Section V Electromagnetic and Other Applications of Aluminum Chapter 15 Capacitor Foil Capacitors both fixed and variable are used today A common economic consideration in capacitor design in almost every electrical system. From great power is to obtain the largest amount of capacitance per unit vol­ generating and distributing networks to electric organs, ume of material used. This is, of course, obtained by using including telephone and radio systems, computers and the thinnest electrodes and the thinnest insulating material motors, elevators and x-ray apparatus and so on, the possessing the highest dielectric constant. How far one use of a capacitor almost always is a fundamental can go in these directions depends upon the circuit voltage necessity. to be withstood, the conductance and!or dielectric loss Any arrangement of electrodes whatsoever upon which that can be tolerated and the stability of the assembly electric cbarges accumulate or move will exhibit the inci­ under the operating conditions. dence of capacitance. Where the electrode geometry is Equivalent Network of a Capacitor extensive in space such as a wire or cable, the charges are distributed likewise and we speak of such a structure as All capacitors possess a certain amount of series resist­ being a distributed capacitance. When the electrodes are ance and inductance as well as shunt capacitance and con­ deliberately concentrated in space, the charges are con­ ductance. Fig. 15-1 shows a simple, unrolled foil capaci­ centrated and this is termed a lumped capacitance. All elec­ tor and its equivalent electrical circuits. The series impe­ trical components used specifically as capacitors are looked dance (r + jwL) is made up of the resistance and in­ upon as providing lumped capacitance. ductance of the capacitor leads plus those inherent in the Aluminum has been and is the preferred metal for electrode material, shape and extent. Usually the induc­ capacitor electrodes whether used in rigid plate form or tance is negligibly small as compared to the other factors. in varying thicknesses of foil for d-., a-c low voltage, Actually, the inductance is approximately the same as a high voltage, high frequency, high or low power, impulse wire loop equal in area to that formed by the two leads discharge, etc. and the capacitor unit itself. The foil electrodes appear as a uniformly distributed series resistance. The overall impedance of the equivalent network shown in Fig. 15-1 (c) is: Capacitor Design Considerations .,C Under given conditions of electrical, physical and en­ -~~-+jwL vironmental factors, a capacitor may be called upon to pro­ g2 +.,2 C' g2 +w'C' vide a precise amount of capacitance, a required time (Eq. 15-1) constant of charge and discharge (with proper circuit re­ where: sistance) or a specified impulse release of stored energy. w=2".f The selection of electrode and insulation materials and the fin Hz design of their electrical and mechanical arrangements ginmhos can be optimized to produce an economical capacitoI" that Cin farads will perform properly in its intended service. Lin becrys In Section Ill, Chapter 8, the general conditions gov­ r in ohms erning the relation of potential and charge to capacitance or since is usually very small compared to "PC', were discussed as well as the influence of the dielectric g2 medium. The equations for capacitance relating to a wide g j(w'CL-I) variety of electrode geometries were given and the nature Z~(r+---) of the dielectric polarization of the insulation discussed. wC All of this is applicable to capacitor design. (Eq. 15-2) 15-1 electromagnetic and other applications of aluminum FO'l elECTRODES lEADS "",....,''''".. D.elECTRIC LEAD INDUCTANCE IL) LEAD INDUCTANCE III lEAD RESISTANCE lEAD RESISTANCE PLUS EFFECTIVE FOIL RESISTANCE ld IDEAL EQUiVALENT PARALlel CAPACiTANCE RESISTANCE CORRESPONDING 1<1 TO CONDUCTANCE (g) OF DIELECTRIC Fig. 15-1. Equivalent electrical circuit of a capaciror. From the above it is obvious that the effective capacl­ Reduction of losses in a capacitor is quite important C from the standpoints of both adequate performance and tance seen across the terminals of a capacitor is ( ) stable life. In capacitors carrying heavy currents, the w'CL-l energy loss is a source of heating which, if not adequately and that this will vary with frequency. For low frequencies reduced or carried off by thermal conduction, can cause it will be equal to C; as frequency increases, the effect of rapid deterioration and failure of the insulation. Control the inductance increasingly reduces the capacitance and of heat loss enters into the design and use of capacitors the capacitive reactance. At a frequency where resonance for low frequency operation in connection with power occurs (usually very high), the overall impedance is en­ factor correction and, at high frequencies, in radio trans­ tirely made up of the effective resistance of the leads and mitting capacitors. In radio frequency circuits, effective the foil electrodes at that frequency. Above the resonant resistance becomes important in series coil and capacitor frequency, the capacitor acts as an inductance coil with combinations required to. have low impedance at the some series capacitance. Although theoretically every resonance frequencies or parallel combinations required to practical capacitor will exhibit resonance at some high have high impedance at the anti-resonance frequency. This frequency, it is always possible to arrange the electrode is because resistance may add appreciably to the desired and terminal wires to obtain the effect of a low impedance, low impedance at the resonance frequency or reduce the long transmission line free of apparent resonance over a desired high impedance at the anti-resonant frequency. wide high-frequency band. Fig. 15-2 shows the reactance In electric wave filters intended to pass a single band vs. frequency effect for a waxed paper insulated capacitor of frequencies and suppress others, the transmission loss designed for audio frequency circuits. is ideally zero over the pass-band and rises sharply be­ Effective Resistance and Loss of a Capacitor: At fre­ yond the edge or edges. Parasitic loss in the reactive ele­ quencies below which parasitiC inductance becomes sig­ ments is unwanted loss which varies over the pass-!Jand nificant, the dissipated watt loss in a capacitor is oc­ and reaches a maximum .at the edges resulting in distorted casioned by both the ohmic loss in the foils and leads and transmission. This sOurce of loss is generally objectionable, the dielectric loss in the insulating materiaL However, as the ohmic loss is almost always insignificant it is usually for example, in carrier-telephone systems where the cumu­ ignored in commerCial practice and power Joss is com­ lative loss of many filters in tandem may result in con­ puted with the following formula: siderable distortion which must be compensated for by means of attenuation-equalizing networks. W = E'",C tan B (Eq. 15-3) Where: E = volts In his efforts to limit the losses in capacitors required to w = 2".f (frequency in Hz) pass alternating current in telephone and electronic cir­ tan B= dissipation factor cuits, the capacitor engineer is usually primarily concerned S= Joss angie with the effect of frequency on series and shunt resistance. 15-2 capacitor foil This is because the effective resistance undergoes large electrodes of wound paper capacitors the effective foil changes with changing frequency and because of the wide resistance is approximately equal to 1/3 of the loop dc frequency-range which circuits are often required to cover. resistance obtained by adding the dc resistance values of Loss in Foil and Leads: At a first approximation, the the two foils. In other words, due to current attenuation along the foils only 33 percent of the total doc foil re­ effective impedance of the foil and leads of a capacitor ap­ sistance is effective with respect to alternating current. pears as a straight-line factor over a wide range of fre­ Fig. 15-3 shows the effect of several laid-in terminals in quency. At higher frequencies, impedance increases due reducing effective foil resistance especiaUy at the higher to eddy-current and other losses including skin effect frequencies. where only the outer portion of the metallic components carry the current. Where, as is more usual in practice, the terminals are From the watt loss (Eq. 15-3) above, it is seen that laid-in at approximately the middle of the foil electrodes, the heat loss in the foil and leads increases as the square the current spreads in opposite directions along the foils. of the frequency for constant applied voltage. In general, The effective resistance of the loop in each direction is this condition applies over the operating frequency-range then R/6 and, since the two loops are in parallel, the total of many capacitors. effective resistance becomes R/12. When "n" terminals are laid-in on a foil of length "L", it may be shown that EIJeClive Resistance of Foil Electrodes the lowest resistance is obtained by spacing the terminals In the case of wound paper capacitors, there is a simple at intervals of L/n, with each end terminal located L/2n relationship between the effective resistance of the foil from the end of the foil. With this arrangement, the effec­ electrodes and their de resistance. With reference to Fig. tive resistance is inversely proportional to the square of 15-1, it is clear that alternating current entering the foil the number of terminals, electrodes at the lead-in wires decreases as it spreads or In the limiting case, the edge of the foil is connected to­ distributes along the foil, and the current flowing at points gether along its entire length. This, kno,,\'n as "extended remote from the lead-in wires may be only a small fraction foil" or "overlapped foil" construction, gives the lowest of the entering current.