Types of Noise
Thermal Noise (Johnson Noise)
Caused by thermal random motion of the current carriers
Spectral Density:
S v ( f ) = 4kTR
Noise Voltage:
Vn 2 = ∫ S v ( f ) df = 4kTRB
B
where Vn2 is the mean-squared thermal noise, T is the absolute temperature
in K, B is the bandwidth, and R is the real part of the impedance.
Example
R = 1 kΩ
B = 100 KHz
T = 300 K
⇒
Vn2 = 1.3 µ V
Flicker Noise ( 1 f noise)
Vn
Sv ( f ) = K f m
f
In semiconductors this noise is mostly due to random trapping and
detrapping of charges at the Si-SiO2 interface and associated changes in
carrier mobility due to Coulombic scattering.
Proposed Hooge’s formula
Sv ( f ) =
αH V 2
N
f
Shot Noise
Shot noise is proportional to the current through the device and is caused by
the random passage of electrons and holes through a potential barrier.
in2, shot = 2qBI
where in2, shot is the root-mean-squared shot noise current, B is the
bandwidth, and I is the current through the device.
Example: Photodiode
I dc = 1 Milli Ampere
B = 100 KHz
T = 300 K
⇒
in2 = 5.6nA
vn2 = 5.6 µV
Noise Bandwidth
+∞
Vn
=
2
in
∫ S ( f ) df
v
0
H
For a system with transfer function
+∞
Vn
+∞
∫
2
out
=
∫
S v ( f ) H ( j 2π f
)
2
0
S v ( f ) H ( j 2π f
)
2
BN
df 0
∫
S v ( f ) H ( 0 ) df
)
df
2
0
If white noise ( Sv ( f ) is constant),
BN =
df
+∞
1
∫
H (0)
2
0
H ( j 2π f
2
Flicker Noise in MOSFET
Two related mechanisms:
• Random trapping/detrapping of carrier at Si-SiO2 interface
• Change in bulk carrier mobility due to trapped charges and hence
additional Coulombic scattering
Assuming uniform trap density at the interface, exponential electron wave
decay in the oxide, n-MOSFET, and ignoring trap states far from EFn
kTq 2
2
1
+
αµ
N
N t ( EFn )
SVg ( f ) =
(
)
2
f γ WLCox
4π
2m*Φ B
h
1
1
1
1
=
+
=
+ α N t ( EFn )
γ=
µ
µn
µox
µn
W = Width of the channel
L = Length of the channel
µn = Electron mobility without oxide charge scattering
µox = Electron mobility limited by oxide charge scattering
SVg ( f ) = Noise Power Spectral Density
γ = Attenuation function of electron wave in oxide ≈ 108 cm -1
m* = Effective mass of carrier in oxide
Φ B = Tunneling barrier height at the interface
h = Plank's constant
α = Scattering coefficient ≈ 1*10-15 Vs
N = Total number of channel carriers per unit area
N t = Number of interface traps
EFn = Electron Quasi-fermi level
For complete derivation see Hung, Ko, Hu, and Cheng, “A Unified Model for the Flicker Noise in MetalOxide-Semiconductor Field Effect Transistors”, IEEE Trans. Elec. Dev., March 1990
Noise Measures
Signal to Noise Ratio
S signal power
=
N noise power
Signal to noise ratio is often expressed in decibels (dB):
⎛S⎞
⎛S⎞
⎜ ⎟ = 10 log10 ⎜ ⎟
⎝ N ⎠dB
⎝N⎠
Noise in Systems
System i
Input xi
Gain Gi (in dB)
Noise Figure NFi (in dB)
S )
(
N
F =
(S N )
input
i
output
(S N )
=
(S N )
xi
xi +1
xi +1 = Gi xi
where x is the signal power in dBm.
Noise Factor (Fi)
NFi = 10 log10 ( Fi )
Effective Noise Temperature (Ti)
Ti = 290 ( Fi − 1)
Output xi+1
Noise Factor of Cascaded Amplifiers
Stage 1
Stage 2
Gain G1
Noise Figure NF1
Gain G2
Noise Figure NF2
x
…
Stage N
Gain GN
Noise Figure NFN
Friis’ Formula
N
Ftotal = F1 + ∑
i=2
Fi − 1
i −1
∏G
j =1
j
Example (N=3)
Ftotal = F1 +
F2 − 1 F3 − 1
+
G1
G1G2
Note: G must be its linear value, ie. not expressed in dB! F being the
noise factor is also linear.
y