EE40 Lecture 10 Venkat Anantharam 2 13 08 Reading Chap 3 EE40 Spring 08 Slide 1 Venkat Anantharam Chapter 3 Outline The capacitor The inductor EE40 Spring 08 Slide 2 Venkat Anantharam The Capacitor Two conductors a b separated by an insulator difference in potential Vab equal opposite charge Q on conductors Q CVab stored charge on each plate in terms of voltage where C is the capacitance of the structure positive charge is on the conductor at higher potential the net charge is zero Parallel plate capacitor area of the plates A m2 separation between plates d m dielectric permittivity of insulator F m capacitance EE40 Spring 08 A C d F F Slide 3 Venkat Anantharam Capacitor Symbol or C Electrolytic polarized 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 supercapacitors up to a few F Current Voltage relationship dvc dQ 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 Spring 08 Slide 4 Venkat Anantharam Voltage in Terms of Current 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 At higher frequencies capacitors become increasingly like short circuits EE40 Spring 08 Slide 5 Venkat Anantharam Stored Energy CAPACITORS STORE ELECTRIC ENERGY During charging the average voltage across the capacitor was only half the final value of V for a linear capacitor Thus the energy needed to build up the charge is 1 QV 2 1 CV 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 EE40 Spring 08 Slide 6 Venkat Anantharam A more rigorous derivation This derivation holds independent of the circuit ic vc v VFinal dQ t t Final v VFinal w v c ic dt dt v c dQ vc v VInitial dt t t Initial v VInitial v VFinal 1 1 2 2 w Cv dv CV CV c c Final Initial 2 2 v VInitial EE40 Spring 08 Slide 7 Venkat Anantharam Example Current Power Energy for a Capacitor t v V 1 0 1 2 i A 0 3 4 5 EE40 Spring 08 2 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 1 i t 1 v t i d v 0 C0 5 Slide 8 t s Note In steady state dc operation time derivatives are zero C is an open circuit Venkat Anantharam p W i t 0 1 2 3 4 5 v t 10 F t s p vi w J 0 t 1 EE40 Spring 08 2 3 4 5 Slide 9 t s 1 2 w pd Cv 2 0 Venkat Anantharam Capacitors in series and parallel v1 t v2 t i t C1 C2 i t Ceq v t v1 t v2 t 1 1 1 Ceq C1 C2 Similarly for capacitors in parallel the capacitance adds EE40 Spring 08 Slide 10 Venkat Anantharam 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 passive configuration 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 Spring 08 Slide 11 Venkat Anantharam Stored Energy INDUCTORS STORE MAGNETIC 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 At higher frequencies inductors behave increasingly like open circuits EE40 Spring 08 Slide 12 Venkat Anantharam Inductors in Series and Parallel Common Current Common Voltage EE40 Spring 08 Slide 13 Venkat Anantharam Summary Capacitor Inductor 1 di v L w Li 2 2 dt dv 1 2 i C w Cv dt 2 v cannot change instantaneously i can change instantaneously Do not short circuit a charged capacitor infinite current n 1 1 n cap s in series Ceq i 1 Ci n n cap s in parallel Ceq Ci i 1 i cannot change instantaneously v can change instantaneously Do not open circuit an inductor with current infinite voltage n ind s in series n Leq Li i 1 n 1 1 n ind s in parallel L i 1 Li eq In steady state not time varying In steady state an inductor a capacitor behaves like an open behaves like a short circuit circuit EE40 Spring 08 Slide 14 Venkat Anantharam
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