Announcements HW 1 Due today at 6pm HW 2 posted online today and due next Tuesday at 6pm Due to scheduling conflicts with some students classes will resume normally this week and next Midterm tentatively 7 12 EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 1 Review Mesh and Nodal Analysis Superposition Equivalent Circuits Thevenin Norton Measuring Voltages and Currents EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 2 1 Review Thevenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a b EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 3 Lecture 4 OUTLINE The capacitor The inductor 1st Order Circuits Transient and Steady State response Reading Chapter 3 Chap 4 1 4 5 EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 4 2 The Capacitor Two conductors a b separated by an insulator difference in potential Vab equal opposite charge Q on conductors 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 A capacitance C F F d EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 5 Capacitor 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 dv dQ dC ic C c vc dt dt dt ic If C geometry is unchanging iC C dvC dt vc Note Q vc must be a continuous function of time EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 6 3 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 EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 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 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 Summer 2005 Lecture 2 Instructor Octavian Florescu 8 4 A more rigorous derivation ic vc t t Final v VFinal dQ v VFinal dt w v c ic dt v c dQ vc dt t t Initial v VInitial v VInitial v VFinal 1 2 2 1 w Cv c dv c CVFinal CVInitial 2 2 v VInitial EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 9 Example Current Power Energy for a Capacitor t 1 i t v t i d v 0 v V C0 v t 10 F 1 0 1 2 i A 0 3 4 5 vc and q must be continuous functions of time however ic can be discontinuous dv i C dt 1 2 3 4 t s 5 t s Note In steady state dc operation time derivatives are zero C is an open circuit EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 10 5 Example Current Power Energy for a Capacitor p W i t 0 1 2 3 4 5 v t 10 F t s p vi w J 0 t 1 2 3 4 EE40 Summer 2005 Lecture 2 5 t s 1 w pd Cv 2 2 0 Instructor Octavian Florescu 11 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 Proof EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 12 6 Capacitors in Parallel i t i1 t i2 t v t i1 t i2 t C2 C1 v t Ceq Ceq C1 C2 Proof EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 13 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 EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 14 7 Inductor Symbol L Units Henrys Volts second Ampere typical range of values H to 10 H Current in terms of voltage 1 vL t dt L t 1 iL t vL d i t0 L t0 diL iL vL Note iL must be a continuous function of time EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 15 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 w t 1 2 1 2 Li Li0 2 2 EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 16 8 Inductors in Series v1 t v2 t L1 i t L2 i t Leq v t v1 t v2 t Leq L1 L2 EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 17 Inductors in Parallel i t i1 t i2 t v t i1 t L1 i2 t L2 v t Leq 1 1 1 L eq L 1 L 2 EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 18 9 First Order Circuits vs A circuit that contains only sources resistors and an inductor is called an RL circuit A circuit that contains only sources resistors and a capacitor is called an RC circuit RL and RC circuits are called first order circuits because their voltages and currents are described by first order differential equations R R i L EE40 Summer 2005 Lecture 2 vs Instructor Octavian Florescu i C 19 Transient vs Steady State Response The momentary behavior of a circuit in response to a change in stimulation is referred to as its transient response The behavior of a circuit a long time many time constants after the change in voltage or current is called the steady state response EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 20 10 Review Conceptual Any first order circuit can be reduced to a Th venin or Norton equivalent connected to either a single equivalent inductor or capacitor RTh RTh L VTh ITh C In steady state an inductor behaves like a short circuit In steady state a capacitor behaves like an open circuit EE40 Summer 2005 Lecture 2 Instructor Octavian Florescu 21 Response The natural response of an RL or RC circuit is its behavior i e current and voltage when …
View Full Document
Unlocking...