RFIC Design and Testing for Wireless Communications
A PragaTI (TI India Technical University) Course
July 18, 21, 22, 2008
Lecture 6: Basic Concepts
– Linearity, noise figure, dynamic range
By
Vishwani D.
D Agrawal
Fa Foster Dai
200 Broun Hall, Auburn University
Auburn, AL 36849-5201, USA
1
RFIC Design and Testing for Wireless Communications
Topics
Monday, July 21, 2008
9:00 – 10:30
Introduction – Semiconductor history, RF characteristics
11:00 – 12:30
Basic Concepts – Linearity, noise figure, dynamic range
2:00 – 3:30
RF front-end design – LNA, mixer
4:00 – 5:30
Frequency synthesizer design I (PLL)
T
Tuesday,
d
July
J l 22,
22 2008
9:00 – 10:30
11:00 – 12:30
2:00 – 3:30
Frequency synthesizer design II
(VCO)
RFIC design for wireless communications
Analog and mixed signal testing
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
2
Units for Microwave and RFIC Design
Peak-to peak voltage: Vpp
Root-mean-square voltage:
Power in Watt :
Pwatt =
V
2
rms
R
Vrms =
=
V pp
2 2
V pp2
8R
Power in dBm :
⎛ Pwatt [mW ] ⎞
PdBm = 10 log 10 ⎜
⎟
1mW
⎝
⎠
On a 50Ohm load, 0dBm=1mW=224mVrms=632mVpp
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 3
Noise Figure
•
•
In RF design, most of the front-end receiver blocks are
characterized in terms of “noise figure” rather than input
referred noise
noise.
Noise factor F is defined as
N o ( total )
N o ( source ) + N o ( added )
N o ( added )
SNRin
Si N i
No
F=
=
=
=
=
= 1+
SNRout S o N o GN i N o ( source )
N o ( source )
N o ( source )
Noise Figure, NF = 10 log 10 F
•
•
Noise figure measures how much the SNR degrades as the
signal passes through a system.
For a noiseless system,
y
, SNRin = SNRout,, namely,
y, F=1,,
NF=0dB, regardless of the gain. This is because both the
input signal and the input noise are amplified (or attenuated)
by the same factor and no additional noise is introduced.
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 4
Thermal Noise
•
Thermall noise
Th
i (Johnson
(J h
noise)
i ) – due
d to
t random
d
thermal
th
l motion
ti off
electrons and is generated by resistors, base and emitter resistance
rb,rE,and rc. of bipolar devices, and channel resistance of MOSFETs.
Thermal noise is a white noise with Gaussian amplitude distribution.
•
Thermal noise floor:
⎛ kT ⎞
0
10 log⎜
⎟ = −174dBm / Hz at 290 K
⎝ 1mW ⎠
Vnb2
_
R
2
n
V
Q
M
+
I n2
+
re
_
V = 4kTRΔf
2
n
rb
2
ne
V
+
_
⎛2 ⎞
I n2 = 4kT ⎜ g m ⎟Δf
⎝3 ⎠
>2/3 for submirocn MOS
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 5
Shot Noise
•
Shot noise (Schottky noise) – due to the particle-like nature of
charge carriers. Only the time-average flow of electrons and holes
appears as constant current. Any fluctuation in the number of
charge carriers produces a random noise current at that instant.
Shot noise is a Gaussian white process associated with the transfer
of charge across an energy barrier (e.g., a p-n junction). This
random process is called shot noise and is expressed in amperes
per root hertz.
Q
I n2 = 2qIΔf
I n2,c
I n2,b
i bn =
2 qI B
i cn =
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
2 qI C
Page 6
Flicker noise
•
Flicker noise (1/f noise) – found in all active devices. In bipolar
transistors, it is caused by traps associated with contamination and
crystal defects in the emitter-base depletion layer. These traps capture
and release carriers in a random fashion with noise energy
concentrated in low frequency. K depends on processing and may vary
by order of magnitude.
a
I
I n2 = K C Δf , a ≈ 0.5~ 2
f
•
In MOSFETs, 1/f noise arises from random trapping of charge at the
oxide-silicon interfaces. Represented as a voltage source in series with
the gate, the noise spectral density is given by
1
K
V =
WLCox f
2
n
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 7
Noise Power Spectral Density
Noisee Power (dBm/Hzz)
-135
-140
10dB/Dec
-145
Total Noise
-150
-155
-160
0.1
Thermal and
Shot Noise
Flicker Noise
1
10
Frequency (kHz)
100
1000
The 1/f corner frequency
can be significantly higher
for MOSFET
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 8
BJT Model with Noise Sources
vb
b
rb
Cμ
b’
4kT b
4kTr
ibn
ibf
rπ
Cπ
c
gmvb’e
icn
ro
2qIC
2qIB
e
1/f noise
Collector
Base
Shot Noise
Shot Noise
c
a)
b
vrb
4kTrb
rb
b’
ibn
2qIB
base
shot noise
c
b)
icn
b rb
2qI
qC
b’
icn
ibn
e
collector
shot noise
e
4kT
rb
2qIB
2qIC
e
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 9
CMOS Model with Noise Sources
CGD
vgs
CGS
Input-referred noise
vo
gmvgs
ro
Ind
2
Vn =
8 kT
3 gm
2
I nd
•
2
⎛2 ⎞
= 4kT ⎜ g m ⎟
⎝3 ⎠
In =
8 kT
3 Zin 2 g m
Gate resistance can be added to the noise model with gate resistivity ρ
1 W
RGATE = ρ
3 L
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 10
Linear vs. Nonlinear Systems
• A system is linear if for any inputs x (t) and
x (t), x1(t) Æ y2(t), x2(t) Æ y2(t) and for all
values of constants a and b, it satisfies
1
2
a x1(t)+bx2(t) Æ ay1(t)+by2(t)
• A system
t
is
i nonlinear
li
if it does
d
nott satisfy
ti f
the superposition law.
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 11
Effects of Nonlinearity
•
•
•
•
Harmonic Distortion
Gain Compression
Desensitization
Intermodulation
• For simplicity, we limit our analysis to
memoryless time invariant system.
memoryless,
system Thus
Thus,
y (t ) = α1 x(t ) + α 2 x 2 (t ) + α 3 x 3 (t ) + ...
(3.1)
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 12
Effects of Nonlinearity -- Harmonics
If a single tone signal is applied to a nonlinear system, the
output generally exhibits fundamental and harmonic
frequencies with respect to the input frequency. In Eq. (3.1), if
x(t) = Acosωt, then
y (t ) = α 1 A cos ωt + α 2 A 2 cos 2 ωt + α 3 A3 cos 3 ωt
= α 1 A cos ωt +
α 2 A2
(1 + cos 2ωt ) +
α 3 A3
(3 cos ωt + cos 3ωt )
2
4
3α 3 A3
α 3 A3
α 2 A2
α 2 A2
=
+ (α 1 A +
) cos ωt +
cos 2ωt +
cos 3ωt
2
4
2
4
Observations:
1. even order harmonics result from αj with even j and vanish if the system
has odd symmetry, i.e., differential circuits.
2. For large A, the nth harmonic grows approximately in proportion to An.
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 13
Effects of Nonlinearity -- Gain Compression
y (t ) =
α 2 A2
2
3α 3 A 3
α 3 A3
α 2 A2
+ (α 1 A +
) cos ωt +
cos 2ωt +
cos 3ωt (3.2)
4
2
4
• Under small-signal
g
assumption, the system
y
is
normally linear and harmonics are negligible. Thus,
α1A dominates Æ small-signal gain = α1.
• For large signal
signal, nonlinearity becomes evident
evident.
large-signal gain = α 1 + 3α 3 A3 / 4 . The gain varies
when input level changes.
• If α3 < 0, the output is a “compressive” or
“saturating” function of the input Æ the gain is
compressed when A increases
increases.
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 14
Output of Bipolar Differential Pair
1
2
7 7
tanh ( x ) = x − x 3 + x 5 −
x +K
3
15
315
⎛ −V
Vod
= α F I EE RC tanh⎜⎜ id
Vid
⎝ 2VT
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
⎞
⎟⎟
⎠
Page 15
Outpu
ut Voltage (dBV)
Effects of Nonlinearity – 1dB Compression Point
20 log
α 1 A1−dB
α 1 A1− dB
1dB
3
3
+ α 3 A1− dB
4
A1−dB
0.145
dB =
A1− dB
= 1dB
α1
α1
= 0.3808
α3
α3
2
20logAin
dB = 10 log
dBm
l
V pp / 8
50Ω × 1mW
• 1-dB compression point is defined as the input signal level that causes
small-signal
small
signal gain to drop 1 dB
dB. It’s
It s a measure of the maximum input
range.
•1-dB compression point occurs around -20 to -25 dBm (63.2 to
35 6 V iin a 50
35.6mVpp
50-Ω
Ω system)
t ) in
i typical
t i l frond-end
f
d d RF amplifiers.
lifi
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 16
Effects of Nonlinearity – Desensitization (Blocking)
• Desensitization -- small signal experiences a vanishingly small gain when coexists with a large signal, even if the small signal itself does not drive the
system into nonlinear range.
A l i two-tone
Applying
t
t
iinputs
t x(t) = A1cosω1t+ A2cosω2t to
t Eq.(3.1),
E (3 1) we h
have
3
3
⎛
⎞
3
y (t ) = ⎜ α 1 A1 + α 3 A1 + α 3 A1 A22 ⎟ cos ω1t + L ,
4
2
⎝
⎠
For A1<< A2, it reduces to
gain of
desired
signal
3
⎛
⎞
y (t ) = ⎜ α 1 + α 3 A22 ⎟ A1 cos ω1t + L ,
2
⎝
⎠
Observations:
• Weak signal
signal’s
s gain decreases as a function of A2 if α3 < 0. For sufficiently large
A2, the gain drops to zero Æ the weak signal is “blocked” by the strong signal.
(Why cannot we see stars during day?)
• Many RF receivers must be able to withstand blocking signals 60 to 70 dB
greater
t th
than the
th wanted
t d signals.
i
l
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 17
Effects of Nonlinearity – Intermodulation
• Harmonic distortion is due to self-mixing of a singlet
tone
signal.
i
l It can be
b suppressed
db
by llow-pass filtering
filt i
the higher order harmonics.
• However,
However there is another type of nonlinearity -intermodulation (IM) distortion, which is normally
determined by a “two tone test”.
• When two signals with different frequencies applied to
a nonlinear system, the output in general exhibits some
components that are not harmonics of the input
frequencies. This phenomenon arises from crossmixing (multiplication) of the two signals
signals.
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 18
Effects of Nonlinearity – Intermodulation
x(t ) = A1 cos ω1t + A2 cos ω 2
1
2
2
y (t ) = α 2 ( A1 + A2 ) +
2
3
⎡
2
2 ⎤
+
+
(
2
α
A
α
A
A
A
2 ) ⎥ cos ω1t +
⎢ 1 1 4 3 1 1
⎣
⎦
3
⎡
2
2 ⎤
α
A
+
α
A
(
2
A
+
A
1
2 ) ⎥ cos ω 2 t +
⎢ 1 2 4 3 2
⎣
⎦
1
α 2 A12 cos 2ω1t + A2 2 cos 2ω 2 t +
2
α 2 A1 A2 [cos(ω1 + ω 2 )t + cos(ω1 − ω 2 )t ] +
[
]
[
]
1
α 3 A13 cos 3ω1t + A2 3 cos 3ω 2 t +
4
2
3 ⎧⎪ A1 A2 [cos(2ω1 + ω 2 )t + cos(2ω1 − ω 2 )t ] + ⎫⎪
α3 ⎨
⎬
4 ⎪⎩ A1 A2 2 [cos(2ω 2 + ω1 )t + cos(2ω 2 − ω1 )t ] ⎪⎭
DC Term
1st Order
Terms
2nd Order
Terms
3rd Order
terms
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 19
Intermodulation – Why do we care about IM3 mostly?
In wireless communication system such as cellular handsets with narrow-band
operating frequencies (i.e., a few tens of MHz), only the IM3 spurious signals
(2w1 - w2) and (2w2 - w1) fall within the filter passband.
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 20
Intermodulation -- Third Order Intercept Point (IP3)
•
Two-tone test: A1=A2=A and A is sufficiently small so that higherorder nonlinear terms are negligible and the gain is relatively
constant and equal to α1.
•
As A increases, the fundamentals increases in proportion to A,
whereas IM3 products increases in proportion to A³.
x(t ) = A cos ω1t + A cos ω 2
y (t ) = α 2 A 2 +
9
4
α 1 >> α 3 A 2 → IIP3 =
4 α1
and OIP3 = α 1 IIP3
3 α3
A1dB
0.145
=
≈ −9.6dB
AIP 3
4/3
9
9
⎡
⎤
⎡
⎤
A⎢α 1 + α 3 A 2 ⎥ cos ω1t + A⎢α 1 + α 3 A 2 ⎥ cos ω 2 t +
4
4
⎣
⎣
⎦
⎦
1
α 2 A 2 [cos 2ω1t + cos 2ω 2 t ] + α 2 A 2 [cos(ω1 + ω 2 )t + cos(ω1 − ω 2 )t ] +
2
[cos(2ω1 + ω 2 )t + cos(2ω1 − ω 2 )t ] + ⎫
1
3
3
3⎧
α 3 A [cos 3ω1t + cos 3ω 2 t ] + α 3 A ⎨
⎬
[
]
cos(
2
ω
+
ω
)
t
+
cos(
2
ω
−
ω
)
t
4
4
1
2
1
2
⎩
⎭
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 21
Intermodulation – IP2 vs. IP3
IP2
40
‰ A1=A2=A
α1
ND
A
=
‰ 2 Order IM Product: IP 2
α1 A
Fundamental
20
20
-20
α 2 A2
Freq
IP3
-40
-60
α1 A
α 2 A2
0
α2
ω1 − ω 2 ω1
1
1
1
3rd
Order IM
Product
3
-80
1
-100
2nd Order
IM Product
2
-120
IIP3
-60
-40
0
-20
0
20
40
ω2
ω1 + ω 2
‰ 3RD Order IM Product
4 α1
AIP 3 =
α1 A
α1 A
3 α3
ΔP
3
α 3 A3
4
2ω1 − ω 2
3
α3 A3
4
ω1
ω2
2ω2 − ω1
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Freq
Page 22
Calculate IIP3 without Extrapolation
α1 A
α1 A
1
A
ΔP
3
α 3 A3
4
3
α3 A3
4
P
3
2ω1 − ω 2
ω1
ω2
2ω2 − ω1
Freq
A3
20logAin
i
P/2
ΔP[dB ]
IIP3 [dBm] =
+ Pin [dBm]
2
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 23
Relationship Between 1-dB Compression and IP3
1-dB compression point with single tone applied:
AIP 3
4 α1
=
3 α3
A1−dB
α
= 0.145 1
α3
AIP 3
= 3.04 = 9.66dB
A1dB
1-dB compression point with two tones applied:
AIP 3
A1dB
α1
3α 3
=
= 5.25 = 14.4dB
α
0.22 1
α3
2
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 24
Determine IIP3 and 1-dB Compression Point from Measurement
•
•
An amplifier operates at 2 GHz with a gain of 10dB. Two-tone test with equal power
applied at the input, one is at 2.01 GHz. At the output, four tones are observed at 1.99,
2.0, 2.01, and 2.02GHz. The power levels of the tones are -70,-20,-20, and -70dBm.
Determine the IIP3 and 1-dB compression point for this amplifier.
Solution: 1.99 and 2.02 GHz are the IP3 tones.
1
[P1 − P3 ] = −20 − 10 + 1 [− 20 + 70] = −5dBm
2
2
= −5 − 9.66 = −14.66dBm
dB
IIP 3 = (P1 − G ) +
P1dB
20 log(
OIP3
1
A)
P
20 log
20logAin
IIP3 [ dBm] =
ΔP[ dB ]
+ Pin [dBm]
2
3
4
3
A3
IIP3
P/2
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 25
Intermodulation of Cascade Nonlinear Stages
y1 (t ) = α 1 x(t ) + α 2 x 2 (t ) + α 3 x 3 (t ) + ...
y 2 (t ) = β1 y1 (t ) + β 2 y1 (t ) + β 3 y1 (t ) + ...
2
3
L
y 2 ((tt )
y1 ((tt )
x(t )
AIP 3,1
AIP 3,3
AIP 3, 2
Linearity is more important for back-end stages.
α 12
α 12 β12
1
1
≈ 2 + 2 + 2 +K
2
AIP 3 AIP 3,1 AIP 3, 2 AIP 3,3
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 26
Noise Figure of Cascade Stages
NF
(
)
=
1
+
−
1
+
NF
NF
G
tot
2
−1
1
p1
−1
−1
NF
NF
+
+ ..... +
G G
G G G
3
p1
m
p1
p1
p ( m −1)
p 2.....
NFtot – total equivalent Noise Figure
NFm
– Noise Figure of mth stage
Gpm – Available
A il bl power gain
i off mth stage
t
Noise figure is more important for front-end stages.
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 27
Sensitivity
•
Sensitivity -- defined as the minimum signal level that the
system can detect with acceptable SNR.
Psig / PRS
SNRin
NF =
=
SNROUT SNROUT
•
The overall signal power is distributed across the channel
bandwidth, B, integrating over the bandwidth to obtain total
mean square power
Psig ,tot = PRS • NF • SNROUT • B
Pin ,min
dBm
= PRS
dBm / Hz
+ NF dB + SNRmin
dB
+ 10 log B
where
here PRS is the so
source
rce resistance noise po
power.
er
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 28
Maximum Input Power
20 log(
OIP3
1
A)
PIIP 3 = Pin +
P
3
20 log
4
20logAin
IIP3
P/2
3
A
3
= Pin +
Pin =
Pout − PIM ,out
2
Pin − PIM ,in
2
=
3Pin − PIM ,in
2
2 PIIP 3 + PIM ,in
3
where PIM,out denotes output-referred power of IM3 products, Pout=Pin+G,
PIM,OUT= PIM,in+G. The input level for which the IM products become equal to
the noise floor F is thus given by
2 PIIP 3 + F 2 PIIP 3 − 174dBm + NF + 10 log B
Pin =
=
3
3
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 29
Dynamic Range
•
•
•
•
Dynamic Range (DR) -- defined as the ratio of the maximum to
minimum input levels that the circuit provides a reasonable
signal quality.
Spurious Free Dynamic Range (SFDR) -- determine the upper
Spurious-Free
end of dynamic range on the intermodulation behavior and the
lower end on sensitivity.
e uppe
upper end
e d of
o the
t e dynamic
dy a c range
a ge is
s de
defined
ed as tthe
e
The
maximum input power in a two tone test for which the 3rd IM
products do not exceed the noise floor F=-174dBm+NF+10logB.
The SFDR is thus given by
2( PIIP 3 − F )
⎛ 2P + F ⎞
− SNRmin
SFDR = Pin , max − Pin , mim = ⎜ IIP 3
⎟ − ( F + SNRmin ) =
3
3
⎝
⎠
• Example: NF=9dB, PIIP3=-15dBm, B=100kHz, SNRmin=12dB Æ
SFDR=(-15-(-174+9+50))/1.5-12=54.7dB.
Basic Concepts – Linearity, noise figure, dynamic range, FDAI, 2008
Page 30