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<DIV><BR><A class=kicker2
href="http://www.mwrf.com/Topics/TopicID/717/717.html"><SPAN
class=kicker2>[Components]</SPAN></A><BR><FONT
class=commonSectionTitle>Technique Trims VCXO Phase Noise</FONT> <BR><FONT
class=deck>This patented circuit approach can improve the phase-noise
performance and frequency stability of even low-cost voltage-controlled crystal
oscillators.</FONT><BR><BR><A class=articleCommonCopy
href="http://www.mwrf.com/Authors/AuthorID/1399/1399.html">Ulrich L.
Rohde,</FONT></A> <A class=articleCommonCopy
href="http://www.mwrf.com/Authors/AuthorID/1659/1659.html">Ajay Kumar
Poddar</FONT></A> | <FONT class=articleCommonCopy>ED Online ID
#16332</FONT> | <A class=articleCommonCopy
href="http://www.mwrf.com/Issues/IssueID/602/602.html">August 2007</A> <BR><BR>
<DIV id=resizable><SPAN id=intelliTxt>
<P>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.<SUP>1-13
</SUP>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.</P>
<P>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.<SUP>1 </SUP>The formula was
modified by Rohde for use with VCOs.<SUP>2</SUP></P>
<P>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.<SUP>3 </SUP>But
by combining Leeson's formula with the contributions of the tuning diode,<SUP>2
</SUP>Eq. 3 results, making it possible to calculate oscillator noise based on a
linear approach:</P>
<P><IMG height=56 src="http://www.mwrf.com/Files/30/16332/Screen_01.gif"
width=400></P>
<P>where: </P>
<P>£(f<SUB>m</SUB>) = the ratio of sideband power in a 1-Hz bandwidth to the
total power (in dB) at the frequency offset (f<SUB>m</SUB>);<BR>f<SUB>0 </SUB>=
the center frequency; <BR>f<SUB>c </SUB>= the flicker frequency;<BR>Q<SUB>L
</SUB>= the loaded quality factor (Q) of the tuned circuit;<BR>F = the noise
factor; <BR>kT = 4.1 10<SUP>–21 </SUP>at 300°K (room temperature);<BR>P<SUB>sav
</SUB>= average power at oscillator output;</P>
<P>R = the equivalent noise resistance of tuning diode (typically 50 Ω to 10
kΩ); and <BR>K<SUB>o </SUB>= the oscillator voltage gain.</P>
<P>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 P<SUB>sav</SUB>, Q<SUB>L</SUB>, and
F, which are required for the Leeson equation. <A
href="http://www.mwrf.com/Files/30/16332/Figure_01.gif">Figure 1</A><I><B>
</B></I>shows the typical simplified Colpitts oscillator giving some insights
into the novel noise calculation approach.<SUP>4</SUP></P>
<P>From ref. 3, the noise factor can be calculated by: </P>
<P><IMG height=60 src="http://www.mwrf.com/Files/30/16332/Screen_02.gif"
width=500></P>
<P>After some small approximation, </P>
<P><IMG height=51 src="http://www.mwrf.com/Files/30/16332/Screen_03.gif"
width=400></P>
<P><A href="http://www.mwrf.com/Files/30/16332/Figure_02.gif">Figure 2</A><I><B>
</B></I>(left) illustrates the dependency of the noise factor on feedback
capacitors C<SUB>1 </SUB>and C<SUB>2</SUB>. 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 C<SUB>1 </SUB>and
C<SUB>2</SUB>.</P>
<P>Equation 4 can be found by substituting 1/r<SUB>e </SUB>for
Y<SUB>21</SUB><SUP>+ </SUP>(+ sign denotes the large-signal Y-parameter). </P>
<P><IMG height=53 src="http://www.mwrf.com/Files/30/16332/Screen_04.gif"
width=500></P>
<P>When an isolating amplifier is added, the noise of an LC oscillator is
determined by Eq. 5.</P>
<P><IMG height=130 src="http://www.mwrf.com/Files/30/16332/Screen_05.gif"
width=300></P>
<P>where:</P>
<P>G = the compressed power gain of the loop amplifier;<BR>F = the noise factor
of the loop amplifier; <BR>k = Boltzmann's constant;<BR>T = the temperature (in
degrees K);<BR>P<SUB>0 </SUB>= the carrier power level (in W) at the output of
the loop amplifier;<BR>F<SUB>0 </SUB>= the carrier frequency (in Hz); f<SUB>m
</SUB>= carrier offset frequency (in Hz);<BR>Q<SUB>L </SUB>=
(πF<SUB>0</SUB>τ<SUB>g</SUB>) = the loaded Q of the resonator in the feedback
loop; and a<SUB>R </SUB>and a<SUB>E </SUB>= the flicker noise constants for the
resonator and loop amplifier, respectively.</P></SPAN></DIV>
<DIV></DIV><BR>
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