The Quality Factor of MicrostripLine Resonators
Key Takeaways

The quality factor describes the ability of a resonator to store energy.

The quality factor is the ratio of stored energy to the energy lost in a resonator circuit.

The quality factor of an unloaded resonator is called the unloaded quality factor. The quality factor of a loaded resonator is called the loaded quality factor.
In microwave circuits, resonators are used for frequency selection
The frequency selectivity property of resonators can be applied in amplifiers, filters, and oscillators. In particular, microstripline resonators are extensively used in microwave circuits, as they provide miniaturization. Microstripline resonators are either openended or shortcircuited and are characterized by the “quality factor”—a parameter that describes the ability of the resonator to store energy. The quality factor of microstripline resonators is high, and this is the advantage of using them in miniaturized amplifiers or filters with narrowband frequency specifications.
The Quality Factor of a MicrostripLine Resonator
A resonator circuit realized using microstrip transmission lines can be modeled with the distributed line elements capacitance, inductance, resistance, and shunt conductance.
A resonator circuit stores electromagnetic energy in it; electric energy is stored in capacitors and magnetic energy is stored in inductors. The energy losses in the resonator circuit are represented by the resistance. At the resonant frequency, the electric energy in the capacitor equals the magnetic energy stored in the inductors. At resonant conditions, microstripline resonators offer purely resistive input impedance.
The quality factor (Qfactor) is the ratio of stored energy to the energy lost in the resonator circuit. It can be generalized as:
Loaded, Unloaded, and External Quality Factors
The fundamental characteristics of a microstripline resonator can be determined from its resonant frequency, coupling coefficient, and unloaded Qfactor. The microstrip lines used for building resonators are either openended, shortcircuited, or connected to other circuits. The Qfactor of a resonator varies with how and to what circuit the resonator is connected or loaded. The Qfactor of the resonator can be described as either a loaded Qfactor, unloaded Qfactor, or external Qfactor.
A microstripline resonator can be a halfwavelength line that shares a capacitive coupling with an input microstrip line or it can be a dielectric resonator that is inductively coupled to the microstrip line. Any microstripline resonator can be modeled using distributed elements.
Next, we will discuss the Qfactor from the generalized model of the microstripline resonator given below.
A Generalized Model of a MicrostripLine Resonator
A model of a microstripline resonator using the distributed elements
The resonator is modeled using C, L, and G_{0} and is connected to an external load, shown as I_{N}, G_{ex}, and B_{ex}. The resonator and load circuit share a common voltage V. The circuit above is impedance matched.
From port 11, the right side circuit forms the resonator and the left side circuit is the external load. The Qfactor of an unloaded resonator is called the unloaded Qfactor Q_{0}. Considering only the resonator circuit, the resonant frequency can be given by:
and the unloaded Qfactor can be given by:
https://drive.google.com/file/d/1V2lG0EJ2GKQtAvP7utYUz4rs_X2I_su/view
G_{0}=1/R_{0}is the conductance, which represents the energy dissipated in the resonator. Conductor losses, dielectric losses, and radiation losses are the ways in which energy is dissipated in the microstrip resonator.
When the external circuit is connected to the resonator, the resonator is considered loaded. In this loaded condition, the Qfactor of the resonator is influenced by external circuit elements. The Qfactor of the resonator in this loaded condition is called the loaded Qfactor, which is different from the unloaded Qfactor. The resonant frequency of the resonator also varies slightly with loading. The loaded Qfactor Q_{L} can be given by the following equation:
Q_{ex} is the external Qfactor:
The Coupling Coefficient and the Qfactor
The quality factor of a microstripline resonator is dependent on the coupling coefficient, k. The loaded and unloaded Qfactors share a relationship with the coupling coefficient as follows:
We have seen that the resonator and external circuit share common voltage, V, so applying the voltageresistance relationship to power, the coupling coefficient can be written as:
Based on the power dissipated in the external circuit and resonator, and depending on the values of Q_{0} and Q_{ex,} we can say that there are different types of coupling in microstripline resonators.
The type of coupling based on Qfactor values
Designing a microstripline resonator with a high Qfactor is essential for increasing the performance of the filters and oscillators using them. Luckily, Cadence’s software can help in the design of microstripline resonators with a high quality factor.
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