Filter Resonators and Quality Factor

The phenomenon of resonance exists in nature in many forms. In response to a suitable excitation, a natural vibration occurs and the frequency of vibration with highest amplitude is the resonant frequency. Humans have early exploited this phenomenon in the form of musical instruments, where the flute is perhaps among the earliest creations. In the flute, resonance manifests itself only if the air in the flute is excited by a sharp blow of air from the mouthpiece. Resonance does not show up unless it is suitably excited. Resonance excitation requires that energy is brought to the resonance capable structure in a suitable way.

Fundamentally, resonance is an exchange of energy from one storage device to another. In the context of rf filters we will of course consider electrical resonance. Electrical resonance exists in lumped element form when a circuit contains an inductance and a capacitance. The inductance is capable of storing energy in a magnetic field whereas the capacitance stores energy in an electric field. LC resonance occurs at a frequency where both elements are having the same energy storage capability. Since stored energy is proportional to the reactance of a lumped element, we can write:

omega*L = 1/omega*C

In distributed element resonators, like coaxial quarterwave resonators (also called: combline resonators) or in waveguide cavity resonators, the energy is stored in either the electric field or in the magnetic field. The energy moves within the resonator between locations where only one of the two field types can exist.

All real-world resonators are characterised by at least two parameters:

  • the resonant frequency f_res
  • the unloaded quality factor Qu

The unloaded quality factor exists because of the losses in the resonator. Hypothetical resonators may well have an infinite Qu (lossless resonators) and during the filter design process (synthesis) we actually deliberately assume this in order to simplify the process.

But what is this Qu ? Let’s start with the Qu of a series resonant lumped element (LC) resonator. Given that a wire wound coil has a certain unavoidable rf resistance besides having a desired inductance, this resistance which can be seen as connected in series with the inductance (equivalence), introduces losses and thereby it dampens the resonance effect. Real capacitors also always have a finite insulation quality and this results in a conductance in parallel to the capacitance (tan(delta) of the dielectric insulation material). Let’s assume here that in comparison to the losses of the coil, the losses in the resonator capacitor are negligible. The Qu of this lumped element resonator is then simply the ratio of the coil reactance and the coil resistance.  This ratio is however also the ratio of the stored energy per cycle at resonance and the lost energy per cycle at resonance.

Qu = omega_res*L/R_loss=2*pi*(energy stored)/(energy dissipated per cycle)

Since the Qu is a ratio of energies, it is a unitless quantity. The higher the Qu, the lower the lost energy in the resonator. For non-LC resonators (distributed element resonators), like coaxial resonators or waveguide cavity resonators, the Qu involves the calculation of the stored energy and the integration of all I^2 * R losses on the metallic surfaces keeping in mind that the current on the metal surfaces varies within the resonator surfaces. This is much more complicated than calculating the Qu of an LC resonator, but luckily, analytic Qu formulas exist for most common types of resonators and if you have access to an electromagnetic simulator like HFSS, CST then the Qu of an arbitrary resonator can be determined with high accuracy using the Eigenmode solver. For standard coaxial filter resonators (combline type) a general rule can be given as: the larger the resonator – the higher the Qu. For a given outer conductor diameter, there is an optimum characteristic impedance for true TEM-mode coaxial resonators of around 70 to 80 Ohms. The maximum Qu is however not very sharp. The Qu of dielectric resonators is a function of the dielectric losses inside the resonator and also the  I^2 * R on the walls of the cavity in which the dielectric resonator is placed.

The bulk of the losses of a coaxial TEM-mode resonator are on the inner conductor in the region of high rf-current and at the base (short circuit end). Here the current density is highest while the total current on the inner conductor has the same magnitude as the total current on the outer conductor. The losses on the outer conductor are lower because of the lower current density there. Coaxial TEM-mode implies that the radius of the outer conductor is small compared to the length of the resonator. If we make the outer diameter deliberately larger so that it is no longer small compared to the length of the resonator then we observe that the Qu of the resonator increases significantly. We have in fact made the transition from true TEM-mode to a ‘loaded cavity mode’ resulting in a higher level of stored energy in the resonator and consequently a higher Qu.

In most resonators there are also hidden losses that exist besides the obvious losses on metal surfaces or inside dielectric structures. A good example for such hidden losses are contact resistances between the inner rod of a TEM-mode coaxial resonator at the cavity surface. These losses escape from simulations unless specific care is taken in the 3D EM model. Quantitative assessment of such hidden losses is best derived from measured data. Depending on the resonant frequency, hidden contact losses can dominate the total loss in a resonator. In such cases suitable contact design is mandatory.

Equivalence between quarterwave TEM-mode resonator and parallel LC resonator

An equivalence exists between TEM-mode quarterwave resonators (a cavity with a center rod of approx. quarterwave length, one end connected to the cavity bottom, the other end open with a significant gap to the top lid) and a parallel LC circuit.

L = 2*Z_0/pi^2*f_res      and      C = 1/8*Z_0*f_res

The equivalence is based on the rate of change of the resonator impedance around the resonant frequency – its admittance slope (the 1st derivative of the admittances of both types of resonators with respect to the angular frequency omega). Therefore, true equivalence only exists over a very limited range around the resonant frequency. This equivalence is useful in  equivalent circuits for bandpass filters  if  internal voltages are to be analysed. The validity of this equivalence is limited to quarterwave long TEM-mode resonators. If the resonator rod is significantly shorter than a quarter wavelength due to capacitive top loading, then the above formula loses its validity. A accurate equivalence for top-loaded TEM-mode resonators can however also be found on the basis of admittance slope identity.

Download pdf of numerical example

Link to loaded quality factor page