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Frequency reference standards are essential to achieving frequency
accuracy and phase stability in electronic systems. Such sources require
the chief characteristics of low phase noise and good frequency stability.1-13 The
best oscillator performance can be expensive, however. Fortunately,
a patented approach has been developed to design and optimize the performance
of voltage-controlled crystal oscillators (VCXOs), even those with
relative low quality-factor (Q) resonators, to achieve excellent phase
noise and frequency stability.
A typical oscillator consists of a tuned circuit and an active device
such as a transistor. Ideally, the tuned circuit provides a high loaded
Q, generally from less than 100 for simple circuits to more than 1
million for crystal-resonator-based circuits. Noise arises from the
active device as well as from resonator losses. Noise from a bipolar
transistor, for example, stems from base and collector contributions
and from device parasitic elements, such as the base-spreading resistor.
The filtering effect of the resonator tends to remove the device noise,
with higher Qs delivering greater filtering effects. The Leeson equation
relates these noise effects.1 The formula was modified by
Rohde for use with VCOs.2
The equation is linear, with many unknowns. Among the more difficult
oscillator performance parameters to predict are output power, noise
figure, operating Q, and flicker corner frequency. The parameters can
not be derived for linear conditions but require large-signal (nonlinear)
analysis.3 But by combining Leeson's formula with the contributions
of the tuning diode,2 Eq. 3 results, making it possible
to calculate oscillator noise based on a linear approach:
where:
£(fm) = the ratio of sideband power in a 1-Hz
bandwidth to the total power (in dB) at the frequency offset (fm);
f0 = the center frequency;
fc = the flicker frequency;
QL = the loaded quality factor (Q) of the tuned circuit;
F = the noise factor;
kT = 4.1 10–21 at 300°K (room temperature);
Psav = average power at oscillator output;
R = the equivalent noise resistance of tuning diode (typically 50 Ω to
10 kΩ); and
Ko = the oscillator voltage gain.
Equation 1 is limited by the fact that loaded Q typically must be
estimated; the same applies to the noise factor. The following equations,
based on this equivalent circuit, are the exact values for Psav,
QL, and F, which are required for the Leeson equation. Figure
1 shows the typical simplified Colpitts oscillator
giving some insights into the novel noise calculation approach.4
From ref. 3, the noise factor can be calculated by:
After some small approximation,
Figure
2 (left) illustrates the dependency of the noise
factor on feedback capacitors C1 and C2. From
Eq. 1, the phase noise of the oscillator circuit can be enhanced
by optimizing the noise factor terms as given in Eq. 3 with respect
to feedback capacitors C1 and C2.
Equation 4 can be found by substituting 1/re for Y21+ (+
sign denotes the large-signal Y-parameter).
When an isolating amplifier is added, the noise of an LC oscillator
is determined by Eq. 5.
where:
G = the compressed power gain of the loop amplifier;
F = the noise factor of the loop amplifier;
k = Boltzmann's constant;
T = the temperature (in degrees K);
P0 = the carrier power level (in W) at the output of the
loop amplifier;
F0 = the carrier frequency (in Hz); fm = carrier
offset frequency (in Hz);
QL = (πF0τg) = the loaded Q
of the resonator in the feedback loop; and aR and aE =
the flicker noise constants for the resonator and loop amplifier, respectively.
From ref. 4, the output power for the Colpitts oscillator circuit
of Fig.
1 can be calculated by:
where:
n = 1 + (C1/C2);
Vce = the collector-emitter voltage (< Vcc);
RL = the load resistance; and
QL = the loaded quality factor.
The factor of 1000 is needed since the result is expressed in dBm
and a function of n and C1.
To calculate the loaded Q (QL), it is necessary to consider
the unloaded Q (Q0) and the loading effect of the transistor.
The influence of Y21+ must also be considered.
The inverse is responsible for the loading and reduction of the Q.
Based on the transformation of the loading of the device emitter's
differential impedance (resistance), it is possible to calculate the
noise factor of the transistor under large-signal conditions. Considering
the large-signal transconductance, Y21+,
which can be determined from the large-signal approach, this new approach
to calculating the noise is based on existing general noise calculations.
The approach applies to bipolar transistors but an equivalent procedure
can be found for FETs.
Given an ideal resonator, with no long-term drift, aging, or variations
due to temperature, tuning circuits would not be needed to correct
for these performance shortcomings. But for practical applications,
resonators require corrective measures to keep them on frequency. This
is typically achieved through the use of tuning diodes with capacitance
values that change as a function of applied voltage. In a Colpitts
oscillator configuration with quartz crystal resonator, a capacitive
voltage divider is used and the crystal acts like a high-Q inductor,
slightly detuned from its series resonant condition. Since quartz crystal
is a mechanical resonator driven by the piezoelectric effect, fundamental
and a variety of overtone frequency modes (3rd, 5th, 7th, and 11th
overtones) are possible. Unfortunately, undesired mode jumping is also
possible even in well-planned circuit designs.
Circuitry must be added for a VCXO to remain in a desired operating
mode. For example, if the fifth overtone mode is desired, a poorly
designed oscillator may jump to the third overtone mode unexpectedly.
Typically, crystal oscillators operate from about 1 to 150 MHz. Higher
frequencies are possible, but the crystal resonators become thinner
for higher-frequency operation and aging effects increase.
To demonstrate the new VCXO optimization approach, a commercial 120-MHz
VCXO was used as an example to apply the concept of conduction angle
as outlined earlier.4 Figure
3 shows the typical measured phase noise of the VCXO
before (left) and after (right) the first modification, respectively.
As another example, a 100-MHz crystal-controlled Colpitts oscillator
was designed according to a set of specifications that included +13
dBm output power, 50-Ω
load, and phase noise of –132 dBc/Hz offset 100 Hz from the carrier,
with the intention of applying the new approach to this basic design
to achieve improved performance (Fig.
4).
The first step in the design process involved calculating the operating
point for a fixed normalized drive of x = 20 (Table 6-1 in ref. 3).
The output voltage at the fundamental frequency, based on the output-power
requirement, can be found from Eq. 8.
The fundamental current can be found from Eq. 9.
The DC operating point is calculated based on the normalized drive
level x = 20. The expression for the emitter DC current can be given
in terms of the Bessel function with respect to the drive level can
be found with Eq. 10.
For the normalized drive level x = 20, the output emitter current
at the fundamental frequency can be given as Eqs. 11 and 12.
Figure
5 shows the oscillator circuit configuration with
DC and RF current distribution components as describes in Eqs. 13
and 14.
For this typical example, an NE68830 transistor from NEC was selected
for validation.
The second step in the design process involved the development of
the biasing circuit. For the best close-in phase noise, a DC/AC feedback
circuit is incorporated to provide the desired operating DC conditions,3 with
IE = 28.3 mA, VCE = 5.5 V, supply voltage, Vcc =
8 V, = 120, and IB ≈ 0.23 mA.
The third step involves calculating the large-signal transconductance
(see Tables 6-1 and 6-2 in ref. 3) using Eqs. 15 and 16.
The fourth step in the procedure involves the calculation of loop
gain, which can be found from Eqs. 17 and 18.
 
where Rp1(f0) is the equivalent resistive load
across the port 1 (Fig.
6).
For practical purpose, the loop gain should be 2.1 to achieve good
starting conditions for oscillation.
From Eq. 10, it is possible to derive Eq. 19.
The fifth step in the design procedure involves calculation of the
feedback capacitor ratio by means of Eq. 20.
The next step involves calculation of the absolute value of the feedback
capacitor by means of Eq. 21. In this case, Zin (looking
into the base of the transistor) can be calculated by means of Eq.
21.3
where:
CP = (CBEPKG + Contribution from layout) = 1.1 pF and
LP = (LB + LBX + the contribution
from layout) = 2.2 nH.
The expression for the negative resistance (Rn, without
parasitics) is found by means of Eqs. 22-24.
For sustained oscillation, the conditions are shown in Eq. 25.
From Eqs. 17 and 18, it is possible to derive Eq. 26.
From Eq. 13, it is possible to derive
Eqs. 27 and 28.
where C2 = 22 pF and, for practical purposes, C1*
= 22 pF.
The final step in the approach involves calculating the phase noise.
From Eqs. 1 through 6, the calculated phase noise offset 100 Hz from
the carrier, using the modified Leeson equation, is –132 dBc/Hz.
Simulated results (not shown here) and calculated data for the example
100-MHz oscillator, along with the measurements, agree within about
2 dB.
By applying the patented technology,5-8 Fig.
7 shows the resulting simplified schematic diagram
for an enhanced-performance 100-MHz VCXO. Figure
8 shows a phase-noise plot for the improved 100-MHz
oscillator after applying noise filtering, controlled RF conduction
angle, and delay feedback as outlined in the patents.5-8 The
new approach can be used to improve the performance of inexpensive
low-Q crystal resonators for cost-effective signal generation.
Figure
8 also compares the simulated
"off-the-shelf" phase noise of the VCXO with a first-pass optimization
and with the best results from these new techniques. The simulated
results from Ansoft Designer were validated with theory and measurements.3 Alternative
VCXO topologies can not compete with these results in terms of cost,
size, power, phase noise, and reconfigurability. The new approach includes
evanescent-mode coupling for optimum group delay to enhance the loaded
Q and suppress mode-jumping phenomena (especially when the crystal
resonates at higher odd-order overtone modes). It also minimizes the
effects of thermal drift without sacrificing phase-noise performance.8
The approach shown at 100 MHz is being applied in the development
of high-performance VCXOs to be used in Synergy Microwave's lines of
advanced frequency synthesizers. The technique can also be extended
to higher frequencies by incorporating mode-injection stubs across
the crystal resonator for the desired operating frequency. The technique
is available for licensing; interested parties are advised to contact
one of the authors.
Synergy Microwave Corp., 201 McLean Blvd., Paterson, NJ 07504;
(973) 881-8800, FAX: (973) 881-8361, Internet: www.synergymwave.com
REFERENCES
1. D.B. Leeson, "A Simple Model of Feedback Oscillator Noise Spectrum," Proceedings
of the IEEE, Vol. 54, 329-330, 1966.
2. U.L. Rohde and T.T.N. Bucher, Communication Receivers Principles &
Design, McGrawHill, New York, 1988, p. 302.
3. U.L. Rohde, A.K. Poddar, and G. Boeck, The Design of Modern Microwave
Oscillators for Wireless Applications: Theory and Optimization,
Wiley, New York, 2005.
4. U.L. Rohde, "A New Efficient Method of Designing Low Noise Microwave
Oscillators,"
Dr.-Ing. Dissertation, TUBerlin, Germany, February 12, 2004, p. 99.
5. U.L. Rohde and A.K. Poddar, "Tunable Frequency, Low Phase Noise
and Low Thermal Drift Oscillator," United States Patent No. 7,196,591.
6. U.L. Rohde and A.K. Poddar, "Wideband voltage controlled oscillators
employing evanescent mode coupled resonators," United States Patent
No. 7,1803,812.
7. U.L. Rohde, A.K. Poddar, and R. Rebel, "Integrated Low Noise Microwave
Wideband Push-Push VCO," United States Patent No. 7,088,189.
8. U.L. Rohde and A.K. Poddar, "Novel Multi-Coupled Line Resonators
Replace Traditional Ceramic Resonators in Oscillators/VCOs," Proceedings
of the 2006 IEEE International Frequency Control Symposium (IFCS),
Florida, June 5-7, 2006.
9. B. Parzen and A. Ballato, Design of Crystal and other Harmonic
Oscillators, Wiley, New York, 1983.
10. N. Nomura, Y. Aoyagi, and Y. Sekine, "1-GHz High Frequency Colpitts
Oscillator For High Frequency Application," Proceedings of the 2005
IEEE International Frequency Control Symposium (IFCS), pp. 526-529.
11. D.V. Bogomolov, R. Boroditsky, "Low Phase Noise UHF TCXO," Proceedings
of the 2005 IEEE International Frequency Control Symposium, pp. 509-511.
12. Y. Watannake, "Phase Noise Measurements in DualMode SC-Cut Crystal
Oscillators,"
IEEE Transactions on UFFC, Vol. 47, No. 2, March 2000.
13. V.F. Kroupa, "Theory of 1/f noise –A New Approach," Phys.
Lett. A, January 2005, pp. 126-132.
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