1363
17.12 Mixers and Balanced Modulators
17.12 MIXERS AND BALANCED MODULATORS
In radio frequency applications, we often need to translate signals from one frequency to another.
This process includes both mixing and modulation, and generally requires some form of nonlinear
multiplication of two signals in order to generate sum and difference frequency components in
the output spectrum. Single-balanced mixers eliminate one of the two input signals from the
output, whereas the outputs of double-balanced circuits do not contain spectral components at
either of the input frequencies.
17.12.1 A Single-Balanced Mixer
One concept for a single-balanced mixer appears in Fig. 17.89(a) with the switch implemented
using a differential pair in Fig. 17.89(b). A signal v1 at frequency ω1 is used to vary the current
supplied to the emitters of the pair:
i E E = I E E + I1 sin ω1 t
(17.198)
The second input is driven by a large-signal square wave at frequency ω2 , which switches current
i E E back and forth between the two collectors (just as in the ECL gate discussed in Chapter 9),
and alternately multiplies the differential output voltage by +1 and −1. This multiplication can
be represented by a unit amplitude square wave with a Fourier series given by
v2 (t) =
4
sin nω2 t
nπ
n odd
(17.199)
Using Eqs. (17.198) and (17.199),
v O (t) = [i C2 (t) − i C1 (t)]RC = (I E E + I1 sin ω1 t)RC
4
sin nω2 t
nπ
n odd
or
(17.200)
4 I 1 RC
I 1 RC
v O (t) =
I E E RC sin nω2 t +
cos(nω2 − ω1 )t −
cos(nω2 + ω1 )t
nπ
2
2
n odd
VCC
VCC
RC
RC
RC
– vO +
RC
– vO +
iC2
iC1
+
v2
Q2
v2
–
iEE = IEE + I1 sin ␻1t
iEE = IEE + I1 sin ␻1t
–VEE
–VEE
(a)
Q1
(b)
Figure 17.89 (a) Basic single-balanced mixer. (b) Differential pair implementation.
1364
Chapter 17
Frequency Response
The output signal has spectral components at ω2 and at frequencies shifted by ω1 above and below
each of the odd harmonics of frequency ω2 . Note that no signal energy appears at frequency ω1 ,
and the circuit is said to be balanced relative to input v1 . On the other hand, ω2 appears in the
output spectrum, so the circuit is termed a single-balanced mixer. The most common applications
focus on the sum and difference components with n = 1. We find energy at ω2 − ω1 and ω2 + ω1 .
A filter is used to select one of these two components. Up-conversion uses ω2 + ω1 and whereas
down-conversion utilizes ω2 − ω1 .
Figure 17.90 shows a simple FM receiver application in which a narrow-band VHF signal at
100 MHz is mixed with a local oscillator (LO) signal at 89.3 MHz. The low-level VHF signal
would be used to generate i 1 (t) in Eq. (17.198), and the LO would be used as the switching
signal v2 (t). The narrow-band VHF spectrum is shifted to both 10.7 MHz, which is selected by
a band-pass filter, and 189.3 MHz, which is rejected by the same filter.
VHF input
(100 MHz)
Mixer
10.7-MHz
band-pass filter
RF output
(10.7 MHz)
LO input
(89.3 MHz)
(a)
iO ( f )
RF
output
VHF
input
Rejected
output
LO
f
10.7 MHz
100 MHz
189.3 MHz
89.3 MHz
(b)
Figure 17.90 (a) Mixer block diagram and (b) spectrum in FM receiver application.
Exercise: The LO signal in Fig. 17.90 could also be positioned above the VHF frequency.
What would then be the local oscillator frequency and the center frequency of the unwanted
output frequency signal?
Answers: 110.7 MHz; 210.7 MHz
Exercise: (a) An FM receiver is to be tuned to receive a station at 104.7 MHz. What must be
the local oscillator frequency to set the output to the 10.7-MHz filter frequency? (b) Repeat
for an input frequency of 88.1 MHz.
Answers: 94.0 MHz or 115.4 MHz; 77.4 MHz or 98.8 MHz
17.12
Mixers and Balanced Modulators
1365
17.12.2 The Gilbert Multiplier as a Double-Balanced
Mixer/Modulator
The Gilbert multiplier introduced in Chapter 16 can be used directly as a double-balanced
modulator or mixer if transistors Q 3 –Q 6 in Fig. 17.91 are driven by the square-wave signal at
input v2 at carrier frequency ωc . The second signal v1 at modulating frequency ωm is applied to
the transconductance stage. (In this case, input v2 no longer acts as a linear input signal in contrast
to the multiplier application in Chapter 16.) For the circuit in Fig. 17.91, we have
i C1 = I B B +
Vm
sin ωm t
2R1
i C2 = I B B −
and
Vm
sin ωm t
2R1
(17.201)
If we take a differential output, the dc current component cancels out, but the signal current at
frequency ωm is switched back and forth by the square-wave input and appears to be multiplied
alternately by +1 and −1. Using Eqs. (17.199) and (17.201), the output signal between the
collectors can be written as
v O (t) = Vm
or
RC 4
sin nωc t sin ωm t
R1 n odd nπ
RC 2
v O (t) = Vm
[cos(nωc − ωm )t − cos(nωc + ωm )t]
R1 n odd nπ
(17.202)
The output signal has spectral components at frequencies shifted ωm above and below each of the
odd harmonics of the carrier frequency ωc as in Fig. 17.92. Note that no signal energy at either
the carrier or modulation signal frequencies ωc or ωm appears at the output, and the circuit is
therefore referred to as a doubly balanced modulator or mixer. A band-pass filter can be used to
select the desired frequencies from the composite spectrum at the output.
In modulator applications, the circuit just described generates a double-sideband suppressedcarrier (DSBSC) output signal. An amplitude-modulated signal (with modulation index M, 0 ≤
M ≤ 1) can also be generated by adding a dc component to the modulating signal
v1 = Vm (1 + M sin ωm t)
(17.203)
VCC
RC
iC4
RC
+
iC3
+
v2
–
Q3
Q4
Q1
iC6
Q5
Q6
iC2
iC1
+
v1
–
–
vO
iC5
Q2
2R1
IBB
IBB
–VEE
Figure 17.91 Double-balanced modulator based on the Gilbert multiplier. Signal v2 is a large-signal
square-wave at the carrier frequency, and v1 is the modulating signal.
1366
Chapter 17
Frequency Response
vm ()
m
(a)
vc ()
c
2c
3c
4c
5c
(b)
vo ()
c – m c + m
2c
4c
3c – m 3c + m
5c – m 5c + m
(c)
Figure 17.92 Spectra for double-balanced modulator (a) modulation input, (b) carrier input, and
(c) output signal.
The dc term unbalances the circuit relative to the carrier frequency thereby injecting a carrier
frequency component into the output. (Note the same effect is caused by offset voltages due to
mismatches in the transistors.) The output voltage becomes
v O (t) = Vm
RC 4
M
M
sin nωc t +
cos(nωc − ωm )t −
cos(nωc + ωm )t
R1 n odd nπ
2
2
(17.204)
In this case, the circuit remains balanced with respect to the modulation signal and is referred to
as a single-balanced modulator.
Exercise: A 20-MHz carrier is modulated with a 10-kHz signal using the double-balanced
modulator in Figs. 17.91 and 17.92. What are the frequencies of the spectral components
in Fig. 17.10(c)?
Answers: 19.99 MHz; 20.01 MHz; 59.99 MHz; 60.01 MHz; 99.99 MHz; 100.01 MHz
Exercise: The amplitude of the signal at 19.99 MHz in the previous exercise in 3 V. What
are the amplitudes of the other components?
Answers: 3 V; 1 V; 1 V; 0.6 V; 0.6 V
17.12
Mixers and Balanced Modulators
1367
17.12.3 Conversion Gain
In the amplifiers that have been discussed up to now, our gain expressions have always involved
signals at the same frequency. We have always assumed that our amplifiers are linear and that the
input and output signals are at the same frequency. The mixer is a nonlinear device in which the
output signals are at frequencies that are different than the input frequencies. A mixer’s conversion
gain is defined as the ratio of the phasor representation of the output signal to that of the input
signal, and we simply ignore the fact that the signals are at two different frequencies.
Exercise: What is the magnitude of the conversion gain for the balanced mixer in Fig. 17.89
if the desired output signal is at frequency ω2 – ω1 ?
Answers: 2/π or −3.92 dB
ELECTRONICS IN ACTION
Mixers in Communications Systems
Mixers circuits are invariably encountered whenever we look at communications receivers and
transmitters. An example of the architecture of a hypothetical direct conversion transceiver is
shown in the block diagram here. This cicuit could represent the RF portion of a device for a
wireless local area network, or the transceiver for a cellular phone depending on the particular
frequencies chosen for the design. In the 5-GHz digital radio system depicted here, the received
radio signal from the antenna is amplified by a low noise amplifier and fed to two mixers, one
for the in-phase (I) data channel and one for the quadrature (Q) data channel. The two 5-GHz
local oscillator (LO) inputs to the two mixers are also quadrature signals, and since the LO
signals are at the same frequency as the received signal, the desired mixer outputs are low
frequency base-band signals. These signals are amplified by variable gain amplifiers (VGAs)
and converted to digital form by the ADCs. The data is then is recovered by the CMOS digital
signal processor (DSP). The unwanted mixer outputs at 10 GHz are rejected by the circuitry.
VGA
0°
LNA
5 GHz
90°
Low noise
amplifier
LOI
5 GHz
LOQ
ADC
RXDI
Variable gain
amplifiers
VGA
ADC
RXDQ
CMOS
DSP
Mixers
Band-pass
filter
Power
amplifier
0°
90°
DAC
TXDI
DAC
TXDQ
LOI
5 GHz
LOQ
1368
Chapter 17
Frequency Response
On the transmitter side, I and Q base-band data signals are generated by the DSP and
converted to analog form by the D/A converters. The analog output signals from the DACs
are up converted to the 5-GHz band by the two mixers whose outputs are then added together,
amplified by the power amplifier, and fed through the bandpass filter to the antenna. More
complex receiver architectures can involve even larger numbers of mixers, and the circuitry
that generates the local oscillator signals often involves the use of mixers as well. In integrated
form, these mixers are most often implemented as either bipolar or MOS versions of the Gilbert
circuits.
SUMMARY
•
Amplifier frequency response can be determined by splitting the circuit into two models,
one valid at low frequencies where coupling and bypass capacitors are most important, and
a second valid at high frequencies in which the internal device capacitances control the
frequency-dependent behavior of the circuit.
•
Direct analysis of these circuits in the frequency domain, although usually possible for
single-transistor amplifiers, becomes impractical for multistage amplifiers. In most cases,
however, we are primarily interested in the midband gain and the upper- and lower-cutoff
frequencies of the amplifier, and estimates of f H and f L can be obtained using the opencircuit and short-circuit time-constant methods. More accurate results can be obtained
using SPICE circuit simulation.
•
The frequency-dependent characteristics of the bipolar transistor are modeled by adding
the base-emitter and base-collector capacitors Cπ and Cµ and base resistance r x to the
hybrid-pi model. The value of Cπ is proportional to collector current IC , whereas Cµ is
weakly dependent on collector-base voltage. The r x Cµ product is one important figure of
merit for the frequency limitations of the bipolar transistor.
•
The frequency dependence of the FET is modeled by adding gate-source and gate-drain
capacitances, C G S and C G D , to the pi-model of the FET. The values of C G S and C G D are
independent of operating point when the FET is operating in the active region.
•
Both the BJT and FET have finite current gain at high frequencies, and the unity gainbandwidth product ωT for both devices is determined by the device capacitances and the
transconductance of the transistor. In the bipolar transistor, the β-cutoff frequency ωβ
represents the frequency at which the current gain is 3 dB below its low-frequency value.
In SPICE, the basic high-frequency behavior of the bipolar transistor is modeled using
these parameters: forward transit-time TF, zero-bias collector-base junction capacitance
CJC, collector junction built-in potential VJC, collector junction grading factor MJC, and
base resistance RB.
•
•
In SPICE, the high-frequency behavior of the MOSFET is modeled using the gate-source
and gate-drain capacitances determined by the gate-source and gate-drain overlap capacitances CGSO and CGDO, as well as TOX, W, and L.
•
If all the poles and zeros of the transfer function can be found from the low- and highfrequency equivalent circuits, then f H and f L can be accurately estimated using Eqs. (17.16)
and (17.23). In many cases, a dominant pole exists in the low- and/or high-frequency
responses, and this pole controls f H or f L . Unfortunately, the complexity of most amplifiers
precludes finding the exact locations of all the poles and zeros except through numerical
means.