Module for estimating the Quality (Q) factor(s) from S-parameters.
This class implements methods for determining loaded and unloaded Q-factor from frequency-domain S-parameters, that can be applied to measurements of transmission or reflection.
Documentation and implementation are adapted from [MAT58]
Q-factor calculation class.
Loaded and Unloaded Q-factor¶
The Quality factor (Q-factor) of a resonator is defined by [Pozar]:
where \(U\) is the average energy stored by the resonator and \(\Delta U\) is the decrease in the average stored energy per wave cycle at the resonant frequency, that is, the average power loss.
The loaded Q-factor, \(Q_L\), describes energy dissipation within the entire resonant system comprising of the resonator itself and the instrument used for observing resonances. The term loading refers to the effect that the external circuit has on measured quantities.
The external circuit consists of the measuring instrument and uncalibrated lines, but not the couplings of microwave resonators. Loading by an instrument that has 50 Ohm impedance, such as a VNA, causes \(Q_L\) to be reduced substantially if strong coupling is used.
For many applications the quantity that is desired is the unloaded Q-factor \(Q_0\), which is determined by energy dissipation associated with the resonator only and therefore gives the best description of the resonant mode(s).
In other words, \(Q_0\) is the Q-factor of the uncoupled resonator. The value of \(Q_0\) can be estimated from measurements of \(Q_L\), but cannot be measured directly. \(Q_0\) is largely governed by ohmic loss arising from surface currents in the metal conductors (walls and loop couplings), and from dielectric loss in any insulating materials that may be present.
Relationships between Loaded and Unloaded Q-factors¶
Energy dissipation in the external circuit is characterised by the external Q-factor, \(Q_e\). For both series and parallel equivalent circuits, the three Q-factors are related by [Pozar]:
The coupling factor \(\beta\) is defined for each port as:
Finding the unloaded Q from measured S-parameters consists in first finding the coupling factor, then measure \(Q_L\) from the 3 dB bandwidth and using the relationships above.
Fortunately, scikit-rf implements methods for determining loaded and unloaded Q-factors from frequency-domain S-parameters. The implemented methods are described in detail in [MAT58], and can be applied to measurements of transmission or reflection.
Q-factor determination through equivalent-circuit models¶
Characterisation of resonances from measurements in the frequency-domain can be achieved through equivalent-circuit models [MAT58]. Resonators can be modelled as an ideal RLC resonator connected to an external circuit, incorporating elements to account for a lossy coupling and coupling reactances.
For high Q-factor resonators (in practice, \(Q_L\) > 100), the S-parameter response of a resonator measured in a calibrated system with reference planes at the resonator couplings can be expressed like [MAT58], [Galwas] :
where \(S_D\) is the detuned S-parameter measured at frequencies far above or below resonance, \(d\) is The diameter of the Q-circle, \(\delta\) is a real-valued constant that defines the orientation of the Q-circle, and \(t\) is the fractional offset frequency given by:
where \(f_L\) is the loaded resonant frequency and \(f\) the frequency at which S is measured. This equation can be applied to measurements by transmission (S21 or S12) or reflection (S11 or S22).
The S-parameters are fitted against a modified expression of the above equations to deduce the resonant frequency, loaded and unloaded Q-factors and other properties.
“Q-factor Measurement by using a Vector Network Analyser”, A. P. Gregory, National Physical Laboratory Report MAT 58 (2021) https://eprintspublications.npl.co.uk/9304/
B. A. Galwas, ‘Scattering Matrix Description of Microwave Resonators’, IEEE Trans. Microwave Theory Techn., vol. 31, no. 8, pp. 669–671, Aug. 1983, doi: 10.1109/TMTT.1983.1131566.