Bài giảng Radio communications circuits - Chapter 5: Mixers - Trịnh Xuân Dũng
Dung Trinh, PhD
HCMUT / 2019
Chapter 5
Mixers
Trinh Xuan Dung, PhD
Department of Telecommunications
Faculty of Electrical and Electronics Engineering
Ho Chi Minh city University of Technology
1
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
Contents
1. Introduction
2. Mixers based on nonlinearities
3. Mixers based on Switches
4. Mixer specifications
5. Exercises
2
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Introduction - Heterodyne
Heterodyning is a radio signal processing technique popularized by Canadian
inventor-engineer Reginald Fessenden in 1901, in which new frequencies are created
by combining or mixing two frequencies.
The two frequencies are combined in a nonlinear signal-processing device such as a
vacuum tube, transistor, or diode, usually called a mixer. In the most common
application, two signals at frequencies f1 and f2 are mixed, creating two new signals,
one at the sum f1 + f2 of the two frequencies, and the other at the difference f1 − f2.
These new frequencies are called heterodynes (dị thể in Vietnamese). Typically only
one of the new frequencies is desired, and the other signal is filtered out of the output
of the mixer.
f1 + f2
f1
f1 − f2
f2
A Minicircuits Mixer
A Marki Mixer
Input and output signals of a mixer
3
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Introduction – Image Frequency
Consider 푣1 푡 = 푉 cos 휔1푡 and 푣2 푡 = 푉 cos 휔2푡. A multiplier can realize the
1
2
mixer function
1
푣표 푡 = 푣1 푡 × 푣2 푡 = 푉 푉 cos 휔1 + 휔2 푡 + cos 휔1 − 휔2 푡
1 2
2
Example: 푓 = 2450푀퐻푧, 푓 = 2400푀퐻푧. The IF frequency: 푓 = 50푀퐻푧.
푅퐹
퐿푂
퐼퐹
Example: 푓 = 2450푀퐻푧, 푓 = 2500푀퐻푧. The IF frequency: 푓 = 50푀퐻푧.
푅퐹
퐿푂
퐼퐹
Example: 푓 = 200푀퐻푧, 푓 = 2400푀퐻푧. The RF frequency: 푓 =
퐼퐹
퐿푂
푅퐹
2200푀퐻푧, 2600MHz.
푓
퐿푂
푓
푅퐹
푓
퐼퐹
푓
퐿푂
푓
푅퐹
푓
푅퐹
푓
퐼퐹
푓
퐼퐹
푓
푅퐹
4
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Introduction – Image Frequency
Example 1: The IS-54 digital cellular telephone system uses a receive frequency band
of 869–894 MHz, with a first IF frequency of 87 MHz and a channel bandwidth of 30
kHz. What are the two possible ranges for the LO frequency? If the upper LO frequency
range is used, determine the image frequency range. Does the image frequency fall
within the receive passband?
Solution:
The two possible LO frequency ranges are:
956 푡표 981푀퐻푧
782 푡표 807푀퐻푧
푓 = 푓 ± 푓 = 869 푡표 894 ± 87 =
퐿푂
푅퐹
퐼퐹
Using the 956 − 981푀퐻푧 LO, the IF frequencies are
푓 = 푓 − 푓 = 869 푡표 894 − 956 푡표 981 = −87푀퐻푧
퐼퐹
푅퐹
퐿푂
The RF image frequency range is
푓 = 푓 − 푓 = 956 푡표 981 + 87 = 1043 푡표 1068푀퐻푧
퐼푀
퐿푂
퐼퐹
which is well outside the receive band.
5
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Introduction – Image Frequency
Signals at the image frequency will be down-converted to IF frequency and corrupted
the desired signal.
• Even if there is no other signal at the image frequency, noise can still be down-
converted, lowering your NF by 3 dB!
• An Image reject filter is usually put in front of the mixer to attenuate the signals
and noise at the image frequency.
Block diagram with a more modern look (super-heterodyne receiver) is based on the
same principle.
Effect of Image Frequency
A simplified superheterodyne receiver
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Introduction – Mixer Implementations
Fundamentally, mixing (heterodyning) can happen by two mechanisms
• Non-linear circuit: use device nonlinearity.
• Linear but time-variant circuit.
Mixers can also be classified by:
• Passive mixers.
• Active mixers.
The nonlinear mixer can be applicable at any frequency. It is the only approach
available at the upper mm-wave frequencies.
When frequencies are low enough that good switches can be built, the switching
mixer is preferred because it generates fewer spurs. In some cases, sampling has been
substituted for switching.
Most mixers in use today are of the switching type, whereby diodes or transistors are
used to switch the connection between the RF input and the IF output at a rate that is
controlled by the local oscillator.
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Mixers based on Nonlinearities
Consider a square-law device (could be a properly biased diode or transistor):
푖표 푡 = 퐾푣푖2푛 (푡)
The output voltage on the load:
푣표 푡 = 푅퐿푖표 푡 = 푅퐿퐾푣푖2푛 푡 = 푅퐿퐾 푣푅퐹 푡 + 푣퐿푂 푡
Spurious signals
2
1
1
푣표 푡 = 푅퐿퐾 푉푅2퐹 cos 2휔푅퐹푡 + 푉2 cos 2휔퐿푂푡
퐿푂
2
2
1
2
1
푉푅2퐹 + 푉2 + 푉푅퐹푉퐿푂 cos 휔퐿푂 + 휔푅퐹 푡 + 푉푅퐹푉퐿푂 cos 휔퐿푂 − 휔푅퐹 푡
퐿푂
2
Mixing signals
DC Components
In general cases, the output current will
contain terms that are proportional to all
powers of the total input voltage:
푖표 푡 = 퐼퐷퐶 + 푘1푣푖푛 푡 + 푘2푣푖2푛 푡 + ⋯
푖표
+
푣푅퐹
푣퐿푂
푣푖푛
푅퐿
-
In general, this produces all possible
mixing products, including: 푓 , 푓 ,
푅퐹
퐿푂
푓 ± 푓 , 2푓 ± 푓 , 푓 ± 푓 , etc.
푅퐹
퐿푂
푅퐹
퐿푂 퐿푂
푅퐹
Principle of mixers based on nonlinerities
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Diode Mixers based on Nonlinearities
Diodes typically exhibit exponential I-V characteristics, which, at small signals, can
be expanded to a polynomial.
2
3
푞푉
푑
푞푣푑
푞푣푑
푞푣푑
푛퐾푇
푖푑 = 퐼푆 푒
− 1 ≃ 퐼푆
+
+
+ ⋯
푛퐾푇
푛퐾푇
푛퐾푇
Mixing by non-linear circuit tend to generate a lot of spurious signals. We are mostly
interested in 푓 = 푓 ± 푓 , the rest of the frequencies are considered as undesired
퐼퐹
푅퐹
퐿푂
intermodulation products. Isolations between the three ports are provided by filters.
Schottky diodes are predominant in mixer applications, particularly at millimeter-
wave frequencies, because their fast transition time. Still, single ended diode mixers
are rarely used because of its bad isolation between the ports.
Non-linear
component
VRF
V
I
+
-
VIF
RL
+
VLO
-
HSCH-9162 mmwave GaAs Schottky diode
A diode mixer
Schottky diode packaging
9
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Diode Mixers based on Nonlinearities
An implementation of an unbalanced diode mixer is shown in belowed figure. There are
a lot of spurious outputs generated. Ideally, we would like to see outputs only at 10MHz
and 210MHz.
So, we prefer the switching type mixer when the RF and LO frequencies are low
enough that we can make decent switches.
This takes us up through much of the mm-wave spectrum.
An implementation of a diode mixer
Output signal of the mixer
10
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. BJT Mixers based on Nonlinearities
In a practical implementation, a matching network would be used between the RF
source and the RF input in order to couple the maximum amount of RF signal to the
base of the transistor.
The matching network would be designed to present a high impedance to the LO
signal. Similarly, a matching network would be used between the LO source and the
LO input, and this network would be designed to present a high impedance to the RF
source. In this way, the LO source is prevented from loading the RF source, and vice
versa. A simpler compromise is to supply the LO voltage through a small-value
coupling capacitor (CLO) such that the LO source does not significantly disturb the
matching for the RF signal. This circuit is attractive because of its simplicity, and it
has been used in inexpensive mass-produced receivers for consumer applications.
In general, a single-ended BJT mixer will have a higher
noise figure than a mixer that employs an FET, or a
properly designed passive mixer based on a diode bridge.
In addition, BJT mixers are subject to severe
intermodulation distortion. To avoid this problem, vRF
must be kept smaller than 10 mV. In general, BJT mixers
of this type have poor large signal characteristics.
A BJT mixer
11
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. FET Mixers based on Nonlinearities
2
If a FET is operated in saturation region, then: 푖퐷 = 퐼퐷푆푆 1 − 푣
.
푔푠
푉
푝
The FET mixers has two advantages over the BJT mixers:
• A much lower 3rd IMD, since there is no cubic term in the 푖퐷 versus 푣푔푠
relationship.
• Much higher RF input voltages are usable, i.e., up to at least 100 mV.
A disadvantage of the FET mixer is smaller conversion gain than the BJT circuit.
VDC
A major drawback of the single-ended BJT and FET
mixers is the presence of a local oscillator component as
L1
well as an RF component at the output of the mixer.
RF
Although the LO and RF signals will be filtered by the
IF
C1
BPF
tuned output circuit, a relatively large LO signal will
inevitably leak through the mixer to the IF stages.
LO
Q1
A large LO component at the input to the IF stages is
obviously undesirable, since it can cause overload at the IF
stages. It can also create undesirable effects such as gain
compression and possible generation of intermodulation
BPF
R1
products in the IF stages.
Dept. of Telecoms Engineering
A FET mixer
12
Dung Trinh, PhD
HCMUT / 2019
1. Introduction – Image Frequency
Example 2: A single-ended FET mixer is to be designed for a wireless local area
network receiver operating at 2.4 GHz. The parameters of the FET are 푅푑 = 300Ω,
푅푖 = 10Ω, 퐶푔푠 = 0.3푝퐹, and 푔1 = 10푚푆. Calculate the maximum possible conversion
gain. Note that the conversion gain of the FET mixer is defined as
2
푉
푍퐿
퐼퐹
2
푅퐿
2
4푅퐿푅푔
푃
푉
퐼퐹
퐼퐹−푎푣푎푖푙
퐺푐 =
=
=
2
2
푉푅퐹
4푅푔
푃푅퐹−푎푣푎푖푙
푍퐿
푉푅퐹
Solution:
The conversion gain is:
푔12푅푑
10 × 10−3 2 × 300
퐺퐶 =
=
= 36.6 = 15.6푑퐵
4휔푅2퐹퐶푔2푠푅푖
2
4 × 2휋 × 2.4 × 109 2 × 10
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Balanced Mixers based on Nonlinearities
A balanced configuration can be used to effectively remove the LO and/or RF signals
from the output of the mixer.
One type of balanced mixer often used in integrated circuits uses the differential-pair
as shown in figure belows.
If the BJT is in active mode:
푣
퐵퐸
푉
푇
푖퐶 = 퐼푆푒
For the emitter-coupled pair with identical transistors:
푣
푖
푣
퐵퐸1
−푣
퐵퐸2
푖퐶1
푖퐶2
푉
푉
푇
푇
= 푒
= 푒
Since the emitters are connected to a constant current source, then
푖퐸1 + 푖퐸2 = 퐼퐸퐸 or
푖퐶1 + 푖퐶2 = 훼퐼퐸퐸
Then:
훼퐼퐸퐸
푖퐶1 =
푖퐶2 =
−푣
푖
푉
푇
1 + 푒
훼퐼퐸퐸
푣
푖
푉
푇
1 + 푒
A differential pair
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Balanced Mixers based on Nonlinearities
If the output is taken between the collectors of the two transistors as in following
figure, the output voltage will be proportional to the difference of the emitter currents:
푣
−푣
푖
푖
2푉
2푉
푇
푇
푇
푇
푒
− 푒
푣푖
Δ푖퐶 = 푖퐶1 − 푖퐶2 = 훼퐼퐸퐸
= 훼퐼퐸퐸 tanh
푣
푖
−푣
푖
2푉푇
2푉
2푉
푒
+ 푒
If the input voltage is small, 푣푖 2푉푇 ≪ 1, then:
푣푖
Δ푖퐶 ≃ 훼퐼
퐸퐸 2푉푇
If a good signal, 푖푖2, is added:
훼
Δ푖퐶 ≃
퐼퐸퐸 + 푖푖2 푣푖
2푉푇
A FET balanced mixer
Current difference Δ푖퐶
15
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Balanced Mixers based on Nonlinearities
If 푣푖1 → 푣푅퐹 and 푣푖2 → 푣퐿푂, the output voltage will proportional to 퐾 + 푣퐿푂 푣푅퐹.
Note that 푣푅퐹 signal is present at the output, along with the desired signal.
If 푣푖1 → 푣퐿푂 and 푣푖2 → 푣푅퐹, the output voltage will proportional to 퐾 + 푣푅퐹 푣퐿푂.
Note that 푣퐿푂 signal is present at the output, along with the desired signal.
The unwanted term can be eliminated as follows:
• We construct two identical multipliers, and drive the differential
inputs of each amplifier with 푣푖1.
• Drive the tail current input of amplifier 1 with 푣푖2 and the tail-
current amplifier of multiplier 2 with −푣푖1.
• The output of multiplier 1 is then proportional to 퐾푣푖2 + 푣푖1푣푖2,
the output of multiplier 2 is proportional to 퐾푣푖2 − 푣푖1푣푖2.
• The difference between the two outputs will contain only a term
proportional to 푣푖1푣푖2 because the unwanted term will be
canceled out.
• Multipliers based on this principle are called Gilbert cell
multipliers, and are named after Barrie Gilbert who described a
practical, and precise, multiplier circuit based on the technique.
A FET balanced mixer
16
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
3. Mixers based on switches
Most mixers used today are based on switches. Let 푣푅퐹 푡 = 푉푅퐹 cos 휔푅퐹푡, then:
∞
푛휋
sin
1
2
2
푣표 푡 = 푣푅퐹 푡 푝 푡 = 푉푅퐹 cos 휔푅퐹푡 푉
+
sin 푛휔퐿푂푡
퐿푂
푛휋
2
푛=1
The coefficient sin 푛휋 is zero if n is even, therefore the output signal has
2
components at the following frequencies: 푓 , 푓 ± 푛푓 , 푛 = 1,3, …
푅퐹
푅퐹
퐿푂
1 2
+ sin 휔퐿푂푡 + sin 3휔퐿푂푡 + sin 5휔퐿푂푡 + ⋯
2 휋
1
1
푣표 푡 = 푉푅퐹 cos 휔푅퐹푡 푉
퐿푂
3
5
1
1
1
= 푉푅퐹푉퐿푂 cos 휔푅퐹푡 + sin 휔푅퐹 + 휔퐿푂 푡 − sin 휔푅퐹 − 휔퐿푂 푡
2
휋
휋
Mixing products
1
1
RF feedthrough
+ sin 휔푅퐹 + 3휔퐿푂 푡 − sin 휔푅퐹 − 3휔퐿푂 푡 + ⋯
3휋 3휋
Spurs
A filter can be used to select the desired component which is usually, 푓 ± 푛푓 .
푅퐹
퐿푂
푣푅퐹
푣푅퐹
푅퐿
푅퐿
푣퐿푂
푣퐿푂
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
3. Mixers based on switches
There are many ways to implement a switch that is controlled by the local oscillator
signal. Diodes are often used as switches.
If a relatively large LO voltage is impressed on a diode, the diode will be forward
biased on positive excursions of the LO signal, and will be in a low-impedance state
for superposed, small RF signals.
On negative excursions of the LO voltage the diode is driven into a high impedance
state and behaves like an open switch.
Transistors can also be used as switches, with the LO controlling the bias current and
determining whether the device is biased into a low-impedance state, or is cutoff.
푣푅퐹
푣푅퐹
푅퐿
푅퐿
푣퐿푂
푣퐿푂
18
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
3. Single Balanced Mixers based on switches
A balanced LO drive can reduce RF feed-through.
If 푣푖1 → 푣퐿푂, 푣푖2 → 푣푅퐹 and allow the amplitude of 푣퐿푂 to be large compared to 푉푇;
푣
푖
then the term tanh
will be driven to +1 on positive excursions of 푣퐿푂 and to
2푉
푇
− 1 on the negative excursions. In this mode of operation, the transistors are driven as
switches.
The output voltage of the circuit is:
∞
푛휋
sin
2
푣표 푡 = − 퐼퐷퐶 + 퐼푅퐹 cos 휔푅퐹푡 푅푉 2
sin 푛휔퐿푂푡
퐿푂
푛휋
2
푛=1
The output signal has components at the following
frequencies: 푛푓 , 푓 ± 푛푓 , 푛 = 1,3, …
퐿푂
푅퐹
퐿푂
A FET balanced mixer
Principle of switching balanced mixer
19
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
3. Single Balanced Mixers based on switches
The DC term which was responsible for the RF feedthrough in the unbalanced mixer
does not appear in the double balanced mixer:
4
1
1
푣표 푡 = − 퐼퐷퐶 + 퐼푅퐹 cos 휔푅퐹푡 푅푉
sin 휔퐿푂푡 + sin 3휔퐿푂푡 + sin 5휔퐿푂푡 + ⋯
퐿푂
휋
3
5
We still get LO feedthrough if we take a single-ended output or if there is a DC
current in the signal path. There is often DC present since the output of the trans-
conductance amplifier will have a DC current component.
Output signals of a single balanced mixer
20
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