Frequency Multiplication Techniques - Crystek

PAGE 1 • APRIL 2013 FEATURE ARTICLE WWW.MPDIGEST.COM

Frequency Multiplication Techniques

by Ramon M. Cerda, VP of Engineering, Crystek Corporation

TAbstract Table 1: PSD of Fluctuations
his paper will review some of the
many ways to achieve frequency
multiplication. Frequency multipli-
cation is commonly done in RF/Microwave
equipment to generate high-stability, low-
noise signals.

How the frequency multiplication is
implemented will affect the quality of the
final frequency (i.e. phase noise/jitter).
What follows is a review of some multipli-
ers, and the reader is encouraged to use
the Reference list for more specific design
details.

A mathematical review of frequency
multiplication will now follow. An ideal
signal has no harmonics and can be char-
acterized by



Eq. 1.1

Where V0 = nominal peak voltage

v0 = nominal signal frequency Table 2: Ideal multiplication by a factor of N

We can expand this ideal signal into a By definition dividing phase fluctua-
realistic signal by adding amplitude and tions by the radiant frequency will give
phase deviation/noise. Eq. 1.1 now becomes



Eq. 1.2 Eq. 1.5 Eq. 1.8

Where ε (t) = instantaneous amplitude Where x(t) is called the instantaneous Allen Deviation
time fluctuations. Allen deviation is a two-sample statistics
fluctuations and standard method to describe short-
Power Spectral Density (PSD) term frequency stability in oscillators. It
ϕ (t) = instantaneous phase In order to understand how a signal is is the square-root of the Allan Variance.
altered by multiplication, it will be use- The Allen deviation is used because the
fluctuations normal variance will not converge for
The main analysis of interest of Eq. 1.2 ful to define ϕ(t), y(t) and x(t) by their large sample sizes of frequency data of
oscillators. The Allen deviation is denoted
is frequency-stability which can be derived power spectral densities (PSD) as listed in by sy(t).
by taking the time derivative of Table 1. It represents the power shape of
the function. Most of us are very familiar Frequency Multiplication by a Noiseless
Which gives the instantaneous frequency: with the PSD of the Gaussian function; a Multiplier
“Bell” curve. Multiplying a signal like Eq. 1.2 by a fac-
Eq. 1.3 tor of N in an ideal noiseless multiplier
The difference between the instanta- Another useful PSD is the single-sid- will affect it as summarized in Table 2.
neous frequency v(t) and nominal frequen- ed sideband (SSB) phase noise to carrier
cy v0, divided by the nominal frequency power ratio, L(ƒ) which has the units of From Table 2, one can note that the
is defined as the fractional frequency (or dBc/Hz. phase noise L(ƒ) went up by 20logN,
normalized frequency). Hence, a well known result. However, Δƒ / ƒ,
The relationships between these densi-
ties are: Sy (ƒ) and σ(τ) are unaffected by frequency
Eq. 1.4
Eq. 1.6 multiplication, which may not be well
known results. The result that Δƒ / ƒ stays
Eq. 1.7 unchanged is the following: if the original
signal has an accuracy of -5.0 ppm, then
after multiplication by N, it is still -5.0
ppm.

PAGE 2 • APRIL 2013 FEATURE ARTICLE WWW.MPDIGEST.COM

Figure 2: Distributed Nonlinear Transmission Line multiplier using
nonlinear capacitors

Figure 1: Typical Step Recovery Diode Multiplier

Figure 3: Lumped element NLTL using inductors and varactors Figure 4: Direct Multiplication AC circuit

Figure 6: General PLL block diagram using a VCSO for the VCO.

Figure 5: Mixer multiplication be considered as a Switching and varactors as the circuit in The bandpass filter and output
Reactance Multiplier (SRM), Figure 3. Varactors are a clever stage are tuned to the desired
Step Recovery Diode (SRD) or a charge-controlled switch. way to implement nonlinear integer multiplication.
Multiplication Figure 1 illustrates a typical capacitors since their capaci- Mixer Multiplication
SRDs are very popular for SRD multiplier. tance will vary nonlinear ver- Another popular technique for
harmonic frequency multipli- sus their reverse voltage. This frequency multiplication is to
cation and frequency comb Nonlinear Transmission Line creates a transmission line in use a mixer as shown in Figure
generation. SRDs are special- Multiplication which the propagation velocity 5. Let
ly designed diodes where the Frequency multiplication can is voltage dependant.
minority carrier’s (electrons on also be accomplished with When VS1(t)=VS2(t) then
the p side and holes on the n Nonlinear Transmission Lines Direct Multiplication
side) lifetime is sufficiently long (NLTL). They are similar to Direct multiplication is a very Hence the frequency and the
so as not to recombine; and SRD only in the fact that they efficient method of frequency phase error are doubled.
hence a charge is stored. There are both comb generators. multiplication. Figure 4 shows PLL Multiplication
are two states of reactance Figure 2 is a distributed version the AC circuit representation Phase Locked Loop (PLL) fre-
during operation of an SRD. using transmission lines and with two transistor stages.
Forward bias corresponds to nonlinear capacitors. A simpler Direct multiplication takes
high capacitance while reversed implementation uses inductors advantage of the nonlinear
bias is low capacitance. Due to nature of a saturated collector.
these two states, an SRD can In the circuit, L1, C1, C2, L2
and C3 form a bandpass filter.
Direct multiplication is usually
used to multiply by small inte-
ger numbers like, 2, 3, 4 or 5.

PAGE 3 • APRIL 2013 FEATURE ARTICLE WWW.MPDIGEST.COM

quency multiplication is proba- Symmetricom, IEEE
bly the most popular technique
mentioned so far. There are International Frequency
many good books and arti-
cles on PLLs. In Figure 6, this Control Symposium, Tampa
particular PLL uses a Voltage
Controlled SAW Oscillator Florida, May 4, 2003.
(VCSO) for the normal VCO.
Sinewave VCSOs can have “Pulse and Waveform
very low phase noise floors as
shown in Figure 7. Because of Generation with Step Recovery
this, PLL multiplication to a
single frequency using a VCSO Diodes,” HP note AN 918.
is good combination. The PLL
will improve the close-in phase “Harmonic Generation using
noise of the VCSO while main-
taining its excellent noise floor. Step Recovery Diodes and SRD

Modules,” HP note 920.

“A New Breed of Comb

Generators Featuring Low

Phase Noise and Low Input

Power,” Microwave Journal,

May 2006 issue.

“Two-Diode Odd-Order

Frequency Multipliers,”

Wenzel Associates, Inc.

Low Noise Schottky Diode Figure 7: 640 MHz VCSO phase noise. Crystek’s model CVCSO-
Odd-Order Multiplier 914-640
Figure 8 is an odd-order low
noise multiplier. It is a good Figure 8: Odd-Order Two-Diode Multiplier
choice for multiplying crystal
oscillators by 3, 5 or 7. When
used with low noise Schottky
diodes, it will add low excess
noise above the theoretical
20logN limit.

References

“The relationship between

phase stability and frequency

stability and method of con-

verting between them,” Peter P.

Bohn (NASA), October 1971.

“ Te c h n i q u e s for

Frequency Stability Analysis”