Crystal Oscillator

Crystal Oscillator:

Some crystals found in nature exhibit the piezoelectric effect i.e. when an ac voltage is applied across them, they vibrate at the frequency of the applied voltage. Conversely, if they are mechanically pressed, they generate an ac voltage. The main substances that produce this piezoelectric effect are Quartz, Rochelle salts, and Tourmaline.

Rochelle salts have greatest piezoelectric activity, for a given ac voltage, they vibrate more than quartz or tourmaline. Mechanically, they are the weakest they break easily. They are used in microphones, phonograph pickups, headsets and loudspeakers.

Tourmaline shows the least piezoelectric activity but is a strongest of the three. It is also the most expensive and used at very high frequencies.

Quartz is a compromise between the piezoelectric activity of Rochelle salts and the strength of tourmaline. It is inexpensive and easily available in nature. It is most widely used for RF oscillators and filters.

The natural shape of a quartz crystal is a hexagonal prism with pyramids at the ends. To get a useable crystal out of this it is sliced in a rectangular slap form of thickness t. The number of slabs we can get from a natural crystal depe­nds on the size of the slabs and the angle of cut.

Fig. 1

For use in electronic circuits, the slab is mounted between two metal plates, as shown in fig. 1. In this circuit the amount of crystal vibration depends upon the frequency of applied voltage. By changing the frequency, one can find resonant frequencies at which the crystal vibrations reach a maximum. Since the energy for the vibrations must be supplied by the ac source, the ac current is maximized at each resonant frequency. Most of the time, the crystal is cut and mounted to vibrate best at one of its resonant frequencies, usually the fundamental or lowest frequency. Higher resonant frequencies, called overtones, are almost exact multiplies of the fundamental frequency e.g. a crystal with a fundamental frequency of 1 MHz has a overtones of 2 MHz, 3 MHz and so on. The formula for the fundamental frequency of a crystal is

f = K / t.

where K is a constant that depends on the cut and other factors, t is the thickness of crystal, f is inversely proportional to thickness t. The thinner the crystal, the more fragile it becomes and the more likely it is to break because of vibrations. Quartz crystals may have fundamental frequency up to 10 MHz. To get higher frequencies, a crystal is mounted to vibrate on overtones; we can reach frequencies up to 100 MHz.

AC Equivalent Circuit:

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

Fig. 2

When the crystal is vibrating, it acts like a tuned circuit. Fig. 2, shows the ac equivalent circuit of a crystal vibrating at or near its fundamental frequency. Typical values are L is henrys, C in fractions of a Pico farad, R in hundreds of ohms and C_m in Pico farads

L_s = 3Hz,     C_s = 0.05 pf, R_s = 2K, C_m = 10 pf.

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

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

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

Since C_loop > C_S, therefore, f_p > f_S.

Since Cm > CS, therefore, Cm || CS is slightly lesser than CS. Therefore fP is slightly greater than fS. Because of the other circuit capacitances that appear across Cm the actual frequency will lie between fS and fP. fS and fP are the upper and lower limits of frequency. The impedance of the crystal oscillator can be plotted as a function of frequency as shown in fig. 3. At frequency fS, the circuit behaves like resistive circuit. At fP the impedance reaches to maximum, beyond fP, the circuit is highly capacitive. The frequency of an oscillator tends to change slightly with time. The drift is produced by temperature, aging and other causes. In a crystal oscillator the frequency drift with time is very small, typically less than 1 part in 106 per day. They can be used in electronic wristwatches. If the drift is 1 part in 1010, a clock with this drift will take 30 years to gain or lose 1 sec.

Fig. 3

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

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

 Example - 1

The parameters of the equivalent circuit of a crystal are given below:

L = 0.4 H, C_S = 0.06 pF, R = 5 kΩ, C_m = 1.0 pF.

Determine the series and parallel resonant frequencies of the crystal.

Solution:

With reference to fig. 2, the admittance of the crystal Y is given by

where, 

and 

The resonant frequencies are obtained by putting B = 0. Thus,

Consider the term C_S R₂ / L_S = C_S R / [L R]. In a crystal, the time constant (L / R) is very much greater than C_S R. Thus the ratio is very much less than 1. For the values given, this ratio is of the order of10^-6. Neglecting this term in comparison with 2, we get two roots as

where, ω_s and ω_p are the series and parallel resonant frequencies respectively. Substituting the values, we get

ω_s = 6.45 M Hz. and ω_p = 6.64 MHz

Crystal Oscillators:

Fig. 4, shows a colpitts crystal oscillator.

Fi. 4

The capacitive voltage divider produces the feedback voltage for the base of transistor. The crystal acts like an inductor that resonates with C₁ and C₂. The oscillation frequency is between the series and parallel resonant frequencies.

Example-2:

If the crystal of example-1 is used in the oscillator circuit as shown fig. 5, determine the values of R for the circuit to oscillate.

Fig. 5

Solution:

The equivalent circuit of crystal (discussed earlier) shows that it has a parallel resonant f_r frequency (ω_p) at which the impedance becomes maximum. The amplified signal output of the circuit is applied across the potential divider consisting of R and the crystal circuit. At the resonant frequency the impedance of crystal becomes maximum (magnitude R) and thus the loop gain will be greater than or equal to unity. At frequencies away from ω_p the loop gain becomes less than unity. The loop base shift is also zero around ω_p. Thus both the conditions required for sustained oscillations are satisfied and the circuit oscillates.

The value of G of the crystal at ω =ω_p is given by

Thus the resistor R should be less than 5x10⁶ Ω.