Calculus review for signals
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| derivative/dɪˈrɪvətɪv/ | đạo hàm |
| integral/ˈɪntɪɡrəl/ | tích phân |
Slope and area, read off signals
- The derivative 导数 is the slope; the integral 积分 is accumulated area. Exponentials are their own slopes: $d/dt e^{at} = a\cdot e^{at}$.
- A rectangular pulse's integral is just area; units are a free check ($F \times V/s = A$).
A 2.5 V/s ramp into 100 µF: $i = C dv/dt = 250 \mu A$, and charge $\int i dt = 500 \mu C = C\cdot ΔV = 100\mu \times 5$ — two methods, one answer.
{exercise:uni-ee2x-calc}
A 12 V ramp over 4 s into 220 µF: what is the charging current?
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
i = C dv/dt = 220µ × 3 V/s = 660 µA, constant because the ramp slope is constant; 2.64 mC of charge follows either way.
A sensor pulse decays as 5e^{-t/0.2} mA. The total charge delivered is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The area of A e^{-t/tau} is A tau = 5 x 0.2 = 1.0 mC - the effective-width rule. The curve never touches zero but its area is finite and exact.
A ramp v = 2.5t volts into C = 100 uF draws —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
i = C dv/dt = 100u x 2.5 = 250 uA, constant while the ramp rises - and 0 on the plateau, where the slope dies. Constant VALUE does not mean constant current; constant SLOPE does.