LC circuits

An LC circuit is a closed loop with just two elements: a capacitor and an inductor. It has a resonance property like mechanical systems such as a pendulum or a mass on a spring: there is a special frequency that it likes to oscillate at, and therefore responds strongly to. LC circuits can be used to tune in to a specific frequency, for example in the station selector of a radio or television.

In an LC circuit, electric charge oscillates back and forth just like the position of a mass on a spring oscillates. To demonstrate the analogy, we list several corresponding equations for a mechanical spring and an LC circuit.


mechanical spring

LC circuit

position:      x charge:      Q
velocity:     wpe5.gif (999 bytes) current:     wpe7.gif (1031 bytes)
kinetic energy:     wpe6.gif (1046 bytes) inductor's energy:     wpe8.gif (1023 bytes)
potential energy:     wpe9.gif (1037 bytes) capacitor's energy:     wpe2.gif (1235 bytes)
Eq. of motion:     wpe5.gif (1172 bytes) Kirchhoff's law:     
frequency:      wpe4.gif (1161 bytes) frequency:     wpe3.gif (1186 bytes)

The parameters that determine the motion of a spring are the mass m, spring constant k, the position x, and the velocity v which is the rate of change of x. The parameters that determine the behavior of an LC circuit are L, C, Q and I which is the rate of change of Q. Thus there is a one-to-one correspondence since the equations of motion are identical given the substitutions:

The characteristic frequency of an LC circuit is the frequency at which large amplitudes are built up when a driving force is applied at that frequency. A child on a swing will be sensitive to a pushing force which comes regularly with the natural frequency of the swing. A force that comes at a different frequency will not build up a large amplitude as it will often be pushing against the child's motion. The magic frequency is called the resonant frequency.

Examples       AC circuits index        Lecture index