There are three main bases for characterizing the properties of microwave ceramics, namely, the relative dielectric constant (εr), the quality factor (Q×f), and the temperature coefficient of resonant frequency (τf)[23]. Generally speaking, in practical applications, the material is generally required to achieve a suitable dielectric constant, a high quality factor and a near-zero temperature coefficient of resonant frequency. The following three parameters are described in detail:

2. Quality factor

Under the action of external field, different forms of bound charges form polarization, which will lead to hysteresis in response to alternating electric field. In this case, when the bound charges move slightly, it may lead to friction or elastic collision in the process of movement, resulting in loss. The quality factor Q, on the other hand, is an important parameter used to measure loss and is inversely proportional to dielectric loss (Q = 1/tan θ)[32]. The occurrence of loss will reduce the carrier energy and weaken the signal transmission capacity. In addition, the power consumed is converted into heat energy, which increases the temperature of the device, resulting in a decrease in signal stability. Since the quality factor Q is related to the frequency f as follows, but the product of Q × f is a fixed value, Q × f is usually used as a measure of the dielectric loss of different materials [33].

In practice, the quality factor Q is defined as the ratio of the resonance frequency f0 to the peak width ∆f0 at the resonance frequency peak -3dB. As shown in Figure 1-4 [34], the larger the quality factor Q, the sharper the resonant peak pattern, the more favorable the frequency selection, and the smaller the probability of interference at the required frequency.

Schematic diagram of the relationship between quality factor and resonance peak related parameters in the project
For dielectric resonators, the on-load loss is mainly divided into four parts: conduction loss, dielectric loss, radiation loss and external loss [35], as shown in Eqs. 1-6,

Where QL,Qd,Qe,Qr,Qext are load, medium, conductivity, radiation and external quality factors, respectively. Essentially, the dielectric loss of a material is divided into intrinsic and extrinsic losses [36]. The intrinsic loss is mainly the loss caused by the interaction between the alternating electric field and the crystal acoustic system, which is mainly determined by the crystal structure and material composition [37]. Extrinsic losses are mainly caused by structural and microstructural defects in the material, such as second phases, pores, grain boundaries, dislocations, ion vacancies, etc. [38]. To some extent, the intrinsic loss represents the lowest loss value that the material can achieve, and in practice, the existence of extrinsic loss will lead to a further increase in dielectric loss.

In the actual situation, there are many factors that affect the non-intrinsic loss, mainly related to the process, microstructure and crystal defects, so the optimization of Q x f value mainly starts from the regulation of non-intrinsic loss. The following is a brief description of commonly used optimization directions and methods:

(1) Optimize the process environment: the sensitivity of microwave dielectric ceramic loss to ambient temperature and humidity reflects the importance of controlling the experimental and test environment [39]. Before preparation, the raw material powder needs to be dried or calcined; in the preparation process, the humidity of the environment should be controlled, and the experimental test should be carried out in a relatively dry environment as far as possible to ensure the feasibility of the experiment and test results.

(2) Ensure the purity of raw materials: Due to the existence of impurities will lead to a substantial increase in dielectric loss, it is necessary to ensure the purity of raw materials in actual production. In an earlier study of BaTi4O9[40], it was found that trace impurities in the raw materials caused the Q×f value to decrease by half; it was also found that impurities introduced in the ball mill also caused the Q×f value to decrease. Therefore, in the actual preparation process, raw materials with higher purity are generally selected, and special raw materials, such as lanthanide oxides that can absorb water to form crystal water, need to be calcined at a specified temperature for pretreatment. In addition, the influence of the particle size and shape of the raw materials on the subsequent sintering performance should be considered.

Control phase composition and microstructure [41]: the existence of the second phase will have a greater impact on the Q×f value, so the design of the material composition is mainly single-phase materials, and the second phase is minimized. Kim[42] found in the study of Ba(Ni1/3Nb 2/3)O3 ceramics that the BaNb2O6 second phase produced by it will lead to the decrease of Q×f value. However, when the Q×f value of the second phase is better than that of the main phase or the densification is promoted, the appearance of the second phase may lead to the increase of the Q×f value. Davies[43] et al. found that the Ba8ZnTa6O24 second phase in Ba(Zn1/3Ta 2/3)O3 ceramics has high Q value, thus improving the Q×F value of the material. Fan[43] et al. found through research that the appropriate amount of second phases (such as CaSmAl3O7,CaO and Sm2O3) produced when Ca2 ions are excessive is beneficial to promote the densification of the microstructure of the material, thus increasing the Q×f value.