The quality factor or ‘Q’ of an inductor or tuned circuit is often used to give an indication of its performance in an RF or other circuit. Values for quality factor are often seen quoted and can be used in defining the performance of an inductor or tuned circuit.

Accordingly this parameter is an important factor in the definition of various RF components and circuits.

**Definition**

The *Q, quality factor,* of a resonant circuit is a measure of the “goodness” or quality of a resonant circuit. A higher value for this figure of merit corresponds to a more narrow bandwith, which is desirable in many applications. More formally, Q is the ratio of reactive power to average power in the circuit reactance and resistance, respectively:

OR

Q = P_{stored}/P_{dissipated} = I^{2}X/I^{2}R

Q = X/R

where: X = Capacitive or Inductive reactance at resonance

R = Series resistance.

This formula is applicable to series resonant circuits, and also parallel resonant circuits if the resistance is in series with the inductor. This is the case in practical applications, as we are mostly concerned with the resistance of the inductor limiting the Q. A practical application of “Q” is that voltage across L or C in a series resonant circuit is Q times total applied voltage. In a parallel resonant circuit, current through L or C is Q times the total applied current.

The Q factor is a dimensionless parameter that indicates the energy losses within a resonant element which could be anything from a mechanical pendulum, an element in a mechanical structure, or within electronic circuit such as a resonant circuit. In particular Q is often used in association with an inductor.

While the Q factor of an element relates the losses, this links directly in to the bandwidth of the resonator with respect to its centre frequency. As such the Q or quality factor is particularly important within RF tuned circuits, filters, etc.

**Other Expressions**

From the definition of quality factor given above, the Q factor can be mathematically expressed as:

When looking at the bandwidth of an RF resonant circuit this translates to:

**Effects of Q factor**

When dealing with RF tuned circuits, there are many reasons why Q factor is important. Usually a high level of Q is beneficial, but in some applications a defined level of Q may be what is required.

Some of the considerations associated with Q in RF tuned circuits are summarised below:

With increasing Q or quality factor, so the bandwidth of the tuned circuit filter is reduced. As losses decrease so the tuned circuit becomes sharper as energy is stored better in the circuit. It can be seen that as the Q increases, so the 3 dB bandwidth decreases and the overall response of the tuned circuit increases.*Bandwidth:*

As the Q of a resonant circuit increases so the losses decrease. This means that any oscillation set up within the circuit will take longer to die away. In other words the circuit will tend to “ring” more. This is actually ideal for use within an oscillator circuit because it is easier to set up and maintain an oscillation as less energy is lost in the tuned circuit.*Ringing:*Any oscillator generates what is known as phase noise. This comprises random shifts in the phase of the signal. This manifests itself as noise that spreads out from the main carrier. As might be expected, this noise is not wanted and therefore needs to be minimised. The oscillator design can be tailored to reduce this in a number of ways, the chief one being by increasing the Q, quality factor of the oscillator tuned circuit.*Oscillator phase noise:*Tuned circuits and filters are often used to remove spurious signals. The sharper the filter and the higher the level of Q, the better the circuit will be able to remove the spurious signals.*General spurious signals:*In many RF applications there is a requirement for wide bandwidth operation. Some forms of modulation require a wide bandwidth, and other applications require fixed filters to provide wide band coverage.*Wide bandwidth:*

The definitions are incorrectly defined here.

The correct ones are:

Damping factor ( represented by alpha or Xi) = R/2L

Damping ratio ( represented by zeta) = Damping Factor/ Resonant Frequency.

The formula that you have mentioned under Damping Factor is actually for Damping Ratio.

Kindly do the needful.