基础高频电路
基本知识：
Q:品质因数，指电路的选频能力
\(\omega\):角速度，关于频率公式：\(\omega = 2 \pi \times f\)
\(\omega_0\):谐振频率的角速度，指该频率下电路阻抗为最小（即既无感性也无容性）
电感元件的高频特性
品质因数\(Q_L\)的计算
- \(Q_{L(Serial)} = \frac{I^2 \omega_0LT}{I^2(r + r_0)T} = \frac{\omega_0 L}{r_{\Sigma}} \rightarrow \frac{X}{R}\)
- \(Q_{L(Parrallel)} = \frac{\frac{U^2}{\omega_0L}}{\frac{U^2T}{r_{\Sigma}}} = \frac{1}{\omega_0Lg_{\Sigma}} \rightarrow \frac{R}{X}\)
\(r_0\):电容损耗电阻，高频范围内会产生损耗
\(g\):电导，即电阻倒数
电容元件的高频特性
- 高频范围内，电容元件等效为理想的无损耗元件
LC谐振回路
LC串联谐振回路
LC电路
- (a)图高频状态下可等效为(b)图，阻抗\(Z = r_0 + j(\omega L - \frac{1}{\omega C})\)
\(\rightarrow \text{当} \omega L = \frac{1}{\omega C} \text{时，为谐振频率的角速度} (\omega \rightarrow \omega_0)\)
\(\omega_0^2 = \frac{1}{LC} \to \omega_0 = \frac{1}{\sqrt{LC}}\) - 得空载品质因数\(Q_0 = \frac{\omega_0 L}{r} = \frac{\omega L}{r_0}\)
LCr电路
- 阻抗
\(Z = r_0 + r_L +j(\omega L - \frac{1}{\omega C})\)
$ \omega_0 = \frac{1}{\sqrt{LC}} $
\(有载品质因数Q_L = \frac{\omega_0 L}{r_\Sigma}= \frac{\omega_0L}{r_0 + r_L} = \frac{1}{\omega_0C(r_0 + r_L)}\)
- 即\(\dot{I}(j\omega) = \frac{U}{r + j(\omega L - \frac{1}{\omega C})} r = r_\Sigma = r_0 + r_L\)
- 谐振电流：\(\frac{\dot{I}(j\omega)}{\dot{I}(j\omega_0)} = \frac{1}{1 + jQ_L(\frac{\omega}{\omega_0} - \frac{\omega_0}{\omega})}\)
- \(\omega \approx \omega_0, \Delta \omega = \omega - \omega_0\)
- \(\rightarrow \text{广义失谐(偏离谐振状态的程度)} \Psi = Q_L (\frac{\omega}{\omega_0} - \frac{\omega_0}{\omega}) = 2Q_L \frac{\Delta \omega}{\omega_0} = 2Q_L \frac{\Delta f}{f_0}\)
\(f_0\): 谐振频率
\(\Delta f\): 当前频率与谐振频率的差值
- 综上所述，相对电流为\(\frac{\dot{I}(j\omega)}{\dot{I}(j\omega_0)} = \frac{1}{j \Psi}\)，其模值为\(\frac{1}{\sqrt{1 + Q_L^2{(\frac{\omega}{\omega_0} - \frac{\omega_0}{\omega})}^2}}\)，相角\(\Phi(\omega) = -arctan[Q_L(\frac{\omega}{\omega_0} - \frac{\omega_0}{\omega})]\)
- 通频带\(B = 2f_{\Delta0.7} = \frac{f_0}{Q_L}\)
通频带通常取谐振状态下信号幅值\(\frac{1}{\sqrt{2}}\)倍的两端，\(\frac{1}{\sqrt{2}} \approx 0.7\)
LC并联谐振回路
LC电路
串并联阻抗变换
- 可将电感的串联等效电阻\(r_0\)转化为并联的等效电阻\(R_0\)，步骤如下
- \(R_0 = (Q_0^2 + 1 )r_0 \approx Q_0^2r_0\)
- \(Y = \frac{1}{r_0 + j\omega L} + j\omega C = G_0(\omega) + jB(\omega)\)
Y : 电导，阻抗倒数，单位为西门子(S) \(Y = \frac{1}{Z}\)
- \(B(\omega) = 0 = \omega C - \frac{\omega L}{r_0^2 + \omega^2L^2}\)
- \(\omega_0 \sqrt{1 - \frac{1}{Q_0^2}} \approx \omega_0\)
\(Q_0 >> 1\)
LCr电路
- 电导\(g = g_0 + g_L = \frac{1}{R_0} + \frac{1}{R_L}\)
- \(\omega_0 = \frac{1}{\sqrt{LC}}\)
- \(Q_L = \frac{1}{\omega_0 Lg} = \frac{R}{\omega L} = \frac{\omega_0C}{g}\)
- 阻抗\(|Z(\omega)| = \frac{1}{Y} = \frac{R}{\sqrt{1 + Q_L^2{(\frac{\omega}{\omega_0} - \frac{\omega_0}{\omega})}^2}}\)
- 相角\(\Phi(\omega) = -arctan Q_L(\frac{\omega}{\omega_0} - \frac{\omega_0}{\omega })\)
串并联变换电路
串并联阻抗等效互换
- 等效即达到\(r_1 + jX_1 = \frac{R_2 \cdot jX_2}{R_2 + jX_2} \to r_1 + jX_1 = \frac{R_2 \cdot X_2^2}{R_2 + X_2^2} + j\frac{X_2 \cdot R_2^2}{R_2^2 + X_2^2}\)
- 由上述公式可得
\[\displaystyle \left\{\begin{array}{l} r_1 = \frac{R_2 \cdot X_2^2}{R_2 + X_2^2}
\\ X_1 = \frac{X_2 \cdot R_2^2}{R_2^2 + X_2^2} \end{array}\right.\]
- 设串联等效电路品质因数为\(Q_1 = \frac{X_1}{r_1}\)，并联等效电路的品质因数为\(Q_2 = \frac{R_2}{X_2}\)
- 得
\(Q_1 = \frac{X_1}{r_1} = \frac{\frac{X_2 \cdot R_2^2}{R_2^2 + X_2^2}}{\frac{R_2 \cdot X_2^2}{R_2 + X_2^2}} = \frac{R_2}{X_2} = Q_2 = Q\)
\(r_1 = \frac{1}{1 + Q^2} R_2\) - 已知\(Q >> 1\)
\(\to 得 r_1 \approx \frac{1}{Q^2} R_2 且X_1 \approx X_2\)