TECHNICAL NOTE 30 - Statek Corporation

TECHNICAL NOTE 30
Design Guidelines for Quartz Crystal Oscillators
Introduction
A CMOS Pierce oscillator circuit is well known and
is widely used for its excellent frequency stability
and the wide range of frequencies over which
they can be used. They are ideal for small, low
current and low voltage battery operated portable
products especially for low frequency applications.
[1,2] When designing with miniaturized quartz
crystals, careful consideration must be given to
the frequency, gain and crystal drive level.
In this paper, the design equations used in a
typical crystal controlled pierce oscillator circuit
design are derived from a closed loop and phase
analysis. The frequency, gain and crystal drive
current equations are derived from this method.
Basic Crystal Oscillator
The basic quartz crystal CMOS Pierce oscillator
circuit configuration is shown on Figure 1. The
crystal oscillator circuit consists of an amplifying
section and a feedback network. For oscillation to
occur, the Barkhausen criteria must be met
V
V1
Figure 1 - Basic Pierce Oscillator Circuit
Crystal Characteristics
In order to analyze the quartz crystal oscillator, we
must first understand the crystal itself. Figure 2
shows the electrical equivalent circuit of a quartz
crystal. The L1, C1 and R1 are generally referred to
as the electrical equivalent of the mechanical
parameters; inertia, restoring force and friction,
respectively. These parameters can be measured
using a crystal impedance meter or a network
analyzer. C0 is the shunt capacitance between
terminals and the sum of the electrode
capacitance of the crystal and package
capacitance.
a) The loop gain must be equal to or
greater than one, and
b) The phase shift around the loop must
be equal to an integral multiple of 2π.
The CMOS inverter provides the amplification and
the two capacitors, CD and CG, and the crystal
work as the feedback network. RA stabilizes the
output voltage of the amplifier and is used to
reduce the crystal drive level.
R1 Motional Resistance L1 Motional Inductance
C1 Motional Capacitance C0 Shunt Capacitance
Figure 2 – Crystal Electrical Equivalent Circuit
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This equivalent circuit can effectively be simplified
as a resistance (Re) in series with a reactance
(Xe) at a frequency f as shown in Figure 3.
Figure 3 - Effective Electrical Circuit of a Quartz
Crystal
From equation (1) and (2), an example of the
magnitude of Re and Xe as a function of frequency
are shown in Figures 4 and 5 respectively for
fs=32.768kHz, C1=2.4fF, and R1=28kΩ. The
frequency is expressed in terms of part per million
(ppm) above the series resonant frequency (fs) of
the crystal (∆f/f). These two graphs are very
useful in the analysis of the crystal oscillator.
1.E+06
Re (ohms)
Re (f) and Xe (f) as a function of frequency are as
follows:
R1
Re(f) =
R1 2
+
XO
( ) (
2
Xm
-1
XO
)
C0=3.5pF
C0=1.4pF
1.E+05
(1)
1.E+04
1
10
100
1000
10000
∆ f/f above fs (ppm)
Figure 4 - Re (ohms) vs. ∆f/f (ppm)
(2)
1.E+08
Co=3.5pF
where
1.E+07
(3)
Xe (ohms)
1.E+06
The series resonant frequency of the crystal is
defined as
Co=1.4pF
1.E+05
(4)
1.E+04
.
The quality factor Q is defined as
Q=
ωSL 1
R1
=
1
ωSR 1C 1
1.E+03
1
(5)
10
100
1000
10000
∆ f/f above fs (ppm)
Figure 5 - Xe (ohms) vs. ∆f/f
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IX
Crystal Oscillator Design
The AC equivalent circuit of the amplifier and
feedback network of a pierce oscillator is shown in
Figure 6. For the following analysis, RA is omitted
and will be reintroduced later.
2
2
Re + Xe’
Re
θ
Xe X e’
’
XG
’
X e’ = Xe - XG
sinθ =
Re
2
’ 2
Xe’
cosθ =
,
2
Re + X e’
’ 2
Re + X e’
Figure 6 - Pierce Oscillator AC Equivalent Circuit
Figure 8 - Impedance Phase Diagram
From Figure 6;
,
,
,
Frequency Equation
and
From the imaginary part of the current phase
diagram (y-axis)
,FRVθ ,;,2VLQθ
The phase and amplitude relationship of the
oscillator voltage, current and impedance are
shown in Figures 7 and 8. Assume that the
oscillator is oscillating at a frequency f and the
amplifier output current ID is 180° out of phase
with the oscillator input voltage V1.
V = IX
2
’ 2
and from the equations derived from the
equivalent circuit, the voltage and impedance
phasor diagram equation (6) becomes
’
X e’
XD
from
VOLTAGE
IX = IX + IX
Re
RO
’
X e’ = Xe - XG
IX
Re + X e’
θ
VXe VXe’
(6)
.
VRe
Then
V1
.
(7)
Assuming
9;H 9;H9
(XR ’) << (XX ’- 1)
1
O
9 2
m
O
2
and
R1
<<
X mX O ’
Xm
-1
XO’
9;H995H
Figure 7 - Current and Voltage Phase Diagram
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s
Equation (2) becomes
Xe(f) =
Xm
X
1- m
XO’
s
(7a)
Then
1
where XO’ = ωC ’ and CO’ = CO + Cs .
O
where CL = CS + CL’ .
Cs is the circuit stray capacitance across the
crystal.
Let
(7b)
and
.
From eq. 7a and 7b one can obtain
X mX O’ = X C ’ (X O’ - Xm)
L
Xm =
XCL ’X O ’
XO’ + XC ’
L
Then
(10)
(8)
Equation 10 is the oscillating frequency of the
crystal oscillator. CL is called the load capacitance
of the oscillator. With a specified CL , the crystal
manufacturer can then match the crystal to the
customer’s circuit to obtain the desired oscillation
frequency. From the CL equation, the relationship
between the other circuit parameters can be
established (i.e. CD, CG, RO and CS) as it relates to
the oscillation frequency of the crystal oscillator.
In a typical CMOS oscillator RO generally
decreases as the supply voltage increases. This
causes a decrease in load capacitance and an
increase in the oscillation frequency.
Figure 9 shows the effective load capacitance
(CL) changes as the output resistance (RO)
changes.
From eq. (3) and (4)
14
CL (pF)
13
CD = 20pF
CG = 30pF
12
11
C&SM
V = 1.0pF
10
CO = 1.4pF
R1 = 28k Ω
9
8
0.01
From equation (8)
0.1
1
10
RO (MΩ)
Figure 9 -
.
Effective Load Capacitance (CL)
vs. Output Resistance (RO)
(9)
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Gain Equation
From the real part of the current phase diagram
(x-axis);
ID = IO cos θ + I1 sin θ
(11)
It is important to note here that in most analyses,
only the first term of equation (12) is used. The
second term must be taken into account
especially for low frequency application were the
second term becomes larger than the first term as
shown in Figure 10, when RO is less than 1.2 MΩ.
and from the equation derived from the voltage,
and impedance phase diagram equation (11)
becomes
6
g m (u mhos)
5
gm2
4
3
2
gm1
1
0
0
1
2
3
4
RO (MΩ)
,
and from X’e = Xe – XG and
Figure 10 - Comparison of minimum gm
requirements vs. Amplifier’s
output resistance (RO).
Where
nd
gm1 = first term and gm2 = 2 term of
equation(12).
For CD=20pF, CG=30pF, CS=1.1pF, CO=1.4pF,
R1=28KΩ, fO=32.768kHz and CL=13pF.
eq. (7)
(12)
(
CO’
where Re ~ R1 1 +
CL ’
)
2
Equation (12) gives the minimum gm required for
the oscillator to maintain oscillation. In practice, 5
to 10 times the calculated value is required to
insure fast start of oscillation. This equation also
aids the designer in selecting the component
values for CD and CG to match the CMOS
amplifier and the crystal.
Using equation (12), Figures 11 and 12 show the
change in the minimum gm requirements due to
change in either CD or CG, while maintaining the
other capacitor constant. For a 32.768kHz
oscillator, as shown in Figure 11, trimming the
output capacitor (CG) will produce more change in
gm than the input capacitor (CD). As shown in
Figure 12, a decrease in the amplifiers’ output
resistance (RO) increases the minimum gm
requirement.
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2.5
Where;
g m (u mhos)
and,
CG = 30pF
2
1.5
1
CD =30pF
0.5
0
0
10
20
30
40
CD or CG (pF)
Figure 11 - For RO= 2.5MΩ gm comparison
between CD and CG, where
CS=1.1pF, CO=1.4pF, R1=28KΩ,
fO=32.768kHz
Figure 13 - Oscillator AC equivalent circuit
with the crystal electrical
equivalent circuit.
The crystal voltage, current and impedance phase
relationships are shown in Figure 14 and 15;
VOLTAGE
CURRENT
6.7
CG = 30pF
g m (u mhos)
5.7
4.7
3.7
CD = 30pF
2.7
1.7
0.7
0
10
20
30
40
CD or CG (pF)
Figure 12 - For RO= 500kΩ gm comparison
between CD and CG, where
CS=1.1pF, CO=1.4pF, R1=28KΩ,
fO=32.768kHz
Figure 14 - Voltage and current phase
relationship with the circuit
equivalent
CRYSTAL IMPEDANCE
Crystal Drive Current
In order to analyze the current flowing through the
crystal, the AC equivalent circuit from Figure 6 is
redrawn to show the crystal’s electrical equivalent
circuit as shown in Figure 13. The crystal drive
current is ib , and ia is the current through the
shunt capacitance CO’.
Figure 15 – Crystal impedance phase
diagram
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V
Typical Effects Of RA In The Oscillator
Circuit
From; |ia| = e and,
XO’
ia = IX
(Re2 + Xe2)
(13)
XO’
where
1
1
=
.
ω(CO ’) ω(CO ’ + C sm ) X O’ =
From the current phase diagram of Figure 14 and
the relationship
.
and from the crystal impedance phase diagram
Figure 15
sinφ =
Re
2
2
;
cosφ =
Re + Xe
Xe
2
2
.
Re + Xe
Substituting sin φ and cos φ and ia from equation
(13)
In many cases, a resistor RA is introduced
between the amplifier output terminal and the
crystal input terminal as shown in Figure 1. The
use of RA will increase the frequency stability,
since it provides a stabilizing effect by reducing
the total percentage change in the amplifier output
resistance RO and also increases the effective
output impedance by RA as shown on Figure 9.
RA also stabilizes the output voltage of the
oscillator and is used to reduce the drive level of
the crystal.
The complete AC equivalent circuit of Figure 1 is
shown in Figure 16, where Xd is the total output
capacitance of the amplifier.
Using the same analytical approach, the
frequency, gain and crystal drive current
equations with RA are derived.
Figure 16 - Pierce oscillator AC equivalent
circuit with RA included
or ib =
From the frequency equation (10);
Substituting;
(10)
(14)
where
X’e = Xe – XG
From eq. (14) the crystal drive can be calculated
from;
(in Watts)
where R1 = crystal’s motional resistance.
CL = CS + CL’
where;
and,
The gain equation is;
2 2
[
gm > 4π f CG (CD + Cd) Re +
Re
(C + R C ) R ]
d
d
A
O
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References:
[1] S.S. Chuang and E. Burnett, “Analysis of
th
CMOS Quartz Oscillator”, Proc. 9 Int. Congress
Chronometry (Stuttgart, W. Germany), Sept. 1974
paper C2.2
[2] E. Vittoz, “High-Performance Crystal Oscillator
circuits: Theory and Application” IEEE Journal of
Solid state circuits, vol. 23, No. June 1988 pp.774783.
where
The crystal drive current;
(1 + XX )+ (XR ’)
R
[R + R (1 - XX ’’)] + [X ’ + R X ]
2
e
|V|
O
ib =
e
e
e
2
O
2
A
e
e
A
O
where
and,
(
th
A version of this paper was presented at the 18
Piezoelectric Devices Conf. in Aug. 1996 by Jim
Varsovia
2
D
CO’
Re ~ R1 1 + C ’
L
) .
2
Summary
By using the closed loop and phase diagram
method, we were able to derive the frequency,
gain and crystal drive current equations for a
simple quartz crystal pierce oscillator. From the
equations derived herein, it can be shown that the
stray capacitance, minimum gain requirements
and the output resistance of the amplifier must be
carefully considered to obtain optimum oscillator
performance. The minimum gain requirements
should include consideration for the full range of
operational temperature and voltage. The stray
capacitance (CS) is especially critical due to
negative feedback effects and will increase the
minimum gain requirements of the oscillator [1].
As crystal manufacturers continue to miniaturize
the crystal resonator, the oscillator designer must
take into account the trade off in the crystal,
amplifier and the circuit layout strays in order to
select the appropriate component values to
achieve proper crystal drive, start up, and a stable
oscillation.
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