**LRC circuits**

Here we consider circuits with an AC power supply and all three fundamental elements: a resistor, an inductor and a capacitor. The same current flows through each element at every instant in time. The voltages across the four elements must sum to zero when travelling around the loop at any instant as well.

Although the voltages across all the
elements rise and fall at the same frequency, they do not all have the same **phase**.
When two signals have the same frequency but different phases, the sine waves
representing the signals begin oscillations at different times. There is an offset of the
patterns. For instance if we say one signal is 90 degrees behind another, that would mean
that the sine wave would be displaced by one fourth of a period.

A consequence of this is that the (rms voltage across the capacitor) +
(rms voltage across the inductor) + (rms voltage across the resistor) add
up to *MORE* (sometimes much more) that the rms voltage supplied
by the generator.

In
the figure on the right, the voltage drop across each element is shown as a function of
time. The green line, which is the voltage drop across the resistor, also demonstrates the
phase of the current. One can see that * V_{L}* preceeds
the current by one fourth of a period. This is because the inductor's
voltage drop is
largest when the current is rising fast, since it is proportional to

The **impedance** is the ratio of
the peak voltage of the source to the peak current. This impedance depends on the
frequency of the source as well as the values of * L*,

The impedance becomes a minimum (for a given
** R**) when the

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