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# Electronic Circuits Lecture 8 Sinusoid Steady-state Analysis

. 3 min read

motivation: 像做小学数学题一样简单去解AC电路

## Phasor Relationships for Resistors Inductors Capacitors

### Resistors

$$\left { \begin{array}{lr} i(t) = I_mcos(\omega t +\phi) \ v(t) = RI_mcos(\omega t +\phi) \end{array} \right. \Rightarrow \left { \begin{array}{lr} \widetilde{I} = I_m \angle\phi \ v(t) = RI_m \angle\phi \end{array} \right.$$

### Inductors

$$v(t) = L\frac{di(t)}{dt} \ \left { \begin{array}{lr} i(t) = I_m cos(\omega t + \phi) \ v(t) = -\omega L I_m sin(\omega t +\phi) = \omega L I_m cos(\omega t + \phi + 90^\omicron) \end{array} \right. \Rightarrow \left { \begin{array}{lr} \widetilde{I} = I_m\angle\phi \ \widetilde{V} = j\omega L \widetilde{I} \end{array} \right.$$

### Capacitors

$$i(t) = C\frac{dv(t)}{dt} \ \left { \begin{array}{lr} v(t) = V_m cos(\omega t + \phi) \ i(t) = -\omega C V_m sin(\omega t +\phi) = \omega C V_m cos(\omega t + \phi + 90^\omicron) \end{array} \right. \Rightarrow \left { \begin{array}{lr} \widetilde{V} = V_m\angle\phi \ \widetilde{I} = \frac{\widetilde{V}}{j\omega C } \end{array} \right.$$

### Impedance/admittance 阻抗/导纳

$$Z = \frac{\widetilde{V}}{\widetilde{I}}\Rightarrow \overbrace{Z = R}^{R},\overbrace{Z = j\omega L}^{L},\overbrace{Z = \frac{1}{j\omega C}}^{C}$$

Z is called impedance , measured in ohms.
$$Z = R + jX, Z = \frac{1}{Y},Y = G+jB$$

ZRXYGB阻抗电阻电抗导纳电导电纳impedanceresistancereactanceadminttanceconductancesusceptance

#### Unit 单位

$Z$ ohm 欧姆，$Y$ siemens 西门子 mho

1. Impedance is not a phasor ! But it is (often) a complex number.
2. Impedance depends on the frequency w.

## Kirchhoff’s Laws in the Phasor Domain

Let $\widetilde{V_1},\widetilde{V_2},\cdots \widetilde{V_n}$be the voltages around a closed loop. Then according to KVL:
$$\widetilde{V_1}+\widetilde{V_2}+\cdots +\widetilde{V_n}=0$$
KCL:
$$\widetilde{I_1}+\widetilde{I_2}+\cdots +\widetilde{I_n}=0$$

## Series Impedance

Once in frequency domain, the impedance elements are generalized, combinations will follow the rules for resistors:

$$Z_{eq} = Z_1+Z_2+\cdots +Z_n$$

*-▲变化还是存在的。

## 解电路的方法

1. Adopt cosme reference
2. Transform circuit into phasor domain
3. Solve the circuit
4. Nodel / Mesh analysis
5. SuperPosition
6. Source transform / Thevenin / Norton
7. Transform solution back into time domain

### Note

1. The method can only be applied to AC steady state
2. the way back to solve the circuit is exactly the same as resistive network