You've successfully subscribed to ムえのBLOG
Great! Next, complete checkout for full access to ムえのBLOG
Welcome back! You've successfully signed in.
Success! Your account is fully activated, you now have access to all content.

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

image-20200514164246526

Phasor Relationships for Resistors Inductors Capacitors

Resistors

image-20200514170202145


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

Inductors

image-20200514171120003
image-20200514171128252


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

Capacitors

image-20200514172115702
image-20200514172124689


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

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.
image-20200514174407838

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:

image-20200514175320700


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

image-20200514175613264
image-20200514175649515

*-▲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


本站总访问量 正在加载今日诗词....