6.776 High Speed Communication Circuits Lecture 11 Noise Figure, Impact of Amplifier Nonlinearities
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6.776 High Speed Communication Circuits Lecture 11 Noise Figure, Impact of Amplifier Nonlinearities Michael Perrott Massachusetts Institute of Technology March 10, 2005 Copyright © 2005 by Michael H. Perrott Noise Factor and Noise Figure (From Lec 10) Rs Equivalent output referred current noise (assumed to be independent of Zout and ZL) enRs vin vx Zin Linear,Time Invariant Circuit (Noiseless) Definitions Calculation of SNRin and SNRout H.-S. Lee & M.H. Perrott iout inout Zout ZL MIT OCW Alternative Noise Factor Expression Rs Equivalent output referred current noise (assumed to be independent of Zout and ZL) enRs vin vx Zin Linear,Time Invariant Circuit (Noiseless) From previous slide Calculation of Noise Factor H.-S. Lee & M.H. Perrott iout inout Zout ZL MIT OCW Thevenin Computation Model For Noise (from Lec 10) iout D Iout G D G S Zg Zg Zgs vgs Cgs gmvgs indg Zdeg S Zdeg H.-S. Lee & M.H. Perrott MIT OCW Thevenin Computation Model For Noise (from Lec 10) iout D Iout G D G S Zg Zg Zgs vgs Cgs gmvgs indg Zdeg S Zdeg We analyzed a non-degenerated CMOS amplifier in Lec 10 - Broadband amplifiers: gate noise not significant - Narrowband amplifiers: Assumed in analysis that we operated at resonance Q turned out to be key design parameter Can we simultaneously design amplifiers for optimal noise match and optimal power match? H.-S. Lee & M.H. Perrott MIT OCW Input Referred Noise Model Equivalent output referred current noise (assumed to be independent of Zout and ZL) enRs Rs vin vx Zin Linear,Time Invariant Circuit (Noiseless) iout inout Zout ZL en iin is Ys vx in Zin Linear,Time Invariant Circuit (Noiseless) iout en Can remove the signal source since Noise Factor can be expressed as the ratio of total output noise to input noise H.-S. Lee & M.H. Perrott is Ys in iin,sc= iout β MIT OCW Input-Referred Noise Figure Expression en is Ys in iin,sc= iout β We know that Let’s express the above in terms of input short circuit current H.-S. Lee & M.H. Perrott MIT OCW Calculation of Noise Factor en is Ys in By inspection of above figure In general, en and in will be correlated -Y c iout iin,sc= β is called the correlation admittance H.-S. Lee & M.H. Perrott MIT OCW Noise Factor Expressed in Terms of Admittances en is Ys Yc iu iout iin,sc= β We can replace voltage and current noise currents with impedances and admittances H.-S. Lee & M.H. Perrott MIT OCW 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 H.-S. Lee & M.H. Perrott MIT OCW 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 ,B ) in the admittance plane - They are also circles on a Smith Chart (see pp 299-302 opt opt of Gonzalez for derivation and examples) How does (Gopt,Bopt) compare to admittance achieving maximum power transfer? *Guillermo Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, Prentice Hall, 1996 H.-S. Lee & M.H. Perrott MIT OCW Optimizing For Noise Figure versus Power Transfer Signal iin Source Source conductance susceptance is Gs en Bs vx in Zin Source noise produced by source conductance iout Bs Example source admittance for maximum Bopt power transfer Circles of constant Noise Factor (Fmin at the center) Bmax Gmax Gopt Linear,Time Invariant Circuit (Noiseless) Gs One cannot generally achieve minimum noise figure if maximum power transfer is desired H.-S. Lee & M.H. Perrott MIT OCW Optimal Noise Factor for MOS Transistor Amp Consider the common source MOS amp (no degeneration) considered in Lecture 10 - In Tom Lee’s book (pp. 272-276 (1 st ed.), pp. 364-369 (2nd ed.)), the noise impedances are derived as - The optimal source admittance values to minimize noise factor are therefore H.-S. Lee & M.H. Perrott MIT OCW 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 Cgs at frequency of operation Minimum noise figure - Exact if one defines w = g /C t H.-S. Lee & M.H. Perrott m gs MIT OCW Recall Noise Factor Comparison Plot From Lecture 10 Noise Factor Scaling Coefficient Versus Q for 0.18 µ NMOS Device 8 Noise Factor Scaling Coefficient 7 6 c = -j0 c = -j0.55 5 Achievable values as a function of Q under the constraint that 1 = wo LgCgs c = -j1 c = -j0 4 c = -j0.55 3 1 0 Minimum across all values of Q and 1 Note: curves meet if we approximate Q2+1 Q2 2 1 2 H.-S. Lee & M.H. Perrott 3 c = -j1 LgCgs 4 5 6 Q 7 8 9 10 MIT OCW Example: Noise Factor Calculation for Resistor Load Source Source Rs Rs enRs enRL vin RL vout RL Total output noise Total output noise due to source Noise Factor H.-S. Lee & M.H. Perrott vnout MIT OCW Comparison of Noise Figure and Power Match Source Source Rs Rs enRs enRL vin RL vout RL To achieve minimum Noise Factor To achieve maximum power transfer H.-S. Lee & M.H. Perrott vnout MIT OCW Example: Noise Factor Calculation for Capacitor Load Source Source Rs Rs vin CL vout Total output noise Total output noise due to source Noise Factor H.-S. Lee & M.H. Perrott enRs CL vnout MIT OCW Example: Noise Factor with Zero Source Resistance Source RL RL vin CL vout Total output noise Total output noise due to source Noise Factor H.-S. Lee & M.H. Perrott enRL CL vnout MIT OCW Example: Noise Factor Calculation for RC Load Source Source Rs Rs enRs enRL vin CL RL vout Total output noise Total output noise due to source Noise Factor H.-S. Lee & M.H. Perrott CL RL vnout MIT OCW Example: Resistive Load with Source Transformer Source RS Vs Rin= N RL Vx 1 2 Vout=NVx 2 RL Rout=N Rs 1:N Source enRs RS Rin= 1 N enRL Vx 2 Rout=N2Rs RL Vnout=NVx RL 1:N For maximum power transfer (as derived in Lecture 3) H.-S. Lee & M.H. Perrott MIT OCW Noise Factor with Transformer Set for Max Power Transfer Source enRs RS enRL Vx Rin= Rs Rout=RL 1:N= RL RL V Rs x RL Rs Total output noise Total output noise due to source Noise Factor H.-S. Lee & M.H. Perrott Vnout= MIT OCW Observations Source RS enRs enRL Vx Rin= Rs Rout=RL 1:N= Vnout= RL RL V Rs x RL Rs 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 H.-S. Lee & M.H. Perrott MIT OCW 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 Vdd Memoryless Nonlinearity RL Lowpass Filter Vout Vin Id Vin M1 CL Id -RL Vout 1+sRLCL Impact of nonlinearities with sine wave input Impact of nonlinearities with several sine wave inputs - Causes harmonic distortion (i.e., creation of harmonics) - Causes harmonic distortion for each input AND intermodulation products H.-S. Lee & M.H. Perrott MIT OCW 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 H.-S. Lee & M.H. Perrott MIT OCW 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 An Very small for small A, very large for large A H.-S. Lee & M.H. Perrott MIT OCW Frequency Domain View of Harmonic Distortion 3 Memoryless Nonlinearity A 0 3c A Afund = c1A + 3 4 w x y 0 w 2w 3w 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! H.-S. Lee & M.H. Perrott MIT OCW 1 dB Compression Point 3 Memoryless Nonlinearity A 0 3c A Afund = c1A + 3 4 w x Definition: input signal level such that the small-signal gain drops by 1 dB y 0 w 2w 3w 20log(Afund) - Input signal level is high! 1 dB A1-dB 20log(A) 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 H.-S. Lee & M.H. Perrott MIT OCW 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! H.-S. Lee & M.H. Perrott MIT OCW Intermodulation Products Second-order intermodulation (IM2) products Third-order intermodulation (IM3) products - These are the troublesome ones for narrowband systems H.-S. Lee & M.H. Perrott MIT OCW Corruption of Narrowband Signals by Interferers Memoryless Nonlinearity X(w) 0 Interferers Desired Narrowband Signal y W w1 w 2 Y(w) x Corruption of desired signal W w1 w2 2w1 2w2 3w1 3w2 0 w2-w1 2w1-w2 2w2-w1 w1+w2 2w1+w2 2w2+w1 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 H.-S. Lee & M.H. Perrott MIT OCW Use Filtering to Remove Undesired Interference Memoryless Nonlinearity X(w) 0 Interferers w1 w 2 Y(w) Desired Narrowband Signal x y Bandpass Filter z W Corruption of desired signal W w1 w2 2w1 2w2 3w1 3w2 0 w2-w1 2w1-w2 2w2-w1 w1+w2 2w1+w2 2w2+w1 Z(w) Corruption of desired signal W 2w1 2w2 3w1 3w2 w1 w2 0 w2-w1 2w1-w2 2w2-w1 w1+w2 2w1+w2 2w2+w1 Ineffective for IM3 term that falls in the desired signal frequency band H.-S. Lee & M.H. Perrott MIT OCW Characterization of Intermodulation Magnitude of third order products is set by c3 and input signal amplitude (for small A) Magnitude of first order term is set by c1 and A (for small A) Relative impact of intermodulation products can be calculated once we know A and the ratio of c3 to c1 - Problem: it’s often hard to extract the polynomial coefficients through direct DC measurements Need an indirect way to measure the ratio of c3 to c1 H.-S. Lee & M.H. Perrott MIT OCW Two Tone Test Input the sum of two equal amplitude sine waves into the amplifier (assume Zin of amplifier = Rs of source) vin(w) Equal Amplitude Sine Waves 2A 0 w 1 w2 Note: v (w) in vx(w) = 2 W first-order output Vout(w) third-order IM term Amplifier Rs Vx Vout vin Vbias Zin=Rs 2 3 Vout=co+c1Vx+c2Vx+c3Vx W w1 w2 2w1 2w2 3w1 3w2 0 w2-w1 2w1-w2 2w2-w1 w1+w2 2w1+w2 2w2+w1 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 H.-S. Lee & M.H. Perrott MIT OCW 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(Afund) 1 dB First-order output = c1A Third-order 3 c A3 IM term = 4 3 A1-dB Aiip3 20log(A) - 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 H.-S. Lee & M.H. Perrott MIT OCW Relationship between IIP3, c1 and c3 Intersection point Solve for A (gives Aiip3) 20log(Afund) 1 dB First-order output = c1A Third-order 3 c A3 IM term = 4 3 A1-dB Aiip3 20log(A) Note that A corresponds to the peak value of the two cosine waves coming into the amplifier input node (Vx) - Would like to instead like to express IIP3 in terms of power H.-S. Lee & M.H. Perrott MIT OCW IIP3 Expressed in Terms of Power at Source IIP3 referenced to Vx (peak voltage) vin(w) 0 Note: v (w) in vx(w) = 2 Equal Amplitude Sine Waves 2A w 1 w2 W Rs Amplifier Vx Vout vin IIP3 referenced to Vx (rms voltage) Vbias Zin=Rs 2 3 Vout=co+c1Vx+c2Vx+c3Vx Power across Zin = Rs H.-S. Lee & M.H. Perrott Note: Power from vin MIT OCW 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 theoretically not necessary – simply curve fit to a third order polynomial H.-S. Lee & M.H. Perrott MIT OCW Impact of Differential Amplifiers on Nonlinearity I1 vid 2 Memoryless Nonlinearity I2 M1 M2 vx -vid 2 vid Idiff = I2-I1 2Ibias Assume vx is approximately incremental ground Second order term removed and IIP3 increased! H.-S. Lee & M.H. Perrott MIT OCW
