linearity 8.1 nonlinearity concept 8.2 physical nonlinearities 8.3 volterra series 8.4 single sige...
TRANSCRIPT
Linearity8.1 Nonlinearity Concept8.2 Physical Nonlinearities8.3 Volterra Series8.4 Single SiGe HBT Amplifier Linearity8.5 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
Linearity8.1 Nonlinearity Concept8.2 Physical Nonlinearities8.3 Volterra Series8.4 Single SiGe HBT Amplifier Linearity8.5 Cascode LNA Linearity
Harmonics (1)
Input:xt Acost
Ouput:yt k1xt k2x2t k3x3t k1Acost k2A2cos2t k3A3cos3t
k2A2
2dcshift
k1A
3k3A3
4cost fundamental
k2A2
2cos2t secondharmonic
k3A3
4cos3t thirdharmonic
Harmonics (2) An “nth-order harmonic term” is proportional to An
HD2(second harmonic distortion) = / =
( neglect 3k3A3/4 term) IHD2 ( the extrapolation of the output at 2ω and ω intersect) obtain
ed by letting HD2 = 1
= 1 A = IHD2 = 2
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 =
k2A2
2k1A
12
k2k1
Ak1k2
12
k2k1
A
2k12
k2k1k2
Gain Compression and Expansion (1)
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
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.
Gain Compression and Expansion (2)
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
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)
ytk1A3k3A3
43k3A3
2cos1t ... fundamental
3k3A3
4cos22 1t ... intermodulation
IM33k3A3
4k1A
34
k3k1
A2
Intermodulation (3) IIP3 ( input third-order intercept point) is obtained by letting
IM3 = 1
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 = A0
2 / 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 )
IM334
k3k1
A2 1 A IIP343
k1k3
Intermodulation (4) OIP3 = k1*IIP3 OIP32 = k1
2*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 low-noise
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
Linearity8.1 Nonlinearity Concept8.2 Physical Nonlinearities8.3 Volterra Series8.4 Single SiGe HBT Amplifier Linearity8.5 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) : the sum of the dc and ac currentsvc(t) : the ac voltage which controls the conductanceVC : the dc controlling (bias) voltage
For small vc(t), considering the first three terms of the power series is usually sufficient.
it fvCt fVC vct fVC
k1
1k
k fvt
vkvVC vc
kt
gfv
vvVC K2g
12
2 fv
v2vVC
K3g 13
3 fv
v3vVC Kng
1n
n fv
vnvVC
The ICE Nonlinearity (2)
The ac current-voltage relation can be rewritteniac(t) = g vc(t) + K2g vc
2(t) + K3g vc3(t) + …
g : the small-signal transconductanceK2g : the second-order nonlinearity coefficientK3g : the third-order nonlinearity coefficient
For an ideal SiGe HBT, ICE increases exponentially with VBE
ICE = IS exp (qVBE/kT) gm
qICEkT
K2gm 12
q2ICEkT2
K3gm 13
q3ICEkT3 Kngm
1n
qnICEkTn
The ICE Nonlinearity (3)
The nonlinear contributions to gm,eff increase with vbe. Improve linearity by decreasing vbe.
gm,eff icvbe
gm1
12
qvbekT
16
q2vbe2kT2 ...
nonlinearcontributions
The IBE Nonlinearity For a constant current gain β
IBE = ICE/βgbe = 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 factorFEarly : Early effect factor
In SiGe HBT, M is modeled using the empirical “Miller equation”
VCBO and m are two fitting parameters
M1
1VCBVCBOm
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).
m, VCBO, ICO, VR are fitting parameters also varies with VCB
)]exp(tanh[1
)exp(13
2
3
1
R
CB
CO
C
CBCBO
CB
V
V
I
I
V
m
V
VM
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 VCB2-D power series
iu = gu uc + K2gu uc2 + K3gu uc
3 + …iv = gv vc + K2gv vc
2 + K3gv vc3 + …
iuv = K2gu&gv uc vc + K32gu&gv uc2 vc + K3gu&2gv uc vc
2 cross-term
The CBE and CBC Nonlinearity (1)
The charge storage associated with a nonlinear capacitor
The first-order, second-order, and third-order nonlinearity coefficients are defined as
Qt fvCt fVC vct fVC
k1
1k
k fvt
vkvVC vc
ktC
fv v
vVC
K2C 12
2 fv
v2vVC
K3C 13
3 fv
v3vVC
The CBE and CBC Nonlinearity (2)
qac(t) = C vc(t) + K2C vc2(t) + K3C vc
3(t) + … The excess minority carrier charge QD in a SiGe HBT is proporti
onal to JC through the transit time τf
QD = τf ICE = τf IS exp (qVBE/kT)
CD fgm fqICEkT
K2CD f K2gm fq2ICE2kT2
K3CD f K3gm fq3ICE6kT3
CD,eff qDvbe
CD1
12
qvbekT
16
q2vbe2kT2 ...
nonlinearcontributions
The CBE and CBC Nonlinearity (3)
The EB and CB junction depletion capacitances are often modeled by
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.
CdepVf C01VfVimj
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.
Linearity8.1 Nonlinearity Concept8.2 Physical Nonlinearities8.3 Volterra Series8.4 Single SiGe HBT Amplifier Linearity8.5 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
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),….
nsss
nn
nn
ddeh
ssH
nn ...),...,,(...
),...,(
1)...(
21
1
2211
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) the admittance matrix at frequency s H1(s) the vector of the first-order transfer functionI1(s) a vector of excitations
YsH1 s I1
s
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
node 2
YsV1 Vs gbeV1 0
where Yss 1Zss
1
Rs 1jCs
gmV1 YLV2 0
where YLs 1ZLs
1RL jLL
First-Order Transfer Functions (3) The corresponding matrix
For an input voltage of unity (Vs = 1) V1 and V2 become the transfer functions at node 1,2
The firs subscript represents the order of the transfer function,and the second subscript represents the node numberH11,H12
Ys gbe 0gm YL
V1V2YsVs
0
Ys gbe 0gm YL
H11sH12sYs0
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
nonlinearityiNL2g(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 : CMNA admittance matrix at a frequency of s1 + s2
H2 (s1,s2) : second-order transfer function vectorI2 : a linear combination of all the second-order nonlinear current sources, and can be obtained by applying Kirchoff’s law at each node
The admittance matrix remains the same, except for the evaluation frequency.
Y H2
s1, s2 I2
Ys gbe 0gm YL
H21s1, s2H22s1, s2iNL2gbeiNL2gm
Third-Order Transfer Functions (1)
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 coefficientK3g(u) the third-order nonlinearity coefficient H1u(s) the first-order transfer functionH2u(s1,s2) the second-order transfer function
Y H3
s1, s2, s3 I3
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 gbe 0gm YL
H31s1, s2, s3H32s1, s2, s3iNL3gbeiNL3gm
Linearity8.1 Nonlinearity Concept8.2 Physical Nonlinearities8.3 Volterra Series8.4 Single SiGe HBT Amplifier Linearity8.5 Cascode LNA Linearity
A Single HBT amplifier for Volterra series analysis
Circuit Analysis
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 usingIIP3dBm = 10 log (103 IIP3)
33213321
221221
11
),,()(
),()(
)()(
IsssHsssY
IssHssY
IsHsY
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.