Transcript Document
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
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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.
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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 jct
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Mixers
Bandpass Input Signal
vin t Re gin t e jct
Mixer Output
v1 t A0 Re gin t e jct cos 0t
A0
gin t e jct gin* t e jct e jct e jct
4
A
j t
j t
j t
j t
0 gin t e c 0 gin* t e c 0 g in t e c 0 g in* t e c 0
4
1
1 *
Re
2
2
A
A
j t
j t
v1 t 0 Re gin 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
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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
fd 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 Reg 2 t e j
c
0 t
A2 Reg 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
* Phase spectrum is inverted
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F g
*
in
t g t e
*
in
jt
j t
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
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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.
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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.
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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(nct n (t ))
Other Terms
vo (t ) CR n (t ) cos(nct n (t ))
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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
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Envelope Detector
Ideal envelope detector: Waveform at the output is a real envelope R(t) of its input
Bandpass input: vin (t ) Rt cosct t
Envelope Detector Output:
vout t KR t
Rt 0
K – Proportionality Constant
Diode Envelope Detector Circuit
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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
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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 2ct t 0
2
2
LPF passes down conversion component:
1
1
A0 R t cos t 0 A0 Re g t e j0
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
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Different Detectors Obtained from Product
Detector
Oscillator phase synchronized with the in-phase component
vout t
if 0 0 :
1
A0 xt
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 Rt
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)
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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
VLc VL K f m(t ) DC AC (Proportional to m(t ))
• The DC output can easily be blocked
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Frequency Detector Using Freq. to Amplitude Conversion
Figure 4–16 Slope detection using a single-tuned circuit for frequency-to
amplitude conversion.
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Balanced Discriminator
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Balanced zero-crossing FM detector.
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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.
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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.
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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
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Phase Locked Loop (PLL)
Various types
of Phase Detector
characteristics used in PLL’s.
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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.
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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.
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