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Chapter 4
Bandpass Circuits
 Limiters
 Mixers, Upconverters and Downconverters
 Detectors, Envelope Detector, Product Detector
 Phase Locked Loops (PLL)
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
Eeng 360 1
Limiters
 Limiter is a nonlinear circuit with an output
saturation characteristic.
 It rejects envelope variations but preserves the
phase variations.
vin(t )  R(t ) cos(ct   (t ))
vout (t )  KVL cos(ct   (t ))
Ideal limiter characteristic with illustrative input and unfiltered
output waveforms.
Eeng 360 2
Mixers
 Ideal mixer is a mathematical multiplier of two input signals. One of the signals is
sinusoidal generated by a local oscillator. Mixing results in frequency translation.
SSB mixer

Bandpass Input Signal vin  t   Re gin  t  e jct

Eeng 360 3
Mixers
Bandpass Input Signal

vin  t   Re gin  t  e jct

Mixer Output


v1  t    A0 Re g in  t  e jct  cos 0t
A0
 gin  t  e jct  gin*  t  e  jct  e jct  e  jct

4
A
j   t
 j   t
j   t
 j   t
 0  gin  t  e  c 0   gin*  t  e  c 0   gin  t  e  c 0   gin*  t  e  c 0  

4 
1
1 *
Re 
  
  

2
2
A
A
j   t
j   t
v1  t   0 Re g in  t  e  c 0   0 Re g in  t  e  c 0 
2
2
fu  f c  f 0
fd  fc  f0



UPCONVERSION
BANDPASS FILTER



DOWNCONVERSION
BASEBAND OR BANDPASS Eeng
FILTE
360 R
4
Choosing LO Frequency of Mixers
v1  t  



A0
A
j   t
j   t
Re gin  t  e  c 0   0 Re gin  t  e  c 0 
2
2
Up-conversion

Down-conversion
f d  fc  f0
fu  fc  f0
Baseband/bandpass
Filter (fc-f0)
Bandpass Filter
 If (fc- f0) = 0  Low Pass Filter gives baseband spectrum
 If (fc- f0 )> 0  Bandpass filter  Modulation is preserved
Filter Output: v2 t   Reg 2 t e j 
c
  0 t
  A2 Reg t e 
j  c   0 t
0
in

 If fc>f0  modulation on the mixer input is preserved
 If fc<f0 



A
A
v1 t   0 Re g in t e j  c   0 t  0 Re g in* t e j  0  c t
2
2

‘’ needs to be
positive
Complex envelope is conjugated ~ sidebands are exchanged
*
dt    gin  t  e
dt   Gin*   f 
 

- f  Upper & lower sidebands are exchanged
F  g
*
in

 t    g  t  e
*
in
 jt
*  Phase spectrum is inverted

 j   t
Eeng 360 5
Mixers (Up Converter and Down Converter)
 Complex envelope of an Up Converter:
g 2 t  
A0
g in t  ;
2
fu  fc  f0  0
- Amplitude is scaled by A0/2
 Complex envelope of a Down Converter:
f d  f c  f 0  0 i.e., f0<fc  down conversion with low-side injection
g 2 t  
A0
g in t 
2
- Amplitude is scaled by A0/2
f d  f 0  f c  0 i.e., f0>fc  down conversion with high-side injection
g2 
A0 *
g in t 
2
- Amplitude is scaled by A0/2
- Sidebands are reversed
from those on the input
Eeng 360 6
Mixer Realizations Without Multipliers
 Multiplication operation needed by mixers can be obtained by using a
nonlinear device together with a summer.
Nonlinear device used as a mixer.
Eeng 360 7
Mixer Realizations Without Multipliers
 Multiplication operation needed by mixers can also be obtained by using an
analog switch.
Linear time-varying device used as a mixer.
Eeng 360 8
Mixer Realizations Without Multipliers
Analysis of a double-balanced mixer
circuit.
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Frequency Multiplier
Frequency Multipliers consists of a nonlinear device together with a tuned circuit. The
frequency of the output is n times the frequency of the input.
vin(t )  R(t ) cos(ct   (t ))
v1 (t )  K n v n in(t )
 K n R n (t ) cos n (ct   (t ))
v1 (t )  CR n (t ) cos(nct  n (t )) 
Other Terms
vo (t )  CR n (t ) cos(nct  n (t ))
Eeng 360 10
Detector Circuits
 Detectors convert input bandpass waveform into an output baseband
waveform.
 Detector circuits can be designed to produce R(t), Θ(t), x(t) or y(t).
• Envelope Detector
• Product Detector
• Frequency Modulation Detector
Information
Signal g (t )
input
processing
m
Carrier
circuits
s (t )
Transmission
medium
(Channel)
r (t )
Carrier
circuits
g~ (t )
Signal
processing
~
m
Detector Circuits
Eeng 360 11
Envelope Detector
 Ideal envelope detector: Waveform at the output is a real envelope R(t) of its input
Bandpass input: vin (t )  Rt cosct   t 
Envelope Detector Output:
vout t   KR t 
Rt   0
K – Proportionality Constant
Diode Envelope Detector Circuit
Eeng 360 12
Envelope Detector
 The Time Constant RC must be chosen so that the envelope variations can be followed.
B 
1
 f c
2 RC
In AM, detected DC is used for Automatic Gain Control (AGC)
vout (t )  KR (t )
 K g (t )
 KAc 1  m(t ) 
 DC  Message
Eeng 360 13
Product Detector
 Product Detector is a Mixer circuit that down converts input to baseband.
fc- Freq. of the oscillator
θ0- Phase of the oscillator
Output of the multiplier:
v1  t   R  t  cos ct    t   A0 cos ct   0  
1
1
A0 R  t  cos   t    0   A0 R  t  cos  2ct   t    0 
2
2
LPF passes down conversion component:


1
1
A0 R  t  cos   t   0   A0 Re g  t  e  j0
g (t )  R (t )e j (t )  x (t )  jy (t )
2
2
Where g(t) is the complex envelope of the input and x(t) & y(t) are the quadrature
components of the input:
vout  t  
Eeng 360 14
Different Detectors Obtained from Product
Detector
 Oscillator phase synchronized with the in-phase component
 vout t  
if  0  0 :
1
A0 xt 
2
We obtain INPHASE DETECTOR.
 We obtain QUADRATURE PHASE DETECTOR
if  0  90
 We obtain ENVELOPE DETECTOR If the input has no
angle modulation and reference phase (θ0) =0
 We obtain PHASE DETECTOR If an angle
if
modulated signal is present at the input and reference
phase (θ0) =90
The product detector output is
If the phase difference is small
vout t  

1
A0 Ac t 
2
if  t   0
1
A0 y t 
2
 vout 
1
A0 Rt 
2
0  90 vin (t )  A cos ct    t  
1
A0 Re Ac e j  t  90 
2
vout t  
 vout 
c

or
vout t  
1
A0 Ac sin  t 
2
sin  t    t 
The output is proportional to the Phase difference (Sinusoidal phase characteristics)
Eeng 360 15
Frequency Modulation Detector
 A ideal FM Detector is a device that produces an output that is
proportional to the instantenous frequency of the input.
Frequency demodulation using slope detection.
vin (t )  A(t ) cos[ct   (t )]
t
 (t )  K f  m( )d

d (t ) 

v1 (t )  VL cos[ct   (t )] v2 (t )  VL c 
sin[ct   (t )]

dt 

d (t ) 
d (t ) 


vout (t )  VL c 

V



L  c


dt 
dt 


VLc  VL K f m(t )  DC  AC (Proportional to m(t ))
• The DC output can easily be blocked
Eeng 360 16
Frequency Detector Using Freq. to Amplitude Conversion
Figure 4–16 Slope detection using a single-tuned circuit for frequency-to
amplitude conversion.
Eeng 360 17
Balanced Discriminator
Eeng 360 18
Balanced zero-crossing FM detector.
Eeng 360 19
Phase Locked Loop (PLL)
 PLL can be used to Track Phase and Frequency of the carrier component of the incoming
signal
 Three basic components:
- Phase Detector : Multiplier (phase comparator)
- VCO : Voltage Controlled Oscillator
- Loop filter: LPF
 Operation is similar to a feedback system
Basic PLL.
Eeng 360 20
PLL, Voltage Controlled Oscillator (VCO)
Voltage Controlled Oscillator (VCO):
 Oscillator frequency is controlled by external voltage
 Oscillation frequency varies linearly with input voltage
 If e0(t) – VCO input voltage, then its output is a sinusoid of frequency
(t)=c+ce0(t)
 c - free-running frequency of the VCO.
 The multiplier output is further low-pass-filtered & then input to VCO
 This voltage changes the frequency of the oscillator & keeps it locked.
Eeng 360 21
Phase Locked Loop (PLL)
Let input signal be :
vin (t )  Ai sin[ ct   i (t )]
t
 o (t )  K v  v2 ( )d
Let the VCO output be: vo (t )  Ao cos[ct   o (t )]

The phase detector output v1(t) is given by :
v1 (t )  K m Ai Ao sin[  ct   i (t )] cos[ ct   0 (t )] 
K m Ai Ao
sin[  i (t ) - 0 (t)]  sin[2 ct   i (t )   0 (t)]
2
The sum frequency term is rejected by LPF so the filter output v2(t) is:
v2 (t )  K d [sin  e (t)]  f (t )
where
 e (t)   i (t) -  o (t)
and
Kd 
K m Ai Ao
2
 e(t) is called the Phase Error. The Phase Error voltage characteristics is SINUSOIDAL.
A PLL can track the incoming frequency only over a finite range  Lock/hold-in range
 The frequency range over which the input will cause the loop to lock  pull-in/capture
range
Eeng 360 22
Phase Locked Loop (PLL)
 Various types
of Phase Detector
characteristics used in PLL’s.
Eeng 360 23
Aplications of PLL
 PLL used for coherent detection of AM signals.
• A synchronized carrier signal is generated by the PLL.
• VCO locks with 90 phase difference so a -90 extra phase shift is needed.
• The generated carrier is used with a product detector to recover the envelope
Figure 4–24 PLL used for coherent detection of AM.
Eeng 360 24
Aplications of PLL
 PLL used as a frequency synthesizer.
Frequency dividers use integer values of M and N.
For M=1 frequency synthesizer acts as a frequency multiplier.
f x f out

M
N
f out 
N
fx
M
Figure 4–25 PLL used in a frequency synthesizer.
Eeng 360 25