Electronic Circuits Lecture 8 Sinusoid Steady-state Analysis

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

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