.6.

TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE Using 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 PHASE Using 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 PHASE Using 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 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.

TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE For the voltage across the capacitor that is decreasing with time, the mathematical expression is: FIG. 10.39 (a) Charging network; (b) discharging configuration.

TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE 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 The Effect of on the Response

TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The 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 PHASE The Effect of on the Response FIG. 10.44 Network to be analyzed in Example 10.8.

TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.45 vC and iC for the network in Fig. 10.44.

TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.47 The charging phase for the network in Fig. 10.46. FIG. 10.46 Network to be analyzed in Example 10.9.

TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The 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 PHASE The Effect of on the Response FIG. 10.49 vC for the network in Fig. 10.47.

TRANSIENTS IN CAPACITIVE NETWORKS: THE DISCHARGING PHASE The Effect of on the Response FIG. 10.50 ic for the network in Fig. 10.47.

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. 10.51 Defining the regions associated with a transient response.

INITIAL CONDITIONS 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.

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.

THÉVENIN EQUIVALENT: t =RThC 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 FIG. 10.59 The resulting waveforms for the network in Fig. 10.56.

THÉVENIN EQUIVALENT: t =RThC FIG. 10.60 Example 10.12. FIG. 10.61 Network in Fig. 10.60 redrawn.

THÉVENIN EQUIVALENT: t =RThC FIG. 10.62 yC for the network in Fig. 10.60.

THÉVENIN EQUIVALENT: t =RThC FIG. 10.63 Example 10.13.

THE CURRENT iC 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: iR = vR/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:

THE CURRENT iC FIG. 10.64 vC for Example 10.14.

THE CURRENT iC FIG. 10.65 The resulting current iC for the applied voltage in Fig. 10.64.

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.

CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.66 Series capacitors.

CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.67 Parallel capacitors.

CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.68 Example 10.15. FIG. 10.69 Example 10.16.

CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.71 Reduced equivalent for the network in Fig. 10.70. FIG. 10.70 Example 10.17.

CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.73 Determining the final (steady-state) value for yC. FIG. 10.72 Example 10.18.

CAPACITORS IN SERIES AND IN PARALLEL FIG. 10.74 Example 10.19.

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. 10.75. 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.

ENERGY STORED BY A CAPACITOR FIG. 10.75 Plotting the power to a capacitive element during the transient phase.

APPLICATIONS Touch Pad FIG. 10.77 Laptop touch pad.

APPLICATIONS Touch Pad FIG. 10.78 Matrix approach to capacitive sensing in a touch pad.

APPLICATIONS Flash Lamp FIG. 10.81 Flash camera: general appearance.

APPLICATIONS Flash Lamp FIG. 10.82 Flash camera: basic circuitry.

APPLICATIONS Flash Lamp FIG. 10.83 Flash camera: internal construction.