Berkeley
BJT High Frequency Distortion
Prof. Ali M. Niknejad
U.C. Berkeley
c 2014 by Ali M. Niknejad
Copyright Niknejad
Advanced IC’s for Comm
High Frequency Distortion in BJTs
Assume current drive
Large Circuit Equivalent Circuit
neglect rb , rπ , ro
Niknejad
Advanced IC’s for Comm
Distortion Due to Diffusion Capacitor
CB = Cπ − Cje
QB = τF IC
CB =
dIC
qIC
IS VBE /VT
dQB
= τF
= τF
= τF
e
dVBE
dVBE
kT
VT
CB is a non-linear diffusion capacitor
Cje is emitter-base depletion capacitor, assume constant
Cje ∝
1
1
(φ + VBE ) n
If fT of a BJT is constant with IC , then we have no high
frequency distortion in the device for current drive
⇒ Linear Differential Equation, fT constant → τF constant
Niknejad
Advanced IC’s for Comm
Kirk Effect
iC = IC −IQ = IQ e v1 /VT − 1
ii = ix + ib + iµ
= Cje
dQµ
dv1 dQB
+
+
dt
dt
dt
dQµ
dQµ
d (v1 − vo )
dv1
dvo
=
= Cµ
− Cµ
dt
d (v1 − vo )
dt
dt
dt
Cµ =
k
(ϕ + VCB )
1
n
=
Niknejad
k
1
(ϕ + VCBQ + VCB ) n
Advanced IC’s for Comm
Governing Differential Eq.
dIc
dic
dQB
= τF
= τF
dt
dt
dt
ii = Cje
dic
dvo
dv1
+ τF
− Cµ
dt
dt
dt
v1 = VT ln
ic + IQ
IQ
dv1
dv1 dic
VT IQ dic
VT dic
=
=
=
dt
dic dt
IQ ic + IQ dt
ic + IQ dt
ii = Cje
VT dic
dic
dvo
+ τF
− Cµ
ic + IQ dt
dt
dt
Niknejad
Advanced IC’s for Comm
BJT Series Expansion
vo = −ic RL
1
ii = −{
RL
=−
Cje VT
1
dvo
[τF +
+ Cµ R L ]
RL
IQ + ic
dt
1
IQ 1 +
Cµ = Cµ o
1+
vo
VQ
ic
IQ
⇒
1 = Cµo
n
Cje VT
dvo
τF +
+ Cµ }
IQ + ic
dt
1
= 1 − x + x 2 + ...
1+x
1 vo
1
1−
+
n VQ
2n
Niknejad
1
+1
n
Advanced IC’s for Comm
vo
VQ
2
!
+ ...
BJT Series Expansion (cont)
Cje VT
Cje VT ic
Cje VT
1
−
+
ii ∼
= − [τF +
RL
IQ
IQ IQ
IQ
1 vo
1
Cµo RL − Cµo RL
+ Cµo RL
n VQ
2n
ic
IQ
+
1
vo 2
dvo
1+
]×
n
VQ
dt
Substitute vo = −ic RL
dvo
ii ∼
= a1 + a2 vo + a3 vo 2
dt
dvo
1 dvo 2 1 dvo 3
∼
+ a2
+ a3
= a1
dt
2
dt
3
dt
Niknejad
2
(∗)
(∗∗)
Advanced IC’s for Comm
Memoryless Terms
Non-linear differential equation for vo vs. ii where
Cje VT
1
τF +
+ Cµo RL
a1 = −
RL
IQ
1 1
1 Cje VT
− Cµo RL
a2 = −
RL RL IQ 2
n VQ
Cje VT
1
1
1
1
a3 = −
+ Cµo RL
1+
RL RL 2 IQ 3
2n
n VQ 2
a2 term has a null but a3 terms always add.
Niknejad
Advanced IC’s for Comm
Memory Terms
In (**), put
vo = A1 (jωa ) ◦ ii + A2 (jωa , jωb ) ◦ ii 2 + ...
ii = a1
1 dvo 2 1 dvo 3
dvo
+ a2
+ a3
dt
2
dt
3
dt
First Order:
ii = a1 jωA1 (jω) ◦ ii
A1 (jω) =
Niknejad
1
jωa1
Advanced IC’s for Comm
Second Order Memory
Second Order:
1
0 = a1 j (ωa + ωb ) A2 (jωa , jωb )+ a2 (jωa + jωb ) A1 (jωa ) A1 (jωb )
2
A2 (jωa , jωb ) = −
1 a2
A1 (jωa ) A1 (jωb )
2 a1
A2 (jωa , jωb ) = −
=
Niknejad
1 a2 1
1
2 a1 jωa a1 jωb a1
1 a2 1
2 a1 3 ωa ωb
Advanced IC’s for Comm
Third Order Memory
Third Order:
1
1
0 = [a1 A3 + a2 2A1 A2 + a3 A1 A1 A1 ] (jωa + jωb + jωc )
2
3
A3 =
− 12 a2 2A1 A2 + 13 a3 A1 A1 A1
a1
A3 =
a3
3
−
a2 2
2a1
jωa ωb ωc a1 4
No null because a3 is negative
Niknejad
Advanced IC’s for Comm
Second-Order Intermodulation (IM2 )
ii = I1 cos ω1 t + I2 cos ω2 t
I M2 =
|A2 | I1 I2 2! 1
v̂o 1!1! 2
v̂o = |A1 (jω1 )| I1 = |A1 (jω2 )| I2
I M2 =
I M2 =
|A2 (jω1 , ±jω2 )|
v̂o
|A1 (jω1 )| |A1 (jω2 )|
1 a2 1
1 a2
a1 ω1 a1 ω2 v̂o =
v̂o
3
2 a1 ω1 ω2
2 a1
IM2 independent of frequency at high frequency
Niknejad
Advanced IC’s for Comm
IM3 in BJT at High Frequency
I M3 =
3! 1 I1 I2 2 |A3 |
2!1! 22
v̂o
Again
|A1 (jω1 )| I1 = |A1 (jω2 )| I2 = v̂o
|A3 (jω2 , jω2 , −jω1 )|
3
3
v̂ 2 = v̂o2
I M3 =
4 |A1 (jω1 )| |A1 (jω2 )| |A1 (jω2 )| o
4
a3
3
−
a3 2
2a1
a1
Cje
1
a1 = −
τF +
+ Cµo RL
RL
gm Q
Cje VT
1
1 1
1
a3 = −
+ Cµo RL
+1
RL RL 2 IQ 3
2n n
VQ 2
Niknejad
Advanced IC’s for Comm
3 a3
≈ v̂o2
4 3a1
Example: 2 GHz LNA Front End
τF = 12ps
Cµo = 37fF
rb = 10Ω
IC = 3mA
Cje = 416fF
RL = 50Ω
η=3
VQ = 2V
Supply ≈ VBE + VQ + 3mA × 50Ω
≈ 750mV + 150mV = 2.9V
Find intercept point:
1
1.6 × 10−10 + 1 × 10−13
a3 = −
RL
1
a1 = − (12 + 3.6 + 1.9) × 10−12
RL
3 a3
I M3 = v̂o2
=1
4 3a1
v̂o = 0.66V (+ 6 dBm output IP3 into 50Ω)
Very close to SPICE and measurements
Niknejad
Advanced IC’s for Comm
LNA Example (cont)
Note:
rπ =
β
150
=
≈ 1.3kΩ
gm
gm
Cπ = Cje + CB
CB = τF gm = 1.38pF
Cje = 0.416pF
1
= 44Ω
ωCπ
Cπ reactance is low compared to rπ
Niknejad
Advanced IC’s for Comm
ial-Pair
Stages
LNA Transconductance
with Emitter Degen
ng Fong, Member, IEEE, and Robert G. Meyer, Fellow, IEEE
he high-frequency nonlinear
fferential-pair transconducons show that transconducration are more linear than
degeneration, and that the
tages are more linear than
e stages with the same bias
nonlinearity equations can
B behavior of the commoninductive degeneration.
FONG AND MEYER: ANALYSIS OF COMMON-EMITTER AND DIFFERENTIAL-PAIR TRA
[5] C. D. Hull, “Analysis and optimization of monolithic RF downconversion receivers,” University of California at Berkeley, Berkeley, CA,
Ph.D. dissertation, 1992.
[6] D. O. Pederson and K. Mayaram, Analog Integrated Circuits for Communication. Norwell, MA: Kluwer, 1991.
Fig. 1. Common-emitter
transconductance
stage.
FONG AND MEYER:
ANALYSIS OF
COMMON-EMITTER AND DIFFERENTIAL-PAIR TRANSCONDUCTANCE STAGES
was born
in Kuala
[5] C. D. Hull, “Analysis and optimization of monolithic RF downcon-Keng Leong Fong (S’93–M’97)
Robert
G. Meyer
(S’64–M’68
6, 1970.
He received
version receivers,” University of California at Berkeley, Berkeley, CA,Lumpur, Malaysia, on March born
in Melbourne,
Australia, on
the B.A.Sc. degree in engineering
(comPh.D. dissertation, 1992.
receivedscience
the B.E.,
M.Eng.Sci.,
researc
the electrical
M.A.Sc. degree
in
[6] D. O. Pederson and K. Mayaram, Analog Integrated Circuits for Com-puter engineering option) and in
engineering
from
electrical engineering, both from
the University
munication. Norwell, MA: Kluwer, 1991.
Melbourne
in 1963,of1965,device
and 1
numera
Toronto, Ontario, in 1992 and 1993,
respectively.
In 1968,
he wasHe
employed
TION
Analysa
received the Ph.D. degree in turer
electrical
engineering
in electrical
engineering
of
the
from the University of California
at Berkeley
in
Melbourne.
Since September
19
e wireless communication
ployed in the Department ofDr.
Ele
Keng Leong Fong (S’93–M’97) was born in Kuala1997.
nd for low-power, high- Fig. 2. Differential-pair transconductance
an
Ass
stage.
During
the
summer
of
1995,
he
worked
at
Rockand Computer Sciences, Unive
Lumpur, Malaysia, on March 6, 1970. He received
IEEE
d RF circuits. The bipolarNewport where
Beach,he
CA,
is now
a Pr
the B.A.Sc. degree in engineering science (com-well International Corporation,Berkeley,
was involved
inindesigning
class
AB mixer
evaluating CAD
tools integrated
research
interests
areand
high-frequency
analog
differential-pair transconputer “High-Frequency
engineering option) where
andNonlinearity
theheM.A.Sc.
degree
Reference: K. L. Fong and R. G. Meyer,
Analysis
of Common-Emitter
and
forfrom
nonlinear
noise analysis.
During
the summerHeofhas
1996,
he
in theon electron
device
fabrication.
acted
asworked
a consultant
electrical engineering, both
the University
of
and 2, respectively, are
same1993,
company,
designing
AB power
amplifiers.
Since
August 1997,
he He is co
numerous
companies
in the
electronics
industry.
Toronto, Ontario, in 1992 and
respectively.
He class
frequency (RF) building
employed
by Philips
Semiconductors,
CA.
His current
Analysis
Design
of Analog
Integrated
Circuits (Wiley
Differential-Pair Transconductance Stages,
” IEEE
J. ofhas
Solid-State
Circuits,
vol.
33,and
no.
4, Sunnyvale,
April
1998.
received the
Ph.D. degree
in been
electrical
engineering
ers (LNA) and mixers. To
research interest
is in the
of book
analogIntegrated
integratedCircuit
circuitOperational
design for Amplifiers
RF
of the
(IE
from the University of California
at Berkeley
in area
applications.
Dr. Meyer was President of the IEEE Solid-State Circu
ductance stages are usually
1997.
an Associate Editor of the IEEE JOURNAL OF SOLID-STATE
During the summer of 1995, he worked at Rockhich can be implemented
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS.
well International Corporation, Newport Beach, CA,
itors, or inductors. It is
where he was involved in designing class AB mixer and evaluating CAD tools
nce stages with reactive Fig. 3. Third-order
for nonlinear
noise analysis.
During
the summer
of 1996, he worked in the
intermodulation
product
corrupts
desired channel.
same company, designing class AB power amplifiers. Since August 1997, he
ration have lower noise
has been employed by Philips Semiconductors, Sunnyvale, CA. His current
sistive degeneration since
research
interest is in the
area of analog
integrated
circuit design for RF
In
this
paper,
high-frequency
nonlinearity
equations
in Volterra
from its loss resistance)
applications.
, analog integrated circuits,
nonlinear circuits, nonlinear
erra series.
Analysis of BJT transconductor as a non-linear Gm cell
Include emitter degeneration.
noise source. Similarly,
stages have a higher noise
onductance stages since the
s. However, there is little
series [1], [2] for both common-emitter and differentialpair transconductance stages are derived. The emphasis
is on interpretation and application of the equations. The
nonlinearity equations are also used
to explain theAdvanced
class AB IC’s for Comm
Niknejad
KCL Equations
FONG AND MEYER: ANALYSIS OF COMMON-EMITTER AND DIFFERENTIAL-PAIR TRANSCOND
phase of
Fig. 4. Model of common-emitter transconductance stage.
Vs = (sCje Vπ + sτF Ic + Ic /β0 )(Zb + Ze ) + Ic Zc + Vπ
Fig. 4 shows the large-signal model used to derive the non#
transconductance
linearity equations
for"
the common-emitter
Vπ
Vπ
1 Vπ 2 1 Vπ 3
Fig.IQ1. This model
Ic = Istage
+ ignores the+effect of base-+ · · ·
Q expshown in =
VT
2 ofVT . Inclusion
6 VT
T
of
collector V
junction
capacitance
greatly complicates the analysis without adding significant
Vπ accuracy
= C1 (s1 )to◦the
Vs results
+ C2 (s1,
s2) ◦ Vs2 + C3 (s1, s2,
s3) ◦ Vs3 + · · ·
is the voltage
in typical situations.
is the impedance at the base of
which
signal source.
,
includes source resistance ( ), base resistance ( ) of
Advanced
IC’s for Comm
shunt impedance of biasNiknejad
circuit, and
impedance
of impedance-
IM3 Calculation
C1 (s) =
1
sCje Z (s) + sτF gm Z (s) + gm Z (s)/β0 + 1 + gm Ze (s)
IQ
C2 (s) = −C1 (s1 + s2 )C1 (s1 )C1 (s2 ) 2V
2 ×
T
h
i
Z (s1 +s2 )
(s1 + s2 )τF Z (s1 + s2 ) + β0 + Ze (s1 + s2 )
2 +s3 )IQ
×
C3 (s1 , s2 , s3 ) = − A1 (s1 +s
6VT3
i
hA1 (s1 )A1 (s2 )A1 (s3 ) + 6VT A1 A2 ×
2 +s3 )
+
Z
(s
+
s
+
s
)
(s1 + s2 + s3 )τF Z (s1 + s2 + s3 ) + Z (s1 +s
e
1
2
3
β0
Ic = A1 (s) ◦ Vs + A2 (s1, s2) ◦ Vs2 + A3 (s1, s2, s3) ◦ Vs3 + · · ·
A1 (s) = gm C1 (s)
A2 (s1 , s2 ) = gm C2 (s1, s2) +
A3 (s1 , s2 , s3 ) = gm C3 (s1 , s2 , s3 ) +
Niknejad
IQ
C1 (s1 )C1 (s2 )
2VT2
IQ
IQ
C1 (s1 )C1 (s2 )C1 (s3 ) + 2 C1 C2
3
6VT
VT
Advanced IC’s for Comm
IM3 Calculation
3 A3 (sa , sa , −sb ) |Vs |2
IM3 = 4 A1 (2sa − sb ) A1 (s) 3 VT
≈ IQ 4 (1 + sCje Z (s))×
n
−1 + A1g(∆s)
(1 + ∆sCje Z (∆s)) +
m
A1 (2s)
2gm (1
o
+ 2sCje Z (2s)) |Vs |2
With Z (s) = Zb (s) + Ze (s), note that these terms can
resonate:
(1 + sCje Zb (s) + sCje Ze (s))
For small offset frequencies (1 + ∆sCje Z (∆s)) ≈ 1
A1 (s) 3 VT
A1 (∆s)
A1 (2s)
|Vs |2
IM3 ≈ (1
+
sC
Z
(s))
×
−1
+
+
(1
+
2sC
Z
(2s))
je
je
IQ
4
gm
2gm
Niknejad
Advanced IC’s for Comm
the same reason, capacitive degeneration would increase the
because the
term is a positive real number
Typical
Example
which adds to the
“1” term in (7).
The
also depends on the magnitude of
Fig. 5. Inductively degenerated common-emitter transconductances stage.
IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 33, NO. 4, APRIL 1998
(8)
where the “ 1” and
terms come from the third-order nonlinearity (
) and
), respectively. Using the
the second-order interaction (
following approximation (for practical design values),
equation (6) can be simplified to
6. Input
5. Inductively degenerated common-emitter transconductancesFig.
stage.
IP3 versus bias current.
The two RF sinusoidal signals used
are at 900 and 910 MHz,
Fig. 5 shows the basic topology of a typical commonemitter transconductance
degeneration.
respectively. The component values
used are:stage
τFwith
= inductive
10.5ps,
is typically implemented by
The degeneration inductor
(9)
serves
as a 3.5nH,
dc blocking capacitor. It is also
wires.
Cje = 1.17pF, β = 73, and Le bond
= 2.4nH,
L1 =
.
is a bias
used to tune out the bond wire inductance
C1of =
term resistor used to isolate the bias circuit from the RF port. At
where the value
the 20pF, and R1 = 150Ω.
is typically small, compared to the value of the
term. However, it is not so small that it can be ignored.
is independent of
if the smallAs shown in (9), the
is kept constant. On the other
signal transconductance
je
for the resistive and capacitive
hand, it increases with
term is a positive
degeneration cases since the
imaginary number and a positive real number, respectively, but
low frequency
, the impedance of
is
The simulation results includenegligible.
the nonlinear
effects
ofis Cdominated
The impedance
of
µ andby the
since the blocking capacitor
has high
bias resistor
C , and neglecting these effects
does
not
seem
to
introduce
impedance at low frequency. In this case, the
term in (9) can be simplified to
significant errors.
Niknejad
Advanced IC’s for Comm
(10)
References
UCB EECS 242 Class Notes, Robert G. Meyer, Spring 1995
K. L. Fong and R. G. Meyer, “High-Frequency Nonlinearity
Analysis of Common-Emitter and Differential Pair
Transconductance Stages,” JSSC, pp. 548-555, vol. 33, no.
4, April 1998.
Niknejad
Advanced IC’s for Comm