Capacitors, inductors and energy
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| capacitor/kəˈpæsɪtə/ | tụ điện |
| capacitance/kəˈpæsɪtəns/ | điện dung |
| inductor/ɪnˈdʌktə/ | cuộn cảm |
| inductance/ɪnˈdʌktəns/ | điện cảm |
| flyback diode/ˈflaɪbæk ˈdaɪəʊd/ | diode giải phóng năng lượng dư |
Two storage elements, two laws
- A capacitor 电容器 stores separated charge:
q = Cvandi = C dv/dt, with capacitance 电容Cin farads. An inductor 电感器 stores magnetic energy:v = L di/dt, with inductance 电感Lin henrys. - At steady DC the capacitor is an open circuit ($dv/dt = 0$ so $i = 0$) and the inductor a short ($di/dt = 0$ so $v = 0$).
A 220 µF capacitor ramping 0 → 10 V in 50 ms carries $i = C dv/dt = 220\mu \times 200 = 44 mA$ during the ramp, then zero.
{exercise:uni-ee16-laws}
At DC steady state, what are a capacitor and an inductor equivalent to?
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Steady DC means dv/dt = 0 and di/dt = 0, so i_C = 0 (open) and v_L = 0 (short).
Combinations are mirror images
- Capacitances in parallel add; in series they divide (
47 ∥ 47 = 23.5 µF). Inductances do the opposite: series adds, parallel divides. - Stored energy depends only on the present state: $w_{C} = ½Cv^{2}$, $w_{L} = ½Li^{2}$.
A 100 µF capacitor at 12 V holds 7.2 mJ. A 20 mH inductor at 3 A holds $½ \times 0.02 \times 9 = 90 mJ$.
{exercise:uni-ee16-energy}
A 100 µF capacitor is charged to 12 V. How much energy does it store?
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
w = ½Cv² = ½ × 100×10⁻⁶ × 144 = 7.2 mJ. Forgetting the ½ doubles it; using v instead of v² gives 1.2 mJ.
Stored quantities cannot jump
- Capacitor voltage cannot jump (that would need infinite current); inductor current cannot jump (that would need infinite voltage). Continuity of the stored quantity is the key to every switching problem.
- Interrupting an inductor is therefore dangerous: stopping 1.5 A in 1 ms through 0.1 H generates $v = 0.1 \times 1500 = 150 V$ — the arc across the opening contact.
A flyback diode 续流二极管 across the coil gives the current a circulating path, so di/dt is set by the coil's own resistance and the spike collapses to about a diode drop.
{exercise:uni-ee16-continuity}
Why can inductor current not jump to zero when a switch opens?
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
v = L di/dt — driving i to zero in ~0 time needs an enormous di/dt, hence an enormous voltage that arcs to keep the current flowing.