Chapter 8
Modulators and Demodulators
1
• 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.
Frequency Mixers
• The most commonly used device for frequency modification is the mixer.
It is basically a multiplier
AA
Vo  ( A1 sin 1t )( A2 sin  2t )  1 2 cos(1   2 )t  cos(1   2 )t 
2
• 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 single-sideband mixer.
• There are 2 main classes of mixers -- nonlinear or switching-type.
2
Switching-Type Mixers
• One or more switches, realized by diodes or transistors, will function
as the time-varying circuit elements.
3
• For ideal center-tapped transformer, the voltages will be indicated as
below
• 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
Vi  VL
Vo  
 Vi  VL
VL  0
VL  0
4
• 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 Vo  VL  Vi *
VL  0
 1
Vi  Vi P(t ) where P(t )  
 1
*
VL  0
• The Fourier series for P(t) and Vi* are
sin[( 2n  1) L t ]
P(t )  
 n 0
2n  1
4

Vi  Vi
*
Vi 
*
4
sin[( 2n  1) L t ]
2n  1
n 0



cos[( 2n  1) L   i ]t  cos[( 2n  1) L   i ]t
 n 0
2n  1
2V


• 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
filtering.
5
• 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  L   i may be close to the oscillator freq.
and will be difficult to separate by filtering. In the new circuit below
•
6
• The local oscillator signal does not appear in the output. For ideal
transformer the voltages are shown below
7
• 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
VL  0
1
output signal will be zero. Thus
Vo  Vi P(t )
P(t )  
VL  0
• If Vi is a sine wave Vi=Vsinit, the output is
0
sin  i t V  cos[( 2n  1) L   i ]t  cos[( 2n  1) L   i ]t
Vo  V
 
2
 n 0
2n  1
• 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 Type Mixer
• The construction of this type of mixer is shown below
8
• 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 below
• rd is the diode on resistance. The loop equations are
Vi  I1  I 2 RL  I1rd  VL
• Thus I1  I 2 
Vi  I1  I 2 RL  I 2 rd  VL
Vi
Vo

RL  rd / 2
RL
• If the local oscillator signal is negative, diodes D1 and D4 conduct and
the equivalent circuit is
9
• In this mixer the output voltage is proportional to the input voltage is
switched at the local oscillator frequency. Therefore Vo (t )  Vi P(t ) RL
if
RL  rd / 2
RL
 2V  cos[( 2n  1) L   i ]t  cos[( 2n  1) L   i ]t 

then Vo (t )  RL  rd / 2   
2n  1
n 0

• Vi (t )  V sin i t
• 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.
10
Conversion Loss
• 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.
V
r
• From Fig. 12.13, and the load impedance seen by Vi is I i  RL  2d
i
• Normally RL>>rd so the input will be matched for maximum power
transfer if RL=Rs. Under this condition Vi=Vs/2 and pi  Vs2 / 4 RL
2Vi Vs
• The output voltage in on sideband, for RL>>rd, is V |


o  
2


the output power is P  Vs
o
 2 RL
L
i
P
4R
4
• So the conversion gain of the double-balanced mixer is G  Po   2 RL   2
i
L
2

• The conversion loss is L  10 log
4
• 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.
•
11
• For the single-balanced mixer, the output voltage of one sideband is
Vo | L  i  Vi / 
• If the port is matched for maximum power transfer
Vo  Vs / 2 ;
Pi  Vs2 / 4 RL ;
Po  Vs2 / 4 2 RL
•
Po
 ( 2 ) 1
The power gain is
. The conversion loss is 4 times (6
Pi
dB) larger than double-balanced mixer
G
Distortion
• 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.
12
Intermodulation Distortion
• Consider a diode-ring mixer with a resistance R in series with each
diode as shown below
• The purpose of the additional resistors will become clear once the IMD
is determined.
• If the local oscillator power is sufficiently large, the circuit during
either half-cycle is as shown below
•
13
• The diode current then consists of a constant component I, due to the
local oscillator, and a small component i, due to the input signal. The
diode current is described by iD  I s exp Vd / VT  where Vd is the voltage
drop across the diode and VT=kT/q
• The input signal Vi causes a signal current 2i to flow through the load.
Because of the circuit symmetry one-half flows through each diode.
That is iD1  I  i and iD2  I  i
• The currents are shown in Fig. 12.18. The voltage equations are
VL  Va  VD1  iDi R;  VL  Va  VD1  iDi R
VL  Va  VT ln
I i
 ( I  i) R
Is


I i
 VL  Va   VT ln
 ( I  i) R
Is


• adding the two equations we get  2Va  VT ln I  i  2iR
I i
2
V

2
(
V

2
iR
)
• and since
the relation between input voltage and
a
i
L
diode current is V  i(2 R  R)  VT ln I  i
i
L
2
I i
14
• This can be expanded for i<<I to
V
Vi  (2 RL  R)i  T
2
2
3

  i 1  i  2 1  i 3

 i 1  i  1  i 

                   

  I 2  I  3  I 
 
 I 2  I  3  I 

• The even order terms cancel out so
3
V  1 i 

Vi   2 RL  R  T i    VT  odd higher - order term s
I  3 I 

• Since the first term of the power series is not zero, the series can be
Vi 3
inverted i  Vi  VT
2 RL  R
3 (2 RL  R) 4 I 3
15
Square-law Mixers
• The square-law characteristic is approximated by several electronic
devices which square the sum of two sine waves
 A1 sin 1t  A2 sin  2t 2  ( A1 sin 1t ) 2  ( A2 sin  2t ) 2  2 A1 A2 sin 1t sin  2t
A12 (1  cos 21t ) A22 (1  cos 2 2t ) 2 A1 A2 [cos(1   2 )t  cos(1   2 )t



2
2
2
• 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 below
16
• Schottky 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 difference frequencies.
BJT Mixers
• This is illustrated
in the figure
17
• 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)
iC  I S exp(Vbe / VT ) since Vbe  VDC  Vi  VL
iC  I S exp(VDC / VT ) exp(Vi / VT ) exp( VL / VT )
• If Vi  V1 cos i t and V2 cos  ot then the current can be expanded
in a series of modified Bessel functions as
iC (t 0  I S exp(VDC / VT )[ I o ( y ) I o ( x)  2 I o ( y ) I1 ( x) cos  ot  2 I1 ( y ) I o ( x) cos  i t
 4 I1 ( y ) I1 ( x) cos  i t cos  ot  higher order term s]
• where y  V1 / VT , x  V2 / 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  o   i , and an infinite number of high-frequency
components. The amplitude of either the upper or lower-sideband
component is
I ( y ) I1 ( x )
I  I S exp(VDC / VT )2 I1 ( y ) I1 ( x)  2 I C 1
I o ( y ) I o ( x)
18
• 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 lim I o ( y )  1 .
y o
•
The mixer should have a linear response to changes in the amplitude
of the input amplitude. The ratio is given as I1 (Y ) Y  Y 2 Y 4  .
I1 (Y )

1 
 
2 
8 16 
• 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
I ( y ) I1 ( x )
I  2IC 1
I o ( y ) I o ( x)
19
FET Mixers
• If an FET is operated in its “constant-current” region, the idealized
FET current transfer characteristics is the square-law relation
2
 Vgs  where Vgs is the gate-to-source voltage and Vp is

iD  I DSS 1 
 V  the transistor pinch-off voltage. Because of the
p 

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 current is
 vi  vL  VDC 

iD  I DSS 1 


V
p


2
20
• where VDC is the gate-to-source bias voltage (or VGS-VT for a
MOSFET). If the applied signals are sine waves
vi  Vi sin i t
vL  VL sin  Lt
•
then the output current is
iD  I DC  K1 (Vi sin  i t  VL sin  Lt )  K 2 (Vi sin 2 i t  VL sin 2 Lt )
 K 3[Vi sin(  i   L )t  VL sin(  i   L )t ]
I VV
K3  DSS 2i L
• The amplitude of the sum and difference frequencies is
Vp
• 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
21
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 constantcurrent region, VL must be less than the magnitude of the pinch off
volgate. If VL  V p / 2 then K3=Vi IDSS/2Vp and the sideband current is
I
•
iD  K 3 sin(  i   2 )t  V1 DSS
2V p
I  V 
I
i
gm 
D
2
1  gs   2 DSS |Vgs 0
VP  VP 
VP
DSS
• Since for a JFET the transconductance is
Vgs
• 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 smallsignal 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 diode-switching mixer.
22
• 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.
• The figure below 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.
23
• The dual-gate MOSFETs is often used as a mixer. A typical dual-gate
MOSFET mixer circuit is shown below
• 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 A  K Vg 2
24
• 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
region.
• The small-signal drain current is
id  g m1Vg1  g m 2Vg 2 ;
g m1  a0  a1Vg1  a2Vg 2 ;
g m 2  b0  b1Vg1  b2Vg 2
• The drain current can be written as
id  a1Vg21  a0Vg1  (a2  b1 )Vg1Vg 2  b0Vg 2  b2Vg22
• 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 region.
25
Amplitude 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 amplitudemodulated wave can be mathematically expressed as S (t )  g (t ) sin  ct
where g(t) is the modulating signal and c is the carrier frequency.
Normally the modulating signal varies slowly compared with the
carrier signal frequency. Conventional AM is in the form of
S (t )  A[1  mf (t )] sin  ct where m is the modulation factor and is
normally less than 1. Consider a simple modulating signal:
m


f (t )  cos  mt then S (t )  Asin  c t  [sin(  c   m )t  sin(  c   m )t 
2


• The frequency spectrum of the modulated signal is shown
26
• 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
Am
S (t ) 
[sin(  c   m )t  sin(  c   m )t ]
which is 2referred 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-toinformation 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 single-sideband-signal (SSB).
27
Amplitude Modulators
• 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 below
28
• The power requirements of the modulator and modulating signal can
be estimated by considering the power in an amplitude-modulated 2
A
waveform S (t )  A[1  m(t )] sin  ct . The peak power is Po  peak  (1  m) 2
2
2
so if the maximum modulation index is unity, ( Po ) peak  2 A  4( Po )car
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 VL  V1 sin  Lt and if a low-pass filter is added to the
output with a bandwidth of B   L  i then the output will be
• S (t )  V1 1  4V sin  i t  sin  Lt . Since the low-pass filter removes the
 V

1


higher-frequency component, the modulation index of the resulting
AM waveform is m  (4 /  )V / V1 . This particular amplitude
modulator functions well only for low indices of modulation.
• 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
29
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. 12-27b. In the figure below
the modulating signal is applied in series with the dc supply voltage, so
the total low-frequency supply for the transistor is
V  VCC  Vm (t );
 VCC (1  m cos  mt )
Vm (t )  mVCC cos  mt
30
• 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
Vo  VCC (1  m cos  mt ) cos  ct . For 100% modulation the peak value
of the voltage Vm(t) must equal VCC. The total output power is
2
3 VCC
Po 
4 RL . 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
31
sidebands with a filter. A block diagram of this form of SSB is shown
below
• Another technique know as phasing method is shown below:
32
• 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
S (t )  A cos  ct sin  ot  A sin  ct cos  ot
 A sin(  c   o )t
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.
Demodulators
• 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 below:
33
Average Envelope Detectors
• A block diagram of the average envelope detector is shown in the fig.
•
The rectifier output
S (t )
Vr (t )  
0
S (t )  0
S (t )  0
• can be written as Vr (t )  S (t ) P(t )
1 2  sin( 2n  1)
P(t )   
 ct
2  n  0 2n  1
If S(t) is periodic with a frequency c, since
•
• If S(t) is the AM wave described by S (t )  A[1  mf (t )] sin  ct
cos 2 c t
 sin  c t

Vr (t )  A[1  mf (t )]
  1 
 higher harmonics of  c 

 2

34
• If the low-pass filter bandwidth is chosen to filter out the component at
A[1  mf (t )]
c and all higher harmonics, the output will be
Vo (t ) 

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 )  sin  mt
• The
cos( c   m )t  cos( c   m )t
 sin  c t m

Vr (t )  A
 sin  mt   1   1
 higher frequency terms 

2
 2

• The output will contain a term at the frequency  c   m , which
must also be removed by the low-pass filter. This is not possible if m
is close to c. To ensure this distortion does not occur the max
c
modulating frequency should be  m max 
and the corresponding
2
low-pass filter bandwidth B must be selected so that Vr (t )  0 if S (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 Acosct is added to the SSB signal, the resultant signal can
35
also be detected with an envelope detector.
V
• A simple diode envelope detector circuit is shown in the figure below
• 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 forwardbiased. 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 figure below
36
• 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  V p exp( t / RL C ) 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
1 1 / 2
modulated signal. The time constant is often selected as [( m c ) ]
37
Angle Modulation
• 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
S (t )  A(t ) cos[ ct   (t )] where (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.
Angle Modulators
• 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 below
38
( K / s )V ( s )
o
M
• The output in response to the modulating signal Vm is  o ( s ) 
1  K o K d F ( s ) /( sN )
• 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.
39
FM Demodulators
• The same type of circuitry is used for detecting both types of angle
modulation, and we will refer to either process as FM detection. FM
detectors are often referred to as frequency discriminators.
• The ideal FM detector produces an output voltage that changes linearly
with changes in the input frequency as shown
• The output voltage is usually 0 at the carrier frequency. Any deviation
from the linear characteristic distorts the detected waveform. Amplitude
modulation caused by noise can also cause distortion in the recovered
signal. Limiting circuitry is usually included in FM detector to reduce the
amount of amplitude modulation. The transfer characteristic of an ideal
40
limiter is shown below
The limiter output is
restricted to the values
that depend only on the
sign of the input. A
single stage differentialpair limiter is shown
41
• The circuit gives a close approximation to the ideal limiter
characteristics. If the input signal is too small, several differential-pair
stages may be cascaded in order for the output to be saturated.
Integrated-circuit limiters frequently contain 3 cascaded stages.
• An analytical basis of FM detection is obtained by considering the
derivative of the FM signal
d
A cos[ ct   (t )]   c  d  A sin[  ct   (t )]
dt
dt 

• The derivative of an angle-modulated signal is an amplitudemodulated FM waveform. All the modulating information is contained
in the amplitude of the differentiated waveform. Normally  c  d / dt
if so the amplitude modulation can be removed with an envelope
detector. The output of the envelope detector will be proportional to
 c  d / dt , which is c  KVm (t ) for a frequency-modulated
waveform. If the output is then high-pass filtered to remove the
constant term  c , the remainder will be proportional to the
modulating signal. This technique has the disadvantage that any dc
components in the modulating signal is lost.
42
• The most often used circuit for realizing the differentiator is the singletuned circuit. The frequency response of an ideal differentiator H ( j )  jK
has a +90° phase shift, and the magnitude increases with increasing
frequency at 6 db per octave. The frequency response of a simple
tuned circuit will approximate this response at frequencies sufficiently
below the circuit’s resonant frequency.
• The frequency response magnitude of the parallel tuned circuit is
R
A( j ) 
[1  Q 2 ( /  0   0 /  ) 2 ]1/ 2
• Values for Q and 0 for a parallel tuned circuit are, in which Rp, C and
L and parallel to each other
Q
Rp
0 L
and  0  [( LC )1/ 2 ]1
43
• The magnitude of the frequency response of the parallel resonant
circuit is shown below
• At frequency  c   ,
A( j ) 
1  Q [(
2
R
2



)
/



/(




]
c
0
0
c

1/ 2
R 0 (   c )
Q[ 02  ( c   ) 2 ]
provided c is close enough to 0 so that

•
 c     02 
Q
  1
  0 ( c   ) 
44
R( c   )
• Also if  c     0 then A( j ) 
Q 0
• The output consists of a constant term corresponding to c plus a
component proportional to the frequency deviation . Balanced
discriminators are often used to eliminate the constant term. A
simplified balanced discriminator is illustrated below
The upper resonant
cirucit is tuned to the
frequency 0- c, and
the output is
proportional to c- .
The differential
output is then Vo  K[ c    ( c   )]  2K
which is proportional to the frequency deviation from the carrier
frequency.
45