Download 13 Signals and Noise..

_____ Notes _____
CONTENTS
13.0 Signals and Noise
13.1 Noise Sources:
13.1.1 Combining Noise Voltages
13.1.2 Noise Figure
13.1.2.1 Friiss’ Formula
13.1.2.2 Noise Reduction
13.1.2.3 Noise Temperature
13.2 Review of RLC Networks
13.2.1 Series RLC Circuits
13.2.2 Parallel RLC Circuits
13.2.3 Self Resonance
13.3 Oscillators
13.3.1 Tuned Circuit Oscillators
13.3.2 Crystal Oscillator
Assignment Questions
For Further Research
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13.0 Signals and Noise
13.1 Noise Sources:
Man made noise < 500 MHz
• Hydro lines
• Ignition systems
• Fluorescent lights
• Electric motors
Therefore deep space networks are placed out in the desert.
Atmospheric noise - lighting < 20 MHz
Solar noise - sun - 11 year sunspot cycle
Cosmic noise - 8 MHz to 1.5 GHz
Thermal or Johnson noise. Due to free electrons striking vibrating ions.
Pn  kTf
where k  1.38  10 23 J/K
This equation applies to copper wire wound resistors, but is close
enough to be used for all resistors.
Maximum power transfer occurs when the source and load
impedance are equal.
Resistor Type
Carbon
Metal film
Wire wound
Cost
Low
Mid
High
Noise
High
Mid
Low
White noise - white noise has a constant spectral density over a specified
range of frequencies. Johnson noise is an example of white noise.
Gaussian noise - Gaussian noise is completely random in nature however,
the probability of any particular amplitude value follows the normal
distribution curve. Johnson noise is Gaussian in nature.
Shot noise - bipolar transistors
in  2qI dcf
where q  electron charge 1.6 10 17 coulombs
Excess noise, flicker, 1/f, and pink noise< 1 KHz also
Inversely proportional to frequency
Directly proportional to temperature and dc current
Transit time noise - occurs when the electron transit time across a junction
is the same period as the signal.
13.1.1 Combining Noise Voltages
The instantaneous value of two noise voltages is simply the sum of their
individual values at the same instant.
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Vtotal inst.  V1inst.  V2 inst.
This result is readily observable on an oscilloscope. However, it is not
particularly helpful, since it does not result in a single stable numerical value
such as one measured by a voltmeter.
If the two voltages are coherent [K = 1], then the total rms voltage value is the
sum of the individual rms voltage values.
Vtotal rms  V1rms  V2 rms
If the two signals are completely random with respect to each other [K = 0],
such as Johnson noise sources, the total power is the sum of all of the
individual powers:
P
 Pn1random  Pn2 random  
 Total random
A Johnson noise of power P = kTB, can be thought of as a noise voltage
applied through a resistor.
Rs
es
A example of such a noise source may be a cable or transmission line. The
amount of noise power transferred from the source to a load, such as an
amplifier input, is a function of the source and load impedances.
Rs
RL
es
Thermal Noise Source
Amplifier Input
If the load impedance is 0 Ω, no power is transferred to it since the voltage is
zero. If the load has infinite input impedance, again no power is transferred to
it since there is no current. Maximum power transfer occurs when the source
and load impedance is equal.
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PL max 
es2
4Rs
The rms noise voltage at maximum power transfer is:
en  4RP  4RkTB
R
en  4RkTB
Thermal Noise Source
under matched load conditions
Observe what happens if the noise resistance is resolved into two components:
en2  4RkTB  4R1  R2 kTB
 4R1 kTB 4R1kTB
2
2
 en1
 en2
From this we observe that random noise resistance can be added directly, but
random noise voltages add vectorially:
ETotal 
2
E2n1  En2
En2
En1
If the noise sources are not quite random, and there is some correlation
between them [0 < K < 1], the combined result is not so easy to calculate:
PTotal not quiterandom 
V12  V22  2KV1V2
Ro
 P1  P2  2K P1  P2
where
K  correlation 0  K  1
Ro  impedance reference level
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13.1.2 Noise Figure
This parameter is specified in all high performance amplifiers and is measure
of how much noise the amplifier itself contributes to the total noise. In a
perfect amplifier or system, NF = 0 dB. This discussion does not take into
account any noise reduction techniques such as filtering or dynamic emphasis.
Sin
Sout
G
Nin
Nout
Signal to noise ratio: It is either unitless or specified in dB. The S/N ratio may
be specified anywhere within a system.
S signal power PS


N noise power PN
S
P

 
  10 log S
 N dB
PN
Noise Ratio: is a unitless quantity
NR 
S N 
S N
in
out
Noise Figure:
NF  10 log NR  10 log
S N 
S N 
in
 
 10log S N
in
 
 10 log S N
out
out
 S 
 S 
  in    out 
N
 in dB Nout dB
13.1.2.1 Friiss’ Formula
It is informative to examine a cascade of amplifiers to see how noise builds up
in a large system.
NR 
Therefore:
Sin Nin
S
N
 in  out
Sout Nout Nin Sout
G
Gain can be defined as:
13 - 4
Sout
Sin
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Sin
G
Therefore the output signal power is:
Sout = G Sin
Sout  GSin
and the noise ratio can be rewritten as:
NR 
Sin
N
N
 out  out
Nin GSin GNin
The output noise power can now be written: Nout  N R GNin
From this we observe that the input noise is increased by the noise ratio and
amplifier gain as it passes through the amplifier. A noiseless amplifier would
have a noise ratio of 1 or noise figure of 0 dB. Consequently, the input noise
would only be amplified by the gain.
The minimum noise that can enter any system is the Johnson Noise:
Nin minimum  kTB
The minimum noise that can appear at the output of any amplifier is:
Nout minimum  NR GkTB
The output noise of a perfect amplifier would be:
Noutperfect  GkTB
The difference between these two values is the noised added by the amplifier
itself:
Noutadded  Nout minimum  Nout perfect
 N R GkTB  GkTB
 N R  1GkTB
This is the added noise, as it appears at the output. The total noise coming out
of the amplifier is then given by:
N Total  GN out perfect  Nout added
 G kTB



input noise
N R  1GkTB






additional noise due to the amp
If a second amplifier were added in series, the total output noise would consist
the first stage noise amplified by the second stage gain, plus the additional
noise of the second amplifier:
NTotal  G1G2kTB  NR1  1G1G2 kTB  NR2 1G2kTB
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If we divide both sides of this expression by the common term: G1G2kTB
we obtain:
G1 G2 kTB  NR1 1G1 G2 kTB  N R2  1G2 kTB
NTotal

G1G2 kTB
G1G2 kTB
NR 
Recall:
Nout
NTotal

GNin G1G2 kTB
NR overall  NR1 
Then:
N R2  1
G1
This process can be repeated for n stages, and the resulting equation is known
as Friiss’ Formula:
NR overall  NR1 

It is more commonly written as:
F  F1 
N R2  1 N R3  1


G1
G1 G2
F2  1 F3  1


G1
G1G2
From an examination of this equation, it is readily apparent that the overall
system noise figure is largely determined by the noise figure of the first stage
in a cascade.
EXAMPLE
An amplifier has a noise figure of 8 dB and a power gain of 10 dB. It is
followed by a mixer with a noise figure of 25 dB and a gain of -20 dB. The
overall noise figure can be determined as follows:
Amp
G1 = 10 dB
F1 = 8 dB
X
Mixer
G2 = -20 dB
F2 = 25 dB
Convert all numbers to ratios:
13 - 6
G1 = 10 dB
10
G2 = -20 dB
0.01
F1 = 8 dB
6.309
F2 = 25 dB
316.22
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Apply Friis’ formula:
F  F1 
F2  1
316.22 1
 6.31
 37.832
G1
10
or in dBs
10log37.832  15.78 dB
Note that the overall noise figure is greater than that of the amplifier, but less
than that of the mixer.
13.1.2.2 Noise Reduction
Inductive and capacitive reactances do not generate thermal noise because
they cannot dissipate power. However, they can affect the noise spectrum.
The noise spectrum density is defined as:
S  kT
but if it passes through a reactance:
2
S  H  kT
13.1.2.3 Noise Temperature
In microwave applications, it is difficult to speak in terms of currents and
voltages since the signals are more aptly described by field equations.
Therefore, temperature is used to characterize noise. The total noise
temperature is equal to the sum of all the individual noise temperatures.
13.2 Review of RLC Networks
13.2.1 Series RLC Circuits
R
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L
C
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Signals and Noise
Voltage
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fo
Frequency
Rs
L
Voltage
R
fo
Frequency
C
The total impedance of the RLC circuit is given by:
Z  R  jX L  jX C  R  j X L  XC 
where
XC 
1
C
and X L  L
When the magnitude of the inductance exceeds that of the capacitor, the
impedance at a specific frequency may be graphically represented as:
+j
XL
XL - X C
Resultant Impedance Z

R
XC
-j
X
2
2
Z  Z   R

X
 tan 1  

R

modulus 
 
phase angle
By definition, the series RLC circuit is resonant when  = 0. This occurs when
X = XL - XC = 0. The magnitude of the output voltage is determined by
applying the voltage divider rule to the source and load resistances. The circuit
impedance is its minimum value at resonance.
The frequency at which resonance occurs can be determined as follows:
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X L  X C  0 or X L  XC
o 
1
LC
or
fo 
or  o L 
1
 oC
1
2 LC
The quality factor or Q is a measure of the amount of energy stored in a
reactor. It can be defined as the ratio of voltage across the inductor to voltage
across the resistor.
Q
Since at resonance
o L 
VL X L  o L


VR
R
R
1
1
, then: Q 
 oC
 o RC
Typical values of Q in RLC circuits are: 10 < Q < 300.
BANDWIDTH
A sketch of the magnitude of the impedance as a function of frequency
resembles:
Z
2R
R
frequency
f1
fo
f2
The impedance is sometimes expressed in terms of Q and a new parameter we
shall call y.


1 
1 

 L
Z  R  j  L 

  R 1  j 
  R 1 
C 

 R RC 




 



  o  


 R 1 j

Q  R1  jyQ 
 o   










 let = y  

   o L  o 1 

j 

 RC 
 o R
Z  R 1  y 2Q 2
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At resonance, the current flowing in the circuit is determined by R. The 3 dB
point is defined when the current drops by 3 dB from its resonant value, at
which point the modulus increases by 2 . The magnitude of the impedance
is therefore given by: Z  2 R
The 3 dB bandwidth can be determined by evaluating the impedance function
at the 3 dB points:
2
2
R 1 y Q  R 2
 y
1
Q
at the 3dB points
At the upper cutoff frequency we obtain:
y
f2 fo 1
 
fo f 2 Q
2
 f 
f
2
 f2  o   o   fo
2Q
2Q
At the lower cutoff frequency we obtain:
y
fo f1 1
 
f1 f o Q
2
 f 
f
2
 f1   o   o   fo
2Q
2Q
The 3 dB bandwidth is the difference between these two frequencies:
B3 dB  f2  f1 
fo
Q
13.2.2 Parallel RLC Circuits
R
C
L
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Parallel circuits are easier to analyze in terms of admittance instead of
impedance.
Y

1
R  jL
 jC  2
 jC
R  jL
R   2 L2
R
L



 j C  2
2 2
2 2 

R  L
R  L
2
At resonance, the admittance is real and the imaginary components cancel.
The resonant frequency can be determined as:
 oC 
 oL
R   2o L2
o 
1
R2
 2
LC L
2
It is interesting to note that in a parallel resonant circuit, the resonant
frequency is influenced by R, but in series resonant circuit, it is not.
Usually
1
R2
 2 , and therefore:  o 
LC
L
1
.
LC
Since the admittance at resonance is purely conductive:
Yo 
R
R   2o L2
2
1
L
RC
, therefore Yo 
. Since Yo 
, the
Zo
C
L
.
L
impedance at resonance is given by: Zo 
 Q2 R .
RC
Recall that
R2   2o L2 
f
f
is similar to that of Z vs
for the series circuit. It
fo
fo
fo
therefore follows that B3 dB 
.
Q
The graph of
Y vs
The impedance of a parallel resonant circuit is high and is often referred to as
the dynamic impedance where:
RD 
1
L
Q


Yo RC  o C
13.2.3 Self Resonance
A real coil has winding resistance and capacitance in addition to its
inductance. As a result, it acts like a parallel tuned circuit. In some
applications, it may be necessary to consider these secondary affects.
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13.3 Oscillators
Many amplifiers and oscillators can be represented by:
Vin
+

G
Vout
-
H
The transfer function of this network can easily be determined as:
Vo
G

 closed loop gain
Vin 1 GH
G  open loop gain
GH  loop gain
This expression is sometimes written:
A
Ao
1 Ao
Ao  open loop gain
  feedback factor
As an example of this, we may consider an opamp.
10 K
1K
Vout
+
Vin
If the open loop gain for this amplifier is: Ao = 104, the closed loop gain is
given by:
ACL 
10 4
1  104
 10.9879
1K
1K  10K
The closed loop gain can be very closely approximated by:
ACL  1 
13 - 12
10K
 11
1K
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Note that the feedback factor is 1/11 of the output signal.
With negative feedback the closed loop gain is less than of open loop gain,
and the circuit is stable. With positive feedback the closed loop gain is greater
than the open loop gain and the circuit is unstable.
For continuous oscillations:
loop must be
GH  1 and the total phase change around the
o
n  360 .
If GH 1 , oscillations build up until something limits the process. Creating
an output with no input means that the closed loop gain is equal to infinity.
This occurs when 1 + GH = 0. The loop gain at this point is GH = -1. This
condition is known as the Barkhausen criteria for oscillation.
13.3.1 Tuned Circuit Oscillators
Note: The following analysis only works for high output impedance amplifiers
such as transistors. In opamp circuits, the low output impedance effectively
shorts out Z1.
Transitor Amplifier
Ro
Vout
Vin
Zin
AVin
Vin 
I1
Z2
Z1
I2
Z3
Writing KVL in the output loop we obtain:
AVin  I1Ro  Z1   I2Z1
Writing KVL in the feedback loop we obtain:
0  I1Z1  I2Z1  Z2  Z3 
From the input loop we obtain:
I2 
Vin 
Z2
The loop gain can be determined as:
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Vin 
Z1 Z2
 A 2
Vin
Z1  Z1  Z2  Z3 Ro  Z2 
In order for oscillations to occur, the loop gain must be greater than 1, therefor
an inversion is necessary in the circuit to change the sign of the gain. Such an
inversion does take place between the base and collector of a transistor.
Recall that at resonance, a series RLC circuit has low impedance. Therefor
Z1  Z2  Z3  0 and the loop gain reduces to:
Vin 
Z
 A 2
Vin
Z1
Oscillator Type
Hartley
Colpitts
Clapp
Z1
L
C
C
Z2
L
C
C
Z3
C
L
L+C
13.3.1.1 Hartley Oscillator
+V
R1
RFC
C2
C4
R2
R3
n2
Z2
C3
n1
L
Z1
C1
Z3
From our previous analysis, the loop gain at resonance is given by:
A  A
Z1
L
A 1
Z2
L2
In order to guarantee oscillations, when the power is first turned on, the loop
gain must be greater than 1.
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EXAMPLE
A worst-case design: the typical gain of a transistor is 35, but at the worst case
operating temperature, it falls 30%. Determine the value of  to ensure
oscillations with a 2:1 safety margin.
Solution
The lowest gain expected is 35 x.7 = 24.5
For a 2:1 safety margin, assume the gain is 24.5/2 = 12
Since A = 1 and A = 12, then  = 1/12
Z2 L2

Z1 L1
 L2  12L1
The operating frequency is determined by:
fo 
1

2 LC 2
1
L1  L2 C1
13.3.1.2 Colpitts Oscillator
This is the same as the Hartley oscillator, but with the inductors and capacitors
reversed.
C2
C1
L
The feedback factor is given by:

X C 2 C1

X C1 C 2
The oscillation frequency is given by:
fo 
1

2 LC
1
 C C 
2 L 1 2 
C1  C2 
13.3.1.3 Clapp Oscillator
This more stable variation of the Colpitts oscillator has a reduced tuning
range.
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+V
RFC
C1
C2
L1
C3
The capacitors C1 and C2 must be much larger than the transistor junction
capacitance. The oscillation frequency is given by:
fo 
1
2 L1C3
13.3.2 Crystal Oscillator
Crystal Oscillators by MXCOM
The equivalent RLC circuit of a crystal is:
Rs
Ls
Cp
Cs
Applying an electric potential across the crystal causes it to deform. But since
the crystal is a piezo-electric device, deforming it causes a slight electrical
potential to develop across it. As a result, the crystal can act as an
electromechanical resonator if the right excitation is applied.
The oscillation frequency is determined by the size and shape of the crystal.
Crystals can be used to make very stable oscillators ranging from 15 KHz to
100 MHz.
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XL
fs
fp
frequency
XC
Placing an inductor in series with the crystal lowers the series resonant
frequency, and placing an inductor in parallel to the crystal increases the
parallel resonant frequency.
13.3.2.1 Crystal Filter
Output
Input
Attenuation
C
Large C
Small C
frequency
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Assignment Questions
Quick Quiz
1.
Friiss’ formula shows that noise figure is proportional to bandwidth.
[True, False]
2.
Johnson noise is also called [pink, white, blue] noise.
3.
Admittance at resonance is purely conductive. [True, False]
Analytical Questions
1.
Determine the total noise power if two thermal noise sources of 12 µW
and 16 µW are connected in series.
2.
A two stage RF amplifier has a 3 dB bandwidth of 150 KHz determined
by an LC circuit at the input and operates at 27o C.
First stage:
PG = 8 dB
NF = 2.4 dB
Second stage:
PG = 40 dB
NF = 6.5 dB
Output Load = 300 Ω
In testing the system, a 100 KΩ resistor is applied at the input. Determine:
a)
Input noise voltage
b) Input noise power
c)
Output noise voltage
d) Output noise power
e)
3.
System noise figure
Prove that under matched load conditions, the rms Johnson noise voltage
is: en  4RkTB
Composition Questions
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For Further Research
http://www.mth.msu.edu/~maccluer/Lna/noisetemp.html
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