NXP Semiconductors UM10301
User Manual PCF85x3, PCA8565 and PCF2123, PCA2125
UM10301_1 © NXP B.V. 2008. All rights reserved.
User manual Rev. 01 — 23 December 2008 13 of 52
Now in order to determine the value of CL resulting from CIN, COUT (plus CT if mounted)
and CSTRAY it is necessary to realize that seen from the crystal, CIN and COUT are
effectively in series; the 32 kHz signal goes from OSCI through CIN to ground, via ground
to COUT and then through COUT to OSCO. In parallel with this series circuit is CSTRAY. For
the remainder of this discussion, whenever in formulas COUT is written this represents
either the value of COUT only, or in case a trimming capacitor CT is present too, the sum of
COUT and CT. Now the load capacitance CL is given by:
STRAY
OUTIN
OUTIN
LC
CC
CC
C+
+
⋅
=
Since C0 is in parallel with CL the total capacitance in parallel with the motional arm
L1-C1-R1 is given by
0
CC
CC
CC
CSTRAY
OUTIN
OUTIN
PAR ++
+
⋅
=
The motional arm is a series circuit, which forms a closed circuit because there is a
capacitance CPAR connected in parallel to this series circuit. Of course the crystal itself
can’t oscillate stand alone, but the equivalent capacitance C which determines together
with L1 the resulting resonance frequency is now given by the series circuit of CPAR and
C1. Thus C is given by
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧++
+
⋅
+
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧++
+
⋅
⋅
=
01
01
CC
CC
CC
C
CC
CC
CC
C
C
STRAY
OUTIN
OUTIN
STRAY
OUTIN
OUTIN
Typical values for crystal parameters are given in Table 4. From these values it is clear
that C1 is several orders of magnitudes smaller than the other capacitances in this
expression and therefore C1 dominates. C will be in the order of magnitude of C1 but it
will be a bit smaller as a result of CPAR in series.
With LC
1
=
ω
and
1
11
RC
Q⋅=
ω the resulting resonance frequency and quality
factor can be calculated.
Because C1 is orders of magnitude smaller than the other capacitances Q can be
approximated by
11