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Chapter 3
Network Noise and
Intermodulation Distortion
1
Introduction
• Noise is one of the most important factors affecting the operations of
IC circuits. This is because noise represents the smallest signal the
circuit can process.
• The principle noise sources are Johnson noise generated in resistors
due to random motion of carriers; shot noise arising from the
discreteness of charge quanta; mixer noise arising from non-ideal
properties of mixers; undesired cross coupling of signals between two
sections of the receiver; flicker noise due to defects in the
semiconductor; and power supply noise.
• Except for Johnson noise and shot noise the other noise sources can be
improved or eliminated through proper design
• The “Noise Figure” measures the noise generated in a network,
together with the “dynamic range” are used to quantify the receiver
performance
2
Noise
• All signals are contaminated with noise
• The noisiness of a signal is specified by the signal-to-noise ratio
defined as
S ( f ) rms signal voltage Average signal power


N ( f ) rms noise voltage
Average noise power
• The last definition will be adopted.
• Noise of human origin is usually the dominant factor in receiver noise.
This can usually be eliminated through proper design, layout, and
shielding. Random noise cannot be eliminated. It sets the theoretical
lower limit on receiver noise
• The mean square noise voltage are referred to as the noise power
• The noise power is normally frequency-dependent and is usually
expressed as a power spectral density function. The total noise power
f
is
2
P   P( f )df
f1
3
Thermal Noise (Johnson Noise)
• Discovered by J.B. Johnson and is therefore commonly known as
Johnson noise
2
• The rms value of thermal noise voltage En is given by En  4kTR( f )f
• Since the noise voltage squared is proportional to f. This implies that
if the interval is infinite, the noise power contributed by the resistor is
also infinite.
• In reality the above equation must be modified above 100 MHz, but is
sufficiently accurate for low frequency
• R(f) is the real part of the impedance Z(f) looking into the two
terminals between which En is measured.
• If a resistor is connected to a frequency-dependent network as shown
4

• then the total noise due to R is E0   4kTRG( f )df
0
• Where G(f) is the magnitude squared of the frequency-dependent
transfer function between the input and the output voltages
2
G( f ) 
E0 ( f )
En ( f )
2
• Since R(f) depends on frequency
• The integral of G(f) is known as the noise bandwidth Bn of the system.

E0  4kTR G( f )df
2
0
• If 2 resistors are connected in series, it is the voltage squared, not the
noise voltages, which are added. En 2  E12  E2 2  4kT ( R1  R2 )f
5
• Example 3.1
• The impedance of the parallel combination of a resistor R and
R
capacitor C is given by z ( ) 
1  jRC
• The real part is given by
• We can calculate E02 using

df
kT
2
E0  4kTR

0 1   2 R 2C 2
C
• Since
E
G( f ) 0
En
2
 (1   2 R 2C 2 ) 1
E0  4kTRBn
2
 Bn  1

df
kT

1   2 R 2C 2 C
0
E0  4kTR
2
4 RC
Hz
6
Current-source representation
• So far we used voltage sources in series with a noiseless resistor to
represent thermal noise. Norton’s theorem shows that the voltage
noise source can also be represented by a current generator in parallel
with a noiseless resistor as show below
Shot Noise
• Shot noise is due to discreteness of the electronic charge arriving at the
anode givingSrise
to pulses
of1current. The current noise power
A2 Hz
I  2qI
spectrum is
7
Flicker Noise
• This type of noise is found in all semiconductor devices under the
application of a current bias
• The mean squared current fluctuation over a frequency range f is
Ia
2
i  K1 b f
f
• Metal films show no or very small flicker noise, thus they should be
used for CKT design if low 1/f noise is desired
8
• K1 may vary by over orders of magnitude because flicker noise is
caused by various unknown mechanisms such crystal imperfections,
contamination.
• Although flicker noise appears to be dominant at low-frequencies, it
may still affect rf applications of the communication circuits through
the nonlinear properties of the oscillators and mixers which mixes the
noise to the carrier frequencies
Avalanche Noise
• This is caused by Zener or avalanche breakdown in a pn junction
• Electrons and holes in the depletion region of a reversed-biased
junction acquire enough energy
• Since additional electrons and holes are generated in the collision
process a random series of large noise spikes will be generated.
• The most common situation is when Zener diodes are used in the
circuit and should therefore be avoided in low noise circuits.
• The magnitude of the noise is hard to predict due to its dependence on
the materials
9
Noise Models of IC Components
I. Junction Diode
• The equivalent circuit for a junction diode the equivalent circuit is
shown to be
• Rs is a physical resistor due to resistivity of the silicon, it exhibits
thermal noise.
• The current noise source is due to shot noise and flicker noise. Thus
v  4kTrs f
2
s
I Da
i  2qI D f  K
f
f
2
10
•
•
•
•
•
II. Bipolar Transistors
In a bipolar transistor in the forward-active region, minority carriers
entering the collector-base depletion region are being accelerated to
the collector. The time of arrival is a random process process, thus IC
shows full shot noise.
The base current IB is due to recombination in the base and baseemitter depletion regions and also due to carrier injection from the base
into the emitter.
Thus IB also shows full shot noise characteristics. The recombination
process in the region also contribute to burst noise and flicker noise.
Transistor base resistor is a physical resistor and thus has thermal noise
Collector rc also shows thermal noise, but since this is in series with
the high-impedance collector node, this noise is usually neglected. r
and rb are fictitious resistors used for modeling and therefore do not
contribute to thermal noise
I Ba
I Bc
2
2
2
vb  4kTrb f
ic  2qI C f
ib  2qI B f  K1 f  K 2
f
2
f
 f 
1   
 f c  11
• The equivalent circuit model for a BJT transistor is shown below
FET Transistor
• FET shows full shot noise for the leakage current at the gate as well as
thermal noise and flicker noise in the channel region.
• Very often in JFETs the dominant type of noise is burst noise instead
and in MOSFETs the dominant type of noise is flicker noise
i  2qI g f
2
g
2
I Da
i  4kT ( g m )f  K
f
3
f
2
d
12
Circuit Noise Calculations
• The device equivalent circuits can be used for calculation of noise
performance. Consider a current noise source i 2  S ( f )f
• if the rms current noise is represented by i, Within a small bandwidth,
f, the effect of the noise current can be calculated by substituting by a
sinusoidal generator and performing circuit analysis in the usual
fashion. When the circuit response to the sinusoid is calculated, the
mean-squared value of the output sinusoid gives the mean squared
value of the output noise in bandwidth f.
• In this way network noise calculations reduce to familiar sinusoidal
circuit analysis calculations.
• When multiple noise sources exists which is the case in most practical
situations, each noise source is represented by a separate sinusoidal
generator, and the output contribution of each source is calculated
separately.
• The total output noise in bandwidth f is calculated as a mean-squared
value by adding the individual mean-squared contributions from each
output sinusoid.
• For example if we have 2 resistors in series the total voltage is
13
vT (t )  v1 (t )  v2 (t )
Thus
vT (t ) 2  v1 (t )  v2 (t )
2
 v1 (t ) 2  v2 (t ) 2  2v1 (t )v2 (t )
• Since the noise sources v1 and v2 are statistically independent of each
other arising from two separate resistors the average of the product v1
v2 will be zero
• Analogous results is true for independent current noise sources placed
in parallel. The spectra are summed together. vT2  4kT ( R1  R2 )f
Bipolar Transistor Noise Performance
• Consider the noise performance of the simple transistor stage as shown
• 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.
14
v1 
Z
vs
Z  rb  Rs
•Consider the noise generator vs due to Rs
where Z is the parallel combination of r and C. The output noise voltage due to vs
vo1   g m RL v1
Z
  g m RL
vs
Z  rb  Rs
vo21  g m2 RL2
Z
2
Z  rb  Rs
2
v
2 s
15
• Similarly it can shown that the output noise voltages by vb and ib are
vo22  g m2 RL2
Z
2
Z  rb  Rs
2
vo23  g m2 RL2
vb2
( Rs  rb ) 2 Z
2
Z  rb  Rs
2
2
2 2
• Noise at the output due to il2 and ic2 is vo 4  il RL
5
• The total output noise is v 2  v 2

o
2
Z
vo2
 g m2 RL2
f
Z  rb  Rs
2
n 1
b
vo25  ic2 RL2
on
4kT ( R  r )  ( R
s
vb2
s
 rb ) 2 2qI B



1
 RL2  4kT
 2qI c 
RL


• Substituting for Z we have
vo2
r2
2 2
 g m RL
f
(r  Rs  rb ) 2
1
2

 4kT ( Rs  rb )  ( Rs  rb )  ( Rs  rb ) 2 2qI B
 f 
1   
 f1 


1
 RL2  4kT
 2qI c 
where
R
L


f1 
1
2 r ( Rs  rb )C

16
• The output noise power spectral density has a frequency-dependent
part, which arises because the gain stage begins to fall above frequency
f1, and noise due to vs2 , vb2 and ib2 which appears amplified in the
output, also begins to fall. The constant term is due to noise generators
il2 and ic2 . Note that this noise contribution would also be frequency
dependent if the effect of C had not been neglected. The noise
voltage spectral density is shown in the following figure
17
Equivalent Input Noise and Minimum Detectable Signal
• The significance of the noise performance of a circuit is the limitation
it places on the smallest input signal. For this reason the noise
performance is usually expressed in terms of an equivalent input noise
signal, which gives the same output noise as the circuit under
consideration.
• Such representation allows one to compare directly with incoming
signals and the effect of the noise on those signals is easily determined.
• Thus the circuit previously studied can be represented by
2
• where viN is an input noise voltage generator that produces the same
output noise as all of the original noise generators. All other source of
noise are considered removed. Thus
18
vo2  g m2 RL2
Z
2
Z  rb  Rs
2
2
viN
2
viN
1 Z  rb  Rs
1
2
2
 4kT ( Rs  rb )  ( Rs  rb ) 2qI B  2 2
R
(
4
kT
 2qI C )
L
2
f
g m RL
R
Z
L
2
• The above equation rises at high frequencies due to variation of |Z|
with frequencies. This is due to the fact that as the gain of the device
falls with frequency, output noise generators 2
have larger
ic and il2
effects when referred back to the input.
• Example: Calculate the total input noise voltage, 2 , for the circuit
viNT
of the following circuit from 0 to 1 MHz
19
• Using the above equation for equivalent input noise
2
viN
1 Z  rb  Rs
1
2
 4kT ( Rs  rb )  ( Rs  rb ) 2 2qI B  2 2
R
(
4
kT
 2qI C )
L
2
f
g m RL
RL
Z
2
2
2 for the calculation of v
• On the other hand we can use voT
iNT
• If Av is the low-frequency gain
r
Av 
g m RL
rb  r  RS
• using the data
I C  100A
  100
r  200
26000
5000
Av 
 18.7
200  26000  500 260
RS  500
2
iNT
v
C  10 pF
RL  5k
2
voT
 2  14.3 1012V 2
Av
viNT  3.78Vrms
20
• The examples shows that from 0 to 1 MHz the noise appears to come
from a 3.78 V rms noise-voltage source in series with the input. This
can be used to estimate the smallest signal that the circuit can
effectively amplify, sometimes called the minimum detectable signal
(MDS). If a sine wave of magnitude 3.78 V were applied to this
circuit, and the output in a 1-MHz bandwidth examined on an
oscilloscope, the sine wave would be barely detectable
Equivalent Input Noise Generators
• Using the equivalent input noise voltage an expression for equivalent
input noise generator dependent on the source resistance can be
determined.
• To extend this to a more general and more useful representation using
2 equivalent input noise generators. The situation is shown below
21
• Here the two-port network containing noise generators is represented
by the same network with internal noise sources removed and with a
2
2
v
noise voltage i and current generator ii connected to the input. It
can be shown that this representation is valid for any source
impedance, provided that the correlation of between the two noise
generators is considered.
• The 2 noise sources are correlated because they are both dependent on
the same set of original noise sources.
• However, correlation my significantly complicate the calculation. If
the correlation is large, it may be simpler to go to the original circuit.
• The need for both voltage and current equivalent input noise
generators to represent the noise performance of the circuit for any
source resistance can be appreciated as follows. Consider the extreme
cases of source resistance RS=, vi2 cannot produce output noise and
ii2
represents the noise performance of the original noisy network.
22
• The values of the equivalent input generators are readily determined.
This is done by first short circuiting the input of both circuits and
equating the output noise in each case to calculate vi2 . The value of i 2
i
is found by open circuiting the input of each circuit and equating the
output noise in each case
Bipolar Transistor Noise Generators
• The equivalent input noise generators for BJT can be calculated from
the equivalent circuit of the following figure
23
• The 2 circuits are equivalent and should give the same output noise for
any source impedance
2
• The value of vi can be calculated by short circuiting the input of each
circuit and equating the output noise,
i0  g m vi
i0
• From 11.23a we have g v  i  i
m b
c
0
• From 11.23b we have
• Here we use rms noise quantities and make no attempts to preserve the
rb  are
r all independent
signs of the noise quantities as the noise generators
ic
g
v

i

g
v
v

v

and have randomm phase.
We
that
.
b
c
m i also assume
i
b
gm
• Thus we have
ib2
2
i
• Since rb is small the effect of is neglected
vi2  vb2  c2
gm
• Using the fact that vb and ic are independent, we obtain

• 2Using previous2 definition of2qI C f
vi2
1 
2
vb  4kTrb f
vi  4kTrb f 
g
2
m
ic  2qI C f

 4kT  rb 
f
2gm 

1
• The equivalent noise-voltage Rspectral
density
thus appears to come

r

eq
b
2gm
from a resistor Req such that
24
Req  rb 
1
2gm
This is known as the “equivalent input noise resistance”
• Here rb is a physical resistor in series with the input, whereas 1/2gm
represents the effect of collector shot noise referred back to the input
• The above equations allows one to compare the relative significance of
2
v
i
noise from r and I in contributing to .
b
C
• Good noise performance requires the minimization of Req. This can be
accomplished by designing the transistor to have a low rb, and running
the devices at a large collector bias current to reduce 1/2gm.
• To calculate the equivalent input noise current, the inputs of both
circuits are open circuited and the output noise currents, i0, are equated
 ( j )ii  ic   ( j )ib
ii  ib   ( j )ib
• which gives
ic2
2
2
ii  ib 
2
• Since ib and ic areindependent
generators,
we
obtain,

(
j

)
0
 ( j ) 

where
1 j

25
• where 0 is the low-frequency, small signal current gain.
• Substituting for ib2 and ic2 gives
a

ii2
IC 
' IB
 2q  I B  K1 

2
f
f
 ( j ) 

•
K1
'
K

where 1 2q
. The last term is due to collector current noise
referred to the input. At low frequencies this becomes I C /  02 and is
negligible compared with IB for typical 0 values. The equivalent input
noise current spectral density appears to come from a current source Ieq
1
I
and
' IB
I eq  I B  K1  C 2
f
( j
• Ieq is minimized by utilizing low bias currents in the transistor, and
using high  transistors. It should be noted that low current
requirement to reduce ii2 contradicts that for reducing v 2
i
2
• Spectral density for ii / f is frequency dependent both at low and
high frequency regime due to flicker noise and collector current noise
referred to the input respectively. fb and fa are defined as in the figure
26
below
27
 ( jf ) 
0
f
• Using the definition
1 j
0
fT
• The collect current noise is
f2
2q
 2qI C 2
2
fT
 ( jf )
IC
• at high frequencies, which increases as f2. Frequency fb is estimated
2q( I B  [ I C /  02 ])
by equating the above equation to the midband noise and is
• For typical values
2qIB We obtain
f b2 of 0 it is approximately
IB
2qI B  2qI C 2
 f b  fT
fT
IC
28
The large signal current gain is  F 
IC
IB
•
fT
f

• Therefore b
F
• Once the input noise generators have been calculated, the transistor
noise performance with any source impedance is readily calculated.
• Consider the following circuit
• with a source resistance RS. The noise performance of this circuit can
2
be represented by the total equivalent noise voltage viN
in series with
the input of the circuit as shown.
• Neglecting noise in Rl and equating the total noise voltage at the base
of the transistor viN  vs  vi  ii RS
29
• If correlation between vi and ii is neglected this equation gives viN2  vs2  vi2  ii2 RS2
• Thus the expression for total equivalent noise voltage is
2

viN
IC 
1
2
 4kTRS  4kT (rb 
)  RS 2q  I B 

2
f
2gm
 ( jf ) 

• Using the data from previous example and neglecting 1/f noise we
calculate the total input noise voltage for the circuit in a bandwidth 0 to
1 MHz. The total input noise in a 1 MHz bandwidth is
2

viN
1 

  RS2 2qI B
 4kT  RS  rb 
f
2gm 

1.66 10
 20
500  200  130  5002  3.2 1019 106 V 2 / Hz
2
viNT
 13.9 1018 106V 2
30
Field-Effect Transistor Noise Generators
• The equivalent noise generators for a field-effect transistor can be
calculated from the equivalent circuit below
• Figure (a) is made equivalent to figure (b).
• The output noise in each case is calculated with a short-circuit load and
Cgd is neglected.
• If the input of each circuit in the figure is short circuited and the
resulting output noise currents i0 are equated we obtain from shorting
fig. a that
31
• Figure (a) is made equivalent to figure (b).
• The output noise in each case is calculated with a short-circuit load and
Cgd is neglected.
• If the input of each circuit in the figure is short circuited and the
resulting output noise currents i0 are equated we obtain from shorting
fig. a that
vg  0  i0  id
• From Fig. b we have
id2
2
vi  vg  g m vi  i0 id  g mvi  vi  2
gm
• Thus
2
id
• Substituting
the
expression
for
into the equation for total noise we
2
a
have vi  4kT 2 1  K I2D
f
3 gm
gm f
vi2
 4kTReq
a
The equivalent
resistance ' Req is defined as f
2 1 input Inoise
•
• where
Req 
3 gm
 K'
D
2
m
g f
in which
K  K / 4kT
• The input noise-voltage generator contains a flicker noise component
which may extend into the Mega Hertz region. The magnitude of
flicker noise depends on the details of the processing procedure,
32
biasing and the area of the device.
• Flicker noise generally increase as 1/A this is because larger devices
contains more defects at the Si-SiO2 interface. An averaging effect
occurs that reduces the overall noise.
• Flicker noise varies inversely with the gate capacitance because
trapping and detrapping lead to variation of the threshold voltage
which is inversely proportional to the gate capacitance. The equivalent
input-referred voltage noise can often be written as vi2  4kT 2 1  K f
f
3 g m WLCox f
• Typical value for Kf is 3 10 24V 2 F
Effect of Feedback on Noise Performance
• The representation of circuit noise performance with two equivalent
input noise generators is extremely useful in the consideration of the
effect of feedback on noise performance.
Effect of Ideal Feedback
• The series-shunt feedback amplifier is shown where the feedback
network is ideal in the signal feedback to the input is a pure voltage
source and the feedback network is unilateral. Noise in the basic
amplifier is represented by input noise generators via2 and iia2
33
• The noise performance of the overall circuit is represented by
equivalent input generators
vi2 and ii2
34
2
• The value of vi can be found by short circuiting the input of each
circuit and equating the output signal. However, since the output of
the feedback network has a zero impedance, the current generators in
each circuit are then short circuited and the two circuits are then
identical if vi2  via2
• If the input terminals are open circuited, both voltage generators have a
floating terminal and thus no effect on the circuit, for equal outputs, it
is necessary that ii2  iia2
• Thus for the case of ideal feedback, the equivalent input noise
generators can be moved unchanged outside the feedback loop and the
feedback has no effect on the circuit noise performance.
• Since the feedback reduces circuit gain and the output noise is reduced
by the feedback, but desired signals are reduced by the same amount
and the signal-to-noise ratio will be unchanged.
Practical Feedback
• Series-shunt feedback circuit is typically realized using a resistive
divider consisting of RE and RF as shown
35
36
• If the noise of the basic amplifier is represented by equivalent input
noise generators iia2 and via2 , and the thermal noise generators in RE and
RF are included in b as shown above. To calculate vi2 consider the
inputs of the circuits of b and c short circuited, and equate the output
noise
RF
RE
vi  via  iia R 
ve 
vf
RF  RE
RF  RE
• where R  RF // RE . Assuming that all noise sources are independent
we have vi2  via2  iia2 R 2  4kTRf where v 2  4kTR f and v 2  4kTR f
e
E
f
F
• Thus in a practical situation the equivalent input noise voltage of the
overall amplifier contains the input noise of the basic amplifier plus
two other terms. The second term is usually negligible, but the third
represents the thermal noise in R and is often significant.
2
• The equivalent input noise current, ii , is calculated by open circuiting
both inputs and equating output noise. For the case of shunt feedback
at the input as shown, opening circuiting the inputs of b and c and
equating the output noise we have
via
via2
1
2
2
ii  iia 
 i f thus ii  iia  2  4kT
f
RF
RF
RF
37
38
• Thus the equivalent input noise current with shunt feedback applied
consists of the input noise current of the basic amplifier together with a
term representing thermal noise in the feedback resistor. The second
term is usually negligible. If the inputs of the circuits of b and c are
short circuited ant the output noise equated it follows that vi2  via2
39
Amplifier Noise Model
• As in the case before, amplifier noise represented by a zero impedance
voltage generator in series with the input port and an infinite
impedance current generator in parallel in the input and by a complex
complex correlation coefficient C.
• The equivalent model is shown in the next page where noise sources
En, Et and In are used. Here Et is the noise generator for the signal
source. Again we determine the equivalent input noise, Eni, to represent
all 3 sources. The levels of signal voltage and noise voltage that reach
Zin in the circuit are multiplied by the noiseless gain Av
40
• The system gain is defined by Kt 
Vso
AV Z
and Vso  v in in
Vin
Rs  Z in
 Kt 
Av Z in
Rs  Z in
• For signal voltage, linear voltage and current division principles can be
applied. However, for the evaluation of noise, we must sum each
contribution in mean square values. The total noise at the output port
2
2 2
E

A
v Ei
is no
•
2
Z in
2
2
2
2
E

(
E

E
)

I
t
n
n Z in // Rs
The noise at the input to the amplifier is i
Z in  Rs
K t2
• The total output noise above
by
the expression for
2 divided
2
2
2 yields
2
Eni  Et  En  I n Rs
equivalent
input
noise
E ni2
•
is independent
of the amplifier’s gain and its input impedance.
E ni2
This makes
the most useful index for comparing the noise
characteristics of various amplifiers and devices. If the individual
noise sources are correlated
2
2 an additional
2
2 2 term must be added to the
above expression Eni  Et  En  I n Rs  2CEn I n Rs
•
41
2
Noise in Feedback Amplifiers
• Feedback is an important technique to alter gains, impedance levels,
frequency response and reduce distortion. When negative feedback is
properly applied the critical performance indexes are improved by a
factor 1+A. However, with noise it was shown that feedback does
not affect the equivalent input noise, but the added feedback resistive
elements themselves will add noise to the system.
• To examine how noise is affected by feedback we consider the block
diagram
42
• The desired input voltage Vin and all the E’s representing the noise
voltages being injected at various critical points in the system. Blocks
A1 and A2 represent amplifiers with voltage gains and  represents the
feedback network. The output voltage V0 is a function of all 5 inputs
according to
V0  E4  A2 [ E3  A1 ( E2  Vin  E1  V0 )]

A2 E3
A1 A2
E4
( E2  Vin  E1 ) 

1  A1 A2 
1  A1 A2  1  A1 A2 
• For comparison consider an open-loop system in which the feedback
loop  is taken out V0  A1 A2' (Vin  E1  E2 )  A2' E3  E4
• To accomplish a meaningful comparison between the 2 cases we set
A2
and we find that V0 for the open loop case is A2' 
1  A1 A2 
A2 E3
A1 A2
V0 
( E2  Vin  E1 ) 
 E4
1  A1 A2 
1  A1 A2 
• Thus feedback does not give any improvement for any noise source
introduced at the input to either amplifier regardless of whether this
noise source exists before or after the summer. Noise injected at the
amplifier’s output is attenuated in the feedback amplifier.
43
• In fact, if the feedback consists of resistive elements will actually increase the
output noise level due to added thermal noise from the feedback resistors.
Amplifier Noise Model for Differential Inputs
• Since operational amplifiers are configured with differential inputs. Users can
configure the feedback network an input signal so as to produce a noninverting
amplifier, an inverting amplifier or a true differential amplifier. Therefore all
op amp model having equivalent noise sources must be able to handle all of
these different configurations.
• The basic amplifier noise model is expanded as below
44
• Noise sources En1 and In1 are noise contributions from the amplifier
reflected to the inverting input terminal referenced to ground. In2 and
En2 are that reflected to the non-inverting terminal. Consider the
typical amplifier circuit shown
45
• Voltages Vp and Vn are the voltages at the respective positive and
negative inputs to the amplifier referenced to ground. The output
voltage for an ideal op amp is
 R4  R1  R2 
R 

Vin2   2 Vin1
Vo  
 R1 
 R3  R4  R1 
• An ideal differential amplifier occurs when we make the coefficients of
Vin1 and Vin2 have identical magnitudes and opposite signs. This
condition is satisfied by choosing the resistors such that R1R4  R2 R3
• Thus the output becomes Vo  R2 / R1 (Vin2  Vin1 )
• Thus the ideal difference mode voltage gain is R2/R1. To examine the
noise behavior of the differential amplifier, first form a Thevenin
equivalent circuit at the noninverting input as shown where Rp=R3//R4
and Vin' 2  Vin 2 R4 / R3  R4 
• Next insert noise voltage and current sources for the op amp and
Thevenin equivalent noise sources for the resistors as shown
46
• Here 7 signal source have arbitrary polarities as shown. Here we
assume the op-amp has a finite open loop voltage gain A but is ideal
otherwise. The four defining equations for this circuit are
Vo  A(V p  Vn )
V p  Vin' 2  R p I 2  Vtp  V2
Vn  Vin1  R1 I in  Vt1  V1
Vin1  R1 I in  Vt1  Vo  Vt 2  R2 ( I in  I1 )
47
• The four equations give
1
R1 
R
  Vin' 2  Vin1  V2  V1  Vtp  Vt1  R p I 2  2 (Vin1  Vt1 )  Vt 2  I1R2
Vo  
R1
 A R1  R2 
• As A  we obtain
 R 
R
Vo  1  2 (Vin' 2  V2  Vtp  I 2 R p  V1 )  2 (Vin1  Vt1 )  Vt 2  I1R2
R1
 R1 
• Previously for clarity, we substituted voltage and current signal sources
for corresponding noise sources. The gain to the output will be the
same for both signal sources and noise sources from the same circuit
position
• The result is
2
2
 R 
R 
2
Eno
 1  2  ( En22  Etp2  En21  I n22 R p2 )   2  Et21  Et22  I n21 R22
 R1 
 R1 
• The equation shows that each noise source contributes to the total
squared output noise. Both equivalent input noise voltages and the
noise from Rp are reflected to the output by the square of the
2
noninverting voltage gain, (1  R2 / R1 ) .
48
• The positive input noise current “flows through” Rp establishing a
noise voltage which, in turn, is reflected to the output by the same gain
2
factor (1  R2 / R1 ) .
• The negative input noise voltage “flows through” the feedback resistor
R2 establishing a noise voltage directly at the output. Finally noise
contribution due to R2 appears directly at the output.
• To determine Eni we first decide which terminal will be the reference.
This is critical since the Kt’s are different for the inverting and noninverting inputs
2
2
• First reflect Eno to the inverting input by dividing Eno
by (R2/R1)2 to
obtain
2
E
2
ni1
 R1 
R1 
2
2
2
2
2 2
2 2
2 2 



 1   ( En 2  Etp  En1 )  Et1  R1 I t 2  R1 I n1  R p I n 2 1  
 R2 
 R2 
2 2
2 2
2
• where R1 I t 2  R1 Et 2 / R2
2
E
• Note that two amplifier noise voltages plus tp are all increased at the
input by (1+R1/R2)2. Usually R1<<R2 for a typical high-gain amplifier
application, so the first 3 noise voltage sources essentially contribute
49
2
• directly to Eni12 as does E2t1. The noise current of the feedback resistor
R2 is multiplied by R12. The In1 noise current “flows through” R1
creating a direct contribution to Eni12. The In2 noise current “flows
through” Rp to produce a noise voltage and then is reflected to the
2
inverting input by the same (1  R1 / R2 ) factor.
• When reflected to the noninverting input, we divide the noise equation
2
(
1

R
/
R
)
2
1
by
50
2
2
 R1  2  R2  2
2
2
2
2
 Et 2  
 Et1  I n21 ( R1 // R2 ) 2  I n22 R 2p
Eni 2  ( En 2  Etp  Eni )  
 R1  R2 
 R1  R2 
• Here the two amplifier noise voltages as well as the noise voltage from
Rp contribute directly to Eni2 2 . The noise voltage in the feedback
resistor is divided by the square of feedback factor. The noise in R1 is
slightly diminished but essentially unchanged when R1<<R2. The
inverting noise flows through the parallel combination of R1 and R2
2
and then contributes directly to E ni1 . The non-inverting noise current
“flows through” Rp and contributes directly to
Eni2 2
51
Noise in Inverting Negative Feedback Circuits
• The inverting amplifier configuration with resistive negative feedback
is the most widely used stage configuration. The input offset voltage
due to bias current will be canceled by making Rp a single resistor
equal to the parallel combination of Rs and R2.
• All noise source are now reflected to the Vin1 input, we obtain
52
2
 R 
R 
Eni2 1  1  s ( En21  En22  Etp2  I n22 R p2 )   s  Et22  Ets2  I n21 Rs2
 R2 
 R2 
2
 R 
Eni2 1  Eni2  Ets2  Rs2 ( I n21  I t22 )  1  s  ( En21  En22  Etp2  I n22 R p2 )
 R2 
• where I t 2  Et 2 / R2
• An op amp specification sheet normally provide En and In which are
defined as En  En21  En22 and I n  I n1  I n 2
• We can now define a new equivalent amplifier noise voltage
2
 Rs 
2
En  1   ( En2  Etp2  I n22 R p2 )  I t22 Rs2
 R2 
• and
2
Eni2  Ets2  Ena
 I n2 Rs2
53
Intermodulation Distortion
54