THIS DOCUMENT CONTAINS
- Introduction to Mixers
- Schottky Barrier Diodes
- Introduction to Sub-Harmonic Mixers
- Sub-Harmonic mixing with anti-parallel diodes
INTRODUCTION TO MIXERS
Mixer was invented by Major Edwin Armstrong. Previously mixing was used earlier
mainly for down-converting received signal to baseband. The techniques for this worked
poorly as stability of local oscillator was inadequate. Armstrong used vaccum tube
mixers to shift received signal to intermediate frequency (IF) which could be amplified
with good selectivity, high gain and low noise. FM’s invention is also credited to
Armstrong. His superheterodyne receiver is still the model for communication receivers
in use today.
Today diode mixers are well established and high quality Schottky barrier diodes are
readily available. Conversion losses of less than 4 dB at frequencies of 50 GHz are
possible. At high frequencies diode mixer is the only device that can be used at all.
A mixer is fundamentally a multiplier. Signal applied to RF port has carrier frequency
ωs and modulation waveform A(t).
Multiplier Block Diagram
The output is found to be modulated components at the sum and difference frequencies.
An ideal multiplier is not the only thing that can realize a mixer. Any non-linear device
can perform multiplying function.
The use of non-ideal multiplier results in the
generation of LO harmonics and in mixing products other than the desired one. In this
case, the desired output frequency component must be filtered from the resulting mess.
I/V characteristic of non-linear device can be described as
I = a0 + a1V + a 2V 2 + a3V 3 + ...
If V is made equal to the sum of two different signals, after some trigonometric
manipulations, it can be shown that the current contains components at frequencies fn =
f0 + nfl, where f0 is the difference frequency fs − fl. The current also contains the
harmonics of the LO, but it is easy to filter out the undesired frequencies and process the
desired difference frequency.
Anti-Parallel Diode Mixer
Schottky barrier diodes are the most popular nonlinear mixing element at millimeter and
sub-millimeter-wave frequencies. They can be incorporated in waveguide or quasi-optical
designs, have instantaneous bandwidths of several gigahertz and can cover the entire
spectral range to 0.1 mm. As pointed out earlier, for many applications, it is extremely
difficult, expensive and inconvenient to generate a fundamental frequency local oscillator
signal at sub-millimeter wavelengths. To overcome this problem, quite often, a nonlinear
mixing element is pumped with half the LO frequency and the RF is mixed with the
second harmonic of the LO generated in the nonlinear device. Though it is possible to
have subharmonic mixing using a single diode, the fundamental mixing response is
greater than the second harmonic response in such mixers. As a result, the conversion
loss in such mixers is greater. Instead, two diode mixers (anti-parallel configuration), as
shown in figure above, give better performance in terms of conversion loss and noise
performance. If the diodes used are identical, this configuration suppresses fundamental
and other harmonic mixing products as well as even harmonics of the LO.
SCHOTTKY BARRIER DIODE
Schottky barrier diodes are made by a metal contact to a semiconductor - the metal
contact end acting as anode and the semiconductor end as cathode. The difference in
work function between the metal contact and the semiconductor gives rise to the
rectification property in the Schottky barrier diodes. It is a majority carrier device,
because the conduction is due to the thermionic emission of the majority carriers over the
barrier formed by the unequal work functions of the metal and the semiconductor. Figure
below shows the band structure of the Schottky junction. Two other figures below show
the band structure for the forward and reverse biased Schottky junction.
Band Structure of Schottky Diode
The current voltage characteristics of diode is given by :
I (V ) = I 0 (e qV / ηKT − 1)
where V is the applied voltage, q is the unit charge, T is the absolute temperature, K is
Boltzmann constant, η is the diode ideality factor - which identifies the strength of the
diode nonlinearity; and
I 0 = A * *WT 2 e − qφb / KT
where A** is the modified Richardson constant and W is the junction area. Figure below
shows the equivalent circuit model of the Schottky diode. This intrinsic diode model has
a nonlinear resistance and capacitance, and a linear series resistance. The series resistance
also varies with the junction voltage, but the variation is not significantly large and for
most practical purposes can be neglected. This model does not show the parasitic
capacitances and inductances which arise from diode metallization or lead geometry.
Forward Biased Schottky Diode Band Structure
Reverse Biased Schottky Diode Band Structures
Schottky Diode Equivalent Circuit
Schottky diodes can be of different kinds, depending on the fabrication methodology. For
high frequency applications, whisker contacted diodes have historically been the most
widely used. The whisker contacted diode has the advantage of minimum parasitics, and
the usefulness of the whisker as a tuning element. However, the whisker is very fragile
and the loss of contact is common under vibration and shock. On the contrary, planar
Schottky diodes are rugged and can easily be integrated into arrays. The disadvantage of
the planar Schottky diode is the added parasitics.
INTRODUCTION TO SUB-HARMONIC MIXERS
Development of high power and inexpensive LO sources is the major challenge faced by
researchers. The power available from solid-state sources drops off with the inverse
square of frequency due to electronic limitations in the material, and
hence, at higher frequencies, higher LO powers come at a much higher cost. Therefore,
one of the main goals of terahertz mixer design has been the reduction of LO power
requirements, with emphasis towards receiver configurations that permit harmonic
mixing. The advantages of harmonic mixing surpass the disadvantages (higher
conversion loss compared to fundamental mixing) when a pair of anti-parallel diodes are
used as mixer element. This has the added benefit of reduced LO noise, suppression of
fundamental and other odd harmonic mixing products, and also the suppression of the
even harmonics of the LO. For the anti-parallel diode pair shown in figure below, the
pump signal must have sufficient power to turn on each diode once in a single RF cycle,
i.e., the VLO must swing from −VTO to VTO. It is clear that a sub-harmonic mixer
employing anti-parallel diodes requires more power than a optimally biased single diode
mixer. One possible improvement would be to design the mixer and associated coupling
structure in a way which permits separate biasing for each of the diodes in the antiparallel pair (see figure). Since each diode is biased near VTO, the VLO does not need to
swing all the way from −VTO to VTO.
I-V Curves for unbiased and biased diodes
Efficient coupling of LO and RF signals to the diodes is one of the essential requirements
for achieving the lowest possible conversion loss and highest receiver sensitivity. At
submillimeter wavelengths, integrated circuit antennas are, perhaps, the most convenient
coupling structures.
SUB-HARMONIC MIXING WITH ANTI-PARALLEL DIODES
The balanced mixers have been one of the main building blocks of microwave
engineering for many years. The symmetrical structure of the balanced mixers have two
major advantages over single diode mixers - i) the down converted AM noise from the
local oscillator (LO) does not appear at the IF output and ii) the signal and the LO power
enter the mixer through separate ports, eliminating any external diplexer. Although twodiode subharmonic mixers have properties similar to balanced mixers (like AM noise
suppression), the basic operating principle is different. In this section, the theory of the
two-diode subharmonic mixer will be described with some mathematical details.
In a conventional single diode mixer, as shown in Figure below, application of a
voltage waveform
V = VLsinωLt + Vssinωst
to the asymmetric diode I-V characteristic results in the diode current having all the
frequencies mfL + nfs. However, in the case of an anti-parallel diode pair, as shown in
Figures below the diode current contains frequencies for which m+ n is an odd integer.
The terms for which m + n is even, (i.e., even harmonics, fundamental mixing products
(ωs −ωL) and (ωs +ωL), and the dc term), flow only within the diode loop. From Figures
below, the instantaneous current through the diodes can be written as :
i1 = −i s (e −αV − 1)
i2 = i s (e −αV − 1)
where α is the diode slope parameter q/ηKT. Similarly the differential conductance for
each diode can be written as :
g1 =
di1
= αi s e −αV
dV
g2 =
di2
= α i s e αV
dV
Single Diode Mixer
Anti-Parallel Diode Pair
The composite time varying differential conductance is given by the sum of these two
individual conductances.
g = g1 + g2
= αis(eαV + e−αV )
= 2αiscoshαV
From the above expression, it is clear that g has even symmetry with V , shown in Figure
above, and the number of conductance pulses per LO cycle in the anti-parallel diode
circuit is twice that for a single diode circuit. When this diode pair is pumped with LO, it
modulates the conductance of the diode and substituting V = VLcosωLt in equation
above, we get :
g = 2αiscosh(αVLcosωLt)
which, upon expansion, gives :
g = 2αis[I0(αVL) + 2I2(αVL)cos2ωLt + 2I4(αVL)cos4ωLt +…]
where In(αVL) are modified Bessel functions of the second kind. It is clear from
the above equations that the conductance terms consist of a dc term and the even
harmonics of the LO frequency, ωL. When the applied voltage is
V = VLcosωLt + Vscosωst
the current will be :
i = g(VLcosωLt + Vscosωst)
i = AcosωLt + Bcosωst + Ccos3ωLt + Dcos5ωLt +
Ecos(2ωL + ωs)t + Fcos(2ωL − ωs)t + Gcos(4ωL + ωs)t + Hcos(4ωL − ωs)t + …
It can be seen from the above that the total current contains only frequency terms
mfL + nfs, where m + n is an odd integer, i.e., m + n = 1, 3, 5,…There is one more
component of current ic, as can be seen in Figure above. This circulating current arises
because the Fourier expansion of individual currents i1 and i2 gives rise to components of
current which are opposite in phase. Because of the opposite polarity, they cancel each
other at the output terminal but circulate within the diode loop. The circulating current
can be written as (from Figure)
ic =
(i2 − i1 )
2
= i s [cosh αV − 1]
Substituting
V = VLcosωLt + Vscosωst
⎡ (V L cos ω L t + Vs cos ω s t )2
⎤
+ ....... − 1⎥
ic = i s ⎢1 +
2!
⎢⎣
⎥⎦
[
]
=
is
2
2
VL cos 2 ω L t + Vs cos 2 ω s t + 2VLVs cos ω L t cos ω s t + .......
2
=
2
2
2
2
⎫⎪
i s ⎧⎪VL + Vs
Vs
VL
ω
t
+
cos
2
+
cos 2ω s t + VLVs [cos(ω L − ω s )t + cos(ω L − ω s )t ] + .......⎬
⎨
L
2 ⎪⎩
2
2
2
⎪⎭
From the above equations it can be seen that the circulating current only contains
frequencies mfL + nfs, where m + n is an even integer.
Thus, the anti-parallel diode pair has the advantage of suppressing fundamental and other
odd harmonic mixing products and also the even harmonics of the LO. However, it
should be kept in mind that the degree of suppression degrades with the imbalance in the
diode pair. It should also be noted that the degradation of receiver noise figure due to LO
noise sidebands (which is the case in single diode mixers) is also reduced in even
harmonic mixing (m = even, n = 1) with anti-parallel diodes. This is because the LO
noise sidebands whose separation from the LO (fL) equals IF (fIF ), generate IF noise
which only circulates within the diode loop when they mix fundamentally with the LO;
but second harmonic mixing of these noise sidebands with the virtual LO (2fL) produces
noise which are not within the IF amplifier pass band (figure below)
Noise Side Band Mixing Products
Finally, the anti-parallel diode circuit has inherent self protection against large peak
inverse voltage, because a reverse biased diode is always in parallel with a forward
biased diode, which limits the reverse bias swing less than the breakdown voltage of the
diodes.