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Designing for Ultra-Low THD+N in Analog Circuits Bruce E. Hofer Chairman & Co-Founder Audio Precision, Inc. Copyright 2013 Audio Precision, Inc. Audio Precision® is a registered trademark of Audio Precision, Inc. Introduction • This presentation is a condensed, but updated version of a design track seminar I gave at the 2011 New York AES convention • Everything I will talk about today is derived from my own experiences designing very high performance and state-of-the-art instruments over the past 44 years – You will not find some of this material in textbooks – Unfortunately I cannot disclose certain revelations for competitive reasons – Our 1-hour time constraint will also limit the number of topics • A copy of today’s slides can be obtained by contacting me at: [email protected] Selected Topics for Today • Taylor Series Model of Non-Linearity – Estimation of 2H & 3H distortion • Resistors and Capacitors – Comparison of technologies – Non-linearity models and distortion estimation • Op-Amps (circa 2013) – Bipolar versus JFET input stages – The 4 major sources of op-amp distortion – Some distortion reduction “tricks” • Noise and Noise Estimation (if time) A Non-Linearity Model that Enables Estimation of 2H and 3H Distortion • Use a Taylor Series to model non-linear behavior – Circuit is modeled as having a voltage dependent gain: Vout / Vs = f(Vs ) = Ao * (1 + k2* Vs + k3* Vs ^2 …) • 2HD and 3HD ratios can be estimated with surprising accuracy using only 3 values for dynamic gain at the positive peak (Ap), negative peak (An) and, zero (Ao) points of an assumed sine-wave signal: – 2HD ≈ (k2 / 2) = |Ap – An| / (8 * Ao) – 3HD ≈ (k3 / 4) = |Ap + An – 2*Ao| / (24 * Ao) • Note that 2HD (as a ratio) is proportional to Vs while 3HD is proportional to Vs^2 Example: Emitter Follower Distortion • A simple emitter follower was constructed using a MPSA18 transistor with RL = 4.02k wired to -15V and the collector to +15V. – Performance with a ±5.0 Vpk (10.0 Vpp) signal is to be determined at +21.8C (≈295K) • Equivalent circuit is shown on next slide – The 100kΩ load resistor represents the input impedance of the audio analyzer (dc coupled) – The element “Re” models the dynamic impedance of the emitter-base junction – “Re” interacts with the total load impedance to give a voltage gain that is slightly less than unity, and that varies as a function to the instantaneous signal voltage Emitter Follower Circuit Note: Re = kT/qIe Follower Distortion Estimation, cont’d • Dynamic emitter impedance, Re ≈ kT/qIe – “k” = Boltzmann’s constant = 1.38065E-23 – “T” = 295K (ignoring self heating within the transistor) – “q” = electron charge = 1.6022E-19 • Gain is calculated at VB = -5.0, 0, and +5.0 V: – VB = -5.0 V: Ie = 2.282 mA, Re = 11.138Ω An = 0.99713 – VB = 0.0 V: Ie = 3.576 mA, Re = 7.1085Ω Ao = 0.99816 – VB = +5.0 V: Ie = 4.870 mA, Re = 5.2200Ω Ap = 0.99865 • 2HD ≈ |Ap – An| / 8*Ao = 0.019% (-74.4 dB) • 3HD ≈ |Ap + An - 2*Ao| / 24*Ao = 0.0023% (-92.8 dB) Actual FFT of the Follower Output • The measured levels of 2H and 3H with a 10.0 Vpp sine-wave at 1 kHz are -75.1 dB and -93.4 dB compared to the estimates of -74.4 dB and -92.8 dB Resistors in Analog Design Linear Resistor Technologies • Composition • Thick Film • Thin Film or Metal Film • Metal Foil • Wire-Wound • Some Comments about Matched Resistor Networks Composition Resistors • The resistive element is a compacted mixture of carbon and ceramic held together in a resin base – Very popular prior to the 1970s, much less popular today – Still useful in some non-audio applications that require high peak power capability, or super low series inductance • Unimpressive performance by today’s standards – Tolerances from 20% to 5% – TCR is typically 150 to 1000 ppm/C (worse at low values) – High modulation noise and voltage coefficient compared to other types • DO NOT USE in high performance analog designs! – One notable exception is in the design of certain guitar amplifiers where certain forms of distortion are desired Thick Film Resistors • The resistive element is a conductive film applied to the surface of a cylindrical or rectangular substrate – Resistance is determined by film composition and etching pattern – Very wide variety of sizes and power ratings • Very popular for general purpose applications – Tolerances of 2% to 0.1%, usually laser trimmed when <1% – TCR is typically 100 to 250 ppm/C – Modulation noise is high compared to thin film, foil and WW types, but much better than carbon composition – Voltage coefficient is rarely specified, and varies considerably from brand to brand, and by value…up to 10 ppm/V is not uncommon Thin Film (Metal Film) Resistors • The resistive element is a more stable conductive film (typically Nichrome or Ta-N) that is sputtered onto the surface of a cylindrical or rectangular substrate – Resistance is determined by film thickness and pattern – Less of a variety of sizes and power ratings versus thick film types • Superior performance, but much more expensive – – – – Tolerances from 1% to 0.02% (usually laser trimmed when <1%) TCR is typically 25, 10, 5, or even 2 ppm/C (very recently) Excellent (i.e. very low) modulation noise Usually much lower voltage coefficient than thick film, but still rarely specified and variable from brand to brand, and by value…typical values are in the range of 0.1 to 1 ppm/V Metal Foil Resistors • The resistive element is a special alloy metal foil that is cemented to an inert substrate – Resistance is determined by the foil characteristics and pattern – Trimming is accomplished by opening links in a carefully designed foil pattern—vastly more stable than “L” cut trimming • The best DC performance, and most expensive of all resistor technologies – Tolerances to 0.001% with TCR as low as 0.05 ppm/C ! – Extremely low modulation noise and thermal EMF – Specified voltage coefficient is typically <0.1 ppm/V at DC • Beware for high-end audio applications! – Low frequency modulation distortion is much worse than expected Wire-Wound (WW) Resistors • The resistive element is wire having a low temperature coefficient and carefully wound on a substrate – Typically appropriate only for lower resistance values – Very high peak and average power ratings are possible Winding Patterns #1 / #3 - Inductive #2 - Bifilar #4 - Ayrton-Perry Resistor Non-Linearity • Resistors exhibit two general forms of non-linearity – Voltage coefficient and power coefficient (“thermal modulation”) • Resistor voltage coefficient non-linearity is best modeled as R(Vs) = Ro * (1 - kv* |Vs|) – kv has units of “ppm/V” and is usually positive – Taylor series model is not appropriate here! • Distortion can be estimated by taking the FFT of a full-wave rectified sine multiplied by the sine – 2HD ≈ 0, assuming no significant dc component – 3HD ≈ |kv* Vs | / 5.9 Note proportionality to Vs not Vs^2 as would be expected using a Taylor series model for non-linearity – 5HD ≈ |kv* Vs | / 41 ≈ 3HD / 7 5HD is about -17 dB below 3HD Resistor Non-Linearity, cont’d • Resistor power coefficient non-linearity is similarly modeled as R(Ps) = Ro * (1 + kP * Ps) – kP has units of ppm/W and can be either positive or negative • However, the real non-linearity mechanism is thermal modulation which leads to a much more useful model: R(Vs) = Ro * (1 + TCR * Z(ω) * (Vs^2 / Ro)) – “TCR” is the dc temperature coefficient (ppm/C) – “Z(ω)” is the device thermal impedance (C/W) which is a very complex function of frequency—instantaneous power dissipation changes in a resistor do NOT cause instantaneous changes in the temperature of the resistive material – As frequency 0, |Z(ω)| θR the dc thermal resistance, which is typically 200-300 C/W for a 1206-size surface mount resistor Resistor Non-Linearity, cont’d • At very low frequencies (<0.2 Hz), the resistor reaches thermal equilibrium as the signal varies – 2HD ≈ 0, assuming no significant dc bias – 3HD ≈ TCR * θR * (Vs ^2 / Ro) / 4 – 5HD ≈ 0 (compared to 3HD / 7 for voltage coefficient distortion) • Within the range of 5-200 Hz the magnitude of Z(ω) is usually much smaller than θR, rolling off to near zero above 1-5 kHz – Beware: Recent experiments by the author (corroborated by another individual) have shown that some metal foil resistors with exceptionally low TCR behave as if |Z(ω)| >> θR at low frequencies, thus contributing higher modulation distortion than a thin film resistor with a larger TCR Recommendations for Audio Circuits • All factors considered, the best resistor technology for audio applications is a low TCR thin film – Avoid the common 25 ppm/C characteristic in critical circuit locations, and pay the premium for either 10 ppm/C or 5 ppm/C – Some manufacturers now offer 2 ppm/C (if you can afford it) • For surface mount resistors, use only 1206 size – Smaller sizes have a lower power rating hence a higher thermal resistance which translates to a higher thermal modulation distortion…sizes larger than 1206 are not commonly available – Limit the signal to 20 mWpk or about 3 Vrms (+12 dBu) for the lowest distortion performance – Consider using series-parallel combinations in circuits requiring higher peak power dissipation or higher voltage Resistor Networks • Resistor networks are especially useful in applications that benefit from ratio matching – Ratio accuracies can be as good as 0.01% for thin film, or an incredible 0.001% for metal foil – Extremely low differential temperature coefficients at dc, however watch out for unexpectedly high thermal modulation effects with metal foil types • Avoid large R ratios (e.g. 10:1 or higher) – Best performance is achieved if all resistors are of equal value • The small size of resistor networks (SOIC-8/-16) will mean a higher thermal impedance, thus causing higher thermal modulation distortion than discrete resistors A True Story… • About 13 years ago a certain manufacturer decided to change their network substrate material from ceramic to passivated silicon without notifying its customers – Ceramic is brittle and more expensive to process and cut to size • Although the resistor DC parameters were unchanged, the AC performance was a total disaster! – The stray C between each resistor and the substrate was not only higher, but NON-LINEAR – It is believed that P-I-N diodes were formed between each resistor and the semi-conducting substrate, thus causing the voltage drop in one resistor to generate distortion products in the other resistors – The manufacturer quickly added the “option” to specify the original ceramic substrate when told they were about to be disqualified Capacitors in Analog Design Linear Dielectric Materials • Polymer Film – A generic term covering many different types of films – Examples include polyester (PE), polyethylene naphtalate (PEN), polyphenylene-sulfide (PPS), polypropylene (PP), polystyrene (PS), and polytetrafluoroethylene (PTFE or Teflon®) • Ceramic – Another generic term covering a very wide variety of compositions and characteristics--beware – Examples include “Z5U”, “X7R”, “NP0”, “Hi-K” • Mica • Glass Polymer Film Capacitors • Films that are widely available from many vendors: – Polyester (PE), aka Mylar® – Polyphenylene-sulfide (PPS), a relatively new film becoming more popular as an alternative to polyester with better characteristics – Polystyrene (PS), very good but has a tempco of about -100 ppm/C; can be easily damaged by soldering--film melts at +85C – Polypropylene (PP), lower cost alternative to PS with very low dissipation factor and a higher melting point (+105C); but it also has a higher tempco (up to -250 ppm/C) • More limited availability films: – Polycarbonate (PC), very hygroscopic (moisture sensitive) must be hermetically sealed for acceptable stability, virtually obsolete today – Polytetrafluoroethylene (PTFE), aka Teflon®, can be problematic due to its porosity (multiple layers are typically required for good reliability)—but many audiophiles believe it just “sounds” better Film Capacitor Construction • Metalized Film – The dielectric film is pre-coated with a conductive surface that is connected to one of the capacitor terminals – Has higher equivalent series resistance, hence higher dissipation factor (tan θ) • Metal Foil Film – The dielectric film is interleaved with real metal foils that are connected to the capacitor terminals – Lower equivalent resistance than metalized film • If possible, use only foil-film construction Ceramic Capacitors • The only ceramic composition that should ever be used in high performance analog design is “NP0” (also known as “COG”) – Now available in values up to >100 nF with tolerances of 1-5% and voltage ratings up to 500V (1 kV for through-hole) – Consider paralleling multiple caps to get higher values – Very low dissipation factor and frequency dependence – ±30 ppm/C specified temperature coefficient, typically ±15 ppm/C – Excellent stability, virtually immune to humidity • Avoid the lowest 25V rating in critical audio designs – 50V and 100V rated caps are not that much larger (perhaps 1206 versus 0603), but they will give superior distortion performance Piezoelectric Effect in Some Ceramics • Capacitor manufacturers are generally very secretive about their ceramic recipes (composition) • Certain “junk” grades of ceramic capacitors exhibit a strong piezoelectric effect—unwanted voltages caused by changes in physical stress within the capacitor – Barium titanate (BaTiO3) is often used to increase the dielectric constant of ceramic dielectrics, thus reducing the size of a capacitor for a given C*V rating; however this substance is highly piezoelectric – Examples include Z5U, Y5Y, and “Hi-K” • Never use these lower grades of ceramic in voltage reference filters or anywhere in the audio signal path Mica Capacitors • 30 years ago mica capacitors were highly regarded in the analog design community…not so today – Commonly available with 1-5% tolerances up to about 3 nF – Temperature coefficient typically 90 ppm/C – Good stability, but mica’s brittleness can sometimes result in abrupt and unexpected value shifts with physical stress • Unfortunately mica is a product of nature, some of its better sources have now been depleted • With the ready availability and lower temperature coefficient of NP0 (COG) ceramic caps, there is no good reason to specify a mica capacitor anymore Glass Capacitors • Glass is among the most stable and inert of dielectrics – Typical values available up to about 2 nF – Extremely stable, almost no aging, near zero voltage coefficient – Some sensitivity to frequency, perhaps a bit worse than “NP0” ceramics and polypropylene (PP) film capacitors—more data welcome – Temperature coefficient is not as good as other types (typ +140 ppm/C) but glass caps can operate up to +200C – Highest immunity to radiation—obvious uses in military and aerospace applications (and perhaps the best choice for the survivalist golden-ears preparing for the post-apocalyptic world) • Unfortunately molten glass is not so easy to form with precise dimensions – 5% tolerance typical, 1-2% available but hyper-expensive Microphonic Effect in all Capacitors • In any capacitor: d(Q) = d(C*V) – The above equation if often simplified as d(Q) = C*d(V) from which the classic equation I = d/dt(Q) = C*d/dt(V) is derived • However, “C” itself is not necessarily constant – C is not only a function of voltage V (non-linearity), but it can also vary with mechanical stress/vibration thus acting as a microphone – The total derivative is really d(Q) = d(C*V) = C*d(V) + V*d(C), thus giving I = d/dt(Q) = C*d/dt(V) + V*d/dt(C) • Obvious Insight Minimize the dc potential across capacitors in series with the signal path – The AC coupling caps in phantom powered microphone input circuits are problematic; should be as small as possible and matched in value Non-Linearity of Capacitors • This is a complex and proprietary subject, thus I can share only some limited comments… • Capacitors also have voltage coefficient effects similar to resistors, that can cause unwanted distortion – Inherently frequency dependent, very difficult to analyze • Film capacitors, in particular, can also exhibit a nonlinearity related to signal current – The metalized surfaces or foils must be electrically connected to the external component leads – These connections are typically physical in nature (e.g. crimped), and they often result in contact resistance (ESR) that is non-linear and variable from unit to unit Op-Amps, circa 2013 Major Categories of Op-Amps • Op-amps are ubiquitous in analog design – They are a fundamental building block enabling high performance amplification, mixing, and frequency contouring of audio signals – They are also useful in signal analysis and generation applications • Op-amps are commonly divided into four categories depending upon their intended application – Precision, optimized for low DC offset and bias current – General purpose, usually dominant-pole compensated, but many newer designs now insert a pole-zero pair into the open loop response to get a higher GBW (gain-bandwidth product) – High speed, higher slew rate, not necessarily stable under unity gain situations – Really high speed and slew rate, typically for video signals A More Useful Classification • Advances in IC processes and circuit techniques now blur these more traditional categories – Indeed, there are a number of op-amps that feature both excellent DC performance and decent slew rate and GBW characteristics, e.g. OPA1611, OPA1641, OPA827, LT1468 (my apologies if your favorite op-amp is not in this brief list) • A much more useful distinction is the input device technology: Bipolar vs. JFET – Both can offer input voltage offset performance to below 200 μV – However, JFET op-amps have near-zero input bias current – An interesting example of a hybrid design (using both bipolar and JFET devices) is the “Butler Amplifier” in the dual OP275 – Forget about CMOS op-amps for high performance applications Bipolar vs JFET Noise Performance • Bipolar inputs offer the lowest noise voltage rating (eN), typically 0.9 nV/√Hz with the AD797 – 0.9 nV/√Hz is equivalent to the noise of a 49Ω resistor! – But super low eN usually comes with the penalty of much higher current noise iN … typically 2.0 pA/√Hz for the AD797 • Today’s best JFET input op-amps have eN as low as 3.8-5.0 nV/√Hz but iN of only 0.0008-0.003 pA/√Hz! – Compare the OPA827 and OPA1642 (dual) with the older bipolar models NE5534 and NE5532 (dual) • Bipolar input op-amps still have a slight advantage for lower “1/f” noise below 1 kHz 4 Distortion Mechanisms in Op-Amps • Input stage trans-conductance non-linearity – ΔIinput = Ccomp * d/dt(Vout), [part of Ccomp may be external] – Typically 3HD and proportional to F^3 in dominant pole designs • Input stage common mode impedance non-linearity – Caused by input capacitance variation with common mode signals, JFET input designs are much worse than bipolar • Output stage or “crossover” non-linearity – Caused by non-linear output impedance versus output current, some designs use a cancellation scheme (e.g. AD797) • Mutual inductance between power supply busses and critical circuit loops Some Distortion Reduction “Tricks” • Output stage non-linearity can often be significantly reduced by forcing the output to behave more like a class-A amplifier by adding a resistor or a biasing dc current source to one of the supplies – Watch out for increased power dissipation in the op-amp! • Op-amps needing an external compensation capacitor can usually benefit from either “2-pole” compensation or “feed-forward” compensation – Instead of using the classic 22 pF between pins 5-8 of a NE5534, use a pair of 47 pF connected in series with a 499-1k resistor connected between the two capacitors and the positive supply – For inverting NE5534 configurations, try connecting a capacitor having a value of about 6.8-12pF between the input and pin 8 Two-Pole Compensation “Feed-Forward” More “Tricks” to Minimize Distortion • Use inverting topologies whenever possible – Input capacitance is usually higher and more non-linear with common mode signals in JFET op-amps – Most op-amps will show dramatically lower THD (particularly 2HD above 5 kHz) when operated with a gain of -1 versus +1. • If an op-amp must be used in a non-inverting topology (e.g. in a Sallen-Key active low-pass filter), arrange for both inputs to “feel” the same source impedance – This usually means adding a complicated RC network in series with the + input to match the impedance seen looking outward from the – input – Try it--the distortion reduction can be quite significant with JFET op-amps! Common-Mode Distortion Reduction Noise in Analog Circuits Sources of Noise • “Thermal” noise of resistors: VN = √(4*k*T*R*BW) • “Shot” noise of dc currents: IN = √(2*q*Idc*BW) • “Op-Amp” noise, usually “eN” and “iN” in datasheets • “1/f” and “Popcorn” noise in op-amps – Mechanisms still not well understood, but under control • “Modulation” noise in resistors – Caused by component material imperfections usually resulting in AM noise sidebands surrounding a pure tone – Carbon composition is terrible, thick film is so-so, thin film is OK, metal foil and WW are best Noise Estimation in Circuits • The residual noise floor of many analog circuits can also be estimated with surprising accuracy using only a well designed spreadsheet! – List all noise sources including resistors, op-amps, bias currents – Calculate the transfer function either from the input or output depending upon the desired result – Express all entries in the same unit (recommend nV/√Hz) – Perform a root-mean-square (rms) summation of all sources – Convert the final result to Volts by multiplying by √BW where BW is the desired measurement bandwidth (e.g. 20 kHz for audio) • The following slide shows an example for a prototype AP analyzer--estimates are blue, measurements are red source resistance input dampers input current limiters MBUF en, ie=146 uA MBUF in post MBUF attenuator atten Rout * in Range Vmin = 24.04 7.60 2.40 0.760 0.240 0 Range Vmax = 85.3 26.99 8.53 2.699 0.853 0.270 173.524 3.524 173.524 3.524 2.855 3.524 3.897 30.912 31.919 152.528 52.172 30.912 31.919 23.473 3.909 3.091 0.311 15.253 5.217 2.855 3.524 3.897 0.395 3.091 0.311 2.347 0.391 2.855 3.524 3.897 0.395 3.091 0.311 2.347 0.391 2.855 3.524 3.897 0.395 3.091 0.311 2.347 0.391 1.561 0.241 1.750 0.583 374 442 698 2.178 0.25 162 preamp en preamp in preamp Rg preamp Rf 1.10 1.70 1000 402 49.377 21.881 15.615 6.919 4.938 2.188 1.561 0.692 98.755 31.229 9.876 3.123 1.561 0.763 2.983 1.844 sum stage en sum stage in sum stage Ri sum stage Rf 2.68 1.60 1000 1000 170.132 71.822 184.425 184.425 53.800 22.712 58.320 58.320 17.013 7.182 18.443 18.443 5.380 2.271 5.832 5.832 1.701 0.718 1.844 1.844 0.538 0.227 0.583 0.583 inv stage en inv stage in inv stage Ri inv stage Rf 2.68 1.60 1400 1400 85.066 50.275 77.151 77.151 26.900 15.898 24.397 24.397 8.507 5.028 7.715 7.715 2.690 1.590 2.440 2.440 0.851 0.503 0.772 0.772 0.269 0.159 0.244 0.244 1.10 1.70 1000 215 3.961 -121.55 3.33 139.520 76.311 92.213 198.871 44.120 24.132 29.160 62.888 13.952 7.631 9.221 19.887 4.412 2.413 2.916 6.289 1.395 0.763 0.922 1.989 0.441 0.241 0.292 0.629 765.689 242.132 76.569 24.213 7.657 2.421 23.0 20.00 924.214 335.071 90.918 29.109 12.037 8.060 predicted uV noise = measured noise = 130.0 129.9 47.1 47.1 12.79 12.80 4.10 4.10 1.694 1.697 1.134 1.134 A/D driver en A/D driver in A/D driver Ri A/D driver Rf A/D 0dBFS A/D noise floor A/D headroom Tambient, C measurement BW Designing for Low Noise • Resistor noise voltage is proportional to √R – Use the lowest possible resistor values that are consistent with power dissipation and distortion requirements – Choose circuit topologies that inherently minimize the value of resistors in the signal path – Series resistor combinations may be good for lower distortion because they reduce the voltage drop across any given resistor; but they do not result in lower noise • Resistor noise is proportional to √T (T in °K) – The temperature of each resistor must be considered • Use only thin-film or metal foil resistors when they must pass significant dc bias currents In Conclusion… • Today we have discussed some selected topics that influence THD+N performance of analog circuits • Ultra-low THD+N does not happen by accident. It is the result of careful attention to detail, clever circuit design, and the use of high quality components • Some issues will continue to challenge engineers well into the future • Let us not allow analog design to become a “lost art” Designing for Ultra-Low THD+N in Analog Circuits Bruce E. Hofer Chairman & Co-Founder Audio Precision, Inc. Copyright 2013 Audio Precision, Inc. Audio Precision® is a registered trademark of Audio Precision, Inc.