New topics energy storage elements Capacitors Inductors EE 42 and 100 Fall 2005 Week 3b 1 Books on Reserve for EE 42 and 100 in the Bechtel Engineering Library The Art of Electronics by Horowitz and Hill 2nd edition A terrific source book on practical electronics Electrical Engineering Uncovered by White and Doering 2nd edition Freshman intro to aspects of engineering and EE in particular Newton s Telecom Dictionary The authoritative resource for Telecommunications by Newton 18th edition he updates it annually A place to find definitions of all terms and acronyms connected with telecommunications TK5102 N486 Shelved with dictionaries to right of entry gate EE 42 and 100 Fall 2005 Week 3b 2 The Capacitor Two conductors a b separated by an insulator difference in potential Vab equal opposite charges Q on conductors Q Q Vab Q CVab stored charge in terms of voltage where C is the capacitance of the structure positive charge is on the conductor at higher potential Parallel plate capacitor area of the plates A m2 separation between plates d m dielectric permittivity of insulator F m capacitance A C d EE 42 and 100 Fall 2005 F F Week 3b 3 or Symbol C C C Electrolytic polarized capacitor Units Farads Coulombs Volt typical range of values 1 pF to 1 F for supercapacitors up to a few F Current Voltage relationship dQ dvc dC ic C vc dt dt dt If C geometry is unchanging iC dvC dt ic vc Note Q vc must be a continuous function of time EE 42 and 100 Fall 2005 Week 3b 4 Voltage in Terms of Current Capacitor Uses t Q t ic t dt Q 0 0 t t 1 Q 0 1 vc t ic t dt ic t dt vc 0 C0 C C0 Uses Capacitors are used to store energy for camera flashbulbs in filters that separate various frequency signals and they appear as undesired parasitic elements in circuits where they usually degrade circuit performance EE 42 and 100 Fall 2005 Week 3b 5 EE 42 and 100 Fall 2005 Week 3b 6 Schematic Symbol and Water Model for a Capacitor EE 42 and 100 Fall 2005 Week 3b 7 Stored Energy CAPACITORS STORE ELECTRIC ENERGY You might think the energy stored on a capacitor is QV CV2 which has the dimension of Joules But during charging the average voltage across the capacitor was only half the final value of V for a linear capacitor Thus energy is 1 QV 1 CV 2 2 2 Example A 1 pF capacitance charged to 5 Volts has 5V 2 1pF 12 5 pJ A 5F supercapacitor charged to 5 volts stores 63 J if it discharged at a constant rate in 1 ms energy is discharged at a 63 kW rate EE 42 and 100 Fall 2005 Week 3b 8 A more rigorous derivation ic vc t t Final v VFinal dQ v VFinal w v c i c dt dt v c v c dQ t t Initial v VInitial dt v VInitial v VFinal 1 1 2 2 w Cv dv CV CV c c Final Initial 2 2 v VInitial EE 42 and 100 Fall 2005 Week 3b 9 Example Current Power Energy for a Capacitor t v V 1 0 1 i A 0 1 2 3 4 5 EE 42 and 100 Fall 2005 3 4 v t 10 F t s vc and q must be continuous functions of time however ic can be discontinuous dv i C dt 2 i t 1 v t i d v 0 C0 5 Week 3b t s Note In steady state dc operation time derivatives are zero C is an open circuit 10 p W i t 0 1 2 3 4 5 v t 10 F t s p vi w J 0 t 1 2 EE 42 and 100 Fall 2005 3 4 5 Week 3b t s 1 2 w pd Cv 2 0 11 Capacitors in Parallel i t i1 t i2 t C1 C2 v t Ceq i t v t Ceq C1 C2 dv i Ceq dt Equivalent capacitance of capacitors in parallel is the sum EE 42 and 100 Fall 2005 Week 3b 12 Capacitors in Series v1 t v2 t i t C1 C2 i t Ceq v t v1 t v2 t 1 1 1 Ceq C1 C2 EE 42 and 100 Fall 2005 Week 3b 13 Capacitive Voltage Divider Q Suppose the voltage applied across a series combination of capacitors is changed by v How will this affect the voltage across each individual capacitor v v1 v2 Q1 C1 v1 Q1 Q1 C1 v v Q1 Q1 Q2 Q2 C2 Q2 Q2 v1 v1 Note that no net charge can can be introduced to this node Therefore Q1 Q2 0 v2 t v2 C1 v1 C2 v2 C1 v2 v C1 C2 Q2 C2 v2 Note Capacitors in series have the same incremental charge EE 42 and 100 Fall 2005 Week 3b 14 MEMS Airbag Deployment Accelerometer Chip about 1 cm2 holding in the middle an electromechanical accelerometer around which are electronic test and calibration circuits Analog Devices Inc Hundreds of millions have been sold EE 42 and 100 Fall 2005 Airbag of car that crashed into the back of a stopped Mercedes Within 0 3 seconds after deceleration the bag is supposed to be empty Driver was not hurt in any way chassis distortion meant that this car was written off Week 3b 15 Application Example MEMS Accelerometer to deploy the airbag in a vehicle collision Capacitive MEMS position sensor used to measure acceleration by measuring force on a proof mass MEMS micro electro mechanical systems g1 g2 FIXED OUTER PLATES EE 42 and 100 Fall 2005 Week 3b 16 Sensing the Differential Capacitance Begin with capacitances electrically discharged Fixed electrodes are then charged to Vs and Vs Movable electrode proof mass is then charged to Vo Circuit model Vs C1 C1 C2 Vo Vs 2Vs Vs C1 C2 C1 C2 C1 Vo C2 Vs EE 42 and 100 Fall 2005 A A Vo g1 g 2 g 2 g1 g 2 g1 Vs A A g 2 g1 const g1 g 2 Week 3b 17 Application Condenser Microphone Condenser microphone G Sound waves Econst x Electret microphone E c onst X1 G Vout x Cylindrical air filled cavity Flexible conducting diaphragm Conducting rigid cup Vout x E c onst EE 42 and 100 Fall 2005 Vout Week 3b X1 Vout x Econst Electret insulator e g teflon that was bombarded with electrons that remain imbedded in it to bias the condenser Widely used in telephone handsets available at RadioShack 18 Practical Capacitors A capacitor can be constructed by interleaving the plates with two dielectric layers and rolling them up to achieve a compact size To achieve a small volume a very thin dielectric with a high dielectric constant is desirable However dielectric materials break down and become conductors when the electric field units V cm is too high Real capacitors have maximum voltage ratings An engineering trade off exists between compact size and high voltage rating EE 42 …
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