# Quality Factor / Q Factor Tutorial

### Quality Factor, Q, Tutorial includes

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.

## Q, quality factor basics

The concept of quality factor is one that is applicable in many areas of physics and engineering and it is denoted by the letter Q and may be referred to as the Q factor.

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..

The Q indicates energy loss relative to the amount of energy stored within the system. Thus the higher the Q the lower the rate of energy loss and hence oscillations will reduce more slowly, i.e. they will have a low level of damping and they will ring for longer.

For electronic circuits, energy losses within the circuit are caused by resistance. Although this can occur anywhere within the circuit, the main cause of resistance occurs within the inductor. Accordingly inductor Q is a major factor within resonant circuits.

## Effects of Q

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:

• Bandwidth:   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.
• Ringing:   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.
• Oscillator phase noise:   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.
• General spurious signals:   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.
• Wide bandwidth:   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. While high rejection of unwanted signals may be required, there is a competing requirement for wide bandwidths. Accordingly in many applications the level of Q required needs to be determined to provide the overall performance that is needed meeting requirements for wide bandwidth and adequate rejection of unwanted signals.

## Quality factor definition

The definition of quality factor is often needed to give a more exact understanding of what this quantity actually is.

For electronic circuits, Q is defined as the ratio of the energy stored in the resonator to the energy supplied by a to it, per cycle, to keep signal amplitude constant, at a frequency where the stored energy is constant with time.

It can also be defined for an inductor as the ratio of its inductive reactance to its resistance at a particular frequency, and it is a measure of its efficiency.

## Q factor equations

The basic Q or quality factor equation is based upon the energy losses within the inductor, circuit or other form of component.

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:

Q of a tuned circuit with respect to its bandwidth

Within any RF or other circuit, each individual component can contribute to the Q or quality factor of the circuit network as a whole. The Q of the components such as inductors and capacitors are often quoted as having a certain Q or quality factor.

## Quality factor and damping

One aspect of Q that is of importance in many circuits is the damping. The Q factor determines the qualitative behaviour of simple damped oscillators and affects other circuits such as the response within filters, etc.

There are three main regimes which can be considered when referring to the damping and Q factor.

• Overdamped (Q < 1/2):   An over-damped system is one where the Q factor is less than 1/2. In this type of system, the losses are high and the system has no overshoot, but instead the system will exponential decay, approaching the steady state value asymptotically after a step impulse is applied. As the quality factor is reduced, so the systems responds more slowly to a step impulse.
• Underdamped (Q > 1/2) :   An under-damped system is one where the Q factor is greater than a half. Those systems where the Q factor is only just over a half may oscillate once or twice when a step impulse is applied before the oscillation falls away. As the quality factor increases, so the damping falls and oscillations will be sustained for longer. In a theoretical system where the Q factor is infinite, the oscillation would be maintained indefinitely without the need for adding any further stimulus. In oscillators some signal is fed back to provide an additional stimulus, but a high Q factor normally produces a much cleaner result. Lower levels of phase noise are present on the signal.
• Critically damped (Q = 1/2) :   Like an over-damped system, the output does not oscillate, and does not overshoot its steady-state output. The system will approach the steady-state asymptote in the fastest time without any overshoot.

When choosing defining the Q factor for a system, it is common to opt for the highest level. In this way the optimum performance is normally achieved. However there are instances where lower levels of Q may be advantageous.

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