In selecting a crystal, a designer should be familiar with the electrical specifications found in a data sheet or catalog (Table 1).
When purchasing a crystal, the designer specifies a particular frequency, along with load capacitance and mode of operation. Notice that shunt capacitance C0 is typically listed as a maximum value and not as an absolute value. Notice also that motional parameters C1, L1, and R1 are not typically provided in the crystal data sheet. They can generally be obtained from a crystal manufacturer, or from measurements. Table 2 shows equivalent-circuit values for an example crystal.
In Table 2, shunt capacitance is provided as an absolute value. However, shunt capacitance can be measured with a capacitance meter at a frequency much less than the fundamental frequency.
A crystal has two resonant frequencies characterized by a zero phase shift. The first is the series resonant frequency, fs, which can be found from:
This is the basic equation for the resonant frequency of an inductor and capacitor in series. Recall that series resonance is that particular frequency which the inductive and capacitive reactances are equal and cancel: XL1 = XC1. When the crystal is operating at its series resonant frequency the impedance will be at a minimum and current flow will be at a maximum. The reactance of the shunt capacitance, XC0, is in parallel with the resistance R1. At resonance, the value of XC0 <<R1 and, as a result, the crystal, appears resistive in the circuit at a value very near R1. Solving for the example crystal, it can be found that fs = 7,997,836.8 Hz. The second resonant frequency is the anti-resonant frequency, fa, which can be found from:
This equation combines the parallel capacitance of C0 and C1. When a crystal is operating at its antiresonant frequency, the impedance will be at its maximum and current flow will be at its minimum. Solving for the example crystal, it can be found that fa = 8,013,816.5 Hz. Note that fs is less than fa and that the specified crystal frequency is between fs and fa so that fs <fXTAL <fa. This area of frequencies between fs and fa is known as the "area of usual parallel resonance" or simply "parallel resonance." The crystal has resistance and reactance and therefore impedance.
Figure 5b has been redrawn in Fig. 6 to show the complex impedances of the equivalent circuit. The complex impedances2 are defined as:
Combining Z0 and Z1 in parallel yields:
Plugging in the values of Table 2 into a spreadsheet program, the Zp over frequency value is solved and a reactance verses frequency plot can be created (Fig. 7).
This plot shows where the crystal is inductive or capacitive in the circuit. Recall that positive reactances are inductive and negative reactances are capacitive. Between the frequencies fs and fa, the impedance of the crystal is inductive, and at frequencies less than fs and frequencies greater than fa the crystal is capacitive.
As mentioned earlier, the equivalent circuit shown in Fig. 5b is a simplified model that represents one oscillation mode. For this example it is the fundamental mode. The plot in Fig. 7 does not show overtone modes and spurious responses and, as a result, the crystal can appear inductive to the circuit at these overtone modes and spurious responses. Care must be taken in the selection of oscillator components, internal and external, to ensure the oscillator does not function at these points.
There is no difference in the construction of a series resonant crystal and a parallel resonant crystal, as they are manufactured exactly alike. The only difference between them is that the desired operating frequency of the parallel resonant crystal is set 100 PPM or so above the series resonant frequency. Parallel resonance means that a small capacitance, known as load capacitance (CL), of 12 to 32 pF (depending on the crystal) should be placed across the crystal terminals to obtain the desired operating frequency.3 Figure 8 depicts load capacitance in series with the crystal-equivalent circuit.
Therefore, when ordering a series resonance crystal, CL is not specified and is implied to be at zero. These crystals are expected to operate at in a circuit designed to take advantage of the crystals' mostly resistive nature at series resonance. On the other hand, a parallel resonant crystal has a specified load capacitance. This is the capacitive load the crystal expects to see in the circuit and thus operate at the specified frequency. If the load capacitance is something other than what the crystal was designed for, the operating frequency will be offset from the specified frequency.