Low Noise Design Methodology

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					         Chapter 8

Modulators and Demodulators

• 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
  Vo  ( A1 sin 1t )( A2 sin  2t )  1 2 cos( 1   2 )t  cos( 1   2 )t 
                                                                
• 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.
                      Switching-Type Mixers
• One or more switches, realized by diodes or transistors, will function
  as the time-varying circuit elements.

• For ideal center-tapped transformer, the voltages will be indicated as

• 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      VL  0
   Vo  
         Vi  VL      VL  0

• 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 *

                               1                    VL  0
  Vi  Vi P(t ) where P(t )  

                               1                    VL  0

• The Fourier series for P(t) and Vi* are
            4   
                  sin[( 2n  1) L t ]                    4   
                                                                   sin[( 2n  1) L t ]
    P(t )                                     Vi  Vi
            n 0       2n  1                               n 0       2n  1

                                                2V   
                                                        cos[(2n  1) L   i ]t  cos[(2n  1) L   i ]t
• If Vi is a sine wave then
                                         Vi 
                                                  n 0                       2n  1
• 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  L   i may be close to the oscillator freq.
  and will be difficult to separate by filtering. In the new circuit below


• The local oscillator signal does not appear in the output. For ideal
  transformer the voltages are shown below

• 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                             1        VL  0
                                      Vo  Vi P(t )    P(t )  
• If Vi is a sine wave Vi=Vsinit, the output is               0         VL  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

• 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   Vi  I1  I 2 RL  I 2 rd  VL

• Thus I1  I 2          Vi          Vo
                       RL  rd / 2    RL
• If the local oscillator signal is negative, diodes D1 and D4 conduct and
  the equivalent circuit is
• 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   
• Vi (t )  V sin  i t                          n0                       2n  1
• 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.
                           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       rd
• From Fig. 12.13, and the load impedance seen by Vi is I i  RL  2

• 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
• The output voltage in on sideband, for RL>>rd, is V |        2V V
                                                              i  s
                                                      o  
                                2                          L   i
                                                                   
   the output power is P  Vs
                             2 RL

                                                                   P   4R   4
• So the conversion gain of the double-balanced mixer is G  Po   2 RL   2
                                    2                        i         L
• The conversion loss is L  10 log
• 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
•                                                                           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
                            G  ( 2 ) 1
•   The power gain is       Pi             . 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.

                   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


• 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 iD 2  I  i
• The currents are shown in Fig. 12.18. The voltage equations are
   VL  Va  VD1  iDi R;  VL  Va  VD1  iDi R
                     I i
   VL  Va  VT ln         ( I  i) R
                        I i              
     VL  Va   VT ln       ( I  i) R 
                         Is               
• adding the two equations we get  2Va  VT ln I  i  2iR
                                                  I i
• and since   2Va  2(Vi  2iRL ) the relation between input voltage and
  diode current is V  i(2R  R)  VT ln I  i
                                                   I i
                            i        L
• This can be expanded for i<<I to
                  V     i 1  i  2 1  i  3
                                                 i 1  i  2 1  i 3 
Vi  (2 RL  R)i  T                      
                   2    I 2  I  3  I 
                                                I 2 I  3 I 
                                                                       
• The even order terms cancel out so
                V  1 i 
 Vi   2RL  R  T i    VT  odd higher - order terms
                 I  3 I 
• Since the first term of the power series is not zero, the series can be
  inverted i  Vi  VT         Vi 3
                  2RL  R     3 (2RL  R) 4 I 3

                                      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 ) A2 (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

• 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

 • 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 ) sinceVbe  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)  2I o ( y) I1 ( x) cosot  2I1 ( y) I o ( x) cosi t
        4I1 ( 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  .
                                                               1    
                                                 I1 (Y )       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)

                             FET Mixers
• If an FET is operated in its “constant-current” region, the idealized
  FET current transfer characteristics is the square-law relation
               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             
                                Vp      
                                        
• 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 )
        K3[Vi sin(i   L )t  VL sin(i   L )t ]                              I VV
                                                                           K 3  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
  the amplitude of the local oscillator signal.                                           21
• 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 VL  Vp / 2 then K3=Vi IDSS/2Vp and the sideband current is
•                                 I
    iD  K 3 sin( i   2 )t  V1 DSS
                                  2V p             i      I  V           I
                                             gm    D
                                                        2   DSS
                                                                1  gs   2 DSS |Vgs 0
• Since for a JFET the transconductance is       Vgs        VP  VP 
                                                                             VP
• 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 diode-switching mixer.
• 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.

• 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

• 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
   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 amplitude-
  modulated 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

• 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
   S (t )      [sin( c   m )t  sin( c   m )t ]
  which is 2 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 single-sideband-signal (SSB).

                      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

• The power requirements of the modulator and modulating signal can
  be estimated by considering the power in an amplitude-modulated 2
  waveform S (t )  A[1  m(t )] sin  c t . The peak power is Po  peak  (1  m) 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
  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

• 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  c t . For 100% modulation the peak value
  of the voltage Vm(t) must equal VCC. The total output power is
        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
   sidebands with a filter. A block diagram of this form of SSB is shown

• Another technique know as phasing method is shown below:

• 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.

• 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:

                   Average Envelope Detectors
• A block diagram of the average envelope detector is shown in the fig.

•    The rectifier output               S (t )          S (t )  0
                              Vr (t )  
                                        0                S (t )  0
• can be written as Vr (t )  S (t ) P(t )
                                                          1 2  sin(2n  1)
                                                   P(t )                  ct
•   If S(t) is periodic with a frequency c, since        2  n  0 2n  1
• If S(t) is the AM wave described by S (t )  A[1  mf (t )] sin  ct
                             sin  c t          cos 2 c t                           
    Vr (t )  A[1  mf (t )]             1              higher harmonics of  c 
                             2                                                      

  • If the low-pass filter bandwidth is chosen to filter out the component at
    c and all higher harmonics, the output will be              A[1  mf (t )]
                                                       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
            sin  c t m                       cos( c   m )t  cos( c   m )t                          
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
    modulating frequency should be  m max  c and the corresponding
    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
    also be detected with an envelope detector.                                 35
• 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 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 figure below
• 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
  modulated signal. The time constant is often selected as [( m c ) ]
                                                                       1 1 / 2

                        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[ c t   (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

                                                                             ( K / s )V ( s )
• The output in response to the modulating signal Vm is  o ( s)           o       M
                                                                    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.

                          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
limiter is shown below

The limiter output is
restricted to the values
that depend only on the
sign of the input. A
single stage differential-
pair limiter is shown

• 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
      A cos[ ct   (t )]   c  d  A sin[ ct   (t )]
                                         
   dt                                 dt 
• The derivative of an angle-modulated signal is an amplitude-
  modulated 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.
• The most often used circuit for realizing the differentiator is the single-
  tuned 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
   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          and  0  [(LC)1/ 2 ]1
        0 L

• The magnitude of the frequency response of the parallel resonant
  circuit is shown below

• At frequency  c   ,
    A( j ) 
                1  Q [(
                             c   ) /  0   0 /( c   ]2   
                                                                  1/ 2

               R 0 (   c )
             Q[ 0  ( c   ) 2 ]

•   provided c is close enough to 0 so that
      c     0 
    Q                   1
       0 ( c   ) 
                                         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   )]  2 K
   which is proportional to the frequency deviation from the carrier


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