Goodarz-
A quartz crystal is modeled to the first order by this simple circuit:
--------------Lm-----Cm-----Rs------------
| |
| |
| |
-----------Cp-----------
Where Lm is the motional inductance (mH)
Cm is the motional capacitance (fF)
Rs is the ESR (10's to 100's of ohms)
Cp is the holder capacitance (pF)
The exact values are dependent on the resonant frequency of the
crystal, the Q of the crystal, the geometric cut and similar factors.
The electrical element values are representative of their mechanical
analogs.
An overtone crystal can be modeled by adding additional series
LRC sections to the model where the resonant frequencies of the additional
sections are equal to the overtone frequencies.
Getting back to the simple fundamental model shown above, notice that
there will be two resonant frequencies associated with the model,
a series resonant frequency that is dependent on Lm and Cm and a
parallel resonant frequency that is primarily dependent on Lm and
the series combination of Cm an Cp. The parallel resonant frequency
will be slightly higher than the series resonant frequency.
Also note that when the crystal model is connected to an oscillator
circuit that has some load capacitance, that load capacitance is effectively
in parallel with Cp. This will effect the parallel resonant frequency of
the crystal by making it lower.
The parallel and series resonant frequencies of a crystal are close
together (typically on the order of kHz). The exact split depends on the
relative values of Lm, Cm, Cp, and Rs .
When you specify a crystal that is parallel resonant with say a 18pF load,
you are asking the vendor to ensure that the crystal oscillates at a
specific frequency when the extra capacitance presented by the oscillator
circuit is 18 pF.
Oscillators can operate in either a series or resonant mode dependant on the
topology of the oscillator circuit. (This involves the phase shift through
the oscillator circuit that in combination with the phase shift through
the crystal adds up such that it supports positive feedback to support
oscillation at the resonant frequency)
For info, here is a spice subcircuit of a crystal that has a
fundamental frequency of 16 MHz and also has a third overtone resonance
of about 48 MHz. The element values were measured on a HP4194A analyzer.
.subckt xtalmod x1 x2
*16 MHz Crystal
* fundamental mode values
Co x1 x2 5pf
L1 x1 mid1 7.74mh
C1 mid1 mid2 12.776ff
RI mid2 x2 12.54
* third overtone values
L1B x1 mid1B 13.0811mh
C1B mid1B mid2B .840ff
RIB mid2B x2 47.22
.ends
You may find it instructive to play around with this model in Spice.
Look at it as a standalone circuit to understand the relationship of
the parallel and series resonant frequencies to the element values.
Then hook it up to your favorite oscillator circuit and see if you can
get it to oscillate. (That is the subject for another conversation :)
Ray Anderson
Sun Microsystems Inc.
raymonda@radium.eng.sun.com