Inductors and Capacitors Energy Storage Devices

Transcription

1 Inuctors an Capacitors Energy Storage Devices Aims: To know: Basics of energy storage evices. Storage leas to time elays. Basic equations for inuctors an capacitors. To be able to o escribe: Energy storage in circuits with a capacitor. Energy storage in circuits with an inuctor. Lecture 8 Energy Storage an Time Delays Changes in resistor networks happen instantaneously No energy is store in a resistor network (only issipate) Devices which store energy introuce time elays Time to store energy Time to release energy Example Flywheel storage Electronic components that store energy will force us to think about how currents an voltages change with time Motor spee Motor with no flywheel Power on Power off Motor with flywheel Lecture 8 3 1

2 Capacitor A evice to store charge. Excess charges generate electrostatic fiels. Electrostatic fiels are associate with energy brass contact glass jar (insulator) brass electroes Capacitors are evices to generate a well-efine electrostatic fiel The Leyen Jar (18 th C) Lecture 8 4 Simplest geometry E You will learn that if you take any close surface surrouning an isolate charge, the electric fiel multiplie by the area of the surface is proportional to the value of the charge: Q = εea (This is Gauss s Theorem) So for this geometry (where we assume that E is constant over the entire area of the plates) Parallel plate capacitor (schematic) Lecture 8 5

3 Simplest geometry E You will learn that if you take any close surface surrouning an isolate charge, the electric fiel multiplie by the area of the surface is proportional to the value of the charge: Q= εea (This is Gauss s Theorem) So for this geometry (where we assume that E is constant over the entire area of the plates) Parallel plate capacitor (schematic) E = /, so ε A Q = or Q = C, where ε A = C coulomb per volt Lecture 8 6 Johann Carl Frierich Gauss ( ) One of the 19th century s great mins Primarily a mathematician, he worke in Göttingen on many funamental aspects of mathematical physics an statistics Lecture 8 7 3

4 Capacitance Q= C, where ε A = C coulomb per volt C is calle the CAPACITANCE of the evice. This is a property of the configuration of the electroes The unit C -1 is calle the FARAD (F). 1 Fara is a very large capacitance an capacitors commonly use range from a few pf through nf an μf to ~1 mf Circuit symbols example: A=1 cm x 1 cm an = 1mm ε A C = F C= = F -3 1 C = 88.5 pf A capacitor stores a well efine amount of charge proportional to the voltage. When it is isconnecte from the battery it will store the charge inefinitely. This is NOT like a battery where the amount of charge GENERATED is inepenent of voltage. You can only take out of a capacitor what you put into it Lecture 8 8 Capacitance Q= C, where ε A = C coulomb per volt C is calle the CAPACITANCE of the evice. This is a property of the configuration of the electroes The unit C -1 is calle the FARAD (F). 1 Fara is a very large capacitance an capacitors commonly use range from a few pf through nf an μf to ~1 mf Circuit symbols example: A=1 cm x 1 cm an = 1mm ε A C = F C= = F -3 1 C = 88.5 pf A capacitor stores a well efine amount of charge proportional to the voltage. When it is isconnecte from the battery it will store the charge inefinitely. This is NOT like a battery where the amount of charge GENERATED is inepenent of voltage. You can only take out of a capacitor what you put into it Lecture 8 9 4

5 Michael Faraay ( ) A great experimentalist an populariser of science. He is most famous for his work on magnetic inuction, but also i funamental work relate to electrolysis He worke at the Royal Institution (one of the first scientific research institutes) an establishe the Christmas Lectures on science for young people which are still running. Lecture 8 1 Dielectrics E eff =ε E If we place an insulating material between the plates of our capacitor, the effective fiel increases*. Eeff = ε E Where ε is a imensionless property of the material calle the ielectric constant or relative permittivity. ε is usually > 1 e.g. for glass ε = 8 εε A This increases the capacitance: C = F * This is because the electron clou roun each atom in the material is istorte by the applie fiel an this generates an aitional fiel (this is calle the isplacement fiel, D) Lecture

6 Practical Capacitors Practical capacitors try to squeeze as much capacitance as possible into the smallest physical volume: Large area Small separation High ielectric constant insulator ~5mm e.g. Ceramic isc capacitor Electroes are metal (Al, Ag) evaporate onto two sies of isk of very high permittivity ceramic Lecture 8 1 Practical Capacitors Plastic film capacitor Electroes are metal (Al, Ag) evaporate onto both sies of a long ribbon of very thin Mylar foil which is stacke in a block or rolle up like a Swiss Roll into a small cyliner Many other types for a wie range of applications Lecture

7 Energy in Capacitors Imagine a capacitor C charge to a voltage If you push into the capacitor a small amount of charge, Q, then the energy increases by an amount W = Q (energy = charge x voltage) At the same time the voltage increases by an amount, where Q = C So W = C. To get the total energy store in a capacitor we nee to integrate this expression: W = C 1 1 W = C or W = Q Compare this with a battery, where W = Q Lecture 8 14 Energy in Capacitors Imagine a capacitor C charge to a voltage If you push into the capacitor a small amount of charge, Q, then the energy increases by an amount W = Q (energy = charge x voltage) At the same time the voltage increases by an amount, where Q = C So W = C. To get the total energy store in a capacitor we nee to integrate this expression: W = C 1 1 W = C or W = Q Compare this with a battery, where W = Q Lecture

8 Charging a capacitor When you first connect a battery to a capacitor: The voltage across the capacitor is ZERO The current is high ( B /R) B I R When the capacitor is fully charge: The voltage across the capacitor is B The current is ZERO C For capacitors: Current leas oltage B /R B Q= C Q i = t so i = C t time oltage Current Lecture 8 16 Series an parallel capacitors C1 C C EQ For parallel capacitors, is the same, so total charge is given by C1 C EQ For series capacitors, the CHARGE on each capacitor must be the same an equal to the net charge. [The centre electroe has a net charge of zero] C Lecture

9 Series an parallel capacitors For parallel capacitors, is the same, so total charge is given by C1 C C EQ Q = C = Q + Q = C + C TOT Hence: EQ C = C + C EQ C1 C C EQ For series capacitors, the CHARGE on each capacitor must be the same an equal to the net charge. [The centre electroe has a net charge of zero] Q Q Q = = + = + C C C Q = Q = Q TOT TOT 1 EQ TOT Hence: = + C C C EQ Lecture 8 18 Inuctors: Energy Storage in Magnetic Fiels Flowing electric currents create The magnetic fiel escribes the magnetic force on MOING charges. Symbol in equations, B, units TESLA, (T). I Check that magnetic fiels B μi Br () = π r Right han rule: current with thumb, fiel with fingers μ is the permeability of free space a funamental constant that relates magnetism to force an energy. μ = 4π 1-7 T A -1 m 1 gives the value of the velocity of light με Lecture

10 Nikola Tesla ( ) Serbian immigrant to the USA. Consiere to be more of an inventor than a scientist an is creite with the iea of using AC for power transmission. Much given to spectacular emonstrations of high voltage sparks, he became one of the first scientific superstars in the US. Lecture 8 Magnetic Flux an Inuctance The total amount of magnetic fiel crossing a surface is calle the flux: If the fiel is uniform, the flux is given by Φ=BA T m Area A B For any general coil of N turns carrying current i the total amount of flux generate is efine as Li Φ= N Where L is a parameter epening only on the shape an number of turns of the coil calle INDUCTANCE. Units: T m A -1 or HENRY (symbol H) N turns Φ Ii Lecture 8 1 1

11 Joseph Henry ( ) Born in upstate New York he worke on electromagnetism an inuctance in Albany an Princeton. Was appointe the first Secretary of the Smithsonian Institution in Washington in 1864 Lecture 8 Solenois The magnetic fiel can be concentrate by forming the wire into a coil or solenoi. For a long solenoi: μni μnai B = so Φ = l l an ΦN μ L = = i l N A Henry Area A Aing a ferromagnetic (e.g. iron) CORE into the coil can increase the flux for a given current an so increase the inuctance l Lecture

12 Practical inuctors L L Circuit symbols Lecture 8 4 Back e.m.f. When we try to change the current passing through an inuctor the increasing magnetic fiel inuces a reverse voltage which tries to oppose the change. This epens on the inuctance an how fast the current is changing: = L I t This is Lenz s Law which is base on Faraay s laws of magnetic inuction. So we have to o work to overcome this back e.m.f. an pass current through an inuctor we are storing energy in the magnetic fiel. L I Lecture 8 5 1

13 Energy in inuctors I = L t So in a short time t we have to o a small amount of work W = It = LII to overcome the back e.m.f. Thus the total energy require to increase the current from to I is L I I 1 W = LII = LI This is the energy store in an inuctor Lecture 8 6 Energy in inuctors I = L t So in a short time t we have to o a small amount of work W = It = LII to overcome the back e.m.f. Thus the total energy require to increase the current from to I is L I I 1 W = LII = LI This is the energy store in an inuctor Lecture

14 Charging an Inuctor When you first connect a battery to an inuctor: The current through the inuctor is ZERO The back e.m.f. is high L I When the inuctor is carrying the full current: The voltage across the inuctor is ZERO The current is high For inuctors: oltage leas Current Current B /R Reverse oltage time Lecture 8 8 Inuctors in Series an Parallel For inuctors in series L1 L For inuctors in parallel L1 L Lecture

15 Inuctors in Series an Parallel For inuctors in series L = L + L EQ 1 L1 L For inuctors in parallel = + L L L EQ 1 L1 L Lecture 8 3 Summary of Capacitor an Inuctor Formulae I- relationship Store energy Dissipate energy Series equivalent Parallel equivalent Current/voltage timing Capacitor I = C t 1 W C Current leas voltage Inuctor I = L t 1 W = LI = 1 1 1/ + C1 C C1+ C L1+ L 1 1 1/ + L1 L oltage leas current Resistor = IR P = I R1+ R 1 1 1/ + R1 R Current in phase with voltage Lecture