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# Quality Factor and Bandwidth

Now we will discuss about a factor that, in effect, measures just how close
to perfect a filter or filter component can be. This same factor affects
**bandwidth** and **selectivity**. The factor is known as ** Q**
(

**quality factor**). The higher the

*Q*, the better the filter; the lower the losses, the closer the filter is to being perfect.

Although *Q* may be defined in several ways, a general definition that
applies to any system is based on the ratio of the energy stored in the system
to the energy dissipated per cycle:

This is the fundamental definition of *Q*, and all other definitions are
derived from it. This equation applies to any type of resonant system including
series-tuned and parallel-tuned circuits comprised of inductors and capacitors,
transmission lines, microwave cavities, acoustic organ pipes, mechanical
pendulums, and RC active circuits.

For an inductor or capacitor, *Q* turns out to be the ratio of the
reactance to the resistance. For an inductor,

and for a capacitor,

where *R* in both instances is an equivalent series resistance.
Applied to inductors and capacitors, *Q* is a measure of the
quality of the component. The higher the *Q*, the more nearly
does an inductor or capacitor approach the ideal component.

In a series-tuned circuit it is possible for the voltage across the inductor
to be considerably greater than the voltage applied to the circuit. In fact,
both the inductor and capacitor voltages will be nearly *Q* times the
applied voltage, where *Q* is the quality factor of the overall circuit.
Similarly, in a parallel-tuned circuit the circulating current will be nearly
*Q* times the current entering the circuit.

*Q*of a resonant circuit shows a sharp rise in gain over a narrow band centered at the resonant frequency

*f*

_{C}.

*Q* can be defined in a way that allows it to be used for tuned circuits
as a measure of the selectivity or sharpness of tuning. The response curve
for a tuned circuit is shown in the figure above, and the quality factor,
*Q*, may be obtained as follows:

where *f*_{C} is the center frequency of the tuned circuit,
*f*_{1} is the upper 3-dB frequency, and *f*_{2}
is the lower 3-dB frequency. Notice that since *Q* is a ratio of two
frequencies, it is a dimensionless quantity, so that *Q* =
*ω*_{C}/(*ω*_{1}-*ω*_{2})
is also valid.

As explained below, *f*_{1} and *f*_{2} are often
referred to as the half-power points: Let the power in a circuit having
resistance *R* be *P*. If the voltage across the circuit is *V*,

If the power is halved, then

Thus the power is halved when the voltage is divided by √2. Expressing this in dB,

Accordingly, the half-power points occur at the frequency where the voltage is
3 dB down from the peak. It can also be shown that *f*_{C}
is the geometric mean of *f*_{1} and *f*_{2}, that is,

**Example**:

A bandpass filter has a center frequency of 1000 Hz and a 3-dB bandwidth of
33.33 Hz. Find the circuit *Q* and the 3-dB frequencies.

**Solution:**

From the equation *f*_{C}^{2} =
*f*_{1}*f*_{2}

Also

Combining the previous two equations, and solving the resulting quadratic equation, there is obtained