|
|
|
|
Q factor
|
| |
|
| |
In physics and engineering the quality factor or Q factor is a dimensionless parameter that compares the time constant for decay of an oscillating physical system's amplitude to its oscillation period. Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy. A higher Q indicates a lower rate of energy dissipation relative to the oscillation frequency, so the oscillations die out more slowly.
Discussion
Ask a question about 'Q factor' Start a new discussion about 'Q factor' Answer questions from other users |
Encyclopedia
In physics and engineering the quality factor or Q factor is a dimensionless parameter that compares the time constant for decay of an oscillating physical system's amplitude to its oscillation period. Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy. A higher Q indicates a lower rate of energy dissipation relative to the oscillation frequency, so the oscillations die out more slowly. For example, a pendulum suspended from a high-quality bearing, oscillating in air, would have a high Q, while a pendulum immersed in oil would have a low one. The concept originated in electronic engineering, as a measure of the 'quality' desired in a good tuned circuit or other resonator.
Generally Q is defined to be
or, more intuitively,
where is defined to be the angular frequency of the circuit (system),
and the energy stored and power loss are properties of a system under consideration.
Usefulness of Q
The Q factor is particularly useful in determining the qualitative behavior of a second-order linear time invariant (LTI) system (i.e., an LTI system with two poles).
- A system with low quality factor (i.e., ) is said to be overdamped. Such a system has two real and distinct poles, and so its impulse response will be the sum of two decaying exponential functions that have different rates of decay corresponding to the different poles. As the quality factor decreases, one pole gets smaller, and the other gets larger. The smaller pole (i.e., the slower decay) dominates the system's response, which results in a relatively slow system. A second-order low-pass filter with a very low quality factor has a nearly first-order step response; the system's output responds to a step input by slowly rising toward an asymptote.
- A system with high quality factor (i.e., ) is said to be underdamped. Such a system has two complex poles that are conjugates of each other. Because the poles are imaginary, the system has an oscillatory character that is damped by the real part of the poles. As the quality factor increases, the amount of damping decreases, and so a purely oscillatory system would have infinite quality factor. Likewise, a high-quality bell rings with a single pure tone endlessly after being struck. More generally, the output of a second-order low-pass filter with a very high quality factor responds to a step input by quickly rising above and oscillating around its eventually steady-state value; these oscillations are subject to exponential decay.
- A system with an intermediate quality factor of is said to be critically damped. Such a system has two real repeated poles, and so it behaves like the cascade connection of two first-order systems. Like an underdamped response, the output of such a system responds quickly to a unit step input. Like an overdamped system, the output does not overshoot its steady-state output (i.e., it approaches a steady-state asymptote). Critical damping results in the fastest response (approach to the final value) possible without overshoot. Real system specifications usually allow some overshoot for a faster initial response or require a slower initial response to provide a safety margin against overshoot.
In negative feedback systems, the dominant closed-loop response is often well-modeled by a second-order system. The phase margin of the open-loop system sets the quality factor Q of the closed-loop system; as the phase margin decreases, the approximate second-order closed-loop system is made more oscillatory (i.e., has a higher quality factor).
Quality factors of common systems
- A Sallen–Key filter topology with equivalent capacitors and equivalent resistors is critically damped (i.e., ).
- A Butterworth filter (i.e., continuous-time filter with the flattest passband frequency response) has an underdamped .
- A Bessel filter (i.e., continuous-time filter with flattest group delay) has an underdamped .
Physical interpretation of Q
Physically speaking, Q is times the ratio of the total energy stored divided by the energy lost in a single cycle or equivalently the ratio of the stored energy to the energy dissipated per one radian of the oscillation.
Equivalently (for large values of Q), the Q factor is approximately the number of oscillations required for a freely oscillating system's energy to fall off to , or about 1/535, of its original energy.
When the system is driven by a sinusoidal drive, its resonant behavior depends strongly on Q.
Resonant systems respond to frequencies close to their natural frequency much more strongly than they respond to other frequencies. A system with a high Q resonates with a greater amplitude (at the resonant frequency) than one with a low Q factor, and its response falls off more rapidly as the frequency moves away from resonance. Thus, a high Q tuned circuit in a radio receiver would be more difficult to tune with the necessary precision, but would have more selectivity; it would do a better job of filtering out signals from other stations that lay nearby on the spectrum. The width (bandwidth) of the resonance is given by
,
where is the resonant frequency, and , the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value.
The factors Q, damping ratio ?, and attenuation a are related such that
So the quality factor can be expressed as
and the exponential attenuation rate can be expressed as
For any 2nd order low-pass filter, the response function of the filter is
For this system, when (i.e, when the system is underdamped), it has two complex conjugate poles that each have a real part of . That is, the attenuation parameter represents the rate of exponential decay of the oscillations (e.g., after an impulse) of the system. A higher quality factor implies a lower attenuation, and so high Q systems oscillate for long times. For example, high quality bells have an approximately pure sinusoidal tone for a long time after being struck by a hammer.
Electrical systems
For an electrically resonant system, the Q factor represents the effect of electrical resistance and, for electromechanical resonators such as quartz crystals, mechanical friction.
RLC circuits
In a series RLC circuit, and in a tuned radio frequency receiver (TRF) the Q factor is:
,
where , and are the resistance, inductance and capacitance of the tuned circuit, respectively.
In a parallel RLC circuit, Q is equal to the reciprocal of the above expression.:
Complex impedances
For a complex impedance
the Q factor is the ratio of the reactance to the resistance (or equivalently, the absolute value of the ratio of reactive power to real power), that is:
Thus, one can also calculate the Q factor for a complex impedance by knowing just the power factor of the circuit
or just the tangent of the phase angle
where is the phase angle and is the power factor of the circuit.
Mechanical systems
For a single damped mass-spring system, the Q factor represents the effect of simplified viscous damping or drag, where the damping force or drag force is proportional to velocity. The formula for the Q factor is:
,
where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation , where is the velocity.
Optical systems
In optics, the Q factor of a resonant cavity is given by
,
where is the resonant frequency, is the stored energy in the cavity, and is the power dissipated. The optical Q is equal to the ratio of the resonant frequency to the bandwidth of the cavity resonance. The average lifetime of a resonant photon in the cavity is proportional to the cavity's Q. If the Q factor of a laser's cavity is abruptly changed from a low value to a high one, the laser will emit a pulse of light that is much more intense than the laser's normal continuous output. This technique is known as Q-switching.
See also
Further reading
External links
|
| |
|
|
