並列共振回路

電源電圧を $v=v_0sin\omega t$ 　抵抗、コイル、コンデンサーの電流を$i_r,i_L,i_c$とする

$$i_R = \frac{{v_0 }}{R}\sin \omega t$$ (32)

$$L\frac{{di}}{{dt}} = v　 Lの回路方程式$$ (33)

$$\frac{{di}}{{dt}} = \frac{v}{L} = \frac{{v_0 \sin \omega t}}{L}$$ (34)

$$i_L = \int {\frac{di}{dt}dt = \frac{1}{L}\int {v_0 \sin \omega t }} dt$$ (35)

$$i_L = - \frac{{v_0 }}{{\omega L}}\cos \omega t = \frac{{v_0 }}{{\omega L}}\sin \left( {\omega t - \frac{\pi }{2}} \right)$$ (36)

$$\frac{q}{C} = v　Cの回路方程式　i = \frac{{dq}}{{dt}}　連続方程式$$ (37)

$$i = C\frac{{dv}}{{dt}} = C\frac{d}{{dt}}\left( {v_0 \sin \omega t} \right)$$ (38)

$$i_C = \omega Cv_0 \cos \omega t = \frac{{v_0 }}{{{\raise0.7ex\hbox{$1$}... ...lower0.7ex\hbox{${\omega C}$}}}}\sin \left( {\omega t + \frac{\pi }{2}} \right)$$ (39)

$$I = i_R + i_C + i_L$$ (40)

$$= \frac{{v_0 }}{R}\sin \omega t + \omega Cv_0 \cos \omega t - \frac{{v_0 }}{{\omega L}}\cos \omega t$$ (41)

$$= \frac{{v_0 }}{R}\sin \omega t + v_0 \left( {\omega C - \frac{1}{{\omega L}}} \right)\cos \omega t$$ (42)

$$I = v_0 \sqrt {\left( {\frac{1}{R}} \right)^2 + \left( {\omega C - \frac{1}{{\omega L}}} \right)^2 } \sin \left( {\omega t + \delta } \right)$$ (43)

$$\tan \delta = \frac{{\omega C - \frac{1}{{\omega L}}}}{{\frac{1}{R}}}$$ (44)

$$I_0 = v_0 \sqrt {\left( {\frac{1}{R}} \right)^2 + \left( {\omega C - \frac{1}{{\omega L}}} \right)^2 }$$ (45)

$$I_{eff} = V_{eff} \sqrt {\left( {\frac{1}{R}} \right)^2 + \left( {\omega C - \frac{1}{{\omega L}}} \right)^2 }$$ (46)

$$Z = \frac{{V_{eff} }}{{I_{eff} }} = \frac{1}{{\sqrt {\left( {\frac{1}{R}} \right)^2 + \left( {\omega C - \frac{1}{{\omega L}}} \right)^2 } }}$$ (47)

$$I_{eff} = \frac{{I_0 }}{{\sqrt 2 }}　　V_{eff} = \frac{{v_0 }}{{\sqrt 2 }}$$ (48)

$$\omega C = \frac{1}{{\omega L}}$$ (49)

$$\omega ^2 = \frac{1}{{LC}}$$ (50)

$$\omega = \frac{1}{{\sqrt {LC} }}$$ (51)

$$f_0 = \frac{1}{2\pi \sqrt {LC}} 　　共振周波数$$ (52)

共振時には電圧が最大になる。