實驗6: RC 和 RLC 電路(課本實驗21) 目的: 利用示波器觀察 RC 和 RLC 電路中電荷對時間之變化 A: RC電路 1. 電鍵 S 接到 a 點, R,C,電池 e0串聯 iR + q/C = R(dq/dt) + q/C = e0 q(t) = Ce0[1 – exp(-t/RC)] 電容器 C 充電, 時間常數 = RC 2. 充電到qmax, 電鍵S 接到b 點, R,C串聯 R(dq/dt) + q/C = 0 q(t) = qmexp(-t/RC) 電容器放電, t = = RC (鬆弛時間) q(t) = qm/e = qm/2.718 q(T1/2) = qm/2
實驗: 1.用信號產生器之方波(週期T)取代 直流電源 e0 及電鍵 S = 0 for 0 < t < T/2 = -0 for T/2 < t < T 2.用示波器Y(垂直)輸入(並聯)觀察 電容器C之電位降VC(t) = q(t)/C q(t) = C·VC(t) 例如 R = 10 k, C = 0.1 mF, t = RC = 1 ms 選擇方波週期 T ~ 100 ms, 10 ms, 1 ms 觀察q(t) qm = q(t = T/2) -qm = q(t = T)
B. LC 和 RLC 電路 原理: 電容器充電到 qmax,電鍵 S 接上, L,C 串聯 L(di/dt) + q/C = L(d2q/dt2) + q/C = 0 q(t) = qmcos[t/(LC)1/2] = qmcos0t (t = 0, q = qm) 電容器 C 電荷簡諧振盪(harmonic oscillation) 振盪角頻率: 0 = 1/(LC)1/2 但 R ≠ 0(電感器 L 線圈有電阻), 為 RLC 電路 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 電容器C電荷阻尼振盪(damped oscillation)
1. b2 < w02 (R2/4L2 < 1/LC) q(t) = [qmexp(-bt)]cos(w1t) 角頻率 w12 = w02 – b2 > 0 振幅 qm(t) = qmexp(-bt) (令q(0) = qm, 相位 d = 0, 鬆弛時間 t = 1/b = 2L/R) 電容器C電荷較長週期(T1 > T0) 之次阻尼振盪 2. b2 = w02 (R2C/4L = 1) w12 = 0 (振盪週期T1 = ) 臨界阻尼(critical damping) 3. b2 > w02 (R2C/4L > 1) w12 < 0 (w1虛數,無振盪) 過阻尼(overdamping) For R, L, C, e0串聯 d2q/dt2 + 2b·dq/dt + w02q = e0/L = A q(t) = (A/w02)[1 - exp(-bt).cos(w1t)] (令d = 0)
實驗 1.用信號產生器之方波取代直流電源e0及電鍵S e = e0 for 0 < t < T/2 e = -e0 for T/2 < t < T d2q/dt2 + 2bdq/dt + w02q = e/L 2.用可變電阻 R, 調 b = R/2L， 令w12 = w02 – b2 > 0 3.用示波器Y輸入觀察電容器C的電位降VC(t) q(t) = CVC(t) 例: C = 0.001 mF = 1 pF L = 10 mH, R ~ 0-25 kW w02 = 1/LC, b = R/2L
C. 強迫振盪(forced oscillation) 原理: L(di/dt) + Ri + q/C = e0coswt d2q/dt2 + 2bdq/dt + w02q = Acoswt q(t) = qA(w)cos[wt – d(w)] 電容器C電荷強迫振盪 振幅 A = e0/L qA(w) = A/[(w02 – w2)2 + 4b2w2]1/2 相位： d(w) = tan-1[2bw/(w02 – w2)]
實驗: 1.用信號產生器之正弦波 2.用可變電阻 R, 調 = R/2L 3.用示波器 Y 輸入觀察電容器 C 電位降VC(t) q(t ) = C ·VC(t ) 4.觀察qA() 共振頻率(resonance frequency) w = wR at dq(w)/dw = 0 (if b2 << w02, wR ~ w0) 頻帶寬(bandwidth) Dw = w+ - w- 品質因素(quality factor) Q Q = wR/2b ~ wR/Dw