EE40 Lec 07 Capacitors and Inductors Prof Nathan Cheung 09 17 2009 R di Reading H Hambley bl Ch Chapter t 3 EE40 Fall 2009 Slide 1 Prof Cheung The Capacitor Two conductors a b separated by an insulator difference in potential Vab equall opposite it charge h Q on conductors d t 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 p plates A separation between plates d dielectric permittivity of insulator capacitance EE40 Fall 2009 A C d Slide 2 Prof Cheung Capacitor Symbol or C Electrolytic y polarized p capacitor C Units Farads Coulombs Volt C These have high capacitance and cannot support voltage drops of the wrong polarity typical range of values 1 pF to 1 F for supercapasupercapa citors up to a few F Current Voltage relationship d c dv dQ d C ic dt dt To write this it is important to have use a passive convention otherwise you need a minus sign ic vc Note vc must be a continuous function of time since the charge stored on each plate cannot change suddenly EE40 Fall 2009 Slide 3 Prof Cheung Voltage in Terms of Current t Q t ic t dt Q 0 0 t 1 Q 0 vc t ic t dt C0 C t 1 ic t dt vc 0 C0 EE40 Fall 2009 Slide 4 Prof Cheung Stored Energy The energy stored Th t d on a capacitor it iis QV QV which hi h h has th the dimension of Joules During charging you might think the average voltage across the capacitor was only half the final value of V for a capacitor Th stored The t d energy 1 Q V 2 1 2 CV 2 Example A 1 pF capacitance charged to 5 Volts Stored energy 5V 2 12 5 12 5 pJ EE40 Fall 2009 Slide 5 Prof Cheung A more rigorous derivation This derivation holds independent of the circuit ic v t Final v v Final Final dQ w v c i c dt v dt v dQ c c dt V t Initial v v Initial Initial v Final 1 1 2 Cv c dv c Cv Final Cv Initial 2 2 2 v Initial EE40 Fall 2009 Slide 6 Prof Cheung vc Example Current Power Energy for a Capacitor t v V 1 0 1 2 4 5 dv i C dt i A 0 3 1 EE40 Fall 2009 2 3 4 i t v t 1 v t i d v 0 C0 10 F t s vc must be a continuous function of time however however ic can be discontinuous 5 Slide 7 t s Note In steady state dc operation time derivatives are zero C is an open circuit Prof Cheung Current Power Energy for a Capacitor p W 0 p vi 1 2 3 4 5 t s i t w J 0 v t 10 F t 1 EE40 Fall 2009 2 3 4 5 Slide 8 t s 1 2 w pd Cv 2 0 Prof Cheung Capacitors in Parallel i t i1 t i2 t C1 C2 v t i t Ceq Ceq C1 C2 v t dv i Ceq dt Equivalent capacitance of capacitors in parallel is the sum EE40 Fall 2009 Slide 9 Prof Cheung 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 EE40 Fall 2009 Slide 10 Prof Cheung 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 Q Q1 Q2 Q2 C2 Q2 Q2 v1 v1 v2 t v2 Note that no net charge can can be introduced to this node Therefore Q1 Q2 0 C1 v1 C2 v2 C1 v2 v C1 C2 Q2 C2 v2 Note Capacitors in series have the same incremental charge EE40 Fall 2009 Slide 11 Prof Cheung Comment on Capacitive Voltage Divider Q Can we always claim Q1 v Q1 C1 Q2 C2 Q2 EE40 Fall 2009 v1 v2 C1 v2 v C1 C 2 A Answer No No Divider formula is valid for total voltage g onlyy if capacitors are uncharged when v is applied Slide 12 Prof Cheung Application Example MEMS Accelerometer Capacitive position sensor used to measure acceleration by measuring force on a proof mass g1 g2 FIXED OUTER PLATES EE40 Fall 2009 Slide 13 Prof Cheung MEMS Accelerometer 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 V Vs EE40 Fall 2009 A A Vo g1 g 2 g 2 g1 g 2 g1 Vs A A g 2 g1 const g1 g 2 Slide 14 Prof Cheung Op Amp Integrator t 1 vo t v IN t dt vC 0 RC 0 C vin vC in vn vp EE40 Fall 2009 R Slide 15 ic vo Prof Cheung 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 A engineering i i ttrade off d ff exists i t between b t compactt size i and d high voltage rating EE40 Fall 2009 Slide 16 Prof Cheung The Inductor An inductor is constructed by coiling a wire around some type of form vL t iL Current flowing through the coil creates a magnetic field and a magnetic flux that links the coil LiL When Wh th the currentt changes h th the magnetic ti flflux changes h a voltage across the coil is induced Note In steady state dc operation time derivatives are zero L is a short circuit EE40 Fall 2009 Slide 17 diL vL t L dt Prof Cheung Inductor Symbol L Units Henrys Volts second Ampere typical range of values H to 10 H Current in terms of voltage iL 1 diL vL t dt L To write this it is important to use t the h passive i configuration fi i 1 iL t vL d i t0 L t0 vL Note iL must be a continuous function of time because magnetic flux cannot change suddenly EE40 Fall 2009 Slide 18 Prof Cheung Stored Energy Consider an inductor having an initial current i t0 i0 p t v t i t t w t p d t0 1 2 1 2 w t Li Li0 2 2 EE40 Fall 2009 Slide 19 Prof Cheung Inductors in Series di v Leq dt v1 t v2 t …
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