2. Noise Models
Noise Models
In the above sections, the various physical sources of
noise in electronic circuits were described.
In this section, these sources of noise are brought
together to form the small-signal equivalent circuits:
•
•
•
•
•
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resistors (already discussed)
diodes,
bipolar (BJT) and
field-effect (FET) transistors and
linear IC (OpAmp)
Resistors
Noise can be modelled as
• a Thevenin equivalent voltage source or
• a Norton equivalent current source.
The noise contributed by the resistor is modeled by
the source, thus the resistor is considered noiseless.
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Two resistors
It is important to note that noise sources:
• Do not have polarity
• Do not add algebraically, but as RMS sums
If the sources are not correlated, then:
vn (rms) = vn2 = vn21 + vn21 = 4kTBR1 + 4kTBR1
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Correlated resistors
If the sources are correlated (derived from the same
physical noise source), then there is an additional term:
C can vary between –1 and 1.
vn2 = vn21 + vn21 + 2Cvn1vn 2
vn (rms) = vn2 = vn21 + vn21 + 2Cvn1vn 2
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Thermal Noise Power
The available noise power can be calculated from the
RMS noise voltage or current:
Pno =
(
E
E /2
= t
RL
RS
)
2
= kTB
That is, the available noise power from the source is
•
•
•
•
2
no
independent of resistance
proportional to temperature
proportional to bandwidth
has no frequency dependence
P=4 x 10-21 watts in a 1 Hz bandwidth at the standard
noise room temperature of 290 K.
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Can a resistor produce
infinite noise voltage?
therefore vt2 → ∞ when B → ∞
vn2 ≡ vt2 = 4kT ⋅ R ⋅ B
Equivalent circuit for noisy resistor
always have some shunt. Therefore
Vno = Vno
1
1 + ω 2C 2 R 2
To find the noise power
∞
v = ∫ Vno dt =
2
no
0
2
kT
C
Therefore total noise power is independent of R !!!
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Junction Diode
The equivalent circuit for a junction diode
was considered briefly in the consideration
of shot noise.
The basic equivalent circuit of diode
(discussed) can be made complete by adding
series resistance rs, as shown.
Since rs, is a physical resistor due to the
resistivity of the silicon, it exhibits thermal
noise.
vt2 = 4kT ⋅ rs ⋅ B
Experimentally it has been found that any flicker noise present
can be represented by a current generator in shunt with rd, and
this is conveniently combined with the shot-noise
α
I
i 2 = 2qI D ⋅ B + K d B
f
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BJT (noise model)
In a BJT in the forward-active region, minority carriers
diffuse and drift across the base region to be collected
at the collector-base junction.
Minority carriers entering the collector-base depletion
region are accelerated by the field existing there and
swept across this region to the collector.
The time of arrival at the collector-base junction of the
diffusing (or drifting) carriers is a purely random
process, and thus the transistor collector current
consists of a series of random current pulses.
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Contribution I
Consequently, collector current Ic shows full shot
noise, and this is represented by a shot noise current
generator ic2 from collector to emitter as shown in the
equivalent circuit of BJT
i = 2qI C ⋅ B
2
c
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Contribution II
Base current IB in a transistor is due to recombination in the base
and to carrier injection from the base into the emitter.
All of these are independent random processes, and thus IB also
shows full shot noise.
Flicker noise and burst noise in a BJT have been found
experimentally. This is represented by shot noise current
generator
c
α
I
I
b
ib2 = 2qI B ⋅ B + K1 b B + K 2
B
2
f
1 + ( f / fc )
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Contribution III
Transistor base resistor rb is a physical resistor and
thus has thermal noise.
Collector series resistor rc also shows thermal noise,
but since this is in series with the high-impedance
collector node, this noise is negligible and is usually
not included in the model.
vb2 = 4kTB
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All Contributions
ic2 = 2qI C ⋅ B
Shot noise in collector current Ic,
Shot, Flicker noise and burst noise in Base current IB
c
α
I
I
b
ib2 = 2qI B ⋅ B + K1 b B + K 2
B
2
f
1 + ( f / fc )
Thermal noise in base resistor rb.
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vb2 = 4kTB
“corner” frequency
The base-current noise spectrum can be plotted where
burst noise has been neglected for simplicity.
The shot noise and flicker noise asymptotes meet at a
frequency fa, which is called the flicker noise "corner"
frequency.
In some transistors using
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careful processing, fa
can be as low as 100 Hz.
In other transistors fa can
be as high as 10 MHz.
FET noise model
In FET the resistive channel joining source and drain, so that the
drain current is controlled by the gate-source voltage.
Since the channel material is resistive, it exhibits thermal noise,
and this is the major source of noise in FETs this noise source
can be represented by a noise-current generator iD2 from drain to
source in the FET small-signal equivalent circuit.
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Contributions
Flicker noise in the FET is also found experimentally to be
represented by a drain-source current generator,
I Da
2 
i = 4kT  g m  + K 2
B
f
3


2
d
The other source of noise in FETs is shot noise generated by the
gate leakage current and is usually very small. It becomes
significant only when the driving-source impedance connected
to the FET gate is very large.
ig2 = 2qI G ⋅ B
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BJT noise performance
Consider the noise performance of the simple transistor stage
with the ac schematic shown below
Consider the noise performance of the simple transistor stage
with the ac schematic shown below
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BJT noise performance (a)
In this equivalent circuit the external input signal vi, has been
ignored so that output signal vo is due to noise generators only.
Cµ is assumed small and is neglected. Output resistance ro is also
neglected. The transistor noise generators are as described
previously and in addition
1
2
2
il = 4kT
B
vs = 4kTRS B
RL
The total output noise can be calculated by considering each
noise source in turn and performing the calculation as if each
noise source were a sinusoid with rms value equal to that of the
noise source being considered.
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BJT noise performance
The spectral density of the noise generator is equal:
2
2
o
Z
v
2 2
= g m RL
B
Z + rb + RS

 4kT
+ R 
+ 2qI C 

 RL
2
L
2
[4kT (R
S
2
where Z is
Substituting for Z gives:
2
]
+ rb ) + (RS + rb ) 2qI B +
1
Z = rπ
jωCπ
[
]
vo2
1
rπ
2
(
)
(
)
4
kT
R
r
R
r
= g m2 RL2
⋅
+
+
+
S
b
S
b 2qI B +
2
2
B
(rπ + rb + RS ) 1 + ( f / f1 )
 4kT


+R 
+ 2qI C 
 RL

2
L
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where
1
f1 =
2π [rπ (rb + RS )]Cπ
BJT noise spectral density
The output noise-voltage spectral density has a frequency-dependent part and
a constant part.
The frequency dependence arises because the gain of the stage begins to fall
above frequency f1, and noise due to generators vs, vb, and ib,which appears
amplified, also begins to fall.
The constant term is due to noise generators il and ic
Assuming IC=100 µA, RS=500 Ω, RL=5 kΩ, β=100, Cπ=10 pF, rb=200Ω
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