Here are the key changes to the noise analysis of the common source amplifier example if: 1) Transistor is in triode region: - gm would be a function of VGS and VDS instead of a constant - inMOS would depend on gm as a function of voltages 2) Include flicker noise: - Add a flicker noise current source inMOS,f - PSD of inMOS,f is Kf/f instead of constant - Integrate PSD from fmin to fmax 3) Replace RL with PMOS load: - Replace RL with the output resistance ro of the PMOS load transistor - Add thermal noise current source of the PMOS load transistor

1. CMOS Device Model
• Objective
– Hand calculations for analog design
– Efficiently and accurately simulation
• CMOS transistor models
– Large signal model
– Small signal model
– Simulation model
– Noise model

2. Large Signal Model
• Nonlinear equations for solving dc values of
device currents given voltages
• Level 1: Shichman-Hodges (VT, K', γ, λ, φ, and
NSUB)
• Level 2: with second-order effects (varying
channel charge, short-channel, weak inversion,
varying surface mobility, etc.)
• Level 3: Semi-empirical short-channel model
• Level 4: BSIM models. Based on automatically
generated parameters from a process
characterization. Good weak-strong inversion
transition.

3. Transconductance when VDS is small

4. Transconductance when VDS is small

5. Transconductance when VDS is small

6. Effect of changing VDS for a large VGS

7. Effect of changing VDS for a given VGS

8. Effect of changing VDS for a given VGS

9. Effect of changing VDS for various VGS
VGS<=VT

10. Effect of changing VDS for various VGS

11. Effect of changing VDS for various VGS

12. MOST Regions of Operation
• Cut-off, or non-conducting: VGS <VT
– ID=0
• Conducting: VGS >=VT
– Saturation: VDS > VGS – VT
μCoxW
iD = (vGS - VT )2
2L
– Triode or linear or ohmic or non-saturation: VDS <= VGS
– VT i = μCoxW ((v - V )V - VDS )
2
D GS T DS 2
L

13. With channel length modulation
μCoxW
iD = (vGS - VT ) ( 1 + λVDS )
2
2L
VT = VT 0 + γ ( 2|φ f | + |v BS| - 2|φ f | )
μCoxW W
β = = K'
L L

14. Capacitors Of The Mosfet

15. CBD and CBS include both the diffusion-bulk
junction capacitance as well as the side wall
junction capacitance. They are highly nonlinear
in bias voltages.
C4 is the capacitance between the channel and
the bulk. It is highly nonlinear and depends on
the operation of the device. C4 is not
measurable from terminals.

16. /2

17. Gate related capacitances

20. Small signal
model

21. Typically: VDB, VSB are in such a way that there is
a reversely biased pn junction.
Therefore: gbd ≈ gbs ≈ 0

22. In saturation:
But

23. In non-saturation region

24. High Frequency Figures of Merit ωT
• AC current source input to G
• AC short S, D, B to gnd
• Measure AC drain current output
• Calculate current gain
• Find frequency at which current gain = 1.
• Ignore rs and rd,  Cbs, Cbd, gds, gbs, gbd all have
zero voltage drop and hence zero current
• Vgs = Iin /jw(Cgs+Cgb+Cgd) ≈ Iin /jwCgs
• Io = − (gm − jw Cgd)Vgs ≈ − gmVgs
• |Io/Iin| ≈ gm/wCgs

25. • At ωT, current gain =1
• ωT ≈ gm/(Cgs+Cgd)≈ gm/Cgs
• or
W
gm
µ nCox (VGS − VT ) 3µ (V − V )
ωT ≈ ≈ L = n GS2 T
C gs 2 2L
WLCox
3
gm µ n (VGS − VT )
ωT = ≈
C gs + C gd L2

26. High Frequency Figures of Merit ωmax
• AC current source input to G
• AC short S, D, B to gnd
• Measure AC power into the gate
• Assume complex conjugate load
• Compute max power delivered by the transistor
• Find maximum power gain
• Find frequency at which power gain = 1.

27. • ωmax: frequency at which power gain
becomes 1
P L=

28. BSIM models
• Non-uniform charge density
• Band bending due to non-uniform gate voltage
• Non-uniform threshold voltage
– Non-uniform channel doping, x, y, z
– Short channel effects
• Charge sharing
• Drain-induced barrier lowering (DIBL)
– Narrow channel effects
– Temperature dependence
• Mobility change due to temp, field (x, y)
• Source drain, gate, bulk resistances

29. “Short Channel” Effects
• VTH decreases for small L
– Large offset for diff pairs with small L
• Mobility reduction:
– Velocity saturation
– Vertical field (small tox=6.5nm)
– Reduced gm: increases slower than root-ID

30. Threshold Voltage VTH
• Strong function of L
– Use long channel for VTH matching
– But this increases cap and decreases speed
• Process variations
– Run-to-run
– How to characterize?
– Slow/nominal/fast
– Both worst-case & optimistic

31. Effect of Velocity Saturation
• Velocity ≈ mobility * field
• Field reaches maximum Emax
– (Vgs-Vt)/L reaches ESAT
• gm become saturated:
– gm ≈ ½µnCoxW*ESAT
• But Cgs still 2/3 WL Cox
• ωT ≈ gm/Cgs = ¾ µnESAT /L
• No longer ~ 1/L^2

32. Threshold Reduction
• When channel is short, effect of Vd extends to S
• Cause barrier to drop, i.e. Vth to drop
• Greatly affects sub-threshold current: 26 mV Vth
drop  current * e
• 100~200 mV Vth drop due to Vd not uncommon
 100s or 1000 times current increase
• Use lower density active near gate but higher
density for contacts

33. Other effects
• Temperature variation
• Normal-Field Mobility Degradation
• Substrate current
– Very nonlinear in Vd
• Drain to source leakage current at Vgs=0
– Big concern for static power
• Gate leakage currents
– Hot electron
– Tunneling
– Very nonlineary
• Transit Time Effects

34. Consequences for Design
• SPICE (HSPICE or Spectre)
– BSIM3, BSIM4 models
– Accurate but inappropriate for hand analysis
– Verification (& optimization)
• Design:
– Small signal parameter design space:
• g m, C L (speed, noise)
• gm/ID, ID (power, output range, speed)
• Av0= gmro (gain)
– Device geometries from SPICE (table, graph);
– may require iteration (e.g. CGS)

35. Intrinsic voltage gain of MOSFET
Sweep V1
Measure vgs
Intrinsic voltage gain = gm/go = ∆vds/∆vgs for constant Id

36. Electronic Noise
• Noise phenomena
• Device noise models
• Representation of noise (2-ports):
– Motivation
– Output spectral density
– Input equivalent spectral density
– Noise figure
– Sampling noise (“kT/C noise”)
• SNR versus Bits
• Noise versus Power Dissipation
– Dynamic range
– Minimum detectable signal

37. Noise in Devices and Circuits
•Noise is any unwanted excitation of a circuit, any
input that is not an information-bearing signal.
• External noise: Unintended coupling with other
parts of the physical world; in principle, can be
virtually eliminated by careful design.
• Intrinsic noise: Unpredictable microscopic events
inherent in the device/circuit; can be reduced, but
never eliminated.
•Noise is especially important to consider when
designing low-power systems because the signal
levels (typically voltages or currents) are small.

38. Noise vs random process variations
• random process variations
– Variations from one device to another
– For any device, it is fixed after fabrication
• Noise
– Unpredictable variations during operation
– Unknown after fabrication
– Remains unknown after measurement during
operation
– May change with environment

39. Time domain description of noise

40. What is signal and what
is noise?

41. Signal and noise power:
x(t ) = s (t ) + n(t )
1 T 2
Ps = ∫ s (t ) dt , S (rms) = Srms = Ps
T 0
1 T 2
Pn = ∫ n (t ) dt , N (rms ) = N rms = Pn
T 0

42. Physical interpretation
If we apply a signal (or noise) as a voltage
source across a one Ohm resistor, the power
delivered by the source is equal to the signal
power.
Signal power can be viewer as a measure of
normalized power.
power

43. Signal to noise ratio
Ps S rms
SNR = 10 log10 ( ) = 20 log10 ( )
Pn N rms
SNR = 0 dB when signal power = noise power
Absolute noise level in dB:
w.r.t. 1 mW of signal power
Pn
Pn in dB m = 10 log
1 mW
= 30 dB + 10 log( Pn )

44. SNR in bits
• A sine wave with magnitude 1 has power
= 1/2.
• Quantize it into N=2n equal levels between
-1 and 1 (with step size = 2/2n)
• Quantization error uniformly distributed
between +–1/2n
• Noise (quantization error) power
=1/3 (1/2n)2
• Signal to noise ratio
= 1/2 ÷ 1/3 (1/2n)2 =1.5(1/2n)2
= 1.76 + 6.02n dB or n bits

45. -1<=C<=+1
C=0: n1 and n2 uncorrelated
C=1: perfectly correlated

46. Adding
uncorrelated
noises
Adding
correlated
noises

47. For independent noises

48. Frequency domain description of
noise
Given n(t) stationary, its autocorrelation is:
1 T
Rn (τ ) = lim
T → ∞ 2T
∫−T n(t )n(t + τ ) dt
The power spectral density of n(t) is:
PSDn ( f ) = S n ( f ) = F ( Rn (τ ))
+∞
Pn = ∫ PSDn ( f ) df
−∞
For real signals, PSD is even.  can use single sided
spectrum: 2x positive side
+∞
Pn = ∫ PSDn ( f ) df
0
↑ single sided PSD

49. Parseval’s Theorem:
If x(t ) ⇔ X ( f )
⇓
+∞ 2 +∞ 2
∫− ∞ x(t ) dt = ∫
−∞
X ( f ) df
If x(t) stationary,
R x (τ ) ⇔ PSDx ( f )
⇓
+T 2 +∞
lim
T →∞ ∫
−T
x(t ) dt = Rx (0) = ∫
−∞
PSDx ( f ) df

50. Interpretation of PSD
Pxf1 = PSDx(f1)
PSDx(f)

53. Types of “Noise”
• “man made”
– Interference
– Supply noise
–…
– Use shielding, careful layout, isolation, …
• “intrinsic” noise
– Associated with current conduction
– “fundamental” –thermal noise
– “manufacturing process related”
– flicker noise

54. Thermal Noise
• Due to thermal excitation of charge carriers in a
conductor. It has a white spectral density and is
proportional to absolute temperature, not
dependent on bias current.
• Random fluctuations of v(t) or i(t)
• Independent of current flow
• Characterization:
– Zero mean, Gaussian pdf
– Power spectral density constant or “white” up to about
80THz

55. Thermal noise dominant in resisters
Example:
R = 1kΩ, B = 1MHz, 4µV rms or 4nA rms

56. HW
Equivalently, we can model a real resistor with an
ideal resistor in parallel with a current noise source.
What rms value should the current source have?
Show that when two resistors are connected in
series, we can model them as ideal series resistors
in series with a single noise voltage source. What’s
the rms value of the voltage source?
Show that two parallel resistors can be modeled as
two ideal parallel resistors in parallel with a single
noise current source. What’s the rms value of the
current source?

57. Noise in Diodes
Shot noise dominant
– DC current is not continuous and smooth but
instead is a result of pulses of current caused by
the individual flow of carriers.
It depends on bias, can be modeled as a
white noise source and typically larger than
thermal noise.
− Zero mean
– Gaussian pdf
– Power spectral density flat
– Proportional to current
– Dependent on temperature

58. Example:
ID= 1mA, B = 1MHz, 17nA rms

59. MOS Noise Model

60. Flicker noise
–Kf,NMOS 6 times larger than Kf,PMOS
–Strongly process dependent
−when referred to as drain current noise, it
is inversely proportional to L2

61. BJT Noise

62. Sampling Noise
• Commonly called “kT/C” noise
• Applications: ADC, SC circuits, …
R
von
C
Used:

63. Filtering of noise
x(t) y(t)
H(s)
|H(f )|2 = H(s)|s=j2πf H(s)|s=-j2πf

64. Noise Calculations
1) Get small-signal model
2) Set all inputs = 0 (linear superposition)
3) Pick output vo or io
4) For each noise source vx, or ix
Calculate Hx(s) = vo(s) / vx(s) (or … io, ix)
5) Total noise at output is
6) Input Referred Noise: Fictitious noise source at
input: 2
vin ,eff = von ,T / A( s )
2 2

65. Example: CS Amplifier
VDD Von=(inRL +inMOS)/goT
RL goT = 1/RL + sCL
1
Μ1 CL
2
i
nRL = 4 k BT
RL
2
2
i
nMOS = 4 k BT g m
3

66. ωo=1/RLCL

67. Some integrals

68. HW
In the previous example, if the transistor is
in triode, how would the solution change?
HW
If we include the flicker noise source, how
would that affect the computation? What
do you suggest we should modify?
HW
In the example, if RL is replaced by a PMOS
transistor in saturation, how would the
solution change? Assume appropriate bias
levels.