# Resonant Circuits

Eddy current probes typically have a frequency or a range of
frequencies that they are designed to operated. When the
probe is operated outside of this range, problems with the data can
occur. When a probe is operated at too high of a frequency,
resonance can occurs in the circuit. In a parallel circuit
with resistance (R), inductance (X_{L}) and capacitance
(X_{C}), as the frequency increases X_{L} decreases
and X_{C} increase. Resonance occurs when
X_{L} and X_{C} are equal but opposite in
strength. At the resonant frequency, the total impedance of
the circuit appears to come only from resistance since
X_{L} and X_{C} cancel out. Every
circuit containing capacitance and inductance has a resonant
frequency that is inversely proportional to the square root of the
product of the capacitance and inductance.

$f_{Resonant}=\frac{1}{2\pi\sqrt{LC}}$

In eddy current probes and cables, it is commonly stated that capacitance is negligible. However, even circuits not containing discreet components for resistance, capacitance, and inductance can still exhibit their effects. When two conductors are placed side by side, there is always some capacitance between them. Thus, when many turns of wire are placed close together in a coil, a certain amount of stray capacitance is produced. Additionally, the cable used to interconnect pieces of electronic equipment or equipment to probes, often has some capacitance, as well as, inductance. This stray capacitance is usually very small and in most cases has no significant effect. However, they are not negligible in sensitive circuits and at high frequencies they become quite important.

The applet below represents an eddy current probe with a default resonant frequency of about 1.0 kHz. An ideal probe might contain just the inductance, but a realistic probe has some resistance and some capacitance. The applet initially shows a single cycle of the 1.0 kHz current passing through the inductor.

**Exercise 1:** Using your mouse,
adjust the resistance by sliding the slide bar. Does the frequency
change?

**Exercise 2:** Note that changing
the inductance and/or the capacitance changes the resonant
frequency of this resonant circuit. Can you find several
combinations of capacitance and inductance that resonate at 1.0
kHz?