1. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

2. 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

3. 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

4. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

5. Harmonics (1)

6. Harmonics (2)  An “nth-order harmonic term” is proportional to A n  HD2(second harmonic distortion) = / = ( neglect 3k 3 A 3 /4 term)  IHD2 ( the extrapolation of the output at 2ω and ω intersect)  obtained 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 =

7. Gain Compression and Expansion (1)  The small-signal gain is obtained by neglecting the harmonics. The small-signal gain ： k 1 The nonlinearity-induced term ： 3k 3 A 3 /4  As the signal amplitude A grows, becomes comparable to or even larger than k 1 A  the variation of gain changes with input  fundamental manifestation of nonlinearity  If k 3 < 0, then 3k 3 A 3 /4 < 0  the gain decreases with increasing input level (A)  “gain compression” in many RF circuits  quantified by the “1 dB compression point,” or P 1dB

8.  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)

9. Intermodulation (1)  A two-tone input voltage x(t) = Acosω 1 t +Acosω 2 t  The output has  not only harmonics of ω 1 and ω 2  but also “intermodution products” at 2ω 1 -ω 2 and 2ω 2 -ω 1 (major concerns, close in frequency to ω 1 and ω 2 )

10. 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)

11. 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 = A 2 / IIP3 2  IIP3 can be measured by A 0, IM3 0 IIP3 2 = A 0 2 / IM3 0  IIP3, A 0  voltage IIP3 2, A 0 2  power ( taking 10 log on both side )  20 log IIP3 = 20 log A 0 – 10 log IM3 0  P IIP3 = P in + ½( P o,1st – P o,3rd )

12. Intermodulation (4)  OIP3 = k 1 *IIP3  OIP3 2 = k 1 2 *IIP3 2  IIP3 2 = OIP3 2 / k 1 2 = A 2 /IM3 2  OIP3 2 = (k 1 A) 2 /IM3 2 ( taking 10 log on both side )  20 log OIP3 = 20 log k 1 A – 10 log IM3  P OIP3 = P o,1st + ½( P o,1st – P o,3rd )  The gain compression at very high input power level can be seen

13. 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 / P dc ( P dc = the dc power dissipation )  excellent linearity efficiency above 10 for first generation HBTs  competitive with Ⅲ - Ⅴ technologies

14. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

15. Physical Nonlinearities in a SiGe HBT  I CE  the collector current transported from the emitter  the I CE -V BE nonlinearity is a nonlinear transconductance  I BE  the hole injection into the emitter  also a nonlinear function of V BE.  I CB  the avalanche multiplication current  a strong nonlinear function of both V BE and V CB  has a 2-D nonlinearity because is has two controlling voltages.  C BE  the EB junction capacitance  includes the diffusion capacitance and depletion capacitance  a strong nonlinear function of V BE when the diffusion capacitance dominates, because diffusion charge is proportional to the I CE  C BC  the CB junction capacitance

16. Equivalent circuit of the HBT 

17. The I CE Nonlinearity (1)   i(t) : the sum of the dc and ac currents v c (t) : the ac voltage which controls the conductance V C : the dc controlling (bias) voltage  For small v c (t), considering the first three terms of the power series is usually sufficient. 

18. The I CE Nonlinearity (2)  The ac current-voltage relation can be rewritten i ac (t) = g v c (t) + K 2g v c 2 (t) + K 3g v c 3 (t) + … g : the small-signal transconductance K 2g : the second-order nonlinearity coefficient K 3g : the third-order nonlinearity coefficient  For an ideal SiGe HBT, I CE increases exponentially with V BE I CE = I S exp (qV BE /kT) 

19. The I CE Nonlinearity (3)   The nonlinear contributions to g m,eff increase with v be.  Improve linearity by decreasing v be.

20. The I BE Nonlinearity  For a constant current gain β I BE = I CE /β g be = g m /β K 2gbe = K 2gm /β K 3gbe = K 3gm /β K ngbe = K ngm /β  For better accuracy, measured I BE -V BE data can be directly used in determining the nonlinearity coefficients.

21. The I CB Nonlinearity (1)  The I CB term represents the impact ionization (avalanche multiplication) current  I CB = I CE (M-1) = I C0 (V BE )F Early (M-1) I C0 : I C measured at zero V CB M : the avalanche multiplication factor F Early : Early effect factor  In SiGe HBT, M is modeled using the empirical “Miller equation”   V CBO and m are two fitting parameters

22. The I CB Nonlinearity (2)  At a given V CB, M is constant at low J C where f T and f max are very low.  At higher J C of practical interest, M decreases with increasing J C, because of decreasing peak electric field in the CB junction (Kirk effect).   m, V CBO, I CO, V R are fitting parameters  also varies with V CB

23. The I CB Nonlinearity (3)   The f T and f max peaks occur near a J C of 1.0-2.0 mA/μm 2, while M-1 starts to decrease at much smaller J C values.  I CB is controlled by two voltages, V BE (J C ) and V CB  2-D power series  i u = g u u c + K 2gu u c 2 + K 3gu u c 3 + … i v = g v v c + K 2gv v c 2 + K 3gv v c 3 + … i uv = K 2gu&gv u c v c + K 32gu&gv u c 2 v c + K 3gu&2gv u c v c 2  cross-term

24. The C BE and C BC Nonlinearity (1)  The charge storage associated with a nonlinear capacitor  The first-order, second-order, and third-order nonlinearity coefficients are defined as

25. The C BE and C BC Nonlinearity (2)  q ac (t) = C v c (t) + K 2C v c 2 (t) + K 3C v c 3 (t) + …  The excess minority carrier charge Q D in a SiGe HBT is proportional to J C through the transit time τ f Q D = τ f I CE = τ f I S exp (qV BE /kT) 

26. The C BE and C BC Nonlinearity (3)  The EB and CB junction depletion capacitances are often modeled by  C 0, V j, and m j 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.

27. The C BE and C BC Nonlinearity (4)   Caution must be exercised in identifying whether the absolute value or the derivative is dominant in determining the transistor overall linearity.

28. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

29. 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 ( H 1, H 2, ……, H n ).

30. Volterra Series - Fundamental Concepts (2)  Time domain  h n (τ 1, τ 2,…., τ n ) is an impulse response Frequency domain  H n ( s 1, …, s n ) is the nth-order transfer function  obtained through a multidimensional Laplace transform  H n takes n frequencies as the input, from s 1 =jω 1 to s n =jω n  H 1 (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 H 1 (s), H 2 (s 1,s 2 ), H 3 (s 1, s 2, s 3 ),….

31. Volterra Series - Fundamental Concepts (3)  To solve H 1 (s)  the nonlinear circuit is first linearized  solved at s = jω  requires first-order derivatives  To solve H 2 (s 1,s 2 ),H 3 (s 1,s 2,s 3 )  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

32. 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 H 1 (s)  the vector of the first-order transfer function I 1 (s)  a vector of excitations

33. 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

34. First-Order Transfer Functions (3)  The corresponding matrix  For an input voltage of unity (Vs = 1)  V 1 and V 2 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 number H 11,H 12

35. 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, s 1 and s 2  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)

36. Second-Order Transfer Functions (2)  The second-order virtual current source for a I-V nonlinearity i NL2g (u) = K 2g (u) H 1u (s 1 ) H 1u (s 2 )  K 2g (u) : second-order nonlinearity coefficient that determines the second-order response of i to u  H 1u (s) : the first-order transfer function of the controlling voltage u

37. Second-Order Transfer Functions (3)  i NL2gbe = K 2gbe H 11 (s 1 ) H 11 (s 2 ) i NL2gm = K 2gm H 11 (s 1 ) H 11 (s 2 )  The controlling voltage v be 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 s 1 + s 2.

38. Second-Order Transfer Functions (4)   Y : CMNA admittance matrix at a frequency of s 1 + s 2 H 2 (s 1,s 2 ) : second-order transfer function vector I 2 : 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.

39. Third-Order Transfer Functions (1)   Y : CMNA admittance matrix at a frequency of s 1 + s 2 + s 3  H 3 (s 1,s 2,s 3 ) : the third-order transfer function  The third-order virtual current source for a I-V nonlinearity i NL3g (u) = K 3g (u) H 1u (s 1 ) H 1u (s 2 ) H 1u (s 3 ) +2/3 K 2g (u) [ H 1u (s 1 ) H 2u (s 2,s 3 ) + H 1u (s 2 ) H 2u (s 1,s 3 ) + H 1u (s 3 ) H 2u (s 1,s 2 ) ] K 2g (u)  the second-order nonlinearity coefficient K 3g (u)  the third-order nonlinearity coefficient H 1u (s)  the first-order transfer function H 2u (s 1,s 2 )  the second-order transfer function

40. Third-Order Transfer Functions (2)  i NL3gbe (u) = K 3gbe (u) H 11 (s 1 ) H 11 (s 2 ) H 11 (s 3 ) +2/3 K 2gbe (u) [ H 11 (s 1 ) H 21 (s 2,s 3 ) + H 11 (s 2 ) H 21 (s 1,s 3 ) + H 11 (s 3 ) H 21 (s 1,s 2 ) ] 

41. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

42. A Single HBT amplifier for Volterra series analysis 

43. 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 using IIP3 dBm = 10 log (10 3 IIP3)

44. Distinguishing Individual Nonlinearities   The value that gives the lowest IIP3 (the highest distortion) can be identified as the dominant nonlinearity.

45. Collector Current Dependence   For I C > 25mA, the overall IIP3 becomes limited and is approximately independent of I C.  Higher I C only increases power consumption, and does not improve the linearity.

46. Collector Voltage Dependence (1)   The optimum I C is at the threshold value.

47. Collector Voltage Dependence (2) 

48. Load Dependence (1)   The load dependence results from the CB feedback, due to the CB capacitance C CB and the avalanche multiplication current I CB.  Collector-substrate capacitance (C CS ) nonlinearity  since V CS is a function of the load condition

49. Load Dependence (2)   C CB = 0, I CB = 0, note that IIP3 becomes virtually independent of load condition for all of the nonlinearities except for the C CS nonlinearity.

50. Dominant Nonlinearity Versus Bias  I CB and C CB nonlinearities are the dominant factors for most of the bias currents and voltages.  Both I CB and C CB nonlinearities can be decreased by reducing the collector doping.  But high collector doping suppresses Kirk effect. 