linearity 8.1 nonlinearity concept 8.2 physical nonlinearities 8.3 volterra series 8.4 single sige...

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

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

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 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 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


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 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


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 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


Caution must be exercised in identifying whether the absolute value or the derivative is dominant in determining the transistor overall 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

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.

linearity 8.1 nonlinearity concept 8.2 physical nonlinearities 8.3 volterra series 8.4 single sige...

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