View detail for Analyzing the Behavior of an Oscillator and Ensuring Good Start-up

Analyzing the Behavior of an Oscillator and
Ensuring Good Start-up
80C51 MCU’s
This application note explains how an oscillator functions and which methods can be
used to check if the oscillation conditions are met in order to ensure a good start-up
when power is applied.
Application Note
Oscillator Fundamentals
A microcontroller integrates on-chip an oscillator to generate a stable clock used to
synchronize the CPU and the peripherals.
Figure 1. Basic Oscillator Architecture
Noise
+
Xtal1
Amplifier
Xtal2
G(f)
Feed-back
Loop
H(f)
The basic architecture of an oscillator (regardless of its structure) is shown in Figure 1
and built around an amplifier, a feed-back and noise applied on Xtal1 input. The role of
each elements is explained hereafter:
•
Amplifier: Used to amplify the signal applied on Xtal1 and to lock the oscillations
exhibit Xtal2. The class A structure is the most popular but new ones are currently
used in order to optimize the consumption or other criterion,
•
Feed-back loop: Used to filter the output signal and to send it to the Xtal1 input.
The oscillator stability is linked to the bandwidth of the loop. The narrower the
filter, the more stable the oscillator. Crystals or ceramic resonators are generally
used because they have the narrowest bandwidth and efficiency for the stability of
the frequency.
Rev. 4363A–80C51–07/04
1
•
Noise: Thanks to the noise an oscillator is able to startup. This noise has different
origins:
–
thermal noise due to the transistor junctions and resistors,
–
RF noise: a wide band noise is present in the air and consequently on all the
pins of the chip and in particular on Xtal1 input of the amplifier. The noise
origin can be industrial, astronomic, semiconductor, ...
–
transient noise during the power-up.
The noise is coupled to the amplifier from the inside and outside of the chip through the
package, the internal power rails, ....
Figure 2 shows the typical oscillator structure used in most microcontroller chips. An onchip amplifier connected to an external feed-back consists in a crystal or a resonator
and two capacitors (a). Sometimes a resistor is inserted (b) between the amplifier output
and the crystal in order to limit the power applied, avoiding the destruction of the crystal.
Figure 2. Typical Oscillator Structures
Xtal1
Xtal2
a)
Xtal2
Xtal1
b)
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4363A–80C51–07/04
Typical Oscillator
Operation
The process involved in start-up and locking of oscillator is explained hereafter (see Figure 3):
•
Biasing process. The power-up is applied and the amplifier output follows the
power until it reaches its biasing level where it can amplify the noise signal on its
input.
•
Oscillation. The amplified noise on the output (Xtal2) is filtered by the feed-back
loop which has a pass-band frequency corresponding to the nominal oscillator
frequency. The filtered output noise is amplified again and starts to increase. The
oscillation level continue to grow and reaches the non-linear area.
•
Lock. In the non-linear area both the gain and the oscillation level starts to reduce.
•
Steady State. A stabilization point is found where the closed-loop gain is
maintained with the unity.
Figure 3. Process Needed to Reach a Stable Oscillation
VDD
Bias Level
Vxtal2
t
Biasing
Start Oscillation
Lock
Steady State
Each element plays a role and their electrical characteristics have to be understood. The
next sections explain this matter.
Crystal Model and
Operation
Crystal and ceramic resonators are piezoelectric devices which transform voltage
energy to mechanical vibrations and vice-versa. At certain vibrational frequencies, there
is a mechanical resonance. Main resonances are called: fundamental, third, fifth, ...
overtones. Overtones are not harmonics but different mechanical vibrational modes.
This crystal is an efficient pass-band filter which exhibits a good frequency stability. The
equivalent model, shown in Figure 4, consists of two resonant circuits:
•
C1, L1 and R1 is a series resonant circuit (fs),
•
In addition the series circuit, C0 in parallel forms a parallel circuit which has a
parallel resonance frequency (fa) .
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4363A–80C51–07/04
Figure 4. Crystal Models.
C0
R1
L1
C1
Q1
Q2
Figure 5 plots the module and phase of the impedance crystal and shows both the
series and parallel resonance frequencies.
Figure 5. Phase and Module Versus the Frequency
Z(f)
dB
96.034
100
Rp
1
:=
⋅
6.28
C1
)
( f )
fs
75
⋅
fs
50
L1
fa := fs ⋅ 1 +
25
R1
0
0
9.998
. 10
6
⋅
9.998
φ(f)
degree
90
) )
⋅
9.999
10
6
6
. 10
1
Series
. 10
C0
fa
C1 L1
C1
C0
f
7
1.0001
. 10
7
. 10
1.0002
7
1.0003
Parallel
. 10
7
1.0004
. 10
7
1.0005
f
. 10
1
×
1
90
45
360
6.28
0
45
−
90
90
f
6
6
7
7
7
7
7
7
The behavior of the crystal depends on the frequency and is summarized in Table 1.
Table 1. Nature of the Impedance Versus the Frequency
Frequency
f < fs
f=fs
fs < f < fa
f=fa
f>fa
Z(f)
Capacitive
C1
Resistance
R1
Inductive
L1
Resistance
Rp
Capacitive
C0
Phase(°)
-90
0
+90
0
-90
4
4363A–80C51–07/04
The impedance phase is related to the frequency and each elements of the model plays
a role in specific frequency ranges. The main electrical characteristics of these elements
are summarized hereafter.
Series resonance
frequency
fs
1
:=
⋅
Quality factor
⋅
6.28
Qs
:=
⋅
⋅
L1 6.28 fs
R1
C1 L1
Parallel resonance
fa := fs ⋅ 1 +
frequency
C1
Quality factor
C0
Qp
1
:=
⋅
⋅ ⋅
C0 6.28 fp R1
With External Load, CL
frequency
fp
:= fs ⋅  1 +

Quality factor
Qp
:=
C1
2⋅ ( C0
+
CL)



ESR
ESR := R1⋅  1 +

C0 

CL 
2
1
⋅
⋅ ⋅
CL 6.28 fp ESR
Table 2 gives some typical crystal characteristics.
Table 2. Examples of Crystal Characteristics
Frequency
MHz
R1
ohms
L1
mH
C1
fF
C0
pF
fs
MHz
fp
MHz
Qs
Qp
32
35
11.25
2.2
7
32
32.005
646k
3.11
30 (2)
20
11
2.6
6
30
30.0065
102k
6.14
30 (1)
40
33.94
0.83
3.8
30
30.00328
160k
3.48
20
50
20
3.2
10
20
20.0032
497k
2.98
16
80
11.641
8.5
3
16
16.022
146k
3.42
10
20
0.025
10
20
10
10.00025
159.2k
80
8
7
0.0862
4.6
40
8
8.00026
618k
17.4
6
8
0.0848
8.3
40
6
6.000356
533k
37
2
100
520
12
4
2
2.003
66K
198
Note:
1. Fundamental Mode
2. Third Overtone Mode
“Series” Versus
“Parallel” Crystal
There is no such thing as a “series cut” crystal as opposed to a “parallel cut” crystal.
Both modes exist in a crystal. Only the oscillator structures (Pierce, Colpitts, ..) will oscillate the crystal close to the fs or between fs and fa resonance frequencies. The first
structure is called a series resonant oscillator and the second a parallel resonant
oscillator. It should be noted that no oscillator structure is able to oscillate at the exact
fa frequency. This is due to the high quality factor at fa and the difficulty to stabilize an
oscillator at this frequency.
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4363A–80C51–07/04
Overtone or
Fundamental Mode
Vibrational mode is used to reduce the crystal cost. Above 20MHz it is costly to produce
such crystals tuned on the fundamental mode. To avoid that, an overtone mode is used
to tune the oscillation frequency. To work properly, this vibrational mode needs a specific schematic where a frequency trap is installed on the oscillator output to short-circuit
the fundamental mode and force the overtone mode. The trap is an LC filter installed
between the Xtal2 and the ground. The frequency on this filter is calculated on the fundamental mode using the Thomson equation (see Figure 6).
Figure 6. A LC trap is Used for an Overtone Oscillator
Xtal2
Xtal1
Ltrap
Ftrap
1
= -----------------------------------------------------------
2×π×
Ltrap × Ctrap
X1
C1
C2
Ctrap
0
Drive Level
The characteristics of quartz crystals are influenced by the drive level. In particular,
when the drive level increases, the frequency and the resistance change through nonlinear effects. In extreme cases an inharmonic mode may replace the main mode as the
selective element and cause the frequency of the oscillator jump to a different frequency. With an overdrive level, the crystal substrate itself may be damaged. Typical
characteristic of frequency vs. drive levels is shown in Figure 7.
Figure 7. Frequency Shift vs. Drive Level
Drive level is a measurement of the total power dissipated through the crystal operating
in the circuit. Typical drive levels are between 50 uW and 1000 uW (1 mW). Drive levels
should be kept at the minimum level that will initiate and maintain oscillation. It should be
less than half of the maximum drive level. Excessive drive may cause correlation difficulties, frequency drift, spurious emissions, "ringing" wave forms, excessive ageing, and/or
fatal structural damage to the crystal.
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4363A–80C51–07/04
The maximum drive, PMax, is specified by the crystal manufacturer. The maximum
RMS current which can flow in the crystal and it is given by the following expression:
PMrms := ESR⋅ IMrms2
IMrms :=
PMrms
ESR
where ESR is equivalent resistance at the parallel frequency, fp.
For example, 0.1 Watt Maximum power with an ESR of 32 ohms gives a 56mA maximum RMS current.
The RMS voltage across the crystal can be evaluated in the same manner:
UMrms := PMrms⋅ ESR
where UMrms is the maximum RMSvalue.
For example, if PMrms is 0.1Watt and ESR =32Ohms, the maximum RMS voltage
accross the crystal is 1.8V. In case of overdrive power, a resistor must be connected
between the amplifier output and the crystal as shown in Table 2.
Class-A Amplifier
Figure 8 gives an example of a class-A amplifier. Resistance Rf is used to bias the output stage to VDD/2. Cxtal1 and Cxtal2 are the parasitic capacitors due to input and
output amplifier pads plus the parasitic capacitances of the package. Rout is the equivalent output resistance of the amplifier. The equivalent schematic is true only for the
linear area of the gain and for small signal conditions. This linear operation occurs during the startup when the power is applied. The transfer function is often first order and
low-pass filter type.
VDD
Xtal1
Xtal1
G0
Rout Xtal2 G(f)
Rf
Xtal2
G0
vin
a)
VSS
vout
Cxtal1
b)
VSS
f
Cxtal2
c)
Figure 8. (a) Typical structure of a class-A amplifier. (b) Equivalent schematic. (c) Gain
response.
Next section explains the two specific amplifier areas needed to startup and lock an
oscillator.
7
4363A–80C51–07/04
The Two Operating Areas Figure 9 illustrates the transfer function of a CMOS amplifier. An amplifier such as that
shown in Figure 8 has two operating regions. These regions determine the oscillator
operation at start-up and during steady state while oscillations are stabilized. Figure 9
shows these two regions:
•
Region A, is the linear region. The gain is constant, and vout is proportional to
vin:
vout ( f )
=
G ( f ) × vin ( f ) )
The dynamic range of this linear region is typically +/- 1 volt around the quiescent
point Q at 5v VDD.
•
Region B, is the non-linear region. The gain is no longer linear, and becomes
dependent on the vout level. The higher the vout, the lower the gain. The
amplification is automatically reduced while the output oscillation increases until a
stabilization point is found (amplitude limitation).
Figure 9. Gain Curve and the Two Amplification Region
vs
B
Non-Linear region
A
VDD/2
Linear region
B
Non-Linear Region
ve
VDD/2
The oscillations start gradually. The noise on its input is amplified until the level reaches
VDD. If conditions (gain and phase) as specified above are fulfilled, startup is normally
guaranteed at circuit power-on time. Indeed, during power-on, noise over a large spectrum appears and is sufficient to start-up the system. Only a few microvolts or millivolts
are needed but the startup time is inversely proportional to this level. Typical waveform
of an oscillation is shown in Figure 10.
Figure 10. Start and Lock of a Feedback Oscillator
Vxtal2
Start and lock
Steady State
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4363A–80C51–07/04
Series and Parallel
Oscillators
Some oscillator architectures force the crystal to operate around the series frequency
and some others to work around the parallel frequency. This section gives information
about these working modes.
series resonant
oscillator
This structure used a non inverted amplifier to force oscillation at its the natural series
resonant frequency fs. The crystal phase is zero, the resistance is minimum (R1) and
the current flow is maximum.
Figure 11. Series Resonant Structure
Xtal1
Xtal2
Q1
Xtal1
Q2
R1
Q1
Xtal2
Q2
X1
The feedback (X1) filters the oscillation frequency and send this signal in phase to Q1
input.
Parallel Resonant
Oscillator
This structure used an inverted amplifier to force oscillation between fs and fa resonance frequencies where the crystal impedance appears inductive (L1). This structure is
called Pierce. To have this frequency resonant, fp, the imaginary part of the crystal
impedance must be zero. So only capacitive reactance can cancel the inductive one.
This is why the C1 and C2 capacitors are added on Xtal1 and Xtal2 (see Figure 12).
Figure 12. Parallel Resonant Structure
Xtal2
Xtal1
Q1
L1
Q2
Xtal1
CL
Q1
Xtal2
Q2
X1
ESR
C1
C2
The resonance frequency is given hereafter:
fp
:= fs ⋅  1 +

C1
2⋅ ( C0
+
CL)



where CL is the capacitive load equivalent to the C1 in parallel to C2.
The equivalent series resistance (ESR) is a little higher than for fs and is given with the
next expression:
2
C1 × C2
ESR = R 1 ×  1 + C-------0 ,CL = --------------------CL
C 1 + C2
Considering the expression of fp, CL plays an important role to have the required oscillation frequency. CL is the loading capacitor used during the crystal calibration by the
crystal manufacturer to tune the oscillator frequency. If an accurate frequency is
9
4363A–80C51–07/04
required CL must be respected. Here are some standard values are 13, 20, 24,30, and
32 pF.
Analysis Method
Two methods of oscillator analysis are considered in this application note. One method
involves the open-loop gain and phase response versus frequency. A second method
considers the amplifier as a one-port with negative real impedance to which the filter is
attached. The second one will be preferred for very low frequency (32KHz).
The next sections explains the basics of these two methods and how to use them.
Open-loop Gain and
Phase
This first method analyzes the product of the gain of the amplifier and the feed-back
loop.
Figure 13. Basic Oscillator Architecture
Amplifier
Noise
+
G(f)
vn(f)
vin(f)
vout(f)
Feed-back
Loop
H(f)
The general equation to start-up the oscillation process is shown hereafter. Let’s
express vout(f):
vout ( f ) = G ( f ) × Hf ( f ) × vout ( f ) + G ( f ) × vn ( f )
the transfer function between vout(f) and vn(f) is:
vout ( f)
G (f )
------------------ = -------------------------------------1 – G (f ) × H( f)
vn ( f )
the start-up condition can now be evaluated with the Barkhausen criteria:
G ( f ) × H ( f)
>1
Φ ( G ( f ) × H ( f) ) = 0
and lock condition can be expressed:
G ( f) × H( f ) =
1
This start-up condition depends on the product of the gain and feed-back but also on the
frequency. The lock condition is controlled by the non-linear area of the amplifier output.
The gain is automatically reduced while the output oscillation increased until a stabilization point is found.
10
4363A–80C51–07/04
To analyze the oscillation conditions, it is useful to use a Spice simulator. Some freeware are available on the Web and only the basic functions of Spice are required. Figure
14 shows a typical oscillator Spice circuit use to demonstrate the AC small signal
analysis.
Figure 14. Typical crystal oscillator structure.
As seen previously, the open-loop gain is analyzed to check the oscillation conditions.
To do that the feed-back loop is broken. The crystal has to be loaded with the same
impedance than the input impedance of the amplifier.
Figure 15 shows the Spice circuit used to analyses the oscillation conditions. A 16MHz
crystal is used for this analysis and CP1 and CP2 are tuned to have the oscillation conditions (G> 0dB, Phase=0).
Figure 15. Spice Circuit Used to Analyze the Oscillation Conditions
Xtal1
Xtal2
38pF
Figure 16 plots the gain and the phase of the open-loop circuit. At 16.001MHZ the gain
is greater than unity (38dB) and the phase is zero. The oscillation conditions are met
ensuring a good oscillator startup.
11
4363A–80C51–07/04
Figure 16. Gain and Phase response for the open-loop gain.
Gain(dB) Phase(°)
180
40
187
30
Phase = 0°
Gain = 38dB
144
Phase
20
108
100
Gain
10
72
0
36
0
16.007MHz
16.001MHz
0
-10
-55
This method allows to check the maximum capacitive loads and the maximum electrical
characteristics of the crystal.
Figure 17 (a) plots the gain and phase when Cp1 and CP2 are too big. The gain is now
too small to guarantee a proper startup. The phase begins to shift and is no longer zero.
Figure 17 (b) plots the gain and phase when the equivalent resistance of the crystal (R1)
is too big. The gain is now negative and the phase is not zero. The oscillation conditions
are not met and this oscillator will not start.
Figure 17. Gain and phase for two conditions
1.1
8.3
Phase > 0
0
G=-3dB
G=0.3dB
0
6.5
.00100MHz
V(VXtal1)/V(VXtal10))
a)
16.00188MHz
-21.4
b)
15.99503MHz
DB(V(VXtal1)/V(VXtal10))
16.00796MHz
a) Cp1 and Cp2 are too big (56pF), b) R1 is too big = 40ohms.
Table 3 resumes the case studies analyze with the spice model and tool.
12
4363A–80C51–07/04
Table 3. Oscillation Conditions versus Cp1, Cp2 and R1
Cp1(pF)
Cp2(pF)
R1(ohms)
Oscillation Conditions
33
33
10
Yes
33
33
40
No
56
56
10
No
CP1 and CP2 are generally chosen to be equal maintaining a gain in closed loop equal
to the unity.
Negative feed-back
resistance
The second method analyzes the real part on the input impedance of the amplifier and
compares it with the real part of the pass-band filter. The impedance seen on the input
amplifier is negative under certain conditions and cancelled the crystal resistance. In
that case there is no more lost of energy and oscillations are stabilized.
Figure 18 shows the equivalent model of an oscillator. The crystal is equivalent to a RLC
filter corresponding to the motional arm. Z3 in the equivalent impedance accross Xtal1
and Xtal2 pins including the C0 crystal capacitor and Cx3. Z1 and Z2 are the input and
output impedances including the two external capacitors Cp1 and Cp2 used to adjust
the oscillator operating point.
Figure 18. a) Oscillator Equivalent model b) Equivalent model around the resonance.
a)
b)
Crystal
R1
L1
Crystal
C1
R1
L1
C1
Z3
Rin
Xtal1
Cin
Xtal2
Amplifier
Z1
0
Z2
0
0
Amplifier
Figure 18 shows in what conditions the oscillator will oscillate. To have an oscillation
stable in steady condition, the lost of energy in the crystal has to be cancelled. This condition occurs when:
Rin = – R 1
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4363A–80C51–07/04
and at the frequency:
1
f = ----------------------------------------------------------C 1 × Cin
6, 28 × L 1 × ----------------------C 1 + Cin
Cin is the equivalent capacitor seen between Xtal1 and Xtal2 and is equal to:
1 × Cx 2
--------------------------Cin = C 0 + Cx 3 + Cx
C 1 x + Cx 2
where Cx1 and Cx2 are the global capacitors seen on the input and output pins. Cx3 is
the capacitor seen between Xtal1 and Xtal2 pins.
To ensure a good startup of the oscillator, Cx1 and Cx2 have to be correctly adjusted. In
order to define them, the amplifier impedance must respect the conditions on Rin and
Cin parameters:
•
Rin: Cx1 and Cx2 has to be adjusted to have Rin > R1:
( Cx 1 × Cx 2 ) × –gm
Rin ( Zc ) = ----------------------------------------------------------------------------------------------------------------------------------------------------------2
2
2
( gm × Cx 3 ) + ω × ( Cx 1 × Cx 2 + Cx 2 × Cx 3 + Cx 1 × Cx 3 )
•
Cin: Cx1 and Cx2 have to be adjusted to obtain a negative imaginary part and
finally a input capacitor.
gm 2 × Cx 3 + ω 2 × ( Cx 1 + Cx 2 ) × ( Cx 1 × Cx 2 + Cx 1 × Cx 3 + Cx 2 × Cx 3 )2Im ( Zc ) = –-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------2
2
2
ω × ( ( gm × Cx 3 ) + ω × ( xC 1 × Cx 2 + Cx 2 × Cx 3 + Cx 1 × Cx 3 ) )
Im ( Zc )
C = -------------------6, 28 × f
gm is the amplifier gain.
An example is given hereafter. The main characteristics of this case study is:
•
Amplifier: gm=0.01A/V, Cxtal1=5pf, Cxtal2=8pF, Cxtal3=5pf
•
Crystal: R1=80, L1=11.64mH, C1=8.5fF, C0=5pF
14
4363A–80C51–07/04
Figure 19. Oscillator Example
Crystal
R1
L1
C1
C0
5p
Cxtal3 5p
Xtal1
Xtal2
Cp1
Cxtal1
Cxtal2
gm
5p
8p
0.01A/V
0
Cp2
0
0
Amplifier
Table 4 shows two cases: first, there is no external additional capacitors and second two
capacitors are adjusted to the oscillation frequency.
When there is no capacitor Rin is less than R1 (80 ohms) and no oscillation occurs.
With Cp1=Cp2=5pf, Rin is -175 ohms and is greater than R1 and the condition to have
oscillations is met. As with the previous method, Cp1 and Cp2 can be tuned and the
electrical characteristics can be checked. Table 4 resumes the case studies.
Table 4. Cp1 and Cp2 capacitors with R1=80ohms.
Conclusions
Cp1(pF)
Cp2(pF)
Rin(ohms)
Cin(pF)
Oscillation Condition
0
0
-60
8.26
No
5
5
-175
9.2
Yes
Two methods have been presented to analyze and to check the oscillation conditions.They have shown the possibility to predict the added capacitors in versus the
electrical characteristics of the crystal or resonator devices. It will help to specify the
margin of the crystal and resonator devices.
15
4363A–80C51–07/04
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