NXP Semiconductors UM10301 PCF2123 - Manual (Page 13)

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 C
L
resulting from C
IN
, C
OUT
(plus C
T
if mounted)
and C
STRAY
it is necessary to realize that seen from the crystal, C
IN
and C
OUT
are
effectively in series; the 32 kHz signal goes from OSCI through C
IN
to ground, via ground
to C
OUT
and then through C
OUT
to OSCO. In parallel with this series circuit is C
STRAY
. For
the remainder of this discussion, whenever in formulas C
OUT
is written this represents
either the value of C
OUT
only, or in case a trimming capacitor C
T
is present too, the sum of
C
OUT
and C
T
. Now the load capacitance C
L
is given by:
STRAY
OUT IN
OUT IN
L
C
C C
C C
C
+
+
⋅
=
Since C
0
is in parallel with C
L
the total capacitance in parallel with the motional arm
L
1
-C
1
-R
1
is given by
0
C C
C C
C C
C
STRAY
OUT IN
OUT IN
PAR
+ +
+
⋅
=
The motional arm is a series circuit, which forms a closed circuit because there is a
capacitance C
PAR
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 L
1
the resulting resonance frequency is now given by the series circuit of C
PAR
and
C
1
. Thus C is given by
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+ +
+
⋅
+
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+ +
+
⋅
⋅
=
0 1
0 1
C C
C C
C C
C
C C
C C
C C
C
C
STRAY
OUT IN
OUT IN
STRAY
OUT IN
OUT IN
Typical values for crystal parameters are given in Table 4. From these values it is clear
that C
1
is several orders of magnitudes smaller than the other capacitances in this
expression and therefore C
1
dominates. C will be in the order of magnitude of C
1
but it
will be a bit smaller as a result of C
PAR
in series.
With
LC
1
= ω
and
1
1 1
R C
Q
⋅ =
ω
the resulting resonance frequency and quality
factor can be calculated.
Because C
1
is orders of magnitude smaller than the other capacitances Q can be
approximated by
1 1
1 1
R C
Q
a
⋅ =
ω