Two Coupled LC Circuits

(182) | ||

(183) |

Differentiating with respect to , and dividing by , we obtain the coupled time evolution equations of the system:

where and .

It is fairly easy to guess that the normal coordinates of the system are

(186) | ||

(187) |

Forming the sum and difference of Equations (184) and (185), we obtain the evolution equations for the two independent normal modes of oscillation,

(188) | ||

(189) |

(We can be sure that we have correctly guessed the normal coordinates because the previous two equations do not couple to one another.) These equations can readily be solved to give

(190) | ||

(191) |

where . Here, , , , and are constants determined by the initial conditions. It follows that

As an example, suppose that and . We obtain

(194) | ||

(195) |

where . This solution is illustrated in Figure 18. [Here, and . Thus, the two normal frequencies are and .] Observe the beat modulation generated by the superposition of two normal modes with similar normal frequencies. (See Section 3.6.)

We can also solve the problem in a more systematic manner by specifically searching for a normal mode of the form

(196) | ||

(197) |

Substitution into the time evolution equations (184) and (185) yields the matrix equation

where . The normal frequencies are determined by setting the determinant of the matrix to zero. This gives

(199) |

or

(200) |

The roots of the preceding equation are and . (Again, we have neglected the negative frequency roots, because they generate the same patterns of motion as the corresponding positive frequency roots.) Hence, the two normal frequencies are and . The characteristic patterns of motion associated with the normal modes can be calculated from the first row of the matrix equation (198), which can be rearranged to give

(201) |

It follows that for the normal mode with , and for the normal mode with . We are thus led to Equations (192) and (193), where and are the amplitude and phase of the higher frequency normal mode, whereas and are the amplitude and phase of the lower frequency mode.