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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.