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Review: Lecture 3 • Linear circuit • Superposition • Equivalent Circuits —Simple Circuits —Y- Transforms —Source transformation —Thevenin's theorem —Norton's theorem • Maximum power transfer Lecture 4 DC Circuits Capacitors and Inductors Objectives • Review the basics of capacitors and inductors • Series and parallel capacitors/inductors • Applications —Integrator —Differentiator —Analog Computer Capacitors Capacitors: fundamental passive components. — Basic form: two conductive plates separated by an insulating dielectric — Capacitance: the ability to store charge (electric potential energy) The charging process… Vs The external source cannot move charge any more: fully charged The physical dimensions of the capacitor determine the charge ‘storage ability’ Conductive plate dielectric VS + + + + + + + + + + A + E B A capacitor with stored charge can act as a temporary battery. q Cv Parallel-plate Capacitors The permittivity of the dielectric q A C v d The surface area of the plates The spacing between plates Circuit behaviour of capacitors • Current and voltage relation t 1 v idt v(t0 ) C t0 — q Cv — Linear element dq dv i C dt dt • v kv0 i C Capacitor voltage depends on the past history of the current Circuit theorems valid Superposition, Thevenin… d kv0 dv kC 0 ki0 dt dt —Passive sign convention: • Being charged: v 0, i 0; v 0, i 0 Passive elements p=vi >0, ‘consume’ energy: charging • A capacitor is an open circuit to dc — v const, dv 0, i 0 dt Current is undefined. • The voltage on a capacitor cannot change abruptly — iC dv dt • Power and energy stored: — — dv Power: p vi Cv dt Energy stored: w t t Ideal capacitors do not dissipate energy v (t ) dv 1 2 p dt Cv dt - dt 2 Cv v ( ) Series and Parallel Capacitors • Parallel-plate capacitors A — C d A 1 d d 1 1 d d d , C , — Series: d d C A A C C (A A ) — Parallel: A A A , C d C C 1 T 1 2 1 2 T 1 T 1 2 T 2 1 2 T 1 2 • Behaviour consideration t — Series: KVL v v1 v2 ... vN 1 t 1 t 1 t v idt v1 (t0 ) idt v2 (t0 ) ... idt v ( t ) N 0 C1 t0 C t0 C N t0 1 v idt v(t0 ) C t0 t t 1 t 1 1 idt idt ... idt v1 (t0 ) v2 (t0 ) ... v N (t0 ) C t0 C N t0 C1 t0 t 1 1 1 ... idt v(t0 ) C N t0 C1 C 1 1 1 1 ... CT C1 C CN — Parallel: KCL i i1 i2 ... iN dv dv dv C2 ... C N dt dt dt dv C1 C2 ... C N dt i C1 CT C1 C2 ... C N 2 iC dv dt Inductors Inductors: fundamental passive components. — Basic form: a length of wire is formed into a coil — Store energy in its magnetic field Inductance is greatly magnified by adding turns and winding them on a magnetic core material. Current change opposite Air core Iron core number of turns of wire inductance in henries N 2 A L l Ferrite core permeability in H/m (same as Wb/At-m) Variable Magnetic field change Induce voltage Circuit behaviour of inductors • Current and voltage relation di vL dt t 1 i v(t )dt i (t0 ) L t0 — — Linear element • i i1 i2 v L d (i1 i2 ) di di L 1 L 2 v1 v2 dt dt dt —Passive sign convention: • Energy stored: v 0, i 0; v 0, i 0 • An inductor is an short circuit to dc — i const, di 0, v 0 dt • The current on an inductor cannot change abruptly — vL di dt • Power and energy stored: di p vi Li dt — Power: — Energy stored: t t Ideal inductors do not dissipate energy i (t ) di 1 w pdt Li dt Li 2 dt 2 i ( ) - Series and Parallel Inductors • Series: KVL v v 1 v2 ... v N di di di L2 ... LN dt dt dt di L1 L2 ... LN dt v L1 Leq L1 L2 ... LN • Parallel: KCL i i1 i2 ... iN 1 t 1 t 1 t i vdt i1 (t0 ) vdt i2 (t0 ) ... vdt iN (t0 ) L1 t0 L2 t0 LN t0 t t 1 t 1 1 vdt vdt ... vdt i1 (t0 ) i2 (t0 ) ... iN (t0 ) L2 t0 LN t0 L1 t0 t 1 1 1 ... vdt i (t0 ) LN t0 L1 L2 N 1 1 Leq n 1 Ln Typical passive elements As long as all the elements are of the same type, the delta-wye transformation can be extended to C and L Integrator iR iC vi iR R iC C vi dvo C R dt dvo dt vi dvo dt RC t 1 vo (t ) vo (0) vi dt RC o t 1 vo (t ) vi dt RC o Assume vo(0)=0 Differentiator iR iC vo iR R dvi vo RC dt iC C dvi dt Analog Computer Analog Computer t vo (t ) 1 vi dt RC o 0.5 Analog Computer t vo (t ) 1 vi dt RC o Analog Computer – after class practice Evaluate the circuit: Appendix Practical issues of capacitors • Leakage resistance • Dielectric breakdown — insulating materials forced to conduct — sudden surge of current —Burn, melt, vaporize: permanent damage • Capacitor types — paper, air, polycarbonate, polyester, polypropylene, polystyrene, mixed, silvered mica, electrolytic(aluminium, tantalum), ceramic… • Capacitor labelling —Small capacitor: 0.01.. Microfarads —103: 10x103 pF: Third digit is the multiplier Practical issues of inductors • Winding resistance and capacitance • Inductor types The winding resistance is in series with the coil; the winding capacitance is in parallel with both. Encapsulated Torroid coil • Applications: —Power supplies, transformers, radios, TVs, radars, and electric motors. CW RW Variable L Applications • Capacitors — temporary voltage source —frequency discrimination • Inductors — temporary current source — spark/arc suppression —Converting pulsating dc voltage into relatively smooth dc — frequency discrimination Lecture 5 DC Circuits Transient Circuits