Damping within materials is an interesting phenomenon, as it can be both a desired and an unwanted effect. Frequency-selective behavior is essential in resonators; however, it complicates the operation of microphones and musical instruments. In this blog post we’ll review acoustic attenuation and its implications on frequency domain behavior, and how it can be described by the quality factor, Q.
What is the quality factor? How is it related to damping and loss?
Two definitions exist for quality factor: one is based on energy loss, while the other uses the resonant frequency and bandwidth. These are not equivalent (they are only so for large Q values) so let’s focus on the energy-loss-based one, which is more general. To facilitate that, a quick overview of energies in a mechanical or electrical system shall follow.
In a mechanical structure, there are generally at least two types of complementary energies: one that is stored in the deformed structure (very much like a lumped spring), called strain/elastic/potential energy, and its counterpart, kinetic energy, which is stored in the movement of the elemental domains (much like the kinetic energy in large scale systems, e.g. a moving car). If there are no losses in the system, these two periodically convert from one to the other. Imagine a metal signpost or flagpole or plastic ruler that is slightly(!) bent from its equilibrium state and released: the bent structure stores elastic energy, and as soon as it is released, the elastic energy is converted to kinetic energy as the object tries to return to its original state. Its momentum drives it over the equilibrium towards the other side, while the kinetic energy is converted back to strain energy. And in the ideal lossless world, this goes on forever.
However, once we add losses, in each cycle a fraction of the stored energy will be dissipated (usually as heat). Now, the ratio of this energy loss and the stored energy is the quality factor:
Note here that the Q does not change during device operation. This implies that as the stored energy decreases due to the dissipation, the losses also decrease. The whole ringdown is an exponential process, approximated as , where , f being the drive frequency. To appreciate this, imagine that materials usually exhibit viscous damping: dissipation is proportional to velocity, which in turn relates to stored kinetic energy; the larger the stored energy, the larger the loss. Real-life examples include riding a bike or mixing ingredients for a cake: in both cases, it’s more difficult to perform the task when trying to be fast. The counteracting forces and losses are larger.
(Reproduced from Ref. 1)
The counterpart of this system in the electrical world is an RLC network. If we have a charged capacitor and connect it to an inductor, the stored energy of these two are continuously converted from one to the other. (To see the lack of a steady state: the capacitor would be in steady state with zero current if there was no change in voltage; the inductor would be in steady state with zero voltage if there was no change in current; these two contradict each other and would only be possible at zero voltage and current.)
Now, the loss mechanism here is a bit more straightforward to see: if we add a resistor to the system, as the energy (and current) flows between the reactive elements (L and C), it passes through the resistor and causes Ohmic losses. Note that the direction of current is irrelevant; the loss is always positive. And the larger the energy, the larger the current flow and loss.
The duality between mechanical and electrical domains is especially clear if we sketch the differential equations, strictly for illustrative purposes:
Mechanical:
where m is mass, b is the damping constant and k the spring constant. The partials are acceleration, velocity and position.
(Reproduced from Ref. 2)
Electrical:
where L is inductance, R is resistance and C capacitance. The variables , , and q are the derivative of the current, current and charge, respectively.
(Reproduced from Ref. 3)
These are clearly equivalent by relating the relevant quantities.
What happens during transients? What happens in steady state?
First, think of an RLC circuit, with initially zero energy. Connecting it to a source causes energy to be gradually stored in the reactive elements, while a fraction of this is dissipated on the resistor. But in general cases, Q is much higher than one, and the energy pumped into the system over a cycle is larger than the losses, causing the system to overall charge and store larger and larger amounts of energy.
At some point, however, the stored energy reaches a saturation point: the dissipation over a cycle is the same as the energy pumped into the system over a period. There is no total energy change; whatever is added from the source is dissipated.
A similar mechanism occurs in a mechanical system. If you were to try to move the aforementioned flagpole as fast as possible (don’t get into trouble!), after a certain threshold the input excess energy you provide to the flagpole would be exactly the same as the losses due to air friction and nothing would change.
Okay, but what is a good Q value?
As always, it depends. But to be a bit more specific, let’s look at some examples, keeping in mind that a high Q means a more frequency-selective operation. (The rigorous proof is omitted here but to give some insight imagine a system excited with a unit energy wideband pulse. To conserve energy, the total integral of losses over the whole frequency range must be the same as the input pulse energy; moreover, it must be the same for all, different Q devices. A higher Q results in a higher peak, but this must be narrower than what a lower Q device has to have the same total integral.)
A high Q factor is good in resonators and watches, where operation at a single frequency is desired. The narrow frequency response ensures that outside the band of interest the output signal is minimal, so no crosstalk or interference happens.
A low Q factor is desired when high levels of damping or a more uniform frequency behavior is wanted. Cars or trucks would ideally have springs with large damping: you wouldn’t notice hitting a pothole! In microphones or speakers generally, a low Q is desired to have uniform operation across frequencies; we would ideally have the same pressure amplitude from a unit voltage excitation (or vice versa) across the audible frequency range. Similarly, musical instruments should amplify sound across the frequency range in a way that is pleasant to the ear, without many sharp peaks; hence the complex shapes of guitars and aerophone instruments, such as tubas.
References
- Chowdhury et al., “An experimental study of bicycle aerodynamics”, International Journal of Mechanical and Materials Engineering 6(2):269-274
- https://ctms.engin.umich.edu/CTMS/index.php?example=Introduction§ion=SystemModeling
- https://commons.wikimedia.org/wiki/File:RLC_series_circuit_v2.svg
