# 串、并联阻抗的等效互换

\begin{align} R_s + jX_s &= \frac{R_p\cdot jX_p}{R_p+jX_p}\\ &=\frac{X_p^2}{R_p^2 + X_p^2} R_p + j\frac{R_p^2}{R_p^2 + X_p^2} X_p \end{align}

$R_s = \frac{X_p^2}{R_p^2 + X_p^2} R_p = \frac{X_p^2}{\vert Z_p \vert^2} R_p\\ X_s = \frac{R_p^2}{R_p^2 + X_p^2} X_p = \frac{R_p^2}{\vert Z_p \vert^2} X_p$

$\frac{R_s - jX_s}{R_s^2 + X_s^2} \leftarrow \frac{1}{R_s + jX_s} = \frac{1}{R_p} + \frac{1}{jX_p}$

$R_p = \frac{R_s^2 + X_s^2}{R_s} = \frac{\vert Z_s\vert^2}{R_s}\\ X_p = \frac{R_s^2 + X_s^2}{Z_s} = \frac{\vert Z_s\vert^2}{X_s}$

$Q_L = \frac{X_s}{R_s} = \frac{R_p}{X_p}$ \begin{align}& \begin{cases} R_s &= \frac{1}{1+Q_L^2} R_p\\ X_s &= \frac{Q_L^2}{1+Q_L^2} X_p \end{cases}\\\\& \begin{cases} R_p &= (1+Q_L^2) R_s\\ X_p &= (1+\frac{1}{Q_L^2}) X_p \end{cases} \end{align}

$R_p \approx Q_L^2 R_s\\ X_p \approx X_s$

# 并联电路的广义形式

$Z_p = \frac{Z_1Z_2}{Z_1+Z_2} = \frac{(R_1+jX_1)(R_2+jX_2)}{(R_1+jX_1)+(R_2+jX_2)}$

\begin{align} Z_p &= \frac{(R_1 +jX_1)(R_2 + jX_2)}{R_1+R_2} \\ &\approx -\frac{X_1X_2}{R_1+R_2}\\ &= \frac{X_1^2}{R_1+R_2} = \frac{X_2^2}{R_1+R_2} \end{align}

注意 注意上式成立的条件是 $X \gg R$ 和 $X_1+X_2=0$

$Z_p = = \frac{(\omega_p L)^2}{R_1+R_2} = \frac{(1/\omega_p C)^2}{R_1+R_2}$

# 抽头式并联电路

$p =\frac{Ij\omega_p \cdot L_1}{Ij\omega_p \cdot (L_1 + L_2)} = \frac{V_{ab}}{V_{db}}$

$Z_{ab} = \frac{(\omega_p L)^2}{R_1+R_2} p^2 = p^2Z_{bd}$

$Z_p = \frac{(R_1+jX_1)(R_2+jX_2)}{(R_1+jX_1)+(R_2+jX_2)}$