Resonance in a Tuned Circuit

For every combination of L and C, there is only ONE frequency (in both series and parallel circuits) that causes XL to exactly equal XC; this is known as the "PERFECTLY-TUNED-FREQUENCY". When this frequency is fed to a series or parallel circuit, XL becomes equal to XC, and the circuit is said to be in perfect tune. The circuit is now called a "PERFECTLY-TUNED-CIRCUIT"; The circuit condition wherein XL becomes equal to XC is known as RESONANCE .

Each LCR circuit responds to its tuned frequency
differently than it does to any other. Because of this, an LCR circuit
has the ability to separate frequencies. For example, suppose the TV or
radio station you want to see or hear is broadcasting at the
resonant-frequency. The LC "tuner" in your set can divide the
frequencies, picking out the "tuned-frequency" and rejecting the other
frequencies. Thus, the tuner selects the station you want and rejects
all other stations. If you decide to select another station, you can
change the frequency by tuning the "circuit" to the desired frequency.

RESONANT-FREQUENCY

As stated before, the frequency at which XL equals XC (in a given circuit) is known as the tuned-frequency of that circuit. Based on this, the following formula has been derived to find the exact tuned-frequency when the values of circuit components are known:

There are two important points to remember about this formula. First, the tuned frequency found when using the formula will cause the reactances (XL and XC) of the L and C components to be equal. Second, any change in the value of either L or C will cause a change in the tuned frequency.

An increase in the value of either L or C, or both L and C, will lower the tuned frequency of a given circuit. A decrease in the value of L or C, or both L and C, will raise the tuned frequency of a given circuit.

The symbol for a tuned frequency used in this text is f. Dif-ferent texts and references may use other symbols for a tuned frequency, such as fo, Fr, and fR. The symbols for many circuit parameters have been standardized while others have been left to the discretion of the writer. When you study, apply the rules given by the writer of the text or reference; by doing so, you should have no trouble with nonstandard symbols and designations.

The tuned-frequency formula in this text is:

By substituting the constant .159 for the quantity

the formula can be simplified to the following:

Let's use this formula to figure the tuned-frequency (fr). The circuit is shown in the practice tank circuit of the figure below.

Practice tank circuit.

The important point here is not the formula nor the mathematics. In fact, you may never have to compute a tuned-frequency. The important point is for you to see that any given combination of L and C can be perfectly tuned at only one frequency; in this case, 205 kHz.

The universal reactance curves of capacitive and inductive reactance are joined in the figure below to show the relative values of XL and XL at resonance, below the perfect tuning spot, and above.

Relationship between XL and XC as frequency increases.

First, note that fr, (the perectly tuned frequency) is that frequency (or point) where the two curves cross. At this point, and ONLY this point, XL equals XC. Therefore, the frequency indicated by fr is the one and only frequency of resonance. Note the resistance symbol which indicates that this perfectly tuned spot all reactance is cancelled and the circuit impedance is effectively purely resistive. Remember, a.c. circuits that are resistive have no phase shift between voltage and current. Therefore, at resonance, phase shift is cancelled. The phase angle is effectively zero.

Second, look at the area of the curves to the left of fr. This area shows the relative reactances of the circuit at frequencies BELOW resonance. To these LOWER frequencies, XC will always be greater than XL. There will always be some capacitive reactance left in the circuit after all inductive reactance has been cancelled. Because the impedance has a reactive component, there will be a phase shift. We can also state that below fr the circuit will appear capacitive.

Lastly, look at the area of the curves to the right of f. This area shows the relative reactances of the circuit at frequencies ABOVE resonance. To these HIGHER frequencies, XL will always be greater than XC. There will always be some inductive reactance left in the circuit after all capacitive reactance has been cancelled. The inductor symbol shows that to these higher frequencies, the circuit will always appear to have some inductance. Because of this, there will be a phase shift.

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