Resonant Circuits

Remember that as frequency increases inductive reactance increases. On the other hand, as frequency increases capacitive reactance decreases. This means that, with any LC circuit, there must be a frequency where the inductive reactance and capacitive reactance are the same. The frequency where inductive reactance and capacitive reactance are equal is called the resonant frequency.

The following graph shows the capacitive reactance of a 100 microfarad capacitor over a range of frequencies from 10Hz to 200Hz. It also shows the inductive reactance of a 100mH inductor over the same range. Notice that he curves cross at 50.4Hz. This is the frequency where the inductive reactance and the capacitive reactance are the same.


At the resonant frequency, the capacitive reactance and inductive reactance are the same. Therefore, the following is true:

At the resonant frequency:

2πƒL   =	1
	—————
	2πƒC

By manipulating this formula according to the rules of the basic algebraic formula we can see that the following is also true:


Applying the commutative property of multiplication we can rearrange the formula as follows.


Applying the rules for the basic algebraic formula again we can get:
 

ƒ2   =	1
	—————
	22π2LC

Finally, taking the square root of the whole formula we can produce the following formula to find the resonant frequency (ƒ_R) of a resonant circuit.:


In the following circuit the resonant frequency would be calculated by plugging the values for the inductor and capacitor into the above formula:
       


The procedure to solve this formula is:
To calculate the resonant frequency for the above circuit on a typical scientific calculator type:

.

2

X

.

0

0

0

2

=

√

X

6

.

2

8

=

1/χ

25.2

This gives a resonant frequency of 25.2 Hz. Therefore, at 25.2 Hz, the capacitive reactance and the inductive reactance are the same.

If either inductance or capacitance increases, the resonant frequency decreases. Likewise, if either the inductance or capacitance decreased, the resonant frequency increases (if anything below the division line in the formula increases, the answer will decrease and vice versa).

Series Resonance

In the following example the inductor and capacitor are connected in series. As shown above, to find the total reactance, the capacitive reactance is subtracted from the inductive reactance. Since the inductive reactance and the capacitive reactance are equal at the resonant frequency, the total reactance will be zero at the resonant frequency.

A series resonant circuit		The impedance of a series resonant circuit with no resistance. Note that the impedance (the total reactance in this case) is zero at the resonant frequency.

At frequencies below the resonant frequency the capacitive reactance will dominate the circuit and the impedance will be greater than zero. At frequencies above the resonant frequency the inductive reactance will dominate the circuit and the impedance will also be greater than zero.

The following circuit has a 20 ohm resistor in series with the capacitor and inductor. Since the capacitive reactance and inductive reactance cancel each other at the resonant frequency, the only thing left is the resistance. Therefore, at the resonant frequency, the impedance is equal to the value of the resistor. Resistors aren't affected by frequency. That is why the resistance is a flat line in the following graph.

Series resonant circuit with resistance		Again, the impedance is higher when the frequency is above or below the resonant frequency.

In a series resonant circuit, the impedance will always be lowest at the resonant frequency. If there is any resistance in the circuit, the value of that resistance will be the impedance at the resonant frequency.

Parallel Resonance

With a parallel circuit at DC, the total impedance will always be lower than the lowest impedance (recall resistors in parallel). However, things are different with a capacitor and inductor in parallel. At DC, the inductor will be a short circuit discounting any resistance of the inductor wire (here, we will assume that the resistance of the wire is zero). At DC, the capacitor will act as an open circuit, discounting any leakage current. The net circuit is a short circuit in parallel with an open circuit. Therefore, at DC, the circuit will have nearly zero impedance. At very high frequencies, the capacitor acts as a short circuit, and the inductor acts as an open circuit. Again we have a short circuit in parallel with an open circuit. Therefore, at very high frequencies, the circuit has nearly zero impedance.

Things are different between these extremes. The inductor and capacitor each store energy during one part of each cycle and release it at the opposite part. The components are also working 180 degrees out of phase, meaning that when the capacitor releases current in one direction, the inductor releases current in the opposite direction. At low frequencies, the inductor dominates, and at high frequencies, the capacitor dominates. Either way, a considerable current flows through the circuit. However, at the resonant frequency, the components attempt to release equal amounts of current in opposite directions. The net result is no current flow. Therefore, at the resonant frequency, the impedance of the circuit approaches infinity.

A Parallel Resonant Circuit		Impedance of a parallel resonant circuit

Another way to look at it is that current flowing one way out of the inductor is sucked up by the capacitor trying to send current in the opposite direction. The net result is no current in that direction. Likewise, current flowing the other way out of the capacitor is sucked up by the inductor trying to send current opposite to the capacitor. Again, there is no net current flow. The external signal feeding the circuit peaks just as the voltage across the circuit peaks with an equal voltage with opposite polarity (180 degrees out of phase). Thus, the circuit appears to be an infinite impedance. If there were no resistance in the inductor, dissipating energy as heat, there would be no current flow through the circuit at the resonant frequency. (I have heard that there are experimental superconducting parallel resonant circuits submerged in liquid nitrogen that have been "ringing" for decades without loss of energy.)

If you add a resistor to a parallel resonant circuit, that resistor will impose a limit on the total impedance of the circuit. At the resonant frequency the capacitive reactance and the inductive reactance combine in such a way that the impedance approaches infinity. Putting a resistor in parallel with that nearly infinite impedance results in a total impedance essentially equal to the value of the resistor.

Over all, the impedance of a parallel resonant circuit is highest at the resonant frequency. If there is any resistance in the circuit, the impedance at the resonant frequency will be the value of that resistor.

Parallel resonant circuit with resistance. R = 20 ohms.		Parallel resonant circuit impedance with 20 ohms of resistance.

Parallel resonant circuit with resistance. R = 90 ohms.		Parallel resonant circuit impedance with 90 ohms of resistance.

Q factor

Notice that with the higher resistance the impedance peaks more sharply. The steepness of the sides of this curve and the narrowness of the area is referred to as "Q" (from the word "quality"). The steeper the sides and the narrower the area, the higher the Q.

In a parallel resonant circuit, the more resistance you have, the higher the Q. In a series resonant circuit, the more resistance you have, the lower the Q. In a parallel resonant circuit with no resistor, the only resistance in the circuit is the inherent resistance of the inductor. This resistance acts in series with the inductor. Therefore, in either a series or parallel resonant circuit, the more resistance inherent in the inductor, the lower the Q. To have resonant circuits with a high Q you need inductors with low resistance.