6.976 High Speed Communication Circuits and Systems Lecture 7 Noise Modeling in Amplifiers
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6.976 High Speed Communication Circuits and Systems Lecture 7 Noise Modeling in Amplifiers Michael Perrott Massachusetts Institute of Technology Copyright © 2003 by Michael H. Perrott Notation for Mean, Variance, and Correlation Consider random variables x and y with probability density functions fx(x) and fy(y) and joint probability function fxy(x,y) - Expected value (mean) of x is - Note: we will often abuse notation and denote as a random variable (i.e., noise) rather than its mean The variance of x (assuming it has zero mean) is - A useful statistic is If the above is zero, x and y are said to be uncorrelated M.H. Perrott MIT OCW Relationship Between Variance and Spectral Density Two-Sided Spectrum One-Sided Spectrum Sx(f) Sx(f) 2A A -f2 -f1 0 f1 f2 f 0 f1 f2 f Two-sided spectrum - Since spectrum is symmetric One-sided spectrum defined over positive frequencies - Magnitude defined as twice that of its corresponding two-sided spectrum M.H. Perrott In the next few lectures, we assume a one-sided spectrum for all noise analysis MIT OCW The Impact of Filtering on Spectral Density |H(f)|2 Sx(f) AB B A 0 f f 0 x(t) Sy(f) H(f) 0 f y(t) For the random signal passing through a linear, time-invariant system with transfer function H(f) - We see that if x(t) is amplified by gain A, we have M.H. Perrott MIT OCW Noise in Resistors Can be described in terms of either voltage or current R R in en k is Boltzmann’s constant T is temperature (in Kelvins) - Usually assume room temperature of 27 degrees Celsius M.H. Perrott MIT OCW Noise In Inductors and Capacitors Ideal capacitors and inductors have no noise! C L In practice, however, they will have parasitic resistance - Induces noise - Parameterized by adding resistances in parallel/series with inductor/capacitor Include parasitic resistor noise sources M.H. Perrott MIT OCW Noise in CMOS Transistors (Assumed in Saturation) ID G D Drain Noise (Thermal and 1/f) S Transistor Noise Sources Gate Noise (Induced and Routing Parasitic) Modeling of noise in transistors must include several noise sources - Drain noise Thermal and 1/f – influenced by transistor size and bias - Gate noise Induced from channel – influenced by transistor size and bias Caused by routing resistance to gate (including resistance of polysilicon gate) Can be made negligible with proper layout such as fingering of devices M.H. Perrott MIT OCW Drain Noise – Thermal (Assume Device in Saturation) ind G VGS VD>∆V S D Thermally agitated carriers in the channel cause a randomly varying current - γ is called excess noise factor = 2/3 in long channel = 2 to 3 (or higher!) in short channel NMOS (less in PMOS) gdo will be discussed shortly M.H. Perrott 2 ind ∆f 4kTγgdo f MIT OCW Drain Noise – 1/f (Assume Device in Saturation) ind G VGS VD>∆V S Traps at channel/oxide interface randomly capture/release carriers D 2 ind ∆f - Parameterized by Kf and n 4kTγgdo Provided by fab (note n ≈ 1) Currently: Kf of PMOS << Kf of NMOS due to buried channel To minimize: want large area (high WL) M.H. Perrott drain 1/f noise drain thermal noise f 1/f noise corner frequency MIT OCW Induced Gate Noise (Assume Device in Saturation) ing indg G VGS VD>∆V S Fluctuating channel potential couples capacitively into the gate terminal, causing a noise gate current D 2 ing ∆f slope = 20 dB/decade 4kTδgdo - δ is gate noise coefficient Typically assumed to be 2γ - Correlated to drain noise! M.H. Perrott f 5 ft α MIT OCW Useful References on MOSFET Noise Thermal Noise - B. Wang et. al., “MOSFET Thermal Noise Modeling for Analog Integrated Circuits”, JSSC, July 1994 Gate Noise - Jung-Suk Goo, “High Frequency Noise in CMOS Low - M.H. Perrott Noise Amplifiers”, PhD Thesis, Stanford University, August 2001 http://www-tcad.stanford.edu/tcad/pubs/theses/goo.pdf Jung-Suk Goo et. al., “The Equivalence of van der Ziel and BSIM4 Models in Modeling the Induced Gate Noise of MOSFETS”, IEDM 2000, 35.2.1-35.2.4 Todd Sepke, “Investigation of Noise Sources in Scaled CMOS Field-Effect Transistors”, MS Thesis, MIT, June 2002 http://www-mtl.mit.edu/research/sodini/sodinitheses.html MIT OCW Drain-Source Conductance: gdo gdo is defined as channel resistance with Vds=0 - Transistor in triode, so that - Equals g m for long channel devices Key parameters for 0.18µ NMOS devices M.H. Perrott MIT OCW Plot of gm and gdο versus Vgs for 0.18µ NMOS Device 4 Vgs M1 1.8µ W = L 0.18µ 3.5 Transconductance (milliAmps/Volts) Id Transconductances gm and gdo versus Gate Voltage Vgs 3 gd0=µnCoxW/L(Vgs-VT) 2.5 2 1.5 gm (simulated in Hspice) 1 0.5 0 0.4 0.6 0.8 1 1.2 1.4 Gate Voltage Vgs (Volts) 1.6 1.8 2 For Vgs bias voltages around 1.2 V: M.H. Perrott MIT OCW Plot of gm and gdο versus Idens for 0.18µ NMOS Device Transconductances g m and g do versus Current Density 4 Vgs M1 1.8µ W = L 0.18µ 3.5 Transconductance (milliAmps/Volts) Id 3 2.5 gd0=µnCoxW/L(Vgs-VT) 2 1.5 gm (simulated in Hspice) 1 0.5 0 0 100 200 300 400 500 600 700 Current Density (microAmps/micron) M.H. Perrott MIT OCW Noise Sources in a CMOS Amplifier RD enD RG enG Rgpar engpar ing ID RG RD 1 gg Cgd vgs Cgs Cdb gmvgs -gmbvs ro ind Vout Csb endeg vs Vin M.H. Perrott RS Rdeg MIT OCW Remove Model Components for Simplicity RD enD RG enG Rgpar engpar ing ID RG RD 1 gg Cgd vgs Cgs Cdb gmvgs -gmbvs ro ind Vout Csb endeg vs Vin M.H. Perrott RS Rdeg MIT OCW Key Noise Sources for Noise Analysis RD enD RG enG ing ID RG vgs Cgs ind gmvgs RD Vout endeg vs Vin Rdeg RS Transistor gate noise Transistor drain noise Thermal noise M.H. Perrott 1/f noise MIT OCW Apply Thevenin Techniques to Simplify Noise Analysis iout G Zg Zgs ing vgs Cgs D gmvgs ind S iout Zdeg G Zg Zgs vgs Cgs gmvgs D indg S Zdeg Assumption: noise independent of load resistor on drain M.H. Perrott MIT OCW Calculation of Equivalent Output Noise for Each Case iout G Zg Zgs ing vgs Cgs D gmvgs ind S iout Zdeg G Zg Zgs vgs Cgs gmvgs D indg S Zdeg M.H. Perrott MIT OCW Calculation of Zgs iout G D itest Zgs vtest Zg vgs Cgs gmvgs S v1 Z deg Write KCL equations After much algebra: M.H. Perrott MIT OCW Calculation of η iout G Zg Zgs vgs Cgs gmvgs D itest S v1 Zdeg Determine Vgs to find iout in terms of itest After much algebra: M.H. Perrott MIT OCW Calculation of Output Current Noise Variance (Power) iout Iout G G D S Zg D Zg Zgs vgs Cgs gmvgs indg Zdeg S Zdeg To find noise variance: M.H. Perrott MIT OCW Variance (i.e., Power) Calc. for Output Current Noise Noise variance calculation Define correlation coefficient c between ing and ind M.H. Perrott MIT OCW Parameterized Expression for Output Noise Variance Key equation from last slide Solve for noise ratio Define parameters Zgsw and χd M.H. Perrott MIT OCW Small Signal Model for Noise Calculations iout Iout G G D S Zg D Zg Zgs vgs Cgs gmvgs indg Zdeg S Zdeg M.H. Perrott MIT OCW Example: Output Current Noise with Zs = Rs, Zdeg = 0 Source iout Rs Vin ens Zgs vgs Cgs gmvgs Step 1: Determine key noise parameters Step 2: calculate η and Zgsw indg - For 0.18µ CMOS, we will assume the following M.H. Perrott MIT OCW Calculation of Output Current Noise (continued) Step 3: Plug values into the previously derived expression Drain Noise Multiplying Factor - For w << 1/(R C s - For w >> 1/(R C s gs): Gate noise contribution gs): Gate noise contribution M.H. Perrott MIT OCW Plot of Drain Noise Multiplying Factor (0.18µ NMOS) Drain Noise Multiplying Factor Versus Frequency for 0.18 µ NMOS Device 1 f << 1/(2πRsCgs) Drain Noise Gain Factor 0.95 0.9 0.85 0.8 f >> 1/(2πRsCgs) 0.75 1/100 1/10 1 10 100 Normalized Frequency --- f/(2πRsCgs) (Hz) Conclusion: gate noise has little effect on common source amp when source impedance is purely resistive! M.H. Perrott MIT OCW Broadband Amplifier Design Considerations for Noise 2 indg ∆f drain 1/f noise 4kTγgdo drain thermal noise gate noise contribution with purely resistive source impedance f 1/f noise corner frequency 1 2πRsCgs Drain thermal noise is the chief issue of concern when designing amplifiers with > 1 GHz bandwidth - 1/f noise corner is usually less than 1 MHz - Gate noise contribution only has influence at high frequencies (such noise will likely be filtered out) Noise performance specification is usually given in terms of input referred voltage noise M.H. Perrott MIT OCW Narrowband Amplifier Noise Requirements 2 indg ∆f drain 1/f noise 4kTγgdo drain thermal noise gate noise contribution with purely resistive source impedance f 1/f noise corner frequency Narrowband amplifier frequency range 1 2πRsCgs Here we focus on a narrowband of operation - Don’t care about noise outside that band since it will be filtered out Gate noise is a significant issue here - Using reactive elements in the source dramatically impacts the influence of gate noise Specification usually given in terms of Noise Figure M.H. Perrott MIT OCW The Impact of Gate Noise with Zs = Rs+sLg Source iout Rs Vin ens Lg Zgs vgs Cgs gmvgs Step 1: Determine key noise parameters Step 2: Note that η =1 , calculate Zgsw indg - For 0.18µ CMOS, again assume the following M.H. Perrott MIT OCW Evaluate Zgsw At Resonance Source iout Rs Vin ens Lg Zgs vgs Cgs gmvgs indg Set Lg such that it resonates with Cgs at the center frequency (wo) of the narrow band of interest Calculate Zgsw at frequency wo M.H. Perrott MIT OCW The Impact of Gate Noise with Zs = Rs+sLg (Cont.) Key noise expression derived earlier Substitute in for Zgsw Gate noise contribution Gate noise contribution is a function of Q! - Rises monotonically with Q M.H. Perrott MIT OCW At What Value of Q Does Gate Noise Exceed Drain Noise? 2 indg ∆f 4kTγgdo Narrowband amplifier frequency range drain thermal noise gate noise contribution Q2 f wo/2π Determine crossover point for Q value =1 - Critical Q value for crossover is primarily set by process M.H. Perrott MIT OCW Calculation of the Signal Spectrum at the Output Source iout Rs ens Vin Lg Zgs vgs Cgs gmvgs indg First calculate relationship between vin and iout At resonance: Spectral density of signal at output at resonant frequency M.H. Perrott MIT OCW Impact of Q on SNR (Ignoring Rs Noise) 2 indg ∆f 4kTγgdo Narrowband amplifier frequency range signal spectrum Q2 drain thermal noise gate noise contribution Q2 f wo/2π SNR (assume constant spectra, ignore noise from Rs): For small Q such that gate noise < drain noise M.H. Perrott - SNR out improves dramatically as Q is increased For large Q such that gate noise > drain noise - SNR out improves very little as Q is increased MIT OCW Noise Factor and Noise Figure 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 M.H. Perrott Zout inout iout ZL MIT OCW Calculate Noise Factor (Part 1) Source iout Rs ens Lg Zgs Vin Cgs gmvgs indg First calculate SNRout (must include Rs noise for this) -R s noise calculation (same as for Vin) - SNR vgs out: Then calculate SNRin: M.H. Perrott MIT OCW Calculate Noise Factor (Part 2) Noise Factor calculation: From previous analysis M.H. Perrott MIT OCW Calculate Noise Factor (Part 3) Modify denominator using expressions for Q and wt Resulting expression for noise factor: Noise Factor scaling coefficient - Noise factor primarily depends on Q, w /w , and process specs o M.H. Perrott t MIT OCW Minimum Noise Factor Noise Factor scaling coefficient We see that the noise factor will be minimized for some value of Q - Could solve analytically by differentiating with respect to Q and solving for peak value (i.e. where deriv. = 0) In Tom Lee’s book (pp 272-277), the minimum noise factor for the MOS common source amplifier (i.e. no degeneration) is found to be: Noise Factor scaling coefficient How do these compare? M.H. Perrott MIT OCW Plot of Minimum Noise Factor and Noise Factor Vs. Q 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 c = -j1 c = -j0 4 c = -j0.55 3 0 Minimum across all values of Q and 1 Note: curves meet if we approximate Q2+1 Q2 2 1 M.H. Perrott Achievable values as a function of Q under the constraint that 1 = wo LgCgs 1 2 3 c = -j1 LgCgs 4 5 Q 6 7 8 9 10 MIT OCW Achieving Minimum Noise Factor For common source amplifier without degeneration - Minimum noise factor can only be achieved at - resonance if gate noise is uncorrelated to drain noise (i.e., if c = 0) – we’ll see this next lecture We typically must operate slightly away from resonance in practice to achieve minimum noise factor since c will be nonzero How do we determine the optimum source impedance to minimize noise figure in classical analysis? - Next lecture! M.H. Perrott MIT OCW
