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# Lecture 5 RC/RL First-Order Circuits

## Motivation

Previously, we only have static analysis of a circuit.

Two assumptions:

1. Responses at time t only depend on inputs at time t.
2. Circuits respond to input change infinitely fast.

What if without this two assumptions?

dynamic circuit analysis

## Capacitors

1. Passive element
2. Storage element that stores energy in electric field

$$\epsilon 为中间绝缘电介质的介电常数$$
$$E为电场$$
$$C = \frac{\epsilon S}{4\pi kd}$$
$$Q = C V$$

How does current flow through a capacitor?

​ Through charging and discharging the two plates, it seems like current flow through a capacitor.

### V-I characteristic

$$q(t_0) = CV(t_0)$$
$$q(t) = CV(t)$$
$$\int_{t_0}^{t} i(\tau) d\tau = q(t)-q(t_0)$$
$$V(t) = \frac{1}{C}\int_{t_0}^ti(\tau)d\tau +V(t_0)$$
$$i(t) = C \frac{dV(t)}{dt}$$

### Stored Energy

$$P(t) = V(t)i(t) = V(t)C\frac{dV(t)}{dt}$$
$$E = \int_{t_0}^t p(\tau)d\tau =\int_{t_0}^t V(\tau)C \frac{dV(\tau)}{d\tau}d\tau$$
$$=\frac{1}{2}C[V^2(t)-V^2(t_0)]$$

The energy stored in a capacitor with voltage $V(t)$ is
$$W = E(start , from , V(t_0)=0) = \frac{1}{2}CV^2(t)$$

### Important Property

$$i(t) = C \frac{dV(t)}{dt}$$

Is it possible?

If at $t=0$ then $i(t)\arrowvert_{ t = 0 }=\infty$ so it is impossible.

#### Property 1

$V(t)$ cannot change instantaneously

#### Property 2

$V(t):DC \Rightarrow \frac{dV(t)}{dt}=0 \Rightarrow i(t)=0 \Rightarrow opencircuit$

### Capacitors in Series or in Parallel 串并联

#### voltage division

$$v_1 = (\frac{C_2}{C_1+C_2})v_s$$
$$v_2 = (\frac{C_1}{C_1+C_2})v_s$$

#### current division

$$i_1 = (\frac{C_1}{C_1+C_2})i_s$$
$$i_2 = (\frac{C_2}{C_1+C_2})i_s$$

## Inductors

1. Passive element

2. A storage element that stores energy in magnetic field

$$L = \frac{N^2\mu S}{l}$$
$$\Lambda=L\cdot i=N\cdot B\cdot S=N\phi$$

$N$匝数，$\Lambda$ 磁通链

### V-I characteristic

$$V(t)=\frac{d\Lambda(t)}{dt}=\frac{d(li(t))}{dt}=L\frac{di(t)}{dt} \$$