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## 2.10 Q, quality factor

The Q or quality factor is used by experts of resonators to rate how good a resonator is. It is similar to the time constant but instead of seconds, it is a measure of the decay time (i.e. ring time) of a shock-excited resonator in terms of radians (i.e. 2*π* times the decay time in cycles). It is particularly useful for comparing resonators that resonate at different frequencies. Table 1 lists a few typical *Q*'s.

Table 1. Typical Q's of various resonators
| |
---|---|

LRC resonant circuits | 10 - 1000 |

balance wheel | 100 - 300 |

tuning fork | 1000 |

pendulum clock | 3000 - 100,000 |

quartz crystals in watches | 10,000 - 1,000,000 |

### Definition

The time constant and Q are related by:

*Q* = *ω _{0}τ*

_{energy}, (1)

where *ω _{0}* is the angular resonant frequency of the resonator and

*τ*

_{energy}is the resonator's time constant for energy decay.

We illustrate this in Figure 1, at right. The particular resonator graphed has an energy decay time in cycles of 3.1 cycles or 20 radians. The 20 radians is the *Q* of this resonator. To further understand this, we can consider Equation (1) to be a unit conversion as:

Radians is a non-unit and is usually ignored, unless it is particularly illuminating to call it out. Thus *Q* is considered to be without units, i.e. a pure number.

**The Q of a resonator is the number of radians required for the resonator's energy to decay to 1/ e or 37% of its initial energy when the resonator is shock excited.**

### Power loss

The power loss of a decaying resonator can also be related to its *Q*. Using Equation (a) of Section 2.6 and Equation (1) above, we get

We see that power loss is inversely proportional to the *Q*. The higher the *Q*, the less power is lost. The equation works the other way also, the more power lost, the lower the *Q*. This means that even if the resonator has very low internal losses (such as those due to friction), if we couple an outside device to it that takes power from it so that its power loss is large, then its *Q* will be lower. High *Q* requires both low internal losses and low coupling losses, i.e. weak coupling to the outside.

### Bandwidth

In the next chapter we will discuss a **resonance curve** and **bandwidth** in detail, however a very short summary is in order here. Figure 2 shows the energy in a resonator that is continuously driven by a sinusoidal source as a function of the frequency of that source. The peak occurs at the resonant frequency *f _{0}*. The bandwidth or width of the curve half way up is

*Δf*

_{½}(in units of Hz, i.e. cycles per second). The resonant frequency and bandwidth in units of radians per second are denoted by

*ω*and

_{0}*Δω*

_{½}. These are related to

*Q*by:

Thus we see that the bandwidth is inversely proportional to the *Q* of a resonator; that the higher the *Q*, the smaller the bandwidth and the narrower the resonance curve.

*P. Ceperley Feb. 2010*

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