Application of Simulation Technology to AT-Cut Quartz Crystal Design

Quartz crystals are one of the important components that make up a reference clock for ICs such as microcontrollers. They are widely used in different products, ranging from mobile communication terminal devices such as cellular phones and smartphones to automobiles and home appliances. There has been a strong need for the miniaturization of components, especially for mobile communication terminal devices. Among the challenges in designing such components is accomplishing the miniaturization of products while maintaining the product characteristics. In this article, we present how we efficiently conduct our design work using a simulation technique that employs the finite element method  (FEM)  .

1. AT-Cut Quartz Crystals and an Oscillator Circuit. 

AT-cut quartz crystals are devices made of synthetic crystal and use the piezoelectric properties (thickness-shear vibration) of crystal. Figure 1 shows a typical structure of Murata’s product. Figure 2 shows its equivalent circuit. This device is one of the important components that constitute an electronic oscillator circuit that serves as a reference clock, which is required for the operation of microcontrollers, for example. Figure 3 presents a typical configuration of an oscillator circuit. The oscillator circuit generates clock signals by amplifying electric signals passing through a quartz crystal. The resistance of the quartz crystal varies by frequency as shown in Fig. 4. The resistance of the quartz crystal takes the smallest value at the frequency of principal vibration of the crystal blank. This smallest value of resistance is called the equivalent series resistance (ESR) . The oscillator circuit oscillates near the principal vibrational frequency of the crystal blank, generating clock signals. 

What is important in the oscillator circuit is the stability of oscillation. One of the indices of oscillation stability is the oscillation margin , which indicates the ratio of the signal gain (the amplification capability of input signals) of the circuit components other than the quartz crystal to the ESR (a damping factor for signals). In theory, the circuit oscillates when the oscillation margin is greater than 1; however, when the oscillation margin is close to 1, the circuit seldom oscillates or it takes an extremely long time to start oscillating, which may sometimes cause malfunction in devices where the oscillator is installed. Suppressing the ESR can improve the oscillation margin. In general, the lower the frequency, the higher the ESR, and the smaller the product size, the higher the ESR. The 2016- and 1612-size products have been installed in mobile terminal devices, but there is an increasing need for an even smaller product. Hence, design work has been becoming increasingly challenging these days.

Fig. 1. The Product Structure of Quartz Crystal

Fig. 1. The Product Structure of Quartz Crystal

Fig. 2. Equivalent Circuit Diagram of a Quartz Crystal

Fig. 2. Equivalent Circuit Diagram of a Quartz Crystal

Fig. 3. Oscillator Circuit Diagram

Fig. 3. Oscillator Circuit Diagram

Fig. 4. Frequency Characteristics of the Resistance of a Quartz Crystal

Fig. 4. Frequency Characteristics of the Resistance of a Quartz Crystal

2. Key Factors in Designing the Product Characteristics

The AT-cut quartz crystals have a number of unnecessary vibrations besides the thickness-shear vibration; the latter is used as the principal vibration of the quartz crystals. In designing the quartz crystals, it is necessary to choose appropriate geometric parameters so that these unnecessary vibrations do not affect the product characteristics within the range of operating temperature. Figure 5 shows the relationship between temperature and the ESR, where the characteristics of two cases are compared: the graph (a) represents a case where appropriate geometry is chosen in design, whereas the graph (b) represents a case of inappropriate geometry. When inappropriate geometric parameters are chosen, unnecessary vibrations superpose and increase the ESR. Therefore, it is important to choose appropriate geometric parameters that will not be affected by the unnecessary vibrations in the design phase. However, there are a huge number of combinations for geometric parameters, and engineers have been struggling to find optimum solutions empirically by trial and error in a number of cases. It has been one of the primary causes of trouble in the reduction of the research and development time and the quality improvement of the products.

Fig. 5. ESR Characteristics within the Operating Temperature Range

Fig. 5. ESR Characteristics within the Operating Temperature Range

3. Application of a Simulation Technique (the Finite Element Method, FEM) and New Challenges

An efficient method that we can think of to help us find optimum solutions is to conduct characteristic simulation using the finite element method (FEM). This method, however, has a drawback: The simulation results are poorly consistent with the actual sample characteristics. Recent research has come to elucidate that the source of this inconsistency lies in the fact that not only the geometry but also the holding material connecting the crystal blank and the substrate greatly affects the frequency relationship between the principal vibration (thickness-shear vibration) and the unnecessary vibrations.

Fig. 6. Relationship between Blank Width and ESR

Fig. 6. Relationship between Blank Width and ESR

Figure 6 compares the actual sample characteristics and the FEM simulation result when the simulation only modeled the crystal blank without giving any consideration of holding onto the substrate. In the simulation result of a geometry without a holding material, the trend of ESR characteristics did not agree with the actual sample characteristics; hence, appropriate geometric parameters cannot be determined.

Thus, our challenge has been to improve the consistency of simulation results with the actual sample characteristics, and thereby establish a better simulation technique to find optimum solutions efficiently. We have now created a new simulation system that can provide a solution to these challenges. This system has enabled us to improve the consistency of simulation results with the actual sample characteristics.

Glossary

Finite Element Method: Abbreviated as FEM. A technique of numerical analysis for finding approximate solutions to a whole structure by dividing the structure into a collection of smaller domains for computation.
ESR: Equivalent series resistance. A resistance primarily due to losses inside a crystal or in a holding material.
Oscillation margin: It indicates the margin that exists from the state of oscillation to the moment when the oscillation ceases. It is one of the most important items in the oscillator circuit where quartz crystals are used.

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