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Equivalent Circuit and Electrical Characteristics

The electric equivalent circuits of a crystal unit is generally represented as shown in Fig.7, and the four equivalent constants are labeled as follows:
L1: Equivalent series inductance
C1 : Equivalent series capacitance
R1: Equivalent series resistance
C0: Shunt capacitance

Fig. 7 Electric Equivalent Circuit of a Crystal Unit
The main characteristics of a crystal unit is derived by using these equivalent constants.
As an example, Fig. 8 shows the frequency characteristic of impedance.
Typical frequencies and performances of crystal units are shown below.

Fig. 8 Impedance of Frequency characteristics

The general crystal oscillation circuit is capacitive, and can be treated as series circuits with a negative resistance (-R) and load capacitance (CL) as shown in Fig. 9
Fig. 9 Relationship between a Crystal Unit and the Oscillation Circuit
When the load capacitance values are CL and series , fL and fS, respectively, the following relationship is established.

Fig. 10 shows an example.
An overtone crystal with a smaller C1 value than a fundamental wave has a smaller frequency change rate with respect to CL than a fundamental wave.

The C1 value of small-size crystal units is smaller than that of large-size crystal units with the same frequency.
For this reason, frequency change rate to the load capacitance is smaller for a small-sized crystal units.

When the load capacitance (CL) is connected to the crystal unit in series as shown in Fig. 9, the equivalent resistance increases according to the following formula.

Fig. 10 Load Capacitance vs. Frequency Change

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