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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 1 tanh[ IC V exp( CB )] I CO VR 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 ) ... h ( , n 1 2 ,..., n )e ( s1 1 s2 2 ... sn n ) d 1...d 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.