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Noise in op-amp (or any analogue)
circuit/system
 Interference Noise (external)
 electric (capacitive), magnetic (inductive),
conducted or radiated EMI (ElectroMagnetic
Interference)
 periodic, intermittent or random
 reduce by filtering, decoupling, guarding,
shielding, etc
 Inherent Noise (internal)
 random
 resistors
 semiconductors
 Need to be able to detect the signal in the
presence of noise (whatever the source)
SNR
signal-to-noise ratio
SNR  20 log10
Vsignal  rms
Vnoise  rms
 Becomes very important when dealing with lowlevel signals or with high resolution ADC or
DAC
Due to the random nature of noise, we deal with
noise statistically
 The rms value of noise voltage en(t):
1
En  
T


2
 en (t )dt 

t0
t0 T
1
2
 Similarly, for noise current in(t), its rms value is
In
2
2
 E n and I n are the mean square values and may
be regarded as the average noise power dissipated
by the corresponding noise signal in a 1 resistor
 In some cases, we need to consider peak values
E peak
Define: Crest factor =
En
 Epeak is statistically distributed but often it has a
Gaussian or normal distribution in which case the
crest factor is commonly regarded as 3 times the
rms value or the peak-to-peak noise is 6 times the
rms value
Useful rule of thumb for estimating rms of
noise from ‘scope display
 When combining noise voltage sources in series
or noise current sources in parallel we ADD THE
NOISE POWERS (provided they are
uncorrelated)
Noise spectra
 With any noise source, the (noise) power is
spread over the whole frequency spectrum.
Hence, it should be represented as
en2 ( t,f ) and in2 ( t,f ) . Therefore:
1
En ( f )  
T





2

e
(
t
,
f
)
dt
n



t0
t0  T
1
2
In general, one is only interested in the noise
power over some frequency band
It will depend on the width of the band and the
position of the band within the whole spectrum
The rate of change of noise power with frequency
is known as noise power density
en2 ( f
dEn2 ( f )
)
df
in2 ( f ) 

dI n2 ( f
)
V2Hz1 and A2Hz1
df
The quantities en ( f ) and in ( f ) are called voltage
and current noise spectral densities [V/Hz and
A/Hz ]
Thus:
 2

E n [ f L , f H ]    en ( f ) df 
 fL

fH
 2

I n [ f L , f H ]    in ( f ) df 
 fL

fH
1
1
2
2

rms
White noise
 is noise that is evenly spread over the whole
spectrum
 en (f) = constant, enw for white noise voltage
in (f) = constant, inw for white noise current
En  enw
I n  inw

 fH  fL
 fH  fL
Obviously, noise power is proportional to
bandwidth of concern (or of measurement)
1/f Noise
Power density varies as f 1
Kv2
e (f)
f
2
n
Ki2
i (f)
f
2
n
On log-log paper, the plots decrease @ 0.5
decade/decade
rms noise voltage / current
 f 
E n  Kv  ln H 
 fL 

 f 
I n  Ki  ln H 
 fL 
1
1
2
2
 This noise is independent of the position in the
frequency spectrum
 It depends only on the upper and lower
frequencies in the bandwidth of interest
 Thus, it is the same for 1 Hz to 10 Hz as for 200
Hz to 2 kHz
 Also, the rms noise voltage for 400 Hz to 400
kHz (3 decades) is 3 times the rms noise voltage
for, say, 50 Hz to 500 Hz (1 decade)
Integrated Circuit (IC) Noise
 Mixture of white (predominant at HIGH
Frequencies) and 1/f noise (predominant at LOW
Frequencies)
en
Plotted on log-log paper
Slope =  0.5 dec/dec (asymptote)
enw
fce
 Same type of graph for in
f
corner freq = fci

2  f ce
en2 ( f )  enw
 1

 f



2  f ci
in2 ( f )  inw

1


 f



f
En  enw  f ce ln H  f H  f L 
fL





f
I n  inw  f ci ln H  f H  f L 
fL


Noise
Any
Electonic/Electrical
System
1
1
2
2
How much noise?
We generally work (analytically and practically) in
the frequency domain.
eni (f)
eno (f)
G (jf)
G (f) is any transfer function (e.g. Noise gain of
amplifier, An (jf); or Filter: Z (jf) or Y (jf) ) and One
could show noise current density in either I/P or
O/P
eno (f) = G (jf) eni (f)
Total rms output noise (voltage) for an amplifier,
say, over some band of interest (fL to fH) is:


2 2
Eno [ f L , f H ]    An ( jf ) eni ( f )df 


 fL

fH
1
2
For example, white noise into an op-amp in Non-Inverting
configuration with resistive feedback components:
An ( jf ) 
Ano
 f 
1  j 
 fA
and Ano  1 
R2
R1
Without loss of generality, let Ano = 1
{Note: A passive Low Pass Filter (LPF) has Ao =
1}
Total rms output noise (voltage) over entire
spectrum is:




1

E no  enw 
2 df
0

f
 1   f 



A

1
2
 enw 157
. fA
 Compare with Eno  enw f H for white noise
without attenuation from fL ( = 0 ) to fH
So the 1st order system behaves like an ideal
“filter” with an equivalent bandwidth 1.57fA
Known as:
(white) Noise equivalent bandwidth : NEB
Sources of Noise
Thermal or Johnson Noise (in passive resistors):
This is due to random thermal agitation in resistive
material. This is present even when the resistor is
unconnected
 Form of white noise
Real resistor:
R
Model:
eR
R
R
(ideal, noiseless)
The noise voltage/current source has power
density:
eR2 ( f )  4kTR
V2Hz1
4 kT
or
A2Hz1
R
where k is Boltzmann’s constant and T is absolute
temperature (K). R in 
iR2 ( f ) 
iR
Shot noise
 Due to random nature of charges crossing
potential barriers in diodes or transistors
 Form of white noise
 Modelled as a noise current source with power
density:
in2 ( f )  2qI A2Hz1
where q is the electronic charge and I is the dc
current through the barrier
Flicker or 1/f noise:
Ia
K
f
Noise in Transistors:
 Number of junctions and bulk regions involved
e.g. BJT
C
en
Tr (ideal, noiseless)
B
in
Real transistor
E

1 
e ( f )  4 kT  rb 

2 gm 

2
n

I Ba
IC 

i ( f )  2q  I B  K1

2
f
( jf ) 

2
n
{These are given for illustration only}
Similar models for JFETs and MOSFETs
Noise in op-amps
Model:
Noiseless Op-amp
inn
en
inp
 noise voltage generator of spectral density en
 2 noise current generators of spectral densities inn
and inp [uncorrelated but have the same
form/expression]
Example of calculation:
 The idea is to convert all noise sources into a
single noise voltage source with a spectral
density eni referred to an input.
iR1
iR2
R1
R2
inn
en
inp
iR3
R3
 Must sum the mean squares (i.e. the spectral
power densities)
2
 en  en
 iR3 and inp flow through R3 producing noise
voltage with power density:


2
2
R32 inp
 i R2 3  R32inp
 4 kTR3
Eno
Similarly iR1 , iR2 and inp flow through R1 //R2
producing noise voltage with power density:
2 2
2
 i R2 1  i R2 2    R1 / / R2  inn
 4kT  R1 / / R2 
 R1 / / R2 2 inn
 Manufacturer quotes in = inn = inp
 R1 or R3 or both will include resistances of input
signal sources
 Define R = R1 //R2 and R+ = R3

eni  en2  R2 


2
R2 in2
 4 kT  R  R 

1
2
 en is due to internal devices (I.C. noise)
 in is due to flow of bias currents (I.C. noise)


2  f ce
2  f ci
en2 ( f )  enw
 1 and in2 ( f )  inw
 1


 f

 f

 The term 4kT R  R is resistive thermal noise.
So the output is then:
 R2 
eno  An  jf  eni  1  
R1 

1
1   f 
 fA 
2
eni
Now must calculate rms value:


2 2
E no [ f L , f H ]    An ( jf ) eno ( f )df 
f

 L

fH
1
2
 Not easy !!!
 However, in general fA >> fci > fce
en
enw
Plotted on log-log paper
f
fce
in
inw
fci
f
An
Ano
fA
f
So for fH < fA we get, as before:
 En2






R2 
Eno   1    

R1   4 kT ( R  R )( f  f ) 




H
L 
where
R2
R2
I n2


fH
En  enw  f ce ln
 fH  f L
f


L


fH
I n  inw  f ci ln
 fH  f L
fL


1
1
1
2
2
2
If total noise above a certain frequency is being
observed (i.e. fH  ) then use the NEB (= 1.57 fA)
instead of fH in the expression above
Total rms input noise = Eni = Eno /A
 Input SNR = 20 log10
Vi  rms
Eni
Example from text:
configuration):
741 op-amp (Inverting
R1 = 100 k
enw = 20 nV/Hz
R2 = 200 k
fce = 200 Hz
R3 = 68 k
inw = 0.5 pA/Hz
GBP = fc =1 MHz
fci = 2 kHz
4kT=1.6810-20
Find total output noise above 0.1 Hz (10 s of
measurement):
Solution:
R2
GBP 10 6
Ao  1 
 3 f A 

 333kHz
R1
3
3


1.57 f A

E n  enw  f ce ln
 1.57 f A  f L 
fL


1
2
1.57  333  10 3
 20  10
200 ln
 1.57  333  10 3  0.1
0.1
 14.5 V rms
9


1.57 f A
I n  i nw  f ci ln
 1.57 f A  f L 
fL


1
2
 0.5  10 12 2000 ln 5.23  10 6  5.23  10 5
 372 pA rms
R = R1 //R2 = 100// 200 k = 67 k
R+ = R3 = 68 k
E no
 E n2






R2 
 1    

R1   4 kT ( R  R )( f  f ) 




H
L 

R2
R2

 

 14.5  10



3 2
 3    68  10
 67  103
 

 20
3

168
.

10

135

10

1
2
6 2


I n2
 3  2.1  10
10
 12.6  10
 372  10 
  5.23  10
10
2
 119
.  10
= 155 V rms
Need to decrease the resistances !!!
12
5
10

1
2


2




V
1
2
Peak-to-peak output noise over same range is:
6  155 V = 0.93 mV