Page 242 - Complete Wireless Design

Oscillator Design

Oscillator Design 241

crystals can reach only up to 20 MHz, yet can attain much higher frequencies
on harmonic and/or overtone oscillator operation. For example, a harmonic
crystal oscillator, with its output tank circuit tuned to one of these harmonics,
can have an output at the crystal’s second, third, fourth, etc., harmonic, yet the
crystal itself is really only operating at its lower fundamental frequency. When
operated as an overtone crystal oscillator, the crystal must actually vibrate at
high harmonic (overtone) frequencies, and will work only at one of its odd har-
monics—such as its third, fifth, or seventh—with the output tank circuit
tuned to this chosen frequency. Overtone crystal oscillators normally require a
special overtone crystal when operated in this mode.
As stated above, the crystal functions as a very high-Q series resonant cir-
cuit with high temperature stability and very narrow bandwidth (as the high-
Q designation would indicate), with the crystal looking to the rest of the
oscillator’s circuit as shown in Fig. 4.9. R is the resistance of the crystal dur-
s
ing series resonance, while L is the motional inductance, C the motional
m m
capacitance, and C is the capacitance between the crystal’s holder or, in a
0
modern crystal, its plated electrodes. In fact, C at VHF and above has so
0
decreased in reactance that it has effectively shorted the output of the crystal
to its input; this problem can be mitigated by resonating a small value coil in
parallel with C .
0
Most oscillators will operate in series resonance mode, with the values of L
m
and C governing the resonant frequency of the crystal. At series resonance the
m
crystal is resistive with no reactances, since X X , and can be described as:
L C
1
f
s 2 L C
m m
As mentioned above, fundamental crystal operation usually peaks out at 20
MHz. This is due not only to the dangerously decreasing thickness of the crys-
tal, but also to its decreasing R . The crystal’s R can decrease to 10 ohms at
S S
20 MHz on its fundamental, while the crystal in seventh overtone mode can
reach 180 MHz with an R of 80 ohms. This demonstrates why many oscilla-
S
tors must run in overtone modes, which allow the crystal to be more easily
impedance matched at higher operational frequencies.
L , whose value is based on the mechanical mass of the quartz crystal, can
m
vary in inductance anywhere between 3600 mH at 1.5 MHz to 10 mH at 20
MHz. C , whose value is based on the actual stiffness of the quartz crystal,
m
the size of the electrodes, and the size of the quartz, can vary anywhere
between 0.007 pF at a fundamental frequency of 1.5 MHz to 0.02 pF at its
fundamental frequency of 20 MHz. But when a crystal is operated on an over-
tone, the C will decrease. The designer can choose the value of the C
m m
desired, and the L will then be:
m
1
L
m 2
4 f C
r m

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