# Electric Resonant Circuits !

Because both inductive and capacitive reactance are frequency-dependent, so is the impedance *Z* of an oscillating circuit. The current *I* in the circuit is at a maximum when the impedance is at a minimum. This happens when the inductive and capacitive reactances are equal, i.e. *X*_{L} = *X*_{C}. The second added component under the root sign becomes zero, leaving the impedance as:

In this case, the two reactances cancel each other out precisely and the series oscillating circuit acts purely as an active resistance. This condition is termed *resonance*, hence resonant circuit. The corresponding *resonant frequency f*_{r} is determined as follows:

The vector diagram below represents the condition of resonance. The voltages across the capacitance and inductance in this instance cancel each other out exactly, so that even a small supply voltage *U* = *U*_{R} = *I*·*R* is sufficient to achieve the high values for *U*_{C} and *U*_{L}. As indicated in the vector diagram, the current *I* at resonance is in phase with the supply voltage *U*.

A series resonant circuit's resistance reaches a minimum at resonance. In this case, the coil and capacitor both experience a voltage peak (voltage resonance). |

The ratio between either of the voltage components and the total voltage at resonance is termed the resonant circuit's *quality Q*:

At resonance, each of the two voltage components here amounts to *Q* times the applied voltage. Because the voltage components in a series circuit are proportional to the associated resistances, the quality is also determined as follows:

Substituting the formula for the resonant frequency results in the following:

This equation permits the quality to be determined directly from component parameters. Series resonant circuits are used, for instance, to suppress the resonant component of a mixture of frequencies. According to the equation above, an oscillating circuit's quality increases as the ohmic (loss) resistance *R* decreases.

**Parallel resonant circuits:**

Connecting an inductor coil and capacitor in parallel results in a *parallel resonant circuit* as illustrated below.

The *conductance* of a parallel circuit comprising a capacitance and inductance is zero when the capacitive and inductive *susceptances* are identical. In this case, the coil and capacitor current are equal and cancel each other out due to their mutual phase displacement of 180°. Consequently, the circuit acts as a pure active resistance *R*_{r} conducting a purely active current *I*_{R}. The right-hand vector diagram below illustrates the *resonant* mode. As in the case of a series resonant circuit, the current *I* at resonance is in phase with the supply voltage *U*. For comparison, the left-hand vector diagram represents a sub-resonant frequency.

At resonance, the coil and capacitor current can therefore attain very high values exceeding the input current by many times.

At resonance, a parallel resonant circuit's resistance attains a maximum and the coil and capacitor experience excessive current (current resonance). |

Because resonance requires inductive and capacitive susceptances to be identical, the resonant frequency is governed by the same equation as for series resonant circuits:

The ratio between either current component and the total current is termed the parallel resonant circuit's quality:

At resonance, each of the two current components is *Q* times the total current. Again using the formula derived previously for series resonant circuits, the quality can be calculated directly based on component parameters.

Parallel resonantcircuits are used, for instance, to filter a certain (resonant) frequency out of a mixture of frequencies. Because the resonant circuit's resistance reaches a maximum at resonance, its amplitude response looks like the following:

This characteristic is inverted as compared to that of a series resonant circuit. Because this kind of filter only allows passage of a particular bandwidth around the resonance frequency, it is termed a *band-pass filter*. Between an antenna and earth, for instance, all the frequencies receivable by the antenna contribute to the output. Connecting a parallel resonant circuit between the antenna and earth effectively short-circuits all frequencies except for the resonant frequency. The voltage source (antenna) therefore operates practically as if it had no load at the resonant frequency and were short-circuited for all other frequencies.

**Series resonant circuits: **

So far, we have merely considered AC circuits containing reactances of a *single* kind (capacitor or coil). Combining both these types of reactance results in an *oscillating circuit*. Due to their selective frequency properties they are usually called *resonant circuits.*

An resonant circuit includes both an inductance and a capacitance. |

Oscillations are produced through periodic build-up and decay of an electric field (capacitance) or magnetic field (inductance). The following animation illustrates this.

The circuit diagram below shows a *series resonant circuit* consisting of a capacitor, inductor and an active resistance. The latter might also represent, for instance, the ohmic resistance of the coil .

Shown below is a vector diagram of voltages in this circuit. The voltages *U*_{C} and *U*_{L} across the capacitor and inductor are effectively in opposite directions.

For this reason, the total reactance of a series circuit is the *difference* between the inductive and capacitive reactance. The series circuit's apparent resistance is therefore as follows:

__Bandwidth:__

As already mentioned, a series resonant circuit can be used to filter out certain signal frequencies. This type of circuit is also termed a *notch filter *or more usually a *band-stop filter*. The diagram below shows a filter circuit with an input voltage *U*_{e} and output voltage *U*_{a}.

This circuit acts as a frequency-dependent voltage divider where the output voltage is tapped via the reactance *X* = *X*_{L} - *X*_{C}. At very low frequencies, the capacitor behaves like a very high resistance, so that the output voltage is nearly identical to the input voltage. The same applies at very high frequencies, due to the correspondingly high inductive reactance. At the resonant frequency *f*_{r}, the total reactance is zero, so that the output voltage is also zero in this instance. The greater the difference between the input voltage frequency and resonant frequency, the higher the reactance and the output voltage. The diagram below shows the amplitude response (i.e. ratio between output and input voltages) of a band-stop filter near the resonant frequency (resonance peak).

Another characteristic parameter of a resonant circuit of this kind is its *bandwidth B* which indicates the breadth of the frequency range blocked by the filter. The bandwidth is the range between the sub-resonance frequency at which the output voltage has dropped to 0.707 (i.e. 1/Ö2) times the maximum value and the corresponding frequency above the resonance level. The diagram above demonstrates how to determine bandwidth from amplitude response. The narrower the bandwidth, the steeper the drop in the characteristic near the resonant frequency, i.e. the more accurate the filter. The quotient of the resonant frequency and bandwidth is equivalent to the quality *Q* of the circuit, i.e.

The higher an oscillating circuit's quality, the steeper and narrower the resonance peak.