??6: RC ? RLC ??(????21)

1
??6 RC ? RLC ??(????21)
?? ??????? RC ? RLC ???????????
A RC?? 1. ?? S ?? a ?, R,C,?? e0?? iR q/C
R(dq/dt) q/C e0 q(t) Ce01
exp(-t/RC) ??? C ??, ????? RC 2. ???qmax,
??S ??b ?, R,C?? R(dq/dt) q/C 0
q(t) qmexp(-t/RC) ?????, t ? RC
(????) q(t) qm/e qm/2.718 q(T1/2)
qm/2

2
?? 1.?????????(??T)?? ???? e0 ??? S ? ?0
for 0 lt t lt T/2 ? -?0 for T/2 lt t
lt T 2.????Y(??)??(??)?? ???C????VC(t) q(t)/C
q(t) CVC(t) ?? R 10 k?, C
0.1 mF, t RC 1 ms ?????? T 100
ms, 10 ms, 1 ms ??q(t)
qm q(t T/2) -qm q(t T)
3
B. LC ? RLC ?? ?? ?????? qmax,?? S ??, L,C ??
L(di/dt) q/C L(d2q/dt2) q/C 0 q(t)
qmcost/(LC)1/2 qmcos?0t (t 0, q qm) ???
C ??????(harmonic oscillation) ????? ?0
1/(LC)1/2 ? R ? 0(??? L ?????), ? RLC ??
L(di/dt) Ri q/C 0 d2q/dt2 (R/L)dq/dt
q/LC 0 d2q/dt2 2?dq/dt ?02q 0
? ? R/2L ???C??????(damped
oscillation)
4
1. b2 lt w02 (R2/4L2 lt 1/LC) q(t)
qmexp(-bt)cos(w1t) ??? w12 w02 b2 gt 0
?? qm(t) qmexp(-bt) (?q(0) qm, ?? d
0, ???? t 1/b 2L/R) ???C??????(T1 gt
T0) ?????? 2. b2 w02 (R2C/4L 1) w12
0 (????T1 ?) ????(critical damping) 3.
b2 gt w02 (R2C/4L gt 1) w12 lt 0 (w1??,???)
???(overdamping) For R, L, C, e0?? d2q/dt2
2bdq/dt w02q e0/L A q(t) (A/w02)1 -
exp(-bt).cos(w1t) (?d 0)
5
??
1.???????????????e0???S e e0 for 0 lt t lt
T/2 e -e0 for T/2 lt t lt T d2q/dt2
2b?dq/dt w02q e/L 2.????? R, ? b R/2L,
?w12 w02 b2 gt 0 3.????Y???????C????VC(t)
q(t) C?VC(t) ? C 0.001 mF 1 pF L
10 mH, R 0-25 kW w02 1/LC, b R/2L
6
C. ????(forced oscillation) ?? L(di/dt) Ri
q/C e0coswt d2q/dt2 2b?dq/dt w02q
A?coswt q(t) qA(w)coswt
d(w) ???C?????? ?? A e0/L qA(w) A/(w02
w2)2 4b2w21/2 ?? d(w) tan-12bw/(w02
w2)
7
?? 1.?????????? 2.????? R, ? ? R/2L 3.???? Y
??????? C ???VC(t) q(t ) C VC(t
) 4.??qA(?) ????(resonance frequency) w
wR at dq(w)/dw 0 (if b2 ltlt w02, wR w0)
???(bandwidth) Dw w - w- ????(quality
factor) Q Q wR/2b wR/Dw