ee265b: communication circuits ii · 2016-03-27 · ee265b: communication circuits ii •device...
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1
EE265B: Communication Circuits II
• Device Modeling (3 lectures)
– BJT (RF Device Model, RF Noise Model)
– MOS (RF Device Model, RF Noise Model)
– Passive Elements: Res., Cap. & Inductors
• LNA Design (5 Lectures)
→ Stability
• RF Circuit Biasing (1 lectures)
– Voltage and Current References
• Mixer Design (3 lectures)
– BJT mixers
– MOS mixers
• VCO Design (3 lectures)
• Systems and Architectures (3 lectures)
2
Mixer References• “Noise in RF-CMOS Mixers: A Simple Physical Model” by Darabi and Abidi, IEEE
Transactions on Solid-State Circuits, Vol. 35, No. 1, 2000.
• “Noise in Current-Commuting CMOS Mixers” by Terrovitis and Meyer, IEEE Journal of Solid-State Circuits, Vol. 34, No. 6, 1999.
• “Behavioral Models for Noise in Bipolar and MOSFET Mixers” by Hu and Mayaram, IEEE Transactions on Circuits and Systems II, Vol. 46, No. 10, 1999.
• “A Class AB Monolithic Mixer for 900MHz Applications” by Fong and et al, IEEE Journal of Solid-State Circuits, Vol. 32, No. 8, 1997.
• “Monolithic RF Active Mixer Design” by Fong and Meyer, IEEE Transactions on Circuits and Systems II, Vol. 46, No. 3, 1999.
• “A 12mW Wide Dynamic Range CMOS FrontEnd for a Portable GPS Receiver” by Shahani et al, IEEE Journal of Solid-State Circuits, Vol. , No. 12, 1997.
• “A Parallel Structure for CMOS Four-Quadrant Analog Multipliers and Its Application to a 2GHz Down-conversion Mixer” by Hsiao and Wu.
• “A Low Voltage Bulk Driven Down-conversion Mixer Core” by Kathiresan and Toumazou, 1999.
• “Low Voltage Mixer Biasing Using Monolithic Integrated Transformer De-coupling” by MacEachern and et.al, 1999.
3
References• “A Charge-Injection Method for Gilbert Cell Biasing” by MacEachern and Manku,
1998.
• “Doubly Balanced Dual-Gate CMOS Mixer” by Sullivan and et al, IEEE Journal of Solid-State Circuits, Vol. 34, No. 6, 1999.
• “A 1.5GHz Highly Linear CMOS Down conversion Mixer” by Crols and Steyaert, IEEE Journal of Solid-State Circuits, Vol. 30, No. 7, 1995.
• “Micro-power CMOS RF Components for distributed wireless sensors” by Lin and et al, IEEE Radio Frequency Integrated Circuits Symposium, 1998.
• “A Zero DC-Power Low Distortion Mixer for Wireless Applications” by Kucera and Lott, IEEE Microwave and Guided Wave Letters, Vol. 9, No. 4, 1999.
• “A 900MHz/1.8GHz CMOS Receiver for Dual-Band Applications” by Wu and Razavi, IEEE Journal of Solid-State Circuits, Vol. 33, No. 12, 1998.
• “The MICROMIXER: A Highly Linear Variant of the Gilbert Mixer Using a BiSymmetric Class-AB Input Stage” by Gilbert, IEEE Journal of Solid-State Circuits, Vol. 32, No. 9, 1997.
• “A 2V, 1.9GHz Si Down-Conversion Mixer with an LC Phase Shifter” by Komurasaki and et al, IEEE Journal of Solid-State Circuits, Vol. 33, No. 5, 1998.
4
References• “A Low Distortion Bipolar Mixer for Low Voltage Direct Up-Conversion and High IF
Systems” by Behbahani and et al, IEEE Journal of Solid-State Circuits, Vol. 32, No. 9, 1997.
• “A 2GHz Balanced Harmonic Mixer for Direct-Conversion Receivers” by Yamaji and Tanimoto, Custom Integrated Circuits Conference, 1997.
Mixer Purpose
• Why do we need mixers?
• Key metrics:– Noise Figure
– Linearity
• What will we explore? – Practical designs for mixers and limits on noise
figure and linearity.
FOM MIX =IIP3×GC × ISLO-RF × f
FSSB -1( ) ×PDC ×PLO
Review: Homodyne versus Heterodyne
• Heterodyne
• Homodyne
©James Buckwalter
rf lo if
0rf lo
General Spurious Tones
• Many different frequencies can fall into the same intermediate frequency
©James Buckwalter
3
1cos cos
2rf lo rf lo rf lov V V t t
if rf lo
3
m n
rf lov v v
if rf lom n
Heterodyne: High-side injection
• LO is above the RF tone
• Image frequency (IM) is above the LO tone
©James Buckwalter
, , 1rf N lo N
, , 1im N lo N
Heterodyne: Low-side injection
• LO is below the RF tone
• Image frequency (IM) is below the LO tone
©James Buckwalter
, , 1rf N lo N
, , 1im N lo N
Mixer Noise
• How do we cascade the noise contribution of a mixer?
• For homodyne,
• For homodyne,
©James Buckwalter
FDSB
= 1+N
added
kTDfGLNA
FSSB
= 1+G
IM - IF
GRF-IF
æ
èç
ö
ø÷ 1+
Nadded
GIM -IF
+ GRF- IF( )kTDf
æ
èç
ö
ø÷
=G
RF- IF+ G
IM -IF
GRF- IF
+N
added
GLNA
kTDf
Notes: This is the single sideband noise figure (SSB NF) and is at least 3dB.Please remember that SSB noise figure is generally 3 dB higher than double sideband (DSB) noise figure.
LO Waveform in Mixer
• Noise factor degraded by images around harmonics.
©James Buckwalter
1
1cosLO pk LO
nodd
V t V n tn
LO Waveform (cont)
• Now our minimum noise figure is 3.8 dB (or 0.8 dB for DSB)
• Nonetheless, the following slides will explain WHY we use square waves for mixing!
©James Buckwalter
21
21
12
12 2.4
o i MIX LO MIX LO MIX
nodd
o i MIX LO MIX i MIX LO MIX LO MIX
nodd
N N G n G Nn
N N G N N G G Nn
FSSB
= 2.4 +N
MIX
Ni
= 2.4 + FMIX
-1( )
FDSB
= 1.2 +N
MIX
Ni
= 1.2 + FMIX
-1( )
13
Mixer Types• Mixer Types:
– Multiplication through non-linearity
– Multiplication through switching
• Active mixers
• Passive mixers
14
Mixers Based On Non-linearity
SRF
SLO
Non-linear System S a S a S a S
b S b S b S
c S S c S S c S S
MIX RF RF RF
LO LO LO
RF LO RF LO RF LO
1 2
2
3
3
1 2
2
3
3
1 2
2
3
2
...
...
...
15
Mixers Based On Non-linearity
VRF
Rb
VBB1
Cl earg
VLO
2
0.ds SQ GSQ TI K V V
2
0
2 2
0 0
.
. 2 .
ds SQ bias RF LO T
SQ bias T RF LO bias T RF LO
I K V V V V
K V V V V V V V V
ids wLO ±wRF( ) = KSQVRFVLO cos wLO -wRF( )t + cos wLO +wRF( )t{ }
16
Practical Square Law Mixers
VRF
Rb
VBB1
Cl earg 2
0.ds SQ GSQ TI K V V
VLO
Cl earg
IBIAS
ids wLO -wRF( ) = KSQVRFVLO cos wLO -wRF( )t
Transconductance conversion gain = Gc =ids w IF = w LO -w RF( )
VRF w RF( )
= KSQVLO =mCoxW
2LVLO
17
Practical Bipolar Mixer
VRF
Rb
VBB1
Cl earg I I eC CO
V
V
BE
T .
VLO
Cl earg
IBIAS
Transconductance conversion gain = Gc =iC w IF = w LO -wRF( )
VRF w RF( )=
ICQ
VT
2VLO
IC = ICQ .e
VRF -VLO
VT = ICQ . 1+VRF -VLO
VT
æ
èçö
ø÷+
VRF -VLO
VT
æ
èçö
ø÷
2
+ ...ìíï
îï
üýï
þï
18
MOSFET Mixer (with impedance matching)• MOSFET mixer (with impedance matching):
VRF
Rb
VBB1
Cl earg
2
0.ds SQ GSQ TI K V V
VLO
Cl earg
Le
LgRS
RLO
VBB2
VDD
Cmatch
RL
IF Filter
Matching
Network
19
Features of Square Law Mixers• Noise Figure: The square law MOSFET mixer can be designed to have very low noise
figure.
• Linearity: By operating the square law MOSFET mixer in the square law region the linearity of the mixer can be improved considerably. Note that the corresponding BJT mixer produces a host of non-linear components due to the exponential nature of the BJT mixer.
• Power Dissipation: The square law mixer can be designed with very low power dissipation.
• Power Gain: Reasonable power gain can be achieved through the use of square law mixers.
• Isolation: Square law mixers offer poor isolation from LO to RF port. This is by far the biggest short coming of the square law mixers.
20
Mixers: Switching Operation
SRF wRF( )
1
1
Sout
LO Input
Sout = SRF cos wRFt( )Ä .........................................}{1
1
Sout
= SRF
cos wRF
t( )Ä4
pcos w
LOt( ) -
4
3pcos 3w
LOt( ) +
4
5pcos 5w
LOt( ) - ...
ìíî
üýþ
SW t( )
21
Mixers: Multiplication Through Switching
Sout = SRF cos wRFt( )Ä .........................................}{1
1
Sout
= SRF
cos wRF
t( )Ä4
pcos w
LOt( ) -
4
3pcos 3w
LOt( ) +
4
5pcos 5w
LOt( ) - ...
ìíî
üýþ
wLO -wRF
LO RF 3 LOwLO +wRF
3wLO -wRF
3wLO +wRF
S
outw
IF( ) =2
pS
RF®G
MIX=
2
p
22
One-Diode Mixer
• Attractive for very high frequency applications where transistors are slow.
• Poor gain
• Poor LO-RF isolation
• Poor LO-IF isolation
LR
VRF
VLO
CL
DI
DV
IFV
LOV
t
IFV
t
23
Two-Diode Mixer
• Attractive for very high frequency applications where transistors are slow.
• Poor gain
• Good LO-IF isolation
• Good LO-RF isolation
• Poor RF-IF isolation
LR
VLO
CL
VRF
LOV
t
IFV
t
IFV
24
Four-Diode Mixer
• Attractive for very high frequency applications where transistors are slow.
• Poor gain
• Good LO-IF isolation
• Good LO-RF isolation
• Good RF-IF isolation
VLOVRF
LOV
t
IFV
t
IFV
25
Simple Switching Mixer (Single Balanced Mixer)
• The transistor M1 converts the RF voltage signal to the current signal.
• Transistors M2 and M3 commute the current between the two branches.
VLO
RL RL
VLO
VRF
Vout
I IDC RF
M1
M2 M3
26
Simple Switching Mixer (Single Balanced Mixer)
IM1
VLO
t
t
VOUT
t
27
Simple Switching Mixer (Single Balanced Mixer)
IF Filter
VOUT t
VOUT
t
28
Simple Switching Mixer (Single Balanced Mixer)
LO
RF IF
wLO -wRF
LO RF wLO +wRF
wLO -wRF
IF Filter
29
Single Balanced Mixer Analysis (Incl. Harmonics)
wLO -wRF
LO RF 2 LO
SLO wLO( )
SRF wRF( ) SMIX
3 LO
30
Single Balanced Mixer Analysis (Incl. Impd. Match)• Single Balanced Mixer (Including impedance matching for RF port)
• In this architecture, without impedance matching for the LO port is very commonly used in many designs.
VLO
RL RL
VLO
Vout
M2 M3
RS
VS Rb
GGVLs
Lg
Cl earg G VM RF
31
Single Balanced Mixer Analysis (Incl. Impd. Match)• Single Balanced Mixer (Including impedance matching for LO port)
• In this architecture, with impedance matching for the LO port maximizes LO power utilization without wasting it.
VLO
RL RL
Vout
M2 M3
RS
VS Rb
1GGVLs
Lg
Cl earg G VM RF
Lm2 Lm3
2GGV
Lg
2GGV
Lg
LOV
32
Mixer Input Match
VLO
RL RL
VLO
Vout
M2 M3
RS
VS Rb
VBB1
Ls
Lg
Cl earg
S g T SR R L 1
g s
gs
L LC
33
Mixer Gain
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
sig M RFI G V
0 : . .2
: . .2
LOout cc DC sig L cc DC sig L
LOLO out cc cc DC sig L DC sig L
TV V I I R V I I R
TT V V V I I R I I R
out DC sig LV I I R SW t
1
2
TM
S
GR
34
Mixer Gain
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
sig M RFI G V
Vout
= GM
RLV
RF´ SW t( )
Vout
=2
pG
MR
LV
RF
1
2
TM
S
GR
35
Single Balanced Mixer Analysis: Linearity
• Linearity Consideration:
• Linearity of the Mixer primarily depends on the linearity of the transducer (I_tail=Gm*V_rf). Inductor Ls helps improve linearity of the transducer.
• The transducer transistor M1 can be biased in the linear law region to improve the linearity of the Mixer. Unfortunately this results in increasing the noise figure of the mixer (as discussed in LNA design).
VLO
RL RL
VLO
Vout
M2 M3
RS
VS Rb
GGV Ls
Lg
Cl eargG VM RF
36
Single Balanced Mixer Analysis: Linearity• Linearity Consideration cont...:
• Using the common gate or common base stage as the transducer improves the linearity of the mixer. Unfortunately the approach reduces the gain and increases the noise figure of the mixer.
VLO
RL RL
VLO
Vout
M2 M3
RS
VSIbias Cc
G VM RF
GGV
37
Single Balanced Mixer Analysis: Isolation• Isolation Consideration (LO-RF Feed through)
• The strong LO easily feeds through and ends up at the RF port in the above architecture if the LO does not have a 50% duty cycle. Why?
VLO
RL RL
VLO
Vout
M2 M3
RS
VS Rb
GGV
Ls
Lg
Cl earg G VM RF
LO-RF Feed through
0.5 LOT
0.5 LOT
0.5 LOT
0.5 LOT
38
Single Balanced Mixer Analysis: Isolation• Isolation Consideration (LO-RF Feed through)
• The amplified RF signal from the transducer is passed to the commuting switches through use of a common gate stage ensuring that the mixer operation is unaffected. Adding the common gate stage suppresses the LO-RF feed through.
VLO VLOM2 M3
RS
VS Rb
VBB1
Ls
Lg
Cl earg
Weak LO-RF Feed through
G VM RF
VBB2
39
Single Balanced Mixer Analysis: Isolation• Isolation Consideration (LO-IF Feed through)
• The strong LO-IF feed-through may cause the mixer or the amplifier following the mixer to saturate. It is therefore important to minimize the LO-IF feed-through.
VLO
RL RL
VLO
Vout
M2 M3
RS
VS Rb
VBB1
Ls
Lg
Cl earg G VM RF
LO-IF Feed through
40
Double Balanced Mixer• Double balanced mixer (also called Gilbert Cell):
• The strong LO-IF feed is suppressed using the double balanced mixer.
• All the even harmonics are also cancelled.
• All the odd harmonics are doubled (including the signal).
VLO
RL RL
VLOM2 M3
VRF
VLOM2 M3
VRF
VOUT
I IDC RF I IDC RF
41
Double Balanced Mixer• Double balanced mixer (Gilbert Cell) cont...:
• The LO feed through cancels.
• The output voltage due to RF signal doubles.
VLO
RL RL
VLO
VoutM2 M3
VRF
VLO
VoutM2 M3
VRF
VOUT
I IDC RF I IDC RF
42
Double Balanced Mixer: Linearity• Measuring Linearity of a double balanced mixer:
• Show that:
VIF
= 2IDC
RL
KSQ
2IDC
æ
èç
ö
ø÷
1/2
VRF
+1
2.
KSQ
2IDC
æ
èç
ö
ø÷
3/2
VRF
3 + ...
ì
íï
îï
ü
ýï
þïIIP3 in - volts( ) =
8IDC
3KSQ
VLO
RL RL
VLOM2 M3
VRF
VLOM2 M3
VRF
VOUT
I IDC RF I IDC RF1M1M
43
Mixer Output Match• Output match for mixers:
– Heterodyne Mixer
– Homodyne Mixer
• Heterodyne Mixer: For IF frequencies of 100-200MHz (signal bandwidth of 4MHz) we do not do impedance matching due to the following reasons:
– The signal bandwidth is comparable to the IF frequency therefore the impedance matching would create gain and phase distortions
– Need large inductors and capacitors to impedance match at 200MHz
• The general approach taken is the following
– Case (1) Output goes to SAW: In this case we need to keep the VSWR close to 1. The output impedance of the mixer at IF (200MHz) tends very high (~5kOhms). Due to this, we stabilize the output impedance by putting a resistor across the collectors of the mixers.
– Case (2) Output goes to amplifier on chip. We don’t care about impedance matching and directly couple the output of the mixer to the IF amplifier. This generates the largest voltage at the input of the IF amplifier.
44
Mixer Output Match (Heterodyne)• Output match for a Heterodyne mixer:
VLO
400LR
VLO
VRF
Vout
M1
M2 M3
3.0CCV V
400
2parL nH
45
Mixer Output Match (Homodyne)• Output match for a Homodyne mixer:
VLO
RL RL
VLO
VoutM2 M3
RS
VS Rb
VBB1
Ls
Lg
Cl earg
LC
46
Mixer Noise Analysis• Noise analysis of a single balanced mixer:
VLO
RL RL
VLO
VRF
Vout
,DC mix RF NoiseI I I
M1
M2 M3
wLO -wRF
LO RF wLO +wRF
VOUT
t
Instantaneous Switching
47
Mixer Noise Analysis• Noise analysis of a single balanced mixer cont...:
• If the switching is not instantaneous, additional noise from the switching pair will be added to the mixer output.
• Let us examine this in more detail.
VLO
RL RL
VLO
VRF
Vout
,DC mix RF NoiseI I I
M1
M2 M3 VOUT
t
Finite Switching Time
48
Mixer Noise Analysis• Noise analysis of a single balanced mixer cont...:
• When M2 is on and M3 is off:
– M2 does not contribute any additional noise (M2 acts as cascode)
– M3 does not contribute any additional noise (M3 is off)
VLO
RL RL
VLO
VRF
Vout
M1
M on2 M off3 VOUT
t
Finite Switching Time
,DC mix RF NoiseI I I
49
Mixer Noise Analysis• Noise analysis of a single balanced mixer cont...:
• When M2 is off and M3 is on:
– M2 does not contribute any additional noise (M2 is off)
– M3 does not contribute any additional noise (M3 acts as cascode)
VLO
RL RL
VLO
VRF
Vout
M1
M off2 M on3VOUT
t
Finite Switching Time
,DC mix RF NoiseI I I
50
Mixer Noise Analysis• Noise analysis of a single balanced mixer cont...:
• When VLO+ = VLO- (i.e. the LO is passing through zero), the noise contribution from the transducer (M1) is zero. Why?
• However, the noise contributed from M2 and M3 is not zero because both transistors are conducting and the noise in M2 and M3 are uncorrelated.
VLO
RL RL
VLO
VRF
Vout
M1
M on2 M on3 VOUT
t
Finite Switching Time
,DC mix RF NoiseI I I
51
Mixer Noise Analysis• Optimizing the mixer (for noise figure):
• Design the transducer for minimum noise figure.
• Noise from M2 and M3 can be minimized through fast switching of M2/M3 by:
– making LO amplitude large
– making M2 and M3 small (i.e. increasing fT of M2 and M3)
• Noise from M2 and M3 can be increased by using large M2/M3 switches.
VLO
RL RL
VLO
VRF
Vout
M1
M on2 M on3
VOUT
t
Trise
...m DCg W fixed I 1
...T DCfixed IW
,DC mix RF NoiseI I I
52
Mixer Noise Analysis• Noise Figure Calculation:
• Let us calculate the noise figure including the contribution of M2/M3 during the switching process.
VLO
RL RL
VLO
VRF
Vout
M1
M on2 M on3VOUT
t
Trise
,DC mix RF NoiseI I I
53
Heterodyne Mixer Noise Analysis: RL Noise• Noise Analysis of Heterodyne Mixer (RL noise):
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
IF RF LO
,DC mix RF NoiseI I I
2 4 2noise RL Lv kT R
54
Heterodyne Mixer Noise Analysis: Transducer Noise• Noise Analysis of Heterodyne Mixer (Transducer noise):
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
inoise- M1-switch
= inoise- M1
t( )SW t( )
= inoise- M1
t( )4
pcos w
LOt{ } -
4
3pcos 3w
LOt{ } +
4
5pcos 5w
LOt{ } - ...
æ
èçö
ø÷
VLO
t,DC mix RF NoiseI I I
55
Heterodyne Mixer Noise Analysis: Transducer Noise• Noise Analysis of Heterodyne Mixer (Trans-conductor noise):
IF LO
inoise-M1
2 f( ) = 4kTg gdo,1
inoise- M1-switch
= inoise- M1
t( )SW t( )
= inoise- M1
t( )4
pcos w
LOt{ } -
4
3pcos 3w
LOt{ } +
4
5pcos 5w
LOt{ } - ...
æ
èçö
ø÷
3 LO
4 4
3 ...3
LO LOSW f
inoise- M1
2 wIF( ) = 2
4
p
æ
èçö
ø÷
2
1+1
32+
1
52+ ..
é
ëê
ù
ûú4kTg g
do1
5 LO
inoise-M1
2 wIF( ) = 4 ×4kTg g
do,1
1
n2
n=1,odd
¥
å =p 2
8
56
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise):
VLO VLOM on2 M on3id ,3
2id ,2
2
id
2 » 4kTg gm
id
2g vm gs g vm gs
vgn
2 =4kTg
gm
57
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise):
• Show that:
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
out outi i
VLO
Gm
VLO
,
2 3 2,3
2. DC mix
m m m m
IG g g g
V
,DC mix RF NoiseI I I
0mG
58
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
Gm
VLO
2,3n mv
iout
2,3.out m n mi t G t v t
2
LOT
T
59
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
Gm t( )
Gm
t( ) =DTG
m0
TLO
/ 2+
DTGm0
TLO
/ 2( )2
sin kDTw
p
2
æ
èç
ö
ø÷
kDTw
p
2
æ
èç
ö
ø÷
k=1
¥
å cos kwpt( )
Gm f( )
p 2 p 3 p
2,3n mv f
p 2 p 3 p
vn-m2,3
2 = 24kTg
gm2,3
2
/ 2p
LOT
2
LOT
T
2 2 2
2,3 2 3n m n m n mv v v
60
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
Gm f( )
p 2 p 3 p
2,3n mv f
p 2 p 3 p
vn-m2,3
2 = 24kTg
gm2,3
2,3.out m n mi t G t v t
inoise- M 2,3
2 wIF( ) = v
n-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
+ 2vn-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
sinc kDTw
p
2
æ
èç
ö
ø÷
æ
èç
ö
ø÷
2
k=1
å
How do we solve this?
61
inoise- M 2,3
2 wIF( ) = v
n-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
+ 2vn-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
sinc kDTw
p
2
æ
èç
ö
ø÷
æ
èç
ö
ø÷
2
k=1
å
Heterodyne Mixer Noise Analysis: Switch Noise
inoise- M 2,3
2 wIF( ) = v
n-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
+ 2vn-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
-1+2p
DTwp
2
inoise- M 2,3
2 wIF( ) = v
n-m2,3
2 Gm0
DT
TLO
2
æ
èçö
ø÷
æ
è
çççç
ö
ø
÷÷÷÷
2
TLO
2DT= v
n-m2,3
2 Gm0
2 DT
TLO
2
æ
èçö
ø÷
sinc kDTw
p
2
æ
èç
ö
ø÷
æ
èç
ö
ø÷
2
k=1
å =
-1+2p
DTwp
2
62
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
Gm f( )
p 2 p 3 p
2,3n mv f
Gm f( )
p 2 p 3 p
2,3n mv f
inoise- M 2,3
2 wIF( ) =
1
TLO
2
æ
èçö
ø÷
Gm0
2 DT vn-m2,3
2
63
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
inoise- M 2,3
2 wIF( ) =
1
TLO
2
æ
èçö
ø÷
Gm0
2 DT vn-m2,3
2
,
0
2 DC mix
m
IG
V
DV = SlopeDT LO LO LOV t A Cos t
90
90LO
LO
LO
LO LOt
t
dV tSlope A
dt
G
m= g
m2= g
m3= g
m2,3»
2IDC ,mix
DV
vn-m2,3
2 = 24kTg
gm2,3
64
Heterodyne Mixer Noise Analysis: Switch Noise• Noise Analysis of Heterodyne Mixer (switch noise) cont...:
inoise- M 2,3
2 wIF( ) =
Gm0
2 DT
TLO
/ 2v
n-m2,3
2 =G
m0
2 DT
TLO
/ 22
4kTg
gm2,3
æ
èç
ö
ø÷
= 4G
m0DT
TLO
4kTg( ) =4DT
TLO
2IDC ,mix
DV4kTg( )
= 42I
DC ,mix
TLO
DT
DV4kTg( ) = 4
2IDC ,mix
TLO
1
ALO
wLO
4kTg( )
= 4 ×4kTgI
DC ,mix
p ALO
æ
èç
ö
ø÷
Total Noise Contribution due to switches M2 and M3
G
m»
2IDC ,mix
DV
vn-m2,3
2 = 24kTg
gm2,3
inoise- M 2,3
2 wIF( ) =
1
TLO
2
æ
èçö
ø÷
Gm0
2 DT vn-m2,3
2
DV
DT= A
LOw
LO
65
Heterodyne Mixer Noise Analysis: Total Noise• Noise Analysis of Heterodyne Mixer (total noise):
inoise- M1
2 wIF( ) = 4 ×g 4kTg
m1= 4 ×g 4kT
IDC ,mix
VGSQ
-VT 0( )
inoise- M 2,3
2 wIF( ) = 4 ×g 4kT
IDC ,mix
p ALO
æ
èç
ö
ø÷
2 4 2noise RL Lv kT R
vnoise- MIX
2 wIF( ) = 4kTR
L2 + 4g I
DC ,mixR
L
1
VGSQ
-VT 0
+1
p ALO
æ
èç
ö
ø÷
ì
íï
îï
ü
ýï
þï
0
1
2
DS short DS shortm short ox sat
GS GSQ T
dI Ig WC v
dV V V
2 2 2 2 2 2
1 2,3noise MIX IF noise RL L noise M L noise Mv v R i R i
66
Heterodyne Mixer Noise Analysis: Total Noise• Noise Analysis of Heterodyne Mixer (total noise):
0 1 &GSQ TV V M linearity Noise
0... min 2 / 3GSQ T LOAs V V A to imize noise contribution from M M
2
noise MIX IFv
1.6GSQV V
0.8GSQV V
VLO
vnoise- MIX
2 wIF( ) = 4kTR
L2 + 4g I
DC ,mixR
L
1
VGSQ
-VT 0
+1
p ALO
æ
èç
ö
ø÷
ì
íï
îï
ü
ýï
þï
Heterodyne Mixer Noise Analysis: Noise Figure
• This assumes that all of the “white noise” from Rs is folded down. In reality, there is some matching and filtering between the generator and mixer.
22
,20
1 11 1 4
noise MIX IF SDC mix L
L T LOGSQ Tnoise RS IF
v RNF I R
R AV Vv
2 2
2 1 416
2
T Tnoise Rs IF S
S S
kTi kT R
R R
Load
No
ise
LO P
ort
No
ise
68
Heterodyne Mixer Noise Analysis: Total Noise• Noise Analysis of Heterodyne Mixer (total noise)--{Terrovitis and Meyer}:
F =a
c2+
g3+ r
g3g
m3( )gm3
a + 2g1G + R
LO+ 2r
g 2( )G2 +1
RL
c2gm3
2 Rs
sin2 2
2
LO
LO
T
cT
41
3LOTf
G =2I
p ALO
G2 = 4.64K
sq
1/2IDC ,mix
3/2
2p ALO
Homodyne Mixers
• Issues– DC offsets: Mixing to DC means tolerance to
mismatch between devices and waveforms is tighter
– LO leakage: LO at RF frequency means it is difficult to distinguish between the two
– Noise: 1/f noise will play a prominent role in the homodyne mixer.
– Resistive load: Reduces voltage headroom and gain
69
Issues with Homodyne
• Homodyne seems so easy…there must be a catch.
– LO phase must track RF (need a PLL to lock to RF)
– LNA must be exceptionally linear (more on this soon)
– LO leakage must be controlled
©James Buckwalter
LO Leakage Issue
• Doesn’t my LNA have excellent return loss?
• Yes but your LO signal is strong…
• LO leakage back into RF port is
where PLO is LO power, ISMIX is the isolation of the LO-to-RF port, and RLLNA is the return loss of the LNA.
• A typical case
©James Buckwalter
,LO LEAKAGE LO MIX LNAP P IS RL
, 0 40 30 70LO LEAKAGEP dBm dB dB dBm
72
Homodyne Mixer Noise Analysis: Transducer Noise• Noise Analysis of Homodyne Mixer (noise from transducer M1):
LO
RF
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
,DC mix RF NoiseI I I
73
Homodyne Mixer Noise Analysis: RL Noise• Noise Analysis of Homodyne Mixer (noise from RL):
LO
RF
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
Noise from RL
,DC mix RF NoiseI I I
74
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)}:
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
IM1( )Ä DC +
4
pcos w
LOt( ) -
4
3pcos 3w
LOt( ) + ...
æ
èçö
ø÷
VLO
t
2
LOTT
2
LOTT
75
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)}:
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
IM1( )Ä DC +
4
pcos w
LOt( ) -
4
3pcos 3w
LOt( ) + ...
æ
èçö
ø÷
VLO
t
2
LOTT
2
LOTT
p t( ) =DTp
0
TLO
/ 2+
DTp0
TLO
/ 22
sin kDTp
TLO
æ
èç
ö
ø÷
kDTp
TLO
æ
èç
ö
ø÷
k=1
¥
å cos2p k
TLO
tæ
èç
ö
ø÷
p t( ) = d + 4sin kdp( )
kpk=1
¥
å cos 2pkfLO
t( )
76
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)--{Noise from M1}:
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3INoise M 1
INoise thermalINoise f1/
, 1/DC mix RF Noise thermal Noise fI I I I
77
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)--{Noise from M1}:
VLO
RL RL
VLO
VRF
Vout
M1
M2 M3
IDC ,mix
+ IRF
+ INoise-thermal
+ INoise-1/ f( ). DC +
4
pcos w
LOt( ) -
4
3pcos 3w
LOt( ) + ...
æ
èçö
ø÷
LO
RF
3 LO
DC-term of LO
78
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)--{Noise from M2/M3}:
VLO VLOM on2 M on3id3id 2
i i id d thermal d f 1/
2
1/
1. .
f
d f m
ox
Ki g
C WL f
g vm gs g vm gs
1/
1.
f
gn f
ox
Kv
C WL f
79
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)--{Noise from M2/M3}:
VLO
RL RL
VLO
Vout
M2 M3
1/gn fv
, 1/DC mix RF Noise thermal Noise fI I I I
VLO
1/gn fv
80
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)--{Noise from M2/M3}:
VLO
1/gn fv
iout
i i iout out no noise noise f 1/
81
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)--{Noise from M2/M3}:
1/gn fv t
T tSlope
Slope ALO LO 2
VLO1/gn fv
iout
i i iout out no noise noise f 1/
T
iout
1/
2
gn f
LO LO
v tT t
A
82
Homodyne Mixer Noise Analysis: non-50% duty LO• Noise Analysis of Homodyne Mixer (M2,M3 mismatched or non-50% duty cycle of
LO)--{Noise from M2/M3}:
1/
,max ,max
0 0
. . . .2 2 2
gn fLO LODC DC
k kLO LO
v tT TNoise Energy T t I t k I t k
A
iout
,DC mixI
,DC mixI
1/gn fv t
iout
0.5 LOT
1/gn fv f
t
t
t
f
f
f
1
0.5 LOT
1
0.5 LOT
1/
1/ ,max.2
gn f
noise f DC
LO
v fi I
A
83
Increasing Headroom in DBM (Option 1)
eL
2parL nH
eL
1Q
2 1Q '
2 1Q
'
1Q
inV
com
gdV
2 2Q
'
2 2Q bR
bV
LOV LOV
cC cCinV
bR
ccV
gndV
84
Increasing Headroom in DBM (Option 2)
200SR
eL
Lb
2parL nH
eL
Lb
BQIBQI
200LR
10C nF 10C nF
1Q
2 1Q '
2 1Q
'
1QSV
SV
inV
inV
com
gdV
2 2Q
'
2 2Q bR bR bRbR
bVbV bV
LOV LOV
3.0CCV V
cC cC
LR LR
ggV
85
Increasing Headroom in DBM (Option 3)
200SR
eL
Lb
2parL nH
eL
Lb
BQIBQI
200LR
10C nF 10C nF
1Q
2 1Q '
2 1Q
'
1QSV
SV
inV
inV
com
gdV
2 2Q
'
2 2Q bR bR bRbR
bVbV bV
LOV LOV
3.0CCV V
cC cC
LR LR
ggV
86
Homodyne Issues: Harmonic Mixers• The LO radiation problem can be partially overcome by the use of harmonic mixers.
A two-level mixing scheme can be employed, as shown in the figure below. The LO frequency is precisely one half the desired frequency, which is easily filtered by the duplex filter. Additionally, a full differential structure will exhibit extremely low second harmonic distortion of the LO. The harmonic mixers must be driven by LO outputs at 45 degrees phase shift to each other.
LR
CCV
LR
2LOV2LOV
2LOV
1LOV1LOV
1LOVRFV
RFV
BIASI
RF
1LOV 2LOV
IF
87
Passive FET Mixer• Passive FET Mixers:
• Alternative for Homodyne Mixers: Lower 1/f noise because no current through devices (This is not exactly true at high-frequency).
VLO VLOM1 M2
VLO M4 VLOM3
RS
VIF
88
Passive FET Mixer• Passive FET Mixers (homodyne operation):
IM1
VLO
t
t
VOUT
t
Vout
= VRF
cos wRF
t( ) ×4
pcos w
LOt( ) -
4
3pcos 3w
LOt( ) +
4
5pcos 5w
LOt( ) - ...
æ
èçö
ø÷
LO
RF
2out IF
C
RF RF
VG
V
89
Passive FET Mixer• Passive homodyne FET Mixers (non-50% duty cycle of LO) result in no DC offsets!!
IM1
VLO
t
t
VOUT
t
LO
RF
DC-term of LO
Vout
= VRF
cos wRF
t( ) × DC +4
pcos w
LOt( ) -
4
3pcos 3w
LOt( ) +
4
5pcos 5w
LOt( ) - ...
æ
èçö
ø÷
90
Passive FET Mixer With Biasing
VLO
1M2M
VLO
'
1M'
2M
200SR
LC
1biasC nF
1biasC nF
ggR
ggR
1biasC nFggV
sdR
sdR
sdV
SV
2LR k
200
LOV
LOV
LOV
91
Passive FET Mixer• Passive Homodyne Mixers (Shahani & Lee):
Vo = Vi
R2
R1 + R2
= Vi
G1
G1 + G2
92
Passive FET Mixer• Passive Homodyne Mixers (Shahani & Lee):
• where:m t( ) =
g t( ) - g t - TLO / 2( )g t( ) + g t - TLO / 2( )
2
2 2
2
2
LO
IF RF
LO LO
LO
IF RF RF
LO
Tg t
g tV t V t
T Tg t g t g t g t
Tg t g t
V t V t V t m tT
g t g t
93
Passive FET Mixer• Passive Homodyne Mixers (Shahani & Lee):
• where:
• The last expression is “approximate” under the consideration that the load is not actually a conductance but a capacitance. Under this condition, the conductance is solved from the linear system and gives the final expression.
,max
,max
T
IF RF
T L
TT
out RF
T T
g tV t m t V t
g t G
gg tV t m t V t
g g
time varying thevenin equivalent conductanceTg t
,max maximum of T Tg g t
average of T Tg g t
2
22
LO
T IF
LOL
T
T IF
L T
Tg t g t
V t V tT
G g t g t
g tV t V t
G g t
94
Passive FET Mixer (II)• Passive Homodyne Mixers (Shahani & Lee):
• Effect of LO waveforms
square wave
sine wave
break before make
make before break
1ideal passive mixerF Power conversion loss
Power Gain
Subharmonic Mixer
• Khatri and Larson, TMTT 2008
95
96
Mixer Noise Analysis (General Approach)• Determining Noise Figure of Mixers (General Approach)--{Hu & Mayaram}:
VLO
RL RL
VLO
VRF
I IDC RF
M1
M2 M3
Non-linear
Circuit
Filter
LNAVRF
I IDC RF
VLO
97
Mixer Noise Analysis (General Approach)• Determining Noise Figure of Mixers (General Approach--the recipe):
• Procedure:
– Step 1: Calculate the small signal gain g(t) for each noise source.
– Step 2: Calculate the Fourier coefficients Gn of g(t).
– Step 3: Calculate the transfer function of the output filter H(w).
– Step 4: Calculate the frequency response |H(w).Gn|.
– Step 5: Determine the output noise density at Wif.
– Step 6: Determine the noise figure.
Non-linear
Circuit
Filter
LNAVRF
VLO
98
Mixer Noise Analysis (General Approach)• Determining Noise Figure of Mixers (General Approach) cont…:
VLO
RL RL
VLOM2 M3
im1irf Ibias
vb
ib
icic
icib
ib
vb
99
Mixer Noise Analysis (General Approach)• Determining Noise Figure of Mixers (General Approach) cont…:
• Small signal gain from vb to delta_Ic:
• Small signal gain from ie to delta_Ic:
• Refer to paper by Hu and Mayaram for detail on calculating noise figure.
DIC = IBIAS tanhVLO
2VT
æ
èçö
ø÷
gvbt( ) =
dDIC
dV V=VLO t( ),IC =IBIAS
=IBIAS
VT
2eVLO t( )/VT
1+ eVLO t( )/VT( )
2
giet( ) =
dDIC
dIee V=VLO t( ),IC =IBIAS
= tanhVLO t( )
2VT
æ
èçö
ø÷
100
Mixer Noise Analysis (General Approach)• Determining Noise Figure of Mixers (General Approach) cont…:
VLO
RL RL
VLOM2 M3
im1irf Ibias
vb
ib
icic
icib
ib
vb
101
Linearity• Measuring Linearity of a double balanced mixer:
VIF
= 2IDC
RL
KSQ
2IDC
æ
èç
ö
ø÷
1/2
VRF
+1
2.
KSQ
2IDC
æ
èç
ö
ø÷
3/2
VRF
3 + ...
ì
íï
îï
ü
ýï
þï
IIP3 in - volts( ) =8IDC
3KSQ
VLO
RL RL
VLOM2 M3
VRF
VLOM2 M3
VRF
VOUT
I IDC RF I IDC RF1M1M
How about linearity of switching pair?
Distortion in Mixers
• Terrovitis and Gray (JSSC 2000)
• Switching of current through single-balance differential pair
• What does this function look like?
102
I
o+ i
o= F V
LO, I
DC+ i
rf( )IDC+irf
Io1 =ISS
2+
ISSb
2vid 1-
bvid
2
4ISS
æ
èç
ö
ø÷
Io2 =ISS
2-
ISSb
2vid 1-
bvid
2
4ISS
æ
èç
ö
ø÷
Distortion in Mixers
• Switching of current through single-balance differential pair
• Linearize expression in terms of mixing
• pk are the coefficients of a Taylor series describing the switching
• However, pk coefficients are a function of time since F depends on VLO
• Example: If M1 is on and M2 is off, p1 = 1, p2 = 0, p3 = 0
103
I
o+ i
o= F V
LO, I
DC+ i
rf( )
io,1
= p1
t( ) × irf
+ p2
t( ) × irf
2 + p3
t( ) × irf
3
where pk
=1
k!
d k F
dIDC
k
IDC+irf
Distortion in Mixers
• We can simplify this by recognizing that we are primarily concerned about the conversion of tones through the 1st harmonic of the LO
104
io,1
= p1
t( ) × irf
+ p2
t( ) × irf
2 + p3
t( ) × irf
3 where pk
=1
k!
d k F
dIDC
k
io,1
= p1,k
× is+ p
2,k× i
s
2 + p3,k
× is
3( )sin kwLO
t( )k
å
IDC+irf
Distortion in Mixers
105
io,1
= p1,k
× is+ p
2,k× i
s
2 + p3,k
× is
3( )sin kwLO
t( )k
å
io,1
= b1× i
s+ b
2× i
s
2 + b3× i
s
3
where bk
=p
k ,1
2=
1
TLO
pk
t( )sin wLO
t( )dt0
T
ò
IDC+irf
• How can we use this polynomial series?
Distortion in Mixers
106
io,1
= b1× i
s+ b
2× i
s
2 + b3× i
s
3 where bk
=p
k ,1
2=
1
TLO
pk
t( )sin wLO
t( )dt0
T
ò
IDC+irf
• Minimizing b3 is good for the linearity of the mixer.
is= I
Ssin w
LO- D( )t( ) + sin w
LO+ D( )t( )( )
IM3 =3
4
b3
b1
IS
2
Solving for LO swing
• If large amplitude swing, LO approaches “square-wave” and
• There is a period of time Δ during which the switches are both on and in during this time
• where λ is the slope of the p3 in this region and Vo is the maximum is the cut-off voltage.
107
3 30
1 sin
T
LO
LO
b p t t dtT
3 32 2 0
4 2
oV
LO LO LO
LO
b p V V dVT
b1=
1
TLO
p1
t( )sin wLO
t( )dt0
T
ò
b
1®
2
p
Distortion in Mixers
108
• Consider sine LO
• Coefficients of switching waveform are shown to the left
• Then the b1 component depends on the “switching threshold” Vx.
• where
• and
b1 »2
p
Vx
Vo
arcsinVx
Vo
æ
è
ççç
ö
ø
÷÷÷
VX =IDCq
2b+
IDCq
2b
æ
èçö
ø÷
2
+IDC
b
3 32 2 0
4 2
oV
LO LO LO
LO
b p V V dVT
IIP3
• DC analysis shows that
– Lower current is more linear
– Higher swing is more linear (to a point)
109
IM3 =3
4
b3
b1
Is
2
Cascaded IIP3
110
io,1
= b1× i
s+ b
2× i
s
2 + b3× i
s
3 where bk
=p
k ,1
2=
1
TLO
pk
t( )sin wLO
t( )dt0
T
ò
is= a
1×v
rf+ a
2×v
rf
2 + a3×v
rf
3
IM3 =3
4
a3
a1
+ a2
2 b3
b1
æ
èç
ö
ø÷ v
RF
2
Frequency Dependent Effects
• Capacitances cause frequency dependence in switching pair
111
io,1
= P1
t, fa( ) i
rf+ P
2t, f
a, f
b( ) irf
2 + P3
t, fa, f
b, f
c( ) irf
3
P1
t, fa( ) = d
1t( ) H
1t, f
a( )P
2t, f
a, f
b( ) = d1
t( ) H2
t, fa, f
b( ) + d2
t( ) H1
t, fa( ) H
1t, f
b( )
P3
t, fa, f
b, f
c( ) = d1
t( ) H3
t, fa, f
b, f
c( ) + 2d2
t( ) H1
t, fa( ) H
2t, f
b, f
c( ) + d3
t( ) H1
t, fa( ) H
1t, f
b( ) H1
t, fc( )
v = H
1t, f
a( ) irf
+ H2
t, fa, f
b( ) irf
2 + H3
t, fa, f
b, f
c( ) irf
3
d1
t( ) =d
dVf1
t( ) - f2
t( )( )
d2
t( ) =1
2
d 2
dV 2f1
t( ) - f2
t( )( )
d3
t( ) =1
6
d 3
dV 3f1
t( ) - f2
t( )( )
Frequency Dependent Effects
• Time-varying performance indicates some “ideal” bias and LO swing behavior.
• Does that make sense?
112
Measurement
• 0.8 um CMOS
• Low bias: Transconductordominates
• High bias: switching pair dominates
• High frequency effects evident in IIP3 versus Vo.
• Note IIP3 drops rather than increases with swing
113
END