Transcript 投影片 1

Linearity
8.1
8.2
8.3
8.4
8.5
Nonlinearity Concept
Physical Nonlinearities
Volterra Series
Single SiGe HBT Amplifier Linearity
Cascode LNA Linearity
Introduction
(1)
 Nonlinearity causes intermodulation of two adjacent
strongly interfering signals at the input of a receiver,
which can corrupt the nearby (desired) weak signal we
are trying to receive.
 Nonlinearity in power amplifiers clips the large
amplitude input.
@ Modern wireless communications systems typically
 have several dB of variation in instantaneous
power as a function of time
 require highly linear amplifiers
Introduction
(2)
 SiGe HBTs exhibit excellent linearity in
 small-signal (e.g., LNA)
 large-signal (e.g.,PA) RF circuits
 despite their strong I-V and C-V nonlinearities
 The overall circuit linearity strongly depends on
 the interaction ( and potential cancellation) between the
various I-V and C-V nonlinearities
 the linear elements in the device : the source (and load)
termination; feedback present
 The response of a linear (dynamic) circuit is characterized by
 an impulse response function in the time domain
 a linear transfer function in the frequency domain
 For larger input signals, an active transistor circuit becomes
a nonlinear dynamic system
Linearity
8.1
8.2
8.3
8.4
8.5
Nonlinearity Concept
Physical Nonlinearities
Volterra Series
Single SiGe HBT Amplifier Linearity
Cascode LNA Linearity
Harmonics (1)
Input : x t
A cos t
Ouput : y t
k1 x t
2
k2 x
t
2
3
k3 x
2
k1 A cos t k2 A cos
t
k2 A 2
k2 A 2
2
k3 A3
4
3
3
k3 A cos
t
dc shift
2
k1 A
t
3 k3 A3
4
cos t
fundamental
cos 2 t
second harmonic
cos 3 t
third harmonic
Harmonics (2)


An “nth-order harmonic term” is proportional to An
1 k2
2
k2
A
HD2(second harmonic distortion) =
/ k1 A = 2 k1 A
2

( neglect 3k3A3/4 term)
IHD2 ( the extrapolation of the output at 2ω and ω intersect)
obtained by letting HD2 = 1
1 k2
2 k1
A =1
 A = IHD2 = 2
k1
k2
 IHD2 is independent of the input signal level (A)
 HD2 = A / IHD2 ( one can calculate HD2 for small-signal input A )
 OHD2 ( output level at the intercept point ), G (small-signal gain)
OHD2 = G*IHD2 = k1*2
k1
k2
= 2
k12
k2
Gain Compression and Expansion
 The small-signal gain is obtained by neglecting the harmonics.
The small-signal gain ：k1
The nonlinearity-induced term： 3k3A3/4
 As the signal amplitude A grows, becomes comparable to or
even larger than k1A
 the variation of gain changes with input
 fundamental manifestation of nonlinearity
 If k3 < 0, then 3k3A3/4 < 0
 the gain decreases with increasing input level (A)
 “gain compression” in many RF circuits
 quantified by the “1 dB compression point,” or P1dB
(1)
Gain Compression and Expansion (2)
 The transformation between voltage and power involves a
reference impedance, usually 50Ω.
 Typically RF front-end amplifiers require -20 to –25 dBm
input power at the 1dB compression point.
Intermodulation (1)
 A two-tone input voltage x(t) = Acosω1t +Acosω2t
 The output has
 not only harmonics of ω1 and ω2
 but also “intermodution products” at 2ω1-ω2 and 2ω2-ω1
(major concerns, close in frequency to ω1 and ω2 )
Intermodulation (2)
 Products output are given by
y t
k1A
3 k3 A3
4
3 k3 A3
3 k3 A3
4
2
cos 2 2
cos 1t ...
1 t ...
 A 1-dB increase in the input results in
 a 1-dB increase of fundamental output
 but a 3-dB increase of IM product
 IM3 (third-order intermodulation distortion)
IM3
3 k3 A3
4
k1A
3 k3
4 k1
A2
fundamental
intermodulation
Intermodulation (3)
 IIP3 ( input third-order intercept point) is obtained by letting
IM3 = 1
IM3
3 k3
4 k1
2
A
1
A
IIP3
4 k1
3 k3
 independent of the input signal level (A)
 IM3 can be calculated for desired small input A
IM3 = A2 / IIP32
 IIP3 can be measured by A0, IM30
IIP32 = A02 / IM30
 IIP3, A0  voltage
IIP32, A02  power ( taking 10 log on both side )
 20 log IIP3 = 20 log A0 – 10 log IM30
 PIIP3 = Pin + ½( Po,1st – Po,3rd )
Intermodulation (4)
 OIP3 = k1*IIP3  OIP32 = k12*IIP32
 IIP32 = OIP32/ k12 = A2/IM32
 OIP32 = (k1A)2/IM32
( taking 10 log on both side )
 20 log OIP3 = 20 log k1A – 10 log IM3
 POIP3 = P o,1st + ½( Po,1st – P o,3rd)
 The gain compression at very high input power level can be seen
Intermodulation (5)
 IIP3 is an important figure for front-end RF/microwave lownoise amplifiers, because they must contend with a variety of
signals coming from the antenna.
 IIP3 is a measure of the ability of a handset, not to “drop” a
phone call in a crowded environment.
 The dc power consumption must also be kept very low because
the LNA continuously listening for transmitted signals and hence
continuously draining power.
 Linearity efficiency = IIP3 / Pdc
( Pdc = the dc power dissipation )
 excellent linearity efficiency above 10 for first generation
HBTs
 competitive with Ⅲ-Ⅴ technologies
Linearity
8.1
8.2
8.3
8.4
8.5
Nonlinearity Concept
Physical Nonlinearities
Volterra Series
Single SiGe HBT Amplifier Linearity
Cascode LNA Linearity
Physical Nonlinearities in a SiGe HBT





ICE  the collector current transported from the emitter
 the ICE-VBE nonlinearity is a nonlinear transconductance
IBE  the hole injection into the emitter
 also a nonlinear function of VBE.
ICB  the avalanche multiplication current
 a strong nonlinear function of both VBE and VCB
 has a 2-D nonlinearity because is has two controlling voltages.
CBE  the EB junction capacitance
 includes the diffusion capacitance and depletion capacitance
 a strong nonlinear function of VBE when the diffusion
capacitance dominates, because diffusion charge is
proportional to the ICE
CBC  the CB junction capacitance
Equivalent circuit of the HBT

The ICE Nonlinearity (1)

i
t
f
f
vC
t
f
VC
k
1
VC
k 1
vc
t
f
v
t
v VC
vk
k
vc k
t
 i(t) : the sum of the dc and ac currents
vc(t) : the ac voltage which controls the conductance
VC : the dc controlling (bias) voltage
 For small vc(t), considering the first three terms of the
power series is usually sufficient.
2
1
f v

f v
K
g
K3 g
1
3
2g
v VC
v
3
f v
v3
v VC
K ng
v VC
v2
2
1
n
n
f
v
vn
v VC
The ICE Nonlinearity (2)
 The ac current-voltage relation can be rewritten
iac(t) = g vc(t) + K2g vc2(t) + K3g vc3(t) + …
g : the small-signal transconductance
K2g : the second-order nonlinearity coefficient
K3g : the third-order nonlinearity coefficient
 For an ideal SiGe HBT, ICE increases exponentially with VBE
ICE = IS exp (qVBE/kT)

qICE
1
q2 I
gm
K3 gm
K2 gm
kT
1
3
q3 ICE
kT
3
K ngm
CE
2
kT
1
n
2
q n ICE
kT
n
The ICE Nonlinearity (3)

g m,eff
ic
v be
gm
1
1 qv be
2
kT
1 q2 v be2
6
kT
2
...
nonlinear contributions
 The nonlinear contributions to gm,eff increase with vbe.
 Improve linearity by decreasing vbe.
The IBE Nonlinearity
 For a constant current gain β
IBE = ICE/β
gbe = gm/β
K2gbe = K2gm/β
K3gbe = K3gm/β
Kngbe = Kngm/β
 For better accuracy, measured IBE-VBE data can be directly
used in determining the nonlinearity coefficients.
The ICB Nonlinearity (1)
 The ICB term represents the impact ionization (avalanche
multiplication) current
 ICB = ICE (M-1) = IC0(VBE)FEarly(M-1)
IC0 : IC measured at zero VCB
M : the avalanche multiplication factor
FEarly : Early effect factor
 In SiGe HBT, M is modeled using the empirical “Miller equation”

M
1
1
VCB
VCBO
m
 VCBO and m are two fitting parameters
The ICB Nonlinearity (2)
 At a given VCB, M is constant at low JC where fT and fmax are
very low.
 At higher JC of practical interest, M decreases with increasing
JC, because of decreasing peak electric field in the CB junction
(Kirk effect).
1


3
M 1 
VCB
m
exp(
)
2
VCBO
VCB 3
IC
V
exp( CB )]
I CO
VR
  1  t anh[
 m, VCBO, ICO, VR
are fitting parameters
 also varies with VCB
The ICB Nonlinearity (3)

 The fT and fmax peaks occur near a JC of 1.0-2.0 mA/μm2,
while M-1 starts to decrease at much smaller JC values.
 ICB is controlled by two voltages, VBE(JC) and VCB
2-D power series
 iu = gu uc + K2gu uc2 + K3gu uc3 + …
iv = gv vc + K2gv vc2 + K3gv vc3 + …
iuv = K2gu&gv uc vc + K32gu&gv uc2 vc + K3gu&2gv uc vc2  cross-term
The CBE and CBC Nonlinearity (1)
 The charge storage associated with a nonlinear capacitor
Q
t
f
vC
f
t
f
VC
1
VC
k
vc
t
f
v
vk
k
k 1
t
v VC
vc k
t
 The first-order, second-order, and third-order nonlinearity
coefficients are defined as
C
K2 C
K3 C
f
v
v
1
v
2
3
f
v
v2
2
1
VC
3
f
v
VC
v
VC
v
v3
The CBE and CBC Nonlinearity (2)
 qac(t) = C vc(t) + K2C vc2(t) + K3C vc3(t) + …
 The excess minority carrier charge QD in a SiGe HBT is
proportional to JC through the transit time τf
QD = τf ICE = τf IS exp (qVBE/kT)
CD
f
K2 C D
K3 C D

C D,eff
f
f
qD
v be
gm
qICE
f
kT
K2 gm
f
K3 gm
CD
1
q2 ICE
f
2
kT
2
q3 ICE
6
kT
3
1
qv be
1
q2 v be2
2
kT
6
kT
2
...
nonlinear contributions
The CBE and CBC Nonlinearity (3)
 The EB and CB junction depletion capacitances are often
modeled by
Cdep
C0
Vf
1
Vf
Vi
mj
 C0, Vj, and mj are model parameters
 The CB depletion capacitance is in general much smaller than
the EB depletions capacitance. However, the CB depletion
capacitance is important in determining linearity, because of
its feedback function.
The CBE and CBC Nonlinearity (4)

 Caution must be exercised in identifying whether the absolute
value or the derivative is dominant in determining the transistor
overall linearity.
Linearity
8.1
8.2
8.3
8.4
8.5
Nonlinearity Concept
Physical Nonlinearities
Volterra Series
Single SiGe HBT Amplifier Linearity
Cascode LNA Linearity
Volterra Series - Fundamental Concepts (1)
 A general mathematical approach for solving systems of
nonlinear integral and integral-differential equations.
 An extension of the theory of linear systems to weakly
nonlinear systems.
 The response of a nonlinear
system to an input x(t) is equal
to the sum of the response of
a series of transfer functions
of different orders
( H1, H2, ……, Hn ).
Volterra Series - Fundamental Concepts (2)

Time domain  hn (τ1, τ2,…., τn) is an impulse response
Frequency domain
 Hn ( s1, … , sn ) is the nth-order transfer function
 obtained through a multidimensional Laplace transform
 Hn takes n frequencies as the input, from s1=jω1 to sn=jωn
H n ( s1 ,...,sn ) 




 ( s1 1  s 2 2 ... s n n )
...
h
(

,

,...,

)
e
d 1...d n
  n 1 2 n


H1(s), the first-order transfer function, is essentially the transfer
function of the small-signal linear circuit at dc bias.
Solving the output of a nonlinear circuit is equivalent to solving the
Volterra series H1(s), H2(s1,s2), H3(s1, s2, s3),….
Volterra Series - Fundamental Concepts (3)
 To solve H1(s)
 the nonlinear circuit is first linearized
 solved at s = jω
 requires first-order derivatives
 To solve H2(s1,s2),H3(s1,s2,s3)
 also need the second-order and third-order nonlinearity
coefficients
 The solution of Volterra series is a straightforward case
 the transfer functions can be solved in increasing order by
repeatedly solving the same linear circuit using different
excitation at each order
First-Order Transfer Functions (1)
 Consider a bipolar transistor amplifier with an RC source and
an RL load
 Neglect all of the nonlinear capacitance in the transistor, the
base and emitter resistance, and the avalanche multiplication
current
 Base node “1”, Collector node “2”
Y s H1 s
I1 s
Y(s)  the admittance matrix at frequency s
H1(s)  the vector of the first-order transfer function
I1(s)  a vector of excitations
First-Order Transfer Functions (2)
 By compact modified nodal analysis (CMNA)
 Fig 8.9 to Fig 8.10
 By Kirchoff’s current law
 node 1
Ys V1 Vs
g be
where Ys s
 node 2
g m V1
Y L V2
where Y L s
V1 0
1
Zs s
1
Rs
1
j Cs
0
1
ZL s
1
RL
j LL
First-Order Transfer Functions (3)
 The corresponding matrix
Ys
g be
gm
0
YL
V1
V2
Ys Vs
0
 For an input voltage of unity (Vs = 1)
 V1 and V2 become the transfer functions at node 1,2
Ys
g be
gm
0
YL
H11
H12
s
s
Ys
0
 The firs subscript represents the order of the transfer function,
and the second subscript represents the node number
H11,H12
Second-Order Transfer Functions (1)
 The so-called second-order “virtual nonlinear current sources”
are applied to excite the circuit.
 The circuit responses (nodal voltages) under these virtual
excitations are the second-order transfer functions.
 The virtual current source
 placed in parallel with the corresponding linearized element
 defined for two input frequencies, s1 and s2
 determined by 1) second-order nonlinearity coefficients of
the specific I-V nonlinearity in question
 determined by 2) the first-order transfer function of the
controlling voltage(s)
Second-Order Transfer Functions (2)
 The second-order virtual current source for a I-V nonlinearity
iNL2g(u) = K2g(u) H1u(s1) H1u(s2)
 K2g(u) : second-order nonlinearity coefficient that determines
the second-order response of i to u
 H1u(s) : the first-order transfer function of the controlling
voltage u
Second-Order Transfer Functions (3)
 iNL2gbe = K2gbe H11(s1) H11(s2)
iNL2gm = K2gm H11(s1) H11(s2)
 The controlling voltage vbe is equal to the voltage at node “1,”
because the emitter is grounded.
 The virtual current sources are used to excite the same
linearized circuit, but at a frequency of s1 + s2.
Second-Order Transfer Functions (4)

Y H2 s1, s2
I2
 Y : CMNA admittance matrix at a frequency of s1 + s2
H2 (s1,s2) : second-order transfer function vector
I2 : a linear combination of all the second-order nonlinear
current sources, and can be obtained by applying
Kirchoff’s law at each node

i NL2gbe
Y
g
0
H
s ,s
s
be
gm
21
YL
H22
1
2
s1, s2
i NL2gm
 The admittance matrix remains the same, except for the
evaluation frequency.
Third-Order Transfer Functions (1)

Y H3 s1 , s2, s3
I3
 Y : CMNA admittance matrix at a frequency of s1 + s2 + s3
 H3(s1,s2,s3) : the third-order transfer function
 The third-order virtual current source for a I-V nonlinearity
iNL3g(u) = K3g(u) H1u(s1) H1u(s2) H1u(s3)
+2/3 K2g(u) [ H1u(s1) H2u(s2,s3) + H1u(s2) H2u(s1,s3) + H1u(s3) H2u(s1,s2) ]
K2g(u)  the second-order nonlinearity coefficient
K3g(u)  the third-order nonlinearity coefficient
H1u(s)  the first-order transfer function
H2u(s1,s2)  the second-order transfer function
Third-Order Transfer Functions (2)

iNL3gbe(u) = K3gbe(u) H11(s1) H11(s2) H11(s3)
+2/3 K2gbe(u) [ H11(s1) H21(s2,s3) +
H11(s2) H21(s1,s3) + H11(s3) H21(s1,s2) ]

iNL3gbe(u) = K3gbe(u) H11(s1) H11(s2) H11(s3)
+2/3 K2gbe(u) [ H11(s1) H21(s2,s3) +
H11(s2) H21(s1,s3) + H11(s3) H21(s1,s2) ]

Ys
g be
gm
0
YL
H31 s1, s2 , s3
H32 s1, s2 , s3
i NL3gbe
i NL3gm
Linearity
8.1
8.2
8.3
8.4
8.5
Nonlinearity Concept
Physical Nonlinearities
Volterra Series
Single SiGe HBT Amplifier Linearity
Cascode LNA Linearity
A Single HBT amplifier for
Volterra series analysis

Circuit Analysis



Y ( s )  H 1 ( s )  I1


Y ( s1  s2 )  H 2 ( s1 , s2 )  I 2


Y ( s1  s2  s3 )  H 3 ( s1 , s2 , s3 )  I 3
 Y and I are obtained by applying the Kirchoff’s current law at
every node.
 IIP3 (third-order input intercept) at which the first-order
and third-order signals have equal power
 IIP3 is often expressed in dBm using
IIP3dBm = 10 log (103 IIP3)
Distinguishing Individual Nonlinearities

 The value that gives the lowest IIP3 (the highest distortion)
can be identified as the dominant nonlinearity.
Collector Current Dependence

 For IC > 25mA, the overall IIP3 becomes limited and is
approximately independent of IC.
 Higher IC only increases power consumption, and does not
improve the linearity.
Collector Voltage Dependence (1)

 The optimum IC is at the threshold value.
Collector Voltage Dependence (2)

Load Dependence (1)

 The load dependence results from the CB feedback, due to
the CB capacitance CCB and the avalanche multiplication
current ICB.
 Collector-substrate capacitance (CCS) nonlinearity
 since VCS is a function of the load condition
Load Dependence (2)

 CCB = 0, ICB = 0, note that IIP3 becomes virtually independent
of load condition for all of the nonlinearities except for the
CCS nonlinearity.
Dominant Nonlinearity Versus Bias
 ICB and CCB nonlinearities are the dominant factors for most
of the bias currents and voltages.
 Both ICB and CCB nonlinearities can be decreased by reducing
the collector doping.
 But high collector doping suppresses Kirk effect.
