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
Notes
- Impedance is not a phasor ! But it is (often) a complex number.
- 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
- Adopt cosme reference
- Transform circuit into phasor domain
- Solve the circuit
- Nodel / Mesh analysis
- SuperPosition
- Source transform / Thevenin / Norton
- Transform solution back into time domain
Note
- The method can only be applied to AC steady state
- the way back to solve the circuit is exactly the same as resistive network