Prof. David R. Jackson ECE Dept. Fall 2014 Notes 25 ECE 2317 Applied Electricity and Magnetism 1

Capacitance Capacitor C [F] Note: The “ A ” conductor has the positive charge (connected to anode). Q = Q A V = V AB Notation: (both positive) Note: The value of C is always positive! +Q -Q + - V A B

Leyden Jar The Leyden Jar was one of the earliest capacitors. It was invented in 1745 by Pieter van Musschenbroek at the University of Leiden in the Netherlands (1746). 3

Typical Capacitors Ceramic capacitors The ceramic capacitor is often manufactured in the shape of a disk. After leads are attached to each side of the capacitor, the capacitor is completely covered with an insulating moisture-proof coating. Ceramic capacitors usually range in value from 1 picofarad to 0.01 microfarad and may be used with voltages as high as 30,000 volts. 4

Typical Capacitors (cont.) Paper capacitors A paper capacitor is made of flat thin strips of metal foil conductors that are separated by waxed paper (the dielectric material). Paper capacitors usually range in value from about 300 picofarads to about 4 microfarads. The working voltage of a paper capacitor rarely exceeds 600 volts. 5

This type of capacitor uses an electrolyte (sometimes wet but often dry) and requires a biasing voltage (there is an anode and a cathode). A thin oxide layer forms on the anode due to an electrochemical process, creating the dielectric. Dry electrolytic capacitors vary in size from about 4 microfarads to several thousand microfarads and have a working voltage of approximately 500 volts. Note: One lead (anode) is often longer than the other to indicate polarization. Electrolytic capacitors Typical Capacitors (cont.) + - 6

Supercapacitors (Ultracapacitors) Typical Capacitors (cont.) Maxwell Technologies "MC" and "BC" series supercapacitors (up to 3000 farad capacitance) Compared to conventional electrolytic capacitors, the energy density is typically on the order of thousands of times greater. 7

MEMS capacitor Typical Capacitors (cont.) Variable capacitor Capacitors for substations 8

Current - Voltage Equation Note: Q is the charge that flows from left to right, into plate A Q -Q i i + - v ( t ) A B The reference directions for voltage and current correspond to “passive sign convention” in circuit theory.

Hence we have “Passive sign convention” Current - Voltage Equation (cont.) 10 i i + - v ( t )

Example Method #1 (start with E ) Ideal parallel plate capacitor Assume: Find C A B x h [m] A [m 2 ] Ex0Ex0 11

Example (cont.) Hence A B x h [m] A [m 2 ] Ex0Ex0 Note: C is only a function of the geometry and the permittivity! 12

Example (cont.) Method #2 (start with  s ) A B x +s0+s0 -s0-s0 13 Hence

Example (cont.) Hence Therefore, 14

Example h [m] A [m 2 ]  r = 6.0 ( mica ) A = 1 [cm 2 ] h = 0.01 [mm] 15

Example r1r1 h1h1 x h2h2 r2r2 2 1 Ex2Ex2 Ex1Ex1 A B Dx0Dx0 B. C. on plate A : Find C Goal: Put Q and V in terms of D x 0. Starting assumption: 16

Example (cont.) 17

Example (cont.) Hence: or where Hence, 18

Example A1A1 A2A2 r1r1 x r2r2 Dx1Dx1 Dx2Dx2 x = h A B Ex0Ex Find C Goal: Put Q and V in terms of E x 0. Starting assumption: 19

Example (cont.) 20 where

Example -l0-l0 l0l0 a b Find C l (capacitance / length) Coaxial cable 21

Example (cont.) Hence we have Therefore We then have 22

Example (cont.) Formula from ECE 3317: (characteristic impedance) 23 The characteristic impedance is one of the most important numbers that characterizes a transmission line. Coax for cable TV: Z 0 = 75 [  ] Twin lead for TV: Z 0 = 300 [  ] Coaxial cable Twin lead

Example (cont.) Formula from ECE 3317: Hence Assume Using our expression for C l for the coax gives: 24

Example (cont.) Hence, the characteristic impedance of the coaxial cable transmission line is: where 25 (intrinsic impedance of free space)