Chapter 8 Modulators and Demodulators
Diode Switching Type Mixer
and Phase Modulation and Demodulation
Modulation is the modification of a high-frequency carrier
signal to include the information present in a relatively low frequency
signal. This is necessary because radio wave propagation is more efficient
at higher frequencies and smaller antennas can be used. A larger bandwidth
can be obtained at higher frequencies, enabling many information-containing
signals to be multiplexed onto one carrier and sent simultaneously.
Frequency conversion, modulation and detection are common
tasks performed in a communication circuit.
- The most commonly used device for frequency modification is the mixer. It
is basically a multiplier
- The output consists of the sum and difference of the two input frequencies,
one of which is the desired component. The other will be filtered out. This
combination of a mixer and filter to remove an output frequency is known as
- There are 2 main classes of mixers -- nonlinear or switching-type.
- One or more switches, realized by diodes or transistors, will function as
the time-varying circuit elements.
- Fig. 8-1
- For ideal center-tapped transformer, the voltages will be indicated as in
- The local oscillator VL is a constant- amplitude signal. VL
>> Vi so that D1 is on when VL is positive
and D2 is on when VL is negative. Thus
- The output consists of the local oscillator plus Vi switched
by 180° at the frequency of the local oscillator.
If the switched form of Vi is represented by Vi* then
- The Fourier series for P(t) and Vi* are
- If Vi is a sine wave then
- Since Vo=VL+Vi* the mixer output consists
of the local oscillator signal plus an infinite number of additional frequencies
created in the mixer. The output frequencies in addition to the upper and
lower sidebands are called spurious. The desired component is obtained by
- The preceding analysis assumed that the local oscillator signal was much
larger than the input signal and sufficiently large to turn on the diodes
instantly. Deviations from these assumptions will result in distortions in
the desired frequency component.
- A disadvantage of the circuit above is that VL appears in the
output. If the oscillator frequency is much larger than the input frequency,
then the desired mixing product may
be close to the oscillator freq. and will be difficult to separate by filtering.
In the new circuit in Fig. 8-3
- The local oscillator signal does not appear in the output. For ideal transformer
the voltages are shown in Fig. 8-4
- If VL is positive and much larger than Vi than both
diodes are conducting . The local oscillator current balance out in the output
transformer Vo=Vi. If VL is negative, the
diodes will be open and the output signal will be zero. Thus
- If Vi is a sine wave Vi=Vsinwit,
the output is
- The output of this mixer differs from the previous one in that it does not
contain the local oscillator signal but it does contain a signal at the same
frequency as the input signal.
Four Diode Switching
- The construction of this type of mixer is shown in Fig.
- Neither the local oscillator signal and the input signal appears at the
output. If the local oscillator, VL, is positive, then diodes D2
and D3 will conduct and the equivalent circuit is shown in Fig.
8-6 and Fig. 8-7
- rd is the diode on resistance. The loop equations are
- If the local oscillator signal is negative, diodes D1 and D4 conduct and
the equivalent circuit is in Fig. 8-8 and Fig.
- In this mixer the output voltage is proportional to the input voltage is
switched at the local oscillator frequency. Therefore if
- A double-balanced mixer with perfectly matched diodes and ideal transformer
coupling will generate the upper and lower sidebands plus an infinite number
of spurious frequencies centered on odd harmonics of the local oscillator
frequency. Their excellent performance is due in part to modern fabrication
techniques to construct closely matched diodes. High frequency Schottky barrier
diodes are often used today.
- Mixer conversion loss is defined as the ratio of output power in one sideband
to signal input power. It is a most important mixer parameter, particularly
for the receiver.
- From Fig. 8-7, and the load impedance seen by Vi is
- Normally RL>>rd so the input will be matched for maximum
power transfer if RL=Rs. Under this condition Vi=Vs/2
- The output voltage in on sideband, for RL>>rd, is
- So the conversion gain of the double-balanced mixer is
- The conversion loss is
- For an ideal double-balanced mixer matched to the source impedance, and
ignoring the power lost in the transformer and switching diodes, approximately
40% of the signal input power will be transferred to the output.
- For the single-balanced mixer, the output voltage of one sideband is
- If the port is matched for maximum power transfer
- The power gain is .
The conversion loss is 4 times (6 dB) larger than double-balanced mixer
- As the mixer input signal power increases, it will reach the level at which
it is larger than the local oscillator.
- The input signal then assumes the switching role, and the output power becomes
proportional to the local oscillator power. Since the local oscillator is
constant the output power will be constant.
- The square-law characteristic is approximated by several electronic devices
which square the sum of two sine waves
- An ideal square-law device will provide the upper and lower sidebands, together
with a dc component and the second harmonic of both input waveforms. The circuit
is frequently used at microwave frequencies for down conversion to the lower
side-band, which is at a lower frequency than either of the input signals.
A simple square law mixer is shown in Fig. 8-12
- chottky barrier diodes are typically used for high speed applications.
- At lower frequencies this form of the diode mixing is normally not used
because of the large conversion loss. Transistors mixers are preferred because
they can provide conversion gain. Transistors are often used to approximate
the square-law characteristic. The input and local oscillator signal voltages
are applied to the transistor so that they effectively add to the dc bias
voltage to produce the total gate-source of base-emitter voltage. The composite
signal is then passed through the device nonlinearity to create the sum and
- This is illustrated in the Fig. 8-13
- The base to emitter voltage is Vbe=VDC+Vi-VL
where VDC is the base-to-emitter bias voltage. The collector current in a
bipolar transistor is described by (Vbe > 0)
- If then
the current can be expanded in a series of modified Bessel functions as
- where y=V1/VT, x=Vx/VT and
In is the nth-order modified Bessel function.
- The collector current consists of a dc component IC, components
at both the input and oscillator frequencies , components at the frequencies
, and an infinite
number of high-frequency components. The amplitude of either the upper or
lower-sideband component is
- The local oscillator voltage amplitude is constant and V2>>V1,
then the collector direct current will not vary with changes in the amplitude
of the input signal since .
- The mixer should have a linear response to changes in the amplitude of the
input amplitude. The ratio is given as
- So if the input amplitude is sufficiently small the mixer upper- and lower-sideband
outputs will be a linear function of the input signal. For y<0.4 (V1<10.5
mV) the response will be within 2 percent of a linear response. The amplitude
of the sideband current is
- If an FET is operated in its ˇ§constant-currentˇ¨ region, the
idealized FET current transfer characteristics is the square-law relation
is the gate-to-source voltage and Vp is the transistor pinch-off
voltage. Because of the square-law characteristic, the FET will not generate
any harmonics higher than second-order intermodulation distortion. However,
in reality, the transfer characteristic deviates from the idealized version,
version and some intermodulation distortion will be produced. Still, a properly
biased and operated FET mixer will produce much smaller high-order mixing
products than a bipolar transistor. This is one reason why an FET is usually
preferred to a bipolar transistor mixer.
- The FET also provides at least 10 times as great an input voltage range
as the BJT. The following figure illustrates an FET mixer circuit. The drain
- Fig. 8-14
- where VDC is the gate-to-source bias voltage (or VGS-VT
for a MOSFET). If the applied signals are sine waves
- then the
output current is
- The amplitude of the sum and difference frequencies is
- where K3/Vi is referred to as the conversion transconductance
gc. In general the device with the lowest pinch-off voltage has
the highest gain, and the conversion transconductance is directly proportional
to the amplitude of the local oscillator signal.
- It would also appear that FETs with high IDSS are preferred,
but IDSS and Vp are related. It is usually the case
that devices selected for high IDSS also have a high Vp
and a lower conversion transconductance that low- IDSS devices.
Since the device is to be operated in the constant-current region, VL
must be less than the magnitude of the pinch off volgate. If then
K3=Vi IDSS/2Vp and the sideband
- Since for a JFET the transconductance is
- The conversion transconductance is one-fourth the small-signal tansconductance
evaluated at Vgs=0 (provided VL=Vp/2). For
a MOSFET it can be shown that the conversion conductance cannot exceed 1/2
of the transconductance of the device when it is used as a small-signal amplifier.
- Although the conversion transconductance is smaller than the small-signal
transconductance, it is large enough that the circuit can be operated as a
mixer with power and voltage gains. This is an important difference from the
- An FET mixer is capable of producing lower intermodulation and harmonic
products than a comparable bipolar or diode mixer. Also, an FET mixer operating
a high level has a larger dynamic range and greater signal-handling capacity
than a diode mixer operated at the same local oscillator level. However, the
noise figure of FET mixers is currently higher than that of diode mixers.
The best intermodulation and cross-modulation performance is obtained with
the FET operated in the common-gate configuration, where the input impedance
is much lower than that for the common-source configuration.
- Fig. 8-15 illustrates double balanced mixer in
which the FET transistors are operated in the common-gate configuration. The
push pull output cancels the even-order output harmonics.
- The dual-gate MOSFETs is often used as a mixer. A typical dual-gate MOSFET
mixer circuit is shown in Fig. 8-16
- If the input signals are sinusoidal, the output will contain frequency components
at both the sum and difference frequencies. Several other frequency components
are also present in the output. The magnitude of either the sum or difference
frequency is proportional to
- so the conversion gain is proportional to the magnitude of the local oscillator
voltage. For maximum conversion gain, the local oscillator amplitude should
be selected so that it drives the gate just to the point of transistor saturation.
- The input signal is normally connected to the lower (closest to the ground)
input gate terminal and the local oscillator signal to the upper gate. If
the input is connected to the upper terminal, then the drain resistance of
the lower transistor section appears as a source resistance to the input signal.
The source resistance will reduce the voltage gain at the collector. Also,
the connection has a larger drain-to-gate capacitance with a lower bandwidth
than is attainable when the input signal is connected to the lower gate. The
device is usually biased so that both transistors are operating in their nonsaturated
- The small-signal drain current is
- The drain current can be written as
- Since the drain current contains the product of the 2 signals, the dual-gate
MOSFET can be used as a mixer when both transistors are operated in the linear
and Phase Modulation and Demodulation
- Amplitude modulation (AM) is the process of varying the amplitude of a constant
frequency signal with a modulating signal. An amplitude-modulated wave can
be mathematically expressed as S(t)=g(t)sinwct
where g(t) is the modulating signal and wc
is the carrier frequency. Normally the modulating signal varies slowly compared
with the carrier signal frequency. Conventional AM is in the form of where
m is the modulation factor and is normally less than 1. Consider a
simple modulating signal:
- The frequency spectrum of the modulated signal is shown in Fig.
- The equation above shows that for m<1 the amplitude of the carrier is at
least twice as large as the amplitude of either sideband component, so at
least 2/3 of the signal power will be in the carrier and at most 1/3 in the
2 sidebands. Because the carrier does not contain any information, it is often
removed or suppressed in the signal which
is referred to as a double-sideband (DSB) suppressed-carrier signal. The carrier
component is not present in the DSB signal. However, as the waveform gets
more efficient in terms of power-to-information content, the detection method
gets more complex. Some means of recovering the carrier component is needed
for the detector to recover the amplitude and frequency of the modulating
signal. The DSB signal, although more efficient in terms of transmitted power,
still occupies the same bandwidth as a normal AM signal. Since both sidebands
contain the same information, one sideband can be removed, resulting in a
- Full-carrier double-sideband amplitude modulation is achieved either modulating
the oscillator signal at a relatively low power level and amplifying the modulated
signal with a cascade of amplifiers or by using the modulating signal to control
the supply voltage o fthe power amplifier. Both methods are illustrated in
- The power requirements of the modulator and modulating signal can be estimated
by considering the power in an amplitude-modulated waveform .
The peak power is so
if the maximum modulation index is unity, The
modulator must be designed to handle 4 times the average carrier power with
100% modulation; the output power will be 4 times the carrier power.
- The diode mixer can be used to realize low-level modulation. If VL
is a sine wave and
if a low-pass filter is added to the output with a bandwidth of B=wL+wi
then the output will be .
Since the low-pass filter removes the higher-frequency component, the modulation
index of the resulting AM waveform is m=(4/p)V/V1.
This particular amplitude modulator functions well only for low indices of
- Both FET and BJT mixers can function as amplitude modulators with a relatively
high modulation index. The final amplifier will need to be linear. The output
will then be linearly related to the input provided the amplifier output circuit
is not current-limited.
- The most frequently used method of amplitude modulation at high power levels
is to modulate the supply voltage to the power amplifier, as shown in Fig.
8-18b. In the Fig. 8-19, the modulating signal is
applied in series with the dc supply voltage, so the total low-frequency supply
for the transistor is
- For Class C power amplifiers the amplitude of the output signal under saturation-limited
conditions equals the power supply voltage. Therefor changing the transistor
supply voltage modulates the output signal amplitude proportionally, and the
output voltage becomes .
For 100% modulation the peak value of the voltage Vm(t) must equal
VCC. The total output power is .
The unmodulated carrier power is supplied by the power supply. The remaining
power must be furnished by the modulator. One reason that output modulation
has been the most frequently used method is that collector modulation results
in less intermodulation distortion.
- All the information in an AM wave is contained in one sideband. It is possible
to eliminate the other sideband without loss of information; thus the required
transmitter power is reduced to one-third of that previously required.
- The simplest method of SSB generation is to generate the DSB signal using
a double-balanced modulator and then remove one of the sidebands with a filter.
A block diagram of this form of SSB is shown in Fig.
- Another technique know as phasing method is shown in Fig.
- Here both the modulating signal and the carrier signal are processed through
phase splitters, which each generate two signals 90° out
of phase with each other. The summing network output is the desired SSBsignal.
The phasing method has the advantage of not requiring the sharp cutoff filters
of the filtering method of SSBgeneration, but it is difficult to realize a
broadband phase-shifting network for the lower frequency modulating signal.
- AM detection can be divided into synchronous and asynchronous detection.
Synchronous detection employs a time-varying or nonlinear element synchronized
with the incoming carrier frequency. Otherwise the detection is asynchronous.
The simplest asynchronous detector, the average envelope detector, is described
Average Envelope Detectors
- A block diagram of the average envelope detector is shown in the Fig.8-22
- The rectifier output
- can be written as Vr(t)=S(t)P(t)
- If S(t) is periodic with a frequency wc,
- If S(t) is the AM wave described by
- If the low-pass filter bandwidth is chosen to filter out the component at
wc and all higher harmonics, the output
will be which
is a dc term plus the modulating information.
- Two additional points will be made to further describe the operation of
the envelope detector. First, consider the case where f(t)=sinwmt
- The output will contain a term at the frequency wc-wm,
which must also be removed by the low-pass filter. This is not possible if
wm is close to wc.
To ensure this distortion does not occur the max modulating frequency should
be and the
corresponding low-pass filter bandwidth B must be selected so that Vr(t)>0
- This is only possible if m is not greater than 1, and the carrier term is
present. Average envelope detection will only work for normal AM with a modulation
index less than 1. However, if a large carrier component Acoswct
is added to the SSB signal, the resultant signal can also be detected with
an envelope detector.
- A simple diode envelope detector circuit is shown in the Fig.
- It is assumed here that the input signal amplitude is large enough that
the diode can be considered either on or off, depending upon the input signal
polarity. The diode can then be replaced by a open circuit when it is reverse-biased
and by a constant resistance when it is forward-biased. The series capacitor
Cc is included to remove the dc component. The purpose of the load capacitor
C in the circuit is to eliminate the high-frequency component from the output
and to increase the average value of the output voltage. The effect of the
load capacitor can be seen from the Fig. 8-24
- which illustrates the input and the output signal waveforms of a diode detector.
As the input signal is applied, the capacitor charges up until the input waveform
begins to decrease. At this time the diode becomes open-circuited and the
capacitor discharges through the load resistance RL as VL=Vpexp(-t/RLC)
where Vp is the peak value of the input signal, and the diode opens
at time t=0. The larger the value of capacitance used, the smaller will be
the output ripple. However, C cannot be too large or it will not be able to
follow the changes in the modulated signal. The time constant is often selected
- Information can also be transmitted by modulating the phase frequency. Angle
modulation occupies a wider bandwidth, but it can provide better discrimination
against noise and other interfering signals. An angle-modulated waveform can
be written as where
q(t) representing the angle modulation. Angle modulation
can be further subdivided into phase and frequency modulation, depending on
whether it is the phase or the derivative of phase that is modulated. Frequency
modulation and phase modulation are not distinct, since changing the frequency
will result in a change in phase and modulating the phase also modulates the
- Frequency modulation can be achieved directly by modulating a VCO (direct
FM) or indirectly by phase-modulating the RF waveform by the integrated audio
input signal (indirect FM). Another method of FM is to use a phase-locked-loop
as shown in Fig. 8-25
- The output in response to the modulating signal Vm is
- where Kd is the phase-detector gain constant and Ko
is the VCO sensitivity (Hertz per volt). In the steady state, the output phase
will be proportional to the modulating voltage. So the PLL can serve either
as a phase modulator or, if VM is the integral of the modulating
signal of interest, as a frequency modulator.