Noise in Analog CMOS ICs
Gábor C. Temes
School of Electrical Engineering and
Computer Science
Oregon State University
March 2013
temes@ece.orst.edu
1/25
Noise = Unwanted Signal
• Intrinsic (inherent) noise:
– generated by random physical effects in the devices.
• Interference (environmental) noise:
– coupled from outside into the circuit considered.
• Switching noise:
– charge injection, clock feedthrough, digital noise.
• Mismatch effects:
– offset, gain, nonuniformity, ADC/DAC nonlinearity errors.
• Quantization (truncation) “noise”:
– in internal ADCs, DSP operations.
temes@ece.orst.edu
2/25
Purpose and Topics Covered
•
Purpose: Fast Estimation of Noise in Analog Integrated
Circuits Before Computer Simulation. Topics Covered:
• The Characterization of Continuous and Sampled Noise;
• Thermal Noise in Opamps;
• Thermal Noise in Feedback Amplifiers;
• Noise in an SC Branch;
• Noise Calculation in Simple SC Stages;
• Sampled Noise in Opamps.
temes@ece.orst.edu
3/25
Characterization of Continuous-Time Random Noise – (1)
• Noise x(t) – e.g., voltage. Must be stationary as well as mean and
variance ergodic.
• Average power defined as mean square value of x(t):
Pav
1
 lim 
T  T

T

0

2
2
x ( t ) dt   E x ( t )



• For uncorrelated zero-mean noises
E {  x1  x 2  }  E { x 1 }  E { x 2 }
2
•
2
2
Power spectral density (PSD) Sx(f) of x(t): Pav contained in a 1Hz
BW at f. Hence dPav = Sx(f).df . Measured in V2/Hz. Even function of
f.
• Filtered noise: if the filter has a voltage transfer function H(s), the
output noise PSD is |H(j2pf)2|Sx(f).
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4/25
Characterization of Continuous-Time Random Noise – (2)
• Average power in f1 < f < f2:
f2
P f1 , f 2 
S
x
( f ) df
f1
Here, Sx is regarded as a one-sided PSD.
Hence, P0,  Pav. So white noise has infinite power!? NO. Quantum
effects occur when h.f ~ kT. Here, h is the Planck constant. This
occurs at tens of THz; parasitics will limit noise much before this!
• Amplitude distribution: probability density function(PDF) px(x).
px(x1)Dx: probability of x1 < x < x1+Dx occurring. E.g., px(q)=1/LSB for
quantization noise q(t) if |q| < LSB/2, 0 otherwise. Gaussian for
thermal noise, often uniform for quantization noise.
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5/25
Characterization of Sampled-Data Random Noise
• Noise x(n) – e.g., voltage samples.

 1
• Ave. power:
P  E  lim 
x(n)

N 1
N
av

N 

n0
2



Pav is the mean square value of x(n).
• For sampled noise, Pav remains invariant!
• Autocorrelation sequence of x(n) – shows periodicities!
rx ( k )  E { x ( n )  x ( n  k )}
rx ( 0 )   Pav
of
x (n )  E{ x (n) }
2
• PSD of x(n): Sx(f) = F [rx(k)]. Periodic even function of f with a period
fc. Real-valued, non-negative.
f
• Power in f1 < f < f2: P f , f   S x ( f ) df
2
1
fc / 2
Pav 
2
f1
2
 S x ( f ) df  P0 , f x / 2  E { x ( n ) }
Sampled noise has finite power!
• PDF of x(n): px(x), defined as before.
0
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6/25
Thermal Noise – (1)
Due to the random motion of carriers with the MS velocity ∝ T.
Dominates over shot noise for high carrier density but low drift velocity,
occurring, e.g., in a MOSFET channel.
Mean value of velocity, V, I is 0.
The power spectral density of thermal noise is PSD = kT. In a resistive
voltage source the maximum available noise power is hence
____
2
E
4R
 k  T  BW
giving
____
2
E
 4  k  T  BW  R
k: Boltzmann constant, 1.38x10^-23 J/K
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7/25
Thermal Noise – (2)
The probability density function of the noise amplitude follows a
Gaussian distribution
2
p(E ) 
e
____
2
E / E
____
2
E  2p
____
2
Here, E is the MS value of E.
In a MOSFET, if it operates in the triode region, R=r can be used in the
drain-source branch.
In the active region, averaging over the tapered channel, R=(3/2)/gm
results. The equivalent circuit is
__________
2
E (f) 
8 kT
( PSD )
gm
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8/25
Noise Bandwidth
Let a white noise x(t) with a PSD Sx enter an LPF with a transfer function:
H (s) 
G0
s  3  dB  1
Here G0 is the dc gain, and 3-dB is the 3-dB BW of the filter. The MS
value of the output noise will be the integral of |H|2. Sx, which gives
___
2
x 
 3  dB
4
G0  S x
2
Assume now that x(t) is entered into an ideal LPF with the gain function:
|H|=G0
if f < fn
and 0 if f > fn. The MS value of the output will then be:
___
2
x  fn  G0  S x
2
Equating the RHSs reveals that the two filters will have equivalent noise
transfer properties if
fn 
 3  dB
4

p
2
f 3  dB
Here, fn is the noise bandwidth of the LPF.
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9/25
Analysis of 1/f Noise in Switched MOSFET Circuits
• 1/f noise can be represented as threshold voltage variation.
• If the switch is part of an SC branch, it is unimportant. In opamps,it
is critical!
• In a chopper or modulator circuit, it may be very important. See the
TCAS paper shown.
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10/25
Thermal Op-Amp Noise – (1)
Simple op-amp input stage [3]:

With device noise PSD: V  (8 / 3 ) kT / g
g  16 kT
16 kT 
Equivalent input noise PSD: V ( f )  3 g 1  g   3 g V / Hz


for gm1 >> gm3. It can be represented by a noisy resistor RN = (8/3)kT/gm1
at one input terminal. Choose gm1 as large as practical!
All noises thermal - white. Opamp dynamics ignored.
2
ni
mi

2
neq
2
m3
m1
m1
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m1
11/25
Thermal Op-Amp Noise – (2)
All devices in active region, [id(f)]2=(8/3)kT·gm. Consider the short-circuit
output current I0,sh of the opamp. The output voltage is Io,sh·Ro.
If the ith device PSD is considered, its contribution to the PSD of I0,sh is
proportional to gmi. Referring it to the input voltage, it needs to be
divided by the square of the input device gm, i.e, by gm12.
Hence, the input-referred noise PSD is proportional to gm/gm12. For the
noise of the input device, this factor becomes 1/gm1.
Conclusions: Choose input transconductances (gm1) as large as
possible. For all non-input devices (loads, current sources, current
mirrors, cascade devices) choose gm as small as possible!
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12/25
Noisy Op-Amp with Compensation
Op-amp has thermal input noise PSD Pni = 16kT/3gm1, where gm1 is
the transconductance of the input devices.
Single-pole model; voltage gain A(s) = A03-dB/(s+3-dB), where A0 is
the DC gain, 3-dB is the 3-dB BW (pole frequency), and u=A03-dB
is the unity-gain BW of the op-amp. For folded-cascode telescopic
and 2-stage OTAs, usually u=gm1/C, where C is the compensation
capacitor. Open-loop noise BW is fn = gm1/4A0C, and the open-loop
noise gain at DC is A0. Hence, the open-loop output noise power is
Pon 
16 kT
3 g m1
A
2
0
g m1
4 A0 C

4 A0 kT
.
3C
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13/25
Noisy Op-Amp with Unity-Gain Feedback
If the op-amp is in a unity-gain configuration, then (for
A0>>1) the noise bandwidth of the stage becomes
A0fn, and the DC noise gain is 1. Hence, the output
(and input) thermal noise power is
Pn 
16 kT g m 1
3 g m1 4C

4 kT
,
3C
(This result is very similar to the kT/C noise power
formula of the simple R-C circuit!)
Vo
V in
Vn
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14/25
Noisy Op-Amp in an SC Gain Stage
A more general feedback stage:
Y2
Y1
VinVin+
Vo
Vn
Let Gi=Y1/Y2 be constant. Then the noise voltage gain is the singlepole function A ( s )  V ( s ) / V ( s )   /( s    ) where  3  dB   u /(1  G i ) is
the 3-dB frequency of An(j). The DC noise gain is  u /  3  dB  1  G i ,
and the noise BW of the stage is f n  g m 1 /[ 4 C (1  G i )] . Hence, the
output thermal noise power is
n
on
Pno 
n
16 kT
3 g m1
3  dB
u
(1  G i )
2
g m1 / 4C
1  Gi

4 (1  G i ) kT
,
3C
and the input-referred thermal noise power is Pni = (4/3) kTb/C,
where b = 1/(Gi+1 ) is the feedback factor.
Pni is smaller for a higher gain Gi, so a higher SNR is possible for
higher stage gain.
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15/25
Switched-Capacitor Noise – (1)
Two situations; example:
in
C-T
AAF
SCF
ADC
DSP
DAC
Sit. 1
SCF
C-T
RCF
out
Sit. 2
Situation 1: only the sampled values of the output waveform matter; the
output spectrum may be limited by the DSP, and hence VRMS,n is
reduced. Find VRMS from kTC charges; adjust for DSP effects. Noise
can be estimated by hand analysis.
Situation 2: the complete output waveform affects the SNR, including the
S/H and direct noise components. Usually the S/H noise dominates at
low frequencies. High-frequency noise is reduced by the
reconstruction filter. Needs CAD analysis.
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16/25
Sampling Noise
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17/25
Switched-Capacitor Noise
Thermal noise in a switched-capacitor branch: (a) circuit diagram; (b) clock signal; (c)
output noise; (d) direct noise component; (e) sampled-and-held noise component.
The noise power is kT/C in every time segment.
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18/25
Switched-Capacitor Noise Spectra
For f << fc, SS/H >> SD . Sampled PSD = 2kT/fsC ; the unsampled PSD
= 4kTRon.
Their ratio is 1/(2fs.Ron.C) >> 1 !! Sampling penalty.
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19/25
Switched-Capacitor Noise
The MS value of samples in VcnS/H is unchanged:
V
S/H
cn

2
 kT / C
Regarding it as a continuous-time signal, at low frequency its one-sided
PSD is
2 (1  m ) kT
2
S
S/H
(f)
f cC
while that of the direct noise is
S (f)
d
mkT
,
f sw C
S
S/H
S
d

2 (1  m )
m
2
f sw
.
fc
Since we must have fsw/fc > 2/m, usually |SS/H| >> |Sd| at low frequencies.
(See also the waveform and spectra.)
See Gregorian-Temes book, pp. 505-510 for derivation.
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20/25
Calculation of SC Noise –
(1)
In the switch-capacitor branch, when the switch is on, the capacitor
charge noise is lowpass-filtered by Ron and C. The resulting charge
noise power in C is kTC. It is a colored noise, with a noisebandwidth fn = 1/(4·Ron ·C). The low-frequency PSD is 4kTRon.
When the switch operates at a rate fc<<fn, the samples of the charge
noise still have the same power kTC, but the spectrum is now white,
with a PSD = 2kTC/fc. For the situation when only discrete samples
of the signal and noise are used, this is all that we need to know.
For continuous-time analysis, we need to find the powers and spectra
of the direct and S/H components when the switch is active. The
direct noise is obtained by windowing the filtered charge noise
stored in C with a periodic window containing unit pulses of length
m/fc. This operation (to a good approximation) simply scales the
PSD, and hence the noise power, by m. The low-frequency PSD is
thus 4mkTRon. For complex circuits, CAD is required to find noise.
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21/25
Calculation of SC Noise (summary) – (2)
To find the PSD of the S/H noise, let the noise charge in C be sampledand-held at fc, and then windowed by a rectangular periodic window
w(t)=0 for n/fc < t< n/fc+m/fc
w(t)=1 for n/fc+m/fc < t < (n+1)/fc
n= 0, 1, 2, …
Note that this windowing reduces the noise power by (1 - m) squared(!),
since the S/H noise is not random within each period.
Usually, at low frequencies the S/H noise dominates, since it has
approximately the same average power as the direct noise, but its
PSD spectrum is concentrated at low frequencies. As a first estimate,
its PSD can be estimated at 2(1-m)2kT/fc·C for frequencies up to fc/2.
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22/25
Circuit Example: Lossy Integrator with Ideal Op-Amp[4]
RMS noise charge delivered into C3
as f2→0, assuming OTA:
RMS noise charges acquired by Ci
during f1 = 1:
qi  C i
j
kT / C i 
kTC i , i , j  1, 2
with Vin set to 0.
q1 
Form C2:

 C3
q 2   ( kTC 2 ) 

 C2  C3
Total:
fia: advanced cutoff phase
C3
From C1:
q3 
2 kTC 1
C3
C2  C3
C2  C3
2

C 2C 3 
  kT


C2  C3 


2

C 
kT  2 C 1  2 C 2  2 
C3 

1/ 2
1/ 2
Input-referred RMS noise voltage:
V in , n  q 3
V in , n 
C2  C3
C 1C 3
1
C1

2
kT 
C2 
2
C

2
C

2
 1

C1 
C3 
kT ( C 1  C 2 ) for
1/ 2
C 2  C 3 .
Vin,n and Vin are both low-pass filtered
by the stage.
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23/25
Sampled Op-Amp Noise Example [4]
•
f1=1
Direct noise output voltage = Vneq
•
f2=1
Charges delivered by C1 and C2:
-C1(Vneq+Vin)+C2(V0-Vneq).
Charge error –(C1+C2)Vneq.
• Input-referred error voltage
Vneq(1+C2/C1).
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24/25
Reference
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