EE 3318 pplied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of EE Notes 25 apacitance 1
apacitance apacitor [F] + V - +Q ++++++++++++++++++ - - - - - - - - - - - - - - - - - Q ε r B Note: The conductor has the positive charge (connected to anode). Notation: Q V [ F] Q Q V V B (both positive) ( 1 [ F] 1[ /V] ) Note: The value of is always positive! 2
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 apacitors eramic capacitors The ceramic capacitor is often manufactured in the shape of a disk. fter leads are attached to each side of the capacitor, the capacitor is completely covered with an insulating moisture-proof coating. eramic capacitors usually range in value from 1 picofarad to.1 microfarad and may be used with voltages as high as 3, volts. 4
Typical apacitors (cont.) Paper capacitors 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 3 picofarads to about 4 microfarads. The working voltage of a paper capacitor rarely exceeds 6 volts. 5
Typical apacitors (cont.) Electrolytic capacitors - + Note: One lead (anode) is often longer than the other to indicate polarization. This type of capacitor uses an electrolyte (sometimes wet but often dry) and requires a biasing voltage (there is an anode and a cathode). 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 5 volts. 6
Typical apacitors (cont.) Supercapacitors (Ultracapacitors) Maxwell Technologies "M" and "B" series supercapacitors (up to 3 farad capacitance) ompared to conventional electrolytic capacitors, the energy density is typically on the order of thousands of times greater. 7
Typical apacitors (cont.) MEMS capacitor Variable capacitor apacitors for substations 8
urrent - Voltage Equation i dq dt d v dt dv dt ( ) + - + Q Q + - i + - + - + - v t ( ) i The reference directions for voltage and current correspond to passive sign convention in circuit theory. Note: Q is the charge that flows from left to right, into plate. B 9
urrent - Voltage Equation (cont.) it Hence we have ( ) dv dt i i + - v( t) Passive sign convention 1
Find Example x m ε r 2 B Ex h Q V Ideal parallel plate capacitor Method #1 (start with E) ssume: E ˆ x xe B x x x B h h V V E dr E dx E dx E h 11
Example (cont.) x Hence m ε r 2 B Ex Q εεe V E h r x x h Q ρ ˆ s D n εε ˆ rn E εε ˆ r x E εε rex εεe r x εε h [ ] r F Note: is only a function of the geometry and the permittivity! 12
Example (cont.) Method #2 (start with ρ s ) +ρ s Q V x ρ ρ s s ρ s B ( xd ˆ ) D nˆ ρ ρ x s s xˆ ρ D εεe x ρ ρ s s r x s Q ρ s Hence E x E x ρs εε r 13
Example (cont.) B h h V V E dr E dx E dx h E B x x x Hence V he h ρ s x εε r Therefore, or ρ s ρs h εε r εε h [ ] r F 14
Example ε r 6. (mica) 1 [cm 2 ] h.1 [mm] ε r m 2 h εε h r 4 1 1.1 1 ( 12 8.854 1 )( 6.) [ ] F 3 [ pf] [ nf] 53.12.5312 15
Find Example Q V ε r 2 ε r1 Starting assumption: E x2 E x1 Dx h2 D D D 2 x1 x2 x 1 h 1 x B B.. on plate : D nˆ ρ D 2 xˆ ρ D s s D ρ x2 x s Q ρ s Goal: Put Q and V in terms of D x. Q D x 16
Example (cont.) V E h + E h x1 1 x2 2 D h + ε D h x1 x2 1 2 1 ε2 D h + ε D h x x 1 2 1 ε2 h1 h 2 Dx + ε1 ε2 h + h 1 2 V Dx ε 1 ε 2 17
Example (cont.) or Hence: Q Dx V h1 h2 + Dx ε1 ε2 1 h1 h2 h1 h2 + + ε ε ε ε 1 2 1 2 1 1 1 + ε1 ε2 h h 1 2 where 1 Hence, ε 1 1 1 + 1 2 ε 1 2 2 h1 h2 1 1 1 + 1 2 18
Find Example 1 2 + + + + + + + + + + + + + + + + Q V B ε r1 Dx1 E x D x 2 ε r 2 x h x Starting assumption: E E E x1 x2 x V Ex h Goal: Put Q and V in terms of E x. ρ D ε E ε E s1 x1 1 x1 1 x ρ D ε E ε E s2 x2 2 x2 2 x 19
Example (cont.) Q ρ + ρ s1 1 s2 2 ε E + ε E 1 x 1 2 x 2 or Q ε E + ε E V E h 1 1 x 2 2 x x ε ε + h h 1 1 2 2 1 where ε h 1 1 + 1 2 2 ε h 2 2 2
Example Find l ε r oaxial cable (capacitance / length) Note ε r a b ρ l ρ l ( ) : ρ ρ / 2π a s l a E V ˆ ρ ρ ˆ ρ ρ l l 2περ 2πεεr ρ B E dr b a E ρ dρ l ρ V l 21
Hence we have Example (cont.) V ρ 2πε ε ρ 1 dρ ρ 2πεεr a b r a b ln Therefore l ρ ρ V ρ ln 2πεε r b a We then have l 2πεεr b ln a [ F/m] 22
Example (cont.) Formula from EE 3317: Z L l l (characteristic impedance) The characteristic impedance is one of the most important numbers that characterizes a transmission line. oax for cable TV: Z 75 [Ω] Twin lead for TV: Z 3 [Ω] oaxial cable Twin lead 23
Example (cont.) Formula from EE 3317: v p 1 1 phase velocity L µε l l ssume µ µ Hence L l µε l ε εε r Using our expression for l for the coax gives: L l µ b ln [ H/m] 2π a 24
Example (cont.) Hence, the characteristic impedance of the coaxial cable transmission line is: Z L l l Z η b ln [ Ω] 2π ε a r l 2πεεr b ln a [ F/m] where η µ Ω ε 376.763 [ ] L l µ b ln [ H/m] 2π a (intrinsic impedance of free space) ε F/m 12 8.854187 1 [ ] µ π H/m 7 4 1 [ ] 25