AC Equivalent Circuit:

When mounted crystal is not vibrating, it is equivalent to the capacitance C_m, because it has 2 metal plates separated by the dielectric, C_m is known as mounting capacitance.

Figure 1

When the crystal is vibrating, it behaves like a tuned circuit. Figure 1, shows the AC equivalent circuit of a crystal vibrating at or near the fundamental frequency of it. Typical values are L is henrys, C in fractions of the Pico farad, R in hundreds of ohms and Cm in Pico farads

Ls = 3Hz,    Cs = 0.05 pf, Rs = 2K, Cm = 10 pf.

The Q of circuit is very high. As compared with the L-C tank circuit. For given values, Q comes out to be 3000. Because of very high Q, the crystal leads to oscillators with quite stable frequency values.

The series resonant frequency f_S of crystal are the sonant frequency of LCR branch. At this frequency, branch current reaches the maximum value because L_s resonant with C_S.

Above f_S, crystal behaves inductively. The parallel resonant frequency is frequency at which the circulating or loop current reaches the maximum value. Since this loop current should flow through series combination of the C_S and C_m, the equivalent C_loop can be given by

As C_loop > C_S, therefore, f_p > f_S.

Since C_m  > C_S, therefore, C_m  || C_S  is a little lesser than C_S. Therefore f_P  is slightly greater than f_S. Because of other circuit capacitances which appear across Cm  the actual frequency will lie in between f_S and f_P. f_S and f_P are upper and lower limits of the frequency. The impedance of crystal oscillator can be plotted as the function of frequency as shown in the figure 2.

At   the frequency f_S, the circuit behaves like a resistive circuit. At fP impedance reaches to maximum, beyond f_P, circuit is highly capacitive.

The frequency of the oscillator tends to change slightly with the time.  The drift is produced  by temperature, aging and other causes. In a crystal oscillator the frequency drift with time is very small, less than 1 part in 10⁶  per day. They can be used in the electronic wristwatches. If the drift is 1 part in 10¹⁰, a clock with this drift will take 30 years to lose or gain 1 sec.

The crystals can be manufactured with values of fs as low as 10 kHz; at these frequencies the crystal is thick relatively. On the high frequency side, fs  can be as high as 1- MHz; here crystal is very thin.

The temperature coefficient of the crystals is usually small and can be made zero. When the extreme temperature stability is required, the crystal can be housed in an oven to maintain it at the constant temperature. The high Q of crystal also contributes to the relatively drift free oscillation of the crystal oscillators.

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