《电路》（英文版）4-2 The capacitor

§4-2 The capacitor i+q 9=CU i=dg /dt do i=C C=4/d 2 dt The displacement current flowing internally between the capacitor plates is exactly equal to the conduction current flowing in the capacitor leads D (t) i=C→db=-itt t d t (t0) u(t) idt +D(to) C Jto ler:tn=0-,andt=0+:0(0)=id+u(0) 0 (0)=D(0 0 The voltage appearing across the terminals of a linear time invariant capacitor must always be a continuous function

§4-2 The capacitor + − C i + q − q C A d dt d i C q C i dq dt / /    = = = = The displacement current flowing internally between the capacitor plates is exactly equal to the conduction current flowing in the capacitor leads. (0 ) (0 ) + −  =   = ( ) ( ) 0 0 t 1 t t t idt C d   idt  C d dt d i C 1 =   =  ( ) 1 ( ) 0 0 idt t C t t t  = +  The voltage appearing across the terminals of a linear time￾invariant capacitor must always be a continuous function. (0 ) 1 (0 ) 0 0 + −  = +  +  − idt  C − + let :t 0 = 0 ,and t = 0 0

open=0 dt:t=0,Uc(07)= (R2+R3)D R R2 (R1+R2+R3) R (R1+R2+R3) C and 0-)=0 t:t=0,Uc(0)=c(0)= (R2+R3)U → a voltage source (R1+R2+R3) U(0+ ic(0)=-2(0) (R2+ R1R,i2(0) i1(0+)+i2(0+) (R1+R2+R3) Doc(oIR, :c(0)=0→0c(0 UC(0=0-Short circuit

(0 ) 0 ( ) (0 ) (0 ) 1 2 3 1 2 = + + = = − − − C s and i R R R i i  short circuit If C C C = = − = → − − + (0 ) 0 : (0 ) 0 (0 )    ( ) ( ) : 0 , (0 ) 1 2 3 2 3 R R R R R at t s C + + + = = − −   a voltage source R R R R R at t s C C → + + + = = = + + − ( ) ( ) : 0 , (0 ) (0 ) 1 2 3 2 3    ( ) ( ) (0 ) (0 ) (0 ) 1 2 3 2 3 2 R R R R R i i s C C + + = − + = − = − + + +   R1 R2  s R3 (0 ) 2 + i (0 ) +  C (0 ) + c (0 ) i 1 + i R1  s R3 2 i C i C − +  C 1 i R2

The power p(accepted) by a capacitor P=U=CU dt The energy. u(t) dwc= pdt=Clu dt=c odu=C(0(t)-0(to), (to0) dt U(t0) or wr(t)-wc(o 2 =C{U(t)-02(t0) JU(t0)=0 C(=CU

The power p (accepted) by a capacitor: dt d p i C  = =  The energy: dt C d C{ (t ) (t )} dt d dw pdt C ( t ) ( t ) t t w ( t ) w ( t ) t t C C C 0 2 2 2 1 0 0 0 0         = = = = −     2 0 2 1 If (t ) = 0 wC (t) = C { ( ) ( )} 2 1 ( ) ( ) 0 2 2 0 or w t w t C t t C − C =  −

Example l u(t=100sin 2tv r=v/R=10 sin 2nA =Ccv/dt=4丌×103cos2m4 100sin 2np QMI 20uF Cv=0.lsin 2nutJ 2 t=0→1 0 physical-capacitor t=1/4s,>wc=0.1J t=1/2s 0 1/2 1/2 R R dt Rindt=2.5m/

Example 1: i C Cdv / dt 4 10 cos 2t A −3 = =  t = 0 → wC = 0 1/ 2 , 0 1/ 4 , 0.1 = → = = → = C C t s w t s w J   = = = 1/ 2 0 1/ 2 0 2 wR pR dt RiR dt 2.5mJ (t) = 100sin2tV i R v / R 10 sin2tA −4 = = wC Cv 0.1sin 2tJ 2 1 2 2 = = physical − capacitor 100sin2tV − +  R i C i  M1 20F

Some of important characteristics of a capacitor are now apparent 1. The current through a capacitor is zero, if the voltage across it is not changing with time. A capacitor is therefore an open circuit to dc 2. A finite amount of energy can be stored in a capacitor even if the current through the capacitor is zero, such as when the voltage across it is constant

Some of important characteristics of a capacitor are now apparent: 2. A finite amount of energy can be stored in a capacitor even if the current through the capacitor is zero, such as when the voltage across it is constant. 1. The current through a capacitor is zero, if the voltage across it is not changing with time. A capacitor is therefore an open circuit to dc