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:
1 = (
2πƒC) X (2πƒL)
Applying the commutative property of multiplication we can rearrange the formula as follows.
1 = 2^{2}π^{2}ƒ^{2}LC
Applying the rules for the basic algebraic formula again we can get:
_{ } ƒ^{2}_{
= }

1

————— 
2^{2}π^{2}LC

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:

Multiply the inductance by the capacitance.

Take the square root of the product.

Multiply that by 6.28.

Take the reciprocal of that product.
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
In a parallel resonant circuit, the total impedance will always be
lower than the lowest impedance (recall resistors in parallel). At
frequencies below the resonant frequency the inductive reactance will
be lower than the capacitive reactance. Therefore, the total impedance
will be something less than the inductive reactance. At very low
frequencies the capacitor has little effect on the total impedance.
This is not counterintuitive if you remember the rules for parallel
circuits. Recall that placing a 1 megaohm resistor in parallel with a
10 ohm resistor will have little effect on the total resistance.
Likewise, a vary large capacitive reactance in parallel with a very
small inductive reactance will have little effect on the total
reactance. The impedance will essential be the inductive reactance.



A Parallel Resonant Circuit


Impedance of a parallel resonant circuit

At frequencies above the resonant frequency the capacitive reactance
will be lower than the inductive reactance. Therefore, the total
impedance will be something less than the capacitive reactance. At very
high frequencies, the inductor has little effect on the total impedance.
At the resonant frequency the current in the capacitor is equal to the
current in the inductor. These currents are also traveling in opposite
directions. This means that in the overall circuit (including that not
shown in these examples) an equal amount of current is attempting to
travel in both directions at the same time. The result is that very
little current actually flows through the parallel resonant circuit at
the resonant frequency. As shown in the graph above, the total
impedance (total reactance) approaches infinity at the resonant
frequency and the circuit looks like an open circuit.
Although the opposing currents in the parallel resonant circuit prevent
current from flowing through the circuit at the resonant frequency,
current does flow between the inductor and the capacitor. This current
flows back and forth cycling at the resonant frequency. In fact, if the
source of alternating current that is driving the circuit is removed,
these currents will continue to travel back and forth between the
inductor and capacitor several times before the energy is dissipates.
Parallel resonant circuits are sometimes called tank circuits because
of this property of holding energy for a short time at the resonant
frequency.
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. Placing
such a very high impedance in parallel with a relatively low resistance
will have little effect on the total impedance. The total impedance
will essentially be the value of the resistor at the resonant frequency.
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.