User Manual

Jennic
© Jennic 2009 JN-DS-JN5148-001 1v2 87
Preliminary
B.1.3 Crystal ESR and Required Transconductance
The resistor in the crystal equivalent circuit represents the energy lost. To maintain oscillation, power must be
supplied by the amplifier, but how much? Firstly, the Pi connected capacitors C
1
and C
2
with C
S
from the crystal,
apply an impedance transformation to Rm, when viewed from the amplifier. This new value is given by:
2
ˆ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
L
LS
mm
C
CC
RR
The amplifier is a transconductance amplifier, which takes a voltage and produces an output current. The amplifier
together with the capacitors C1 and C2, form a circuit, which provides a negative resistance, when viewed from the
crystal. The value of which is given by:
2
21
ω ××
=
TT
m
NEG
CC
g
R
Where
m
g
is the transconductance
ω
is the frequency in rad/s
Derivations of these formulas can be easily found in textbooks.
In order to give quick and reliable oscillator start-up, a common rule of thumb is to set the amplifier negative
resistance to be a minimum of 4 times the effective crystal resistance. This gives
2
21
ω ×× TT
m
CC
g
≥
2
4
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
L
LS
m
C
CC
R
This can be used to give an equation for the required transconductance.
21
2
2121
2
])([4
TT
TTTTSm
m
CC
CCCCCR
g
×
×
+
+
×
≥
ω
Example: Using typical 32MHz crystal parameters of
mR =40Ω, SC =1pF and 1TC = 2TC =18pF ( for a load
capacitance of 9pF), the equation above gives the required transconductance (
m
g
) as 2.59mA/V. The JN5148 has a
typical value for transconductance of 4.3mA/V
The example and equation illustrate the trade-off that exists between the load capacitance and crystal ESR. For
example, a crystal with a higher load capacitance can be used, but the value of max. ESR that can be tolerated is
reduced. Also note, that the circuit sensitivity to external capacitance [ C
1
, C
2
] is a square law.
Meeting the criteria for start-up is only one aspect of the way these parameters affect performance, they also affect
the time taken during start-up to reach a given, (or full), amplitude. Unfortunately, there is no simple mathematical
model for this, but the trend is the same. Therefore, both a larger load capacitance and larger crystal ESR will give a
longer start-up time, which has the disadvantages of reduced battery life and increased latency.