The Quality Factor of Microstrip-Line 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, microstrip-line resonators are extensively used in microwave circuits, as they provide miniaturization. Microstrip-line resonators are either open-ended or short-circuited and are characterized by the “quality factor”—a parameter that describes the ability of the resonator to store energy. The quality factor of microstrip-line resonators is high, and this is the advantage of using them in miniaturized amplifiers or filters with narrow-band frequency specifications. 

The Quality Factor of a Microstrip-Line 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, microstrip-line resonators offer purely resistive input impedance.  

The quality factor (Q-factor) 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 microstrip-line resonator can be determined from its resonant frequency, coupling coefficient, and unloaded Q-factor. The microstrip lines used for building resonators are either open-ended, short-circuited, or connected to other circuits. The Q-factor of a resonator varies with how and to what circuit the resonator is connected or loaded. The Q-factor of the resonator can be described as either a loaded Q-factor, unloaded Q-factor, or external Q-factor.

A microstrip-line resonator can be a half-wavelength 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 microstrip-line resonator can be modeled using distributed elements.

Next, we will discuss the Q-factor from the generalized model of the microstrip-line resonator given below.

A Generalized Model of a Microstrip-Line Resonator

A model of a microstrip-line resonator using the distributed elements

The resonator is modeled using C, L, and G₀ 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 1-1, the right side circuit forms the resonator and the left side circuit is the external load. The Q-factor of an unloaded resonator is called the unloaded Q-factor Q₀. Considering only the resonator circuit, the resonant frequency can be given by:

and the unloaded Q-factor can be given by:

https://drive.google.com/file/d/1V2lG0EJ2GKQt-AvP7utYUz4rs_X2I_su/view

G₀=1/R₀is 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 Q-factor of the resonator is influenced by external circuit elements. The Q-factor of the resonator in this loaded condition is called the loaded Q-factor, which is different from the unloaded Q-factor. The resonant frequency of the resonator also varies slightly with loading. The loaded Q-factor Q_L can be given by the following equation:

Q_ex is the external Q-factor:

The Coupling Coefficient and the Q-factor

The quality factor of a microstrip-line resonator is dependent on the coupling coefficient, k. The loaded and unloaded Q-factors 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 voltage-resistance 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 Q₀ and Q_ex,  we can say that there are different types of coupling in microstrip-line resonators. 

The type of coupling based on Q-factor values

Designing a microstrip-line resonator with a high Q-factor is essential for increasing the performance of the filters and oscillators using them. Luckily, Cadence’s software can help in the design of microstrip-line resonators with a high quality factor.

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