LC谐振状态轨迹图，相平面分析

理想LC谐振对于典型的LC谐振电路列写动态方程并推导相平面的状态轨迹图{ur−LdirdtirCdurdt→{urir−LdirCdur→CurdurLirdir0→0.5Cur20.5Lir2Const→ur2LCir2ur02LCir02\left\{ \begin{array}{l} {u_r} - L\frac{{d{i_r}}}{{dt}}\\ {i_r} C\frac{{d{u_r}}}{{dt}} \end{array} \right. \to \left\{ \begin{array}{l} \frac{{{u_r}}}{{{i_r}}} - \frac{{Ld{i_r}}}{{Cd{u_r}}}\\ \to C{u_r}d{u_r} L{i_r}d{i_r} 0\\ \to 0.5C{u_r}^2 0.5L{i_r}^2 Const\\ \to {u_r}^2 {\textstyle{L \over C}}{i_r}^2 {u_{r0}}^2 {\textstyle{L \over C}}{i_{r0}}^2 \end{array} \right.{ur​−Ldtdir​​ir​Cdtdur​​​→⎩⎨⎧​ir​ur​​−Cdur​Ldir​​→Cur​dur​Lir​dir​0→0.5Cur​20.5Lir​2Const→ur​2CL​ir​2ur0​2CL​ir0​2​也就是说谐振电压与谐振电流呈现椭圆形轨迹。考虑直流偏置的LC谐振对于含有直流电压偏置的LC谐振系统对应动态方程修改为{Eur−LdirdtirCdurdt→{Eurir−LdirCdur→C(Eur)durLirdir0→0.5Cur2CEur0.5Lir2const→(urE)2LCir2(ur0E)2LCir02\left\{ \begin{array}{l} E {u_r} - L\frac{{d{i_r}}}{{dt}}\\ {i_r} C\frac{{d{u_r}}}{{dt}} \end{array} \right. \to \left\{ \begin{array}{l} \frac{{E {u_r}}}{{{i_r}}} - \frac{{Ld{i_r}}}{{Cd{u_r}}}\\ \to C\left( {E {u_r}} \right)d{u_r} L{i_r}d{i_r} 0\\ \to 0.5C{u_r}^2 CE{u_r} 0.5L{i_r}^2 const\\ \to {\left( {{u_r} E} \right)^2} {\textstyle{L \over C}}{i_r}^2 {\left( {{u_{r0}} E} \right)^2} {\textstyle{L \over C}}{i_{r0}}^2 \end{array} \right.{Eur​−Ldtdir​​ir​Cdtdur​​​→⎩⎨⎧​ir​Eur​​−Cdur​Ldir​​→C(Eur​)dur​Lir​dir​0→0.5Cur​2CEur​0.5Lir​2const→(ur​E)2CL​ir​2(ur0​E)2CL​ir0​2​包含直流偏置以后谐振圆发生平移半径增大振幅增大。