《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

3. It is impossible to change the voltage across a capacitor by a finite amount in zero time, for this requires an infinite current through the capacitor. It will be advantageous later to hypothesize that such a current may be generated or applied to a capacitor, but for the present we shall avoid such a forcing function or response. A capacitor resists an abrupt change in the voltage across it in a manner analogous to the way a spring resists an abrupt change in its displacement. 4. The capacitor never dissipated energy, but only stores it. Although this is true for the mathematical model, it is not true for a physical capacitor

3. It is impossible to change the voltage across a capacitor by a finite amount in zero time, for this requires an infinite current through the capacitor. It will be advantageous later to hypothesize that such a current may be generated or applied to a capacitor, but for the present we shall avoid such a forcing function or response. A capacitor resists an abrupt change in the voltage across it in a manner analogous to the way a spring resists an abrupt change in its displacement. 4. The capacitor never dissipated energy, but only stores it. Although this is true for the mathematical model, it is not true for a physical capacitor