6.976 High Speed Communication Circuits and Systems Lecture 8 Noise Figure, Impact of Amplifier Nonlinearities

Transcription

1 6.976 High Speed Communication Circuits and Systems Lecture 8 Noise Figure, Impact of Amplifier Nonlinearities Michael Perrott Massachusetts Institute of Technology Copyright 2003 by Michael H. Perrott

2 Noise Factor and Noise Figure (From Lec 7) R s e nrs Equivalent output referred current noise (assumed to be independent of Z out and Z L ) Definitions v in Linear,Time Invariant Circuit (Noiseless) Z out i nout Z in v x Z L i out Calculation of SNR in and SNR out

3 Alternative Noise Factor Expression R s e nrs Equivalent output referred current noise (assumed to be independent of Z out and Z L ) v in Linear,Time Invariant Circuit (Noiseless) Z out i nout Z in v x Z L i out From previous slide Calculation of Noise Factor

4 Input Referred Noise Model R s e nrs Equivalent output referred current noise (assumed to be independent of Z out and Z L ) v in Linear,Time Invariant v x Circuit i nout Z in (Noiseless) Z out Z L i out e n i in i s Y s i n Z in v x Linear,Time Invariant Circuit (Noiseless) i out e n Can remove the signal source since Noise Factor can be expressed as the ratio of total output noise to input noise i s Y s i n i in,sc = i out β

5 Input-Referred Noise Figure Expression e n i s Y s i n i in,sc = i out β We know that Let s express the above in terms of input short circuit current

6 Calculation of Noise Factor e n i s Y s i n i in,sc = i out β By inspection of above figure In general, e n and i n will be correlated - Y c is called the correlation admittance

7 Noise Factor Expressed in Terms of Admittances e n i s Y s Y c i u i in,sc = i out β We can replace voltage and current noise currents with impedances and admittances

8 Optimal Source Admittance for Minimum Noise Factor Express admittances as the sum of conductance, G, and susceptance, B Take the derivative with respect to source admittance and set to zero (to find minimum F), which yields Plug these values into expression above to obtain

9 Optimal Source Admittance for Minimum Noise Factor After much algebra (see Appendix L of Gonzalez book for derivation), we can derive - Contours of constant noise factor are circles centered about (G opt,b opt ) in the admittance plane - They are also circles on a Smith Chart (see pp of Gonzalez for derivation and examples) How does (G opt,b opt ) compare to admittance achieving maximum power transfer?

10 Optimizing For Noise Figure versus Power Transfer Signal Source conductance Source susceptance e n i in i s G s B s i n Z in v x Linear,Time Invariant Circuit (Noiseless) i out Source noise produced by source conductance B s Example source admittance for maximum power transfer B opt B max Circles of constant Noise Factor (F min at the center) G max G opt G s One cannot generally achieve minimum noise figure if maximum power transfer is desired

11 Optimal Noise Factor for MOS Transistor Amp Consider the common source MOS amp (no degeneration) considered in Lecture 7 - In Tom Lee s book (pp ), the noise impedances are derived as - The optimal source admittance values to minimize noise factor are therefore

12 Optimal Noise Factor for MOS Transistor Amp (Cont.) Optimal admittance consists of a resistor and inductor (wrong frequency behavior broadband match fundamentally difficult) - If there is zero correlation, inductor value should be set to resonate with C gs at frequency of operation Minimum noise figure - Exact if one defines w t = g m /C gs

13 Recall Noise Factor Comparison Plot From Lecture 7 Noise Factor Scaling Coefficient Versus Q for 0.18 µ NMOS Device 8 Noise Factor Scaling Coefficient c = -j0 c = -j0.55 c = -j1 Note: curves meet if we approximate Q 2 +1 Q 2 Achievable values as a function of Q under the constraint that 1 = w o L g C gs Minimum across all values of Q and 1 L g C gs c = -j0 c = -j0.55 c = -j Q

14 Example: Noise Factor Calculation for Resistor Load Source R s Source R s e nrs e nrl v in R L v out v nout R L Total output noise Total output noise due to source Noise Factor

15 Comparison of Noise Figure and Power Match Source R s Source R s e nrs e nrl v in R L v out v nout R L To achieve minimum Noise Factor To achieve maximum power transfer

16 Example: Noise Factor Calculation for Capacitor Load Source R s Source R s e nrs v in C L v out C L v nout Total output noise Total output noise due to source Noise Factor

17 Example: Noise Factor with Zero Source Resistance Source R L R L e nrl v in C L v out C L v nout Total output noise Total output noise due to source Noise Factor

18 Example: Noise Factor Calculation for RC Load Source R s Source R s e nrs e nrl v in C L R L v out C L v nout R L Total output noise Total output noise due to source Noise Factor

19 Example: Resistive Load with Source Transformer R S Source V s V x V out =NV x 1 R in = RL 2 R out =N 2 R s N R L 1:N Source R S e nrs V x 1:N e nrl 1 R in = RL 2 R out =N 2 V nout =NV x R s N For maximum power transfer (as derived in Lecture 3) R L

20 Noise Factor with Transformer Set for Max Power Transfer Source R S e nrs V x e nrl R in =R s R out =R L V nout = V x R L R L R s Total output noise 1:N= R L R s Total output noise due to source Noise Factor

21 Observations Source R S e nrs V x e nrl R in =R s R out =R L V nout = V x R L R L R s 1:N= R L R s If you need to power match to a resistive load, you must pay a 3 db penalty in Noise Figure - A transformer does not alleviate this issue What value does a transformer provide? - Almost-true answer: maximizes voltage gain given the power match constraint, thereby reducing effect of noise of following amplifiers - Accurate answer: we need to wait until we talk about cascaded noise factor calculations

22 Nonlinearities in Amplifiers We can generally break up an amplifier into the cascade of a memoryless nonlinearity and an input and/or output transfer function V dd R L V out Memoryless Nonlinearity Lowpass Filter V in I d -RL V out V in M 1 I d C L 1+sR L C L Impact of nonlinearities with sine wave input - Causes harmonic distortion (i.e., creation of harmonics) Impact of nonlinearities with several sine wave inputs - Causes harmonic distortion for each input AND intermodulation products

23 Analysis of Amplifier Nonlinearities Focus on memoryless nonlinearity block - The impact of filtering can be added later Memoryless Nonlinearity x y Model nonlinearity as a Taylor series expansion up to its third order term (assumes small signal variation) - For harmonic distortion, consider - For intermodulation, consider

24 Harmonic Distortion Substitute x(t) into polynomial expression Fundamental Harmonics Notice that each harmonic term, cos(nwt), has an amplitude that grows in proportion to A n - Very small for small A, very large for large A

25 Frequency Domain View of Harmonic Distortion A Memoryless Nonlinearity A fund = c 1 A + 3c 3 A 3 4 x 0 w 0 w 2w 3w y Harmonics cause noise - Their impact depends highly on application LNA typically not of consequence Power amp can degrade spectral mask Audio amp depends on your listening preference! Gain for fundamental component depends on input amplitude!

26 1 db Compression Point A Memoryless Nonlinearity A fund = c 1 A + 3c 3 A 3 4 x 0 w 0 w 2w 3w y 20log(A fund ) Definition: input signal level 1 db such that the small-signal gain drops by 1 db - Input signal level is high! 20log(A) A 1-dB Typically calculated from simulation or measurement rather than analytically - Analytical model must include many more terms in Taylor series to be accurate in this context

27 Harmonic Products with An Input of Two Sine Waves DC and fundamental components Second and third harmonic terms Similar result as having an input with one sine wave - But, we haven t yet considered cross terms!

28 Intermodulation Products Second-order intermodulation (IM2) products Third-order intermodulation (IM3) products - These are the troublesome ones for narrowband systems

29 Corruption of Narrowband Signals by Interferers Memoryless Nonlinearity X(w) Interferers Desired Narrowband Signal x y W 0 w 1 w 2 Y(w) Corruption of desired signal 0 w 2 -w 1 w 1 w 2 2w 1 2w 2 3w 1 3w 2 2w 1 -w 2 2w 2 -w 1 w 1 +w 2 2w 1 +w 2 2w 2 +w 1 W Wireless receivers must select a desired signal that is accompanied by interferers that are often much larger - LNA nonlinearity causes the creation of harmonic and intermodulation products - Must remove interference and its products to retrieve desired signal

30 Use Filtering to Remove Undesired Interference Memoryless Nonlinearity X(w) Interferers Desired Narrowband Signal W 0 w 1 w 2 x y Bandpass Filter z Y(w) Corruption of desired signal 0 w 2 -w 1 w 1 w 2 2w 1 2w 2 3w 1 3w 2 2w 1 -w 2 2w 2 -w 1 w 1 +w 2 2w 1 +w 2 2w 2 +w 1 W Z(w) Corruption of desired signal 0 w 2 -w 1 w 1 w 2 2w 1 2w 2 3w 1 3w 2 2w 1 -w 2 2w 2 -w 1 w 1 +w 2 2w 1 +w 2 2w 2 +w 1 Ineffective for IM3 term that falls in the desired signal frequency band W

31 Characterization of Intermodulation Magnitude of third order products is set by c 3 and input signal amplitude (for small A) Magnitude of first order term is set by c 1 and A (for small A) Relative impact of intermodulation products can be calculated once we know A and the ratio of c 3 to c 1 - Problem: it s often hard to extract the polynomial coefficients through direct DC measurements Need an indirect way to measure the ratio of c 3 to c 1

32 Two Tone Test Input the sum of two equal amplitude sine waves into the amplifier (assume Z in of amplifier = R s of source) v in (w) Equal Amplitude Sine Waves 2A Note: v x (w) = v in(w) 2 R s V x Amplifier V out w 1 0 w 2 W V out (w) first-order output third-order IM term v in V bias Z in =R s 2 3 V out =c o +c 1 V x +c 2 V x +c 3 V x 0 w 2 -w 1 w 1 w 2 2w 1 2w 2 3w 1 3w 2 2w 1 -w 2 2w 2 -w 1 w 1 +w 2 2w 1 +w 2 2w 2 +w 1 On a spectrum analyzer, measure first order and third order terms as A is varied (A must remain small) - First order term will increase linearly - Third order IM term will increase as the cube of A W

33 Input-Referred Third Order Intercept Point (IIP3) Plot the results of the two-tone test over a range of A (where A remains small) on a log scale (i.e., db) - Extrapolate the results to find the intersection of the first and third order terms 20log(A fund ) First-order 1 db output = c 1 A Third-order 3 c IM term = 3 A log(A) A 1-dB A iip3 - IIP3 defined as the input power at which the extrapolated lines intersect (higher value is better) Note that IIP3 is a small signal parameter based on extrapolation, in contrast to the 1-dB compression point

34 Relationship between IIP3, c 1 and c 3 Intersection point 20log(A fund ) Solve for A (gives A iip3 ) First-order 1 db output = c 1 A Third-order 3 c IM term = 3 A log(A) A 1-dB A iip3 Note that A corresponds to the peak value of the two cosine waves coming into the amplifier input node (V x ) - Would like to instead like to express IIP3 in terms of power

35 IIP3 Expressed in Terms of Power at Source IIP3 referenced to V x (peak voltage) v in (w) Equal Amplitude Sine Waves 2A Note: v x (w) = v in(w) 2 Amplifier W 0 w 1 w 2 R s V x V out IIP3 referenced to V x (rms voltage) v in V bias Z in =R s 2 3 V out =c o +c 1 V x +c 2 V x +c 3 V x Power across Z in = R s Note: Power from v in

36 IIP3 as a Benchmark Specification Since IIP3 is a convenient parameter to describe the level of third order nonlinearity in an amplifier, it is often quoted as a benchmark spec Measurement of IIP3 on a discrete amplifier would be done using the two-tone method described earlier - This is rarely done on integrated amplifiers due to poor access to the key nodes - Instead, for a radio receiver for instance, one would simply put in interferers and see how the receiver does Note: performance in the presence of interferers is not just a function of the amplifier nonlinearity Calculation of IIP3 is most easily done using a simulator such as Hspice or Spectre - Two-tone method is not necessary simply curve fit to a third order polynomial - Note: two-tone can be done in CppSim

37 Impact of Differential Amplifiers on Nonlinearity v id I 1 I 2 2 M 1 M 2 v x -v id 2 Memoryless Nonlinearity v id I diff = I 2 -I 1 2I bias Assume v x is approximately incremental ground Second order term removed and IIP3 increased!