1. Designing Analog Circuits with Ultra-Low THD (<-120 dB)
Bruce E. Hofer Ex-Chairman & Co-Founder Audio Precision, Inc. (now retired)

2. Foreword
This seminar is derived from my own experiences in the design of high performance and state-of-the-art test equipment over the past 48 years
You will not find some of this material in textbooks
Certain topics are considered proprietary and will be addressed only superficially; other topics literally beg for more research and data
Any errors or misrepresentations are solely my own
In many ways this seminar reflects my passion for analog circuit design, and for “getting things right”… Your feedback is welcome!
A copy of this slide set is available by sending your address to

3. Introduction
All too often, it is what you do not see on a schematic that really matters in achieving ultra-low distortion
Clever circuit design is an important first step…
However the analog design engineer must also be aware of the many forms of component imperfection and non-linearity that can limit performance
Today we will focus on some of those factors
Models of component non-linearity, estimation of distortion
The good, the bad, and the ugly of resistors and capacitors
Op-amp non-linearity
Some circuit suggestions and tips

4. Taylor Series Model of Non-Linearity
Many forms of non-linearity can be modeled using a Taylor Series expansion, for example a dynamic gain: Vout = Vs * A(Vs ) = Vs * A0 * (1 + k2 * Vs + k3 * Vs ^2 …)
Harmonic distortion is related to the series coefficients. If one neglects the effect of higher order terms:
2HD ratio ≈ k2 /  2HD ratio is proportional to Vs
3HD ratio ≈ k3 /  3HD ratio is proportional to Vs^2
2HD & 3HD can be estimated from the dynamic gains at the positive peak (AP), negative peak (AN), and zero crossing (A0) points of an assumed sine-wave signal
2HD ratio ≈ |AP – AN| / (8*A0)  2HD ≈ 0 if AP = AN
3HD ratio ≈ |AP + AN – 2*A0| / (24*A0)

5. Emitter Follower Example
An emitter follower was built using a MPSA18 with 4.02kΩ from emitter to -15V, and collector to +15V
Task: Estimate the distortion in the output signal with a ±5 Vpk (10 Vpp) 1 kHz sine applied to the base…
Assume Ta = +22C (+295K) and Vbe = 0.64V
Equivalent circuit is shown on next slide
The element “re” models the dynamic impedance of the transistor’s emitter-base junction at audio frequencies
The 100 kΩ resistor represents the input impedance of the audio analyzer (dc coupled) that will be used to measure the actual distortion
“re” interacts with the total load impedance to give a voltage gain that is slightly less than unity, and that varies as a function of the emitter current

6. Emitter Follower Circuit
Note: re = kT/qIe

7. Distortion Estimation
Dynamic emitter impedance, re ≈ kT/qIe
“k” = Boltzmann’s constant = E-23
“T” ≈ 295K ignoring self heating (This is often not a good assumption!)
“q” = electron charge = E-19
Calculate dynamic gains at VB = -5.0, 0.0, +5.0:
VB = -5.0 V: Ie = mA, re = Ω  AN =
VB = V: Ie = mA, re = Ω  A0 =
VB = +5.0 V: Ie = mA, re = Ω  AP =
2HD estimate ≈ |AP – AN| / (8*A0) = dB
3HD estimate ≈ |AP + AN - 2*A0| / (24*A0) = dB
Actual measured 2HD = dB; 3HD = dB

8. Frequency Dependent Non-Linearity
Some forms of non-linearity are frequency dependent, in which case one or more of the coefficients of the Taylor Series model become functions of frequency: Vout = Vs * A0 * (1 + k2(ω) * Vs + k3(ω) * Vs ^2 …)
Distortion can still be estimated from the coefficient functions k2(ω) and k3(ω) evaluated at the frequency of interest, ωo. Again, neglecting higher order terms:
2HD ratio ≈ k2(ωo) / 2
3HD ratio ≈ k3(ωo) / 4
One particularly interesting example of this type of distortion is caused by resistor thermal modulation

9. Resistor Thermal Modulation
Instantaneous changes in power dissipation do NOT cause instantaneous (or even identical) changes in temperature through-out the 3D volume of a resistor
The magnitudes of temperature change within a resistor in response to an applied sine-wave signal are inherently frequency dependent
A proposed model for resistor thermal modulation: R(VR, ω) = Ro * [1 + TCR * Z(ω) * (VR^2 / Ro)]
Note Taylor Series form with k2(ω) = 0 and k3(ω) = TCR * Z(ω) / Ro
“TCR” is the DC temperature coefficient (ppm/C) of the resistor
“Z(ω)” is the device thermal impedance (C/W)…a very complex function of frequency not easily described by simple poles and zeros
As ω  0 , Z(ω)  θRA which is the DC thermal resistance to ambient: typically 340 C/W for a 1206 thick film resistor, or C/W for a thin-film resistor of the same size depending upon its design

10. Resistor Thermal Modulation, cont’d
At very low frequencies (typically <0.2 Hz), a resistor can approach thermal equilibrium as the signal varies
2HD ratio ≈ 0, assuming no significant DC component
3HD ratio ≈ TCR * θRA * (VR ^2 / Ro) / 4
As signal frequency increases the magnitude of Z(ω), hence 3HD, usually rolls off monotonically and drops to a negligible value above 1-2 kHz
As ω  ∞ , Z(ω)  0
Experiments by the author suggest |Z(20 Hz)| ≈ (25%-50%) * θRA for a 1206 surface mount resistor [more data is welcome]
Other experiments by the author (corroborated by another individual) show that metal foil resistors behave very strangely, as if |Z(ω)| > θRA in the range of about Hz, before rolling off at higher frequencies
TCR=100, theta=340, 20/dc = 35%, Vp=30, R=10k  3HD (20 Hz) = 270 ppm = 0.027% = dB

11. Limitations of the Taylor Series Model
The Taylor Series model generally works well with a non-linearity that monotonically increases in response to increasing signal voltage or current
However several forms of non-linearity cannot be modeled with a Taylor Series:
Cross-over distortion where non-linearity increases as signal decreases
Contact resistance (often very bizarre with dead-band behavior)
Abrupt or nearly discontinuous functions arising from clipping or slew rate limiting (think of slew rate limiting as “clipping” of the first derivative)
Time varying distortions such as those occurring in compressors or AGC circuits due to non-zero attack and decay time constants
Passive component voltage coefficient effects

12. Resistor Voltage Coefficient
The “voltage coefficient” of a resistor requires a very different model: R(VR) = Ro * (1 + kV* |VR|)
kV has units of “ppm/V”
kV is usually negative (R decreases as |VR| increases), and it is believed to be frequency independent [more data is welcome]
kV has been recognized for decades in high voltage attenuator design
The form of this model may look similar to a Taylor Series, but it has very different characteristics due to the dependence upon |VR|
Distortion ratios can be estimated by calculating the harmonic amplitude coefficients of the FFT of a full-wave rectified sine multiplied by the sine
2HD (and all even harmonics) ≈ 0, assuming no significant DC component
3HD ratio ≈ |kV*VR|* (8/15π)  Note proportionality to |VR| not VR^2
5HD ratio ≈ |kV*VR|* (8/105π)  5HD ≈ (3HD / 7) regardless of VR

13. Circuit Sensitivity to a Non-Linearity
When investigating the impact of a component non-linearity, one must also consider the sensitivity of a circuit to that component
For example, a series combination of ‘N’ identical resistors reduces both thermal modulation and voltage coefficient distortion by dividing the signal amplitude
Thermal modulation 3HD ratio is reduced by a factor of N^2
Voltage coefficient 3HD ratio is reduced by a factor of N
Parallel combinations benefit only from the lower power in each resistor, and do nothing to reduce voltage coefficient distortion
In some cases distortion cancellation may be possible
Consider a voltage divider composed of identical, equal-value resistors…

14. Linear Resistor Technologies
Composition
Thick Film
Thin Film
Axial-leaded “Metal” Film
Axial-leaded “Carbon” Film (will not be covered in this seminar)
Metal Foil
Wire-Wound

15. 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 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 found in older guitar and musical instrument amplifiers where certain forms of distortion are actually desirable

16. Thick Film Resistors
The resistive element is a conductive paste (often RuO2 plus other materials), typically 100 μm thick, applied to the surface of a cylindrical or rectangular substrate then fused at ≈850C into a glass-like film
Resistance is determined by film composition and pattern
Offers the greatest variety of values, sizes, and power ratings
Most popular for general purpose applications today
Tolerances of 5% to 0.1%, usually laser trimmed when <1%
Standard TCR is 100 ppm/C increasing at value extremes
Modulation noise much better than carbon composition, but still high in comparison to thin film, WW, and foil types
Voltage coefficient is rarely specified; and it can vary considerably from brand to brand, lot to lot, and value…up to 10 ppm/V is not uncommon

17. Thin Film Resistors
The resistive element is a more stable metallic film (e.g. Ni-Cr, Si-Cr, or Ta-N), typically 0.1 μm thick, deposited onto a cylindrical or rectangular substrate
Far fewer choices of size and power ratings versus thick film
The designation “metal film resistor” typically refers to an axial leaded design in which the resistive element is cut into a spiral form (helix)
“Carbon film” is still a further sub-set of axial leaded resistors…
Superior performance, but much more expensive
Tolerances from 1% to 0.01% (usually laser trimmed)
Standard TCR is typically 25, 10, 5, or even 2 ppm/C (very recently)
Excellent (i.e. very low) modulation noise
Voltage coefficient is usually much lower than thick film, but still rarely specified and variable from brand to brand, lot to lot, and by R value… typically in the range of 0.1 to 1 ppm/V n

18. Metal Foil Resistors
The resistive element is a special metal alloy foil that is cemented to an inert substrate
Resistance is determined by the foil characteristics and a cleverly designed pattern that allows trimming by opening links instead of “L” cuts
Superb DC performance, but very expensive
Tolerances to 0.001% with TCR as low as 0.05 ppm/C
Extremely low modulation noise and thermal EMF
Specified voltage coefficient is <0.1 ppm/V at DC
Beware for high-end audio applications!
Low frequency thermal modulation distortion in the range of Hz is typically much higher than that implied by its ultra-low TCR specification. This phenomenon is currently unexplained (perhaps due to the foil and the substrate having different thermal responses?) [more data is welcome] 18

19. Wire-Wound (WW) Resistors
Wire-wound resistors are typically used in high power applications and metrology as secondary standards
The resistive element is a special alloy wire having a low TCR and wound on a physically stable substrate
TCR can be vanishingly small with proper design and annealing to relieve the residual physical stress in the wire from the winding process
Superb DC performance, but both large and expensive
Significant parasitic shunt capacitance and series inductance, however the inductance can be minimized with dual opposed windings (Ayrton-Perry)
Virtually non-existent modulation noise
Usually very low voltage coefficient, but still rarely specified
Can tolerate serious short term power overloads without shifting value n

20. Recommendations
In my experience, the best resistor technology for most high performance analog circuits is thin-film with the lowest TCR that one can afford
Avoid the more common 25 ppm/C grade in critical circuit locations such as gain setting elements…pay the premium for 10 ppm/C or 5 ppm/C
[Update 2016] At least one manufacturer now offers a 2 ppm/C family
For surface mount resistors, use only the 1206 size
1206 is a “sweet” spot. Smaller sizes have a lower power rating hence a higher thermal resistance which will increase thermal modulation effects. Sizes larger than 1206 are not commonly available in thin film
Limit the signal across critical resistors to about mWpk or 3-5 Vpk for the lowest distortion performance
Use series combinations in circuits requiring higher peak power dissipation or higher voltage (e.g. the feedback resistor in an audio power amplifier)

21. Capacitor Dielectric Materials
Polymer Film
A broadly generic term covering many different types of films
Ceramic
Another very broadly generic term covering a wide variety of compositions and performance characteristics. Indeed, beware! Some types of ceramic capacitors are grossly non-linear, and should never be used in the signal path of high performance analog circuits
Mica
Glass
Electrolytic (will not be covered in this seminar)

22. Polymer Films Widely available from many major vendors:
Polyester (PET, or just PE depending upon one’s age), aka Mylar®
Polyphenylene-sulfide (PPS), rapidly gaining popularity as an alternative or replacement for polyester and polycarbonate with better characteristics
Polystyrene (PS), very good but has a TCC of about -100 ppm/C; can be easily damaged by soldering because the film melts at +85C
Polypropylene (PP), very low dissipation factor and a lower cost alternative to PS with a higher melting point (+105C); higher TCC up to -250 ppm/C
Less common, typically from “boutique” vendors:
Polytetrafluoroethylene (PTFE), aka Teflon®, can be problematic due to its porosity; many audiophiles believe PTFE capacitors sound better
Others: Polycarbonate (PC), once popular but virtually obsolete today; Polyethylene Naphthalate (PEN), similar to polyester but better >100C; Polyimide (PI), can potentially operate in environments up to +400C 22

23. Film Capacitor Construction
Metalized Film
The dielectric film is pre-coated with a conductive surface that is connected to the capacitor terminals
Has higher equivalent series resistance (ESR) due to the relatively small thickness of the metallization, 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 ESR and dissipation factor than metalized film
The nature of the electrical connection between the conductive plates of the capacitor and the external leads can result in non-linear ESR 23

24. Ceramic Capacitors
There are many types of ceramic capacitors, but the only one that should ever be used in high performance analog design is “C0G” (also known as “NP0”)
Available in values to >100nF with tolerances of 1-5% and voltage ratings up to 500V (up to 1kV for through-hole)
TCC is ±30 ppm/C, typically <15 ppm/C  better than many resistors!
Low dissipation factor and frequency dependence
Excellent stability, virtually immune to humidity
Lower grades of ceramic caps can exhibit profound non-linearity with applied voltage…some can also be strongly piezoelectric and microphonic
Examples include Z5U, Y5Y, and Hi-K® 24

25. Mica Capacitors
Mica capacitors were once highly regarded for analog design in the past…but much less so today…
Still commonly available with 1-5% tolerances in values up to 10,000 pF
TCC varies by brand and value, usually around +90 ppm/C
Relatively stable, however the typical epoxy coating is very easy to fracture near the leads allowing moisture intrusion during board wash or long term exposure to high humidity (caps so affected behave very strangely)
Potentially abrupt small shifts in value when exposed to vibration or shock
Unfortunately mica is a product of nature, some of its better sources have been depleted
With the widespread availability and lower guaranteed TCC of C0G (NP0) ceramic caps, there are very few reasons to specify a mica capacitor anymore 25

26. Glass Capacitors
Glass is relatively inert and potentially one of the most stable of all capacitor dielectrics
Values commonly available up to 2,200 pF (some vendors to 100,000 pF)
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 is welcome]
Curiously, TCC is not as good as other types: typically +140 ppm/C
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 much more expensive 26

27. Non-Linearity of Capacitors
A complex, fascinating, and often neglected subject…
Capacitors can also have a voltage coefficient effect
The E-field between the capacitor plates will cause a compressive force on the dielectric that can modulate its thickness (especially non-rigid films)
Unlike resistors, the voltage coefficient is usually positive
A good design practice is to avoid peak signal levels that are greater than about 5-10% of the capacitor’s voltage rating
As mentioned earlier, film capacitors can also have a nasty form of non-linearity related to signal current
The metalized surfaces or foils of a film capacitor are often electrically connected to the external leads via physical crimping, not welding
The resulting contact resistance (ESR) can be non-linear causing unwanted distortion voltages in series with the capacitor 27

28. An Interesting Distortion Comparison
Audio Precision recently replaced a custom 333 nF PP-Foil tubular capacitor with a module containing multiple 100V C0G ceramic caps wired in parallel 28

29. An Interesting Distortion Comparison
The following data was taken at 15 Hz and 5.30 Vac for 9 random samples:
Original PP-Foil Design Multiple 100V C0G Ceramic HD HD HD 3HD dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB dB _________ _________ _________ _________ μ = μ = μ = μ = σ = σ = σ = σ = 4.50 29

30. Op-Amps, The Traditional Categories
Precision
Optimized for low DC offset and bias current. Older designs tended to be slow with a very low slew rate, some were “chopper” stabilized…
General Purpose
Often dominant-pole compensated and stable under unity gain situations. Slew rates tend to be in the range of 4-20 V/μs. Many newer designs insert a pole-zero pair into the open loop response to increase GBW
High Speed
Not necessarily stable under unity gain situations. Slew rates tend to be >20 V/μs. Op-amps optimized for ultra-low noise and distortion tend to fall more into this category
Really High Speed
Optimized for video signal processing and specialized signal generation

31. Op-Amps, A More Useful Classification
Advances in IC processes and circuit techniques now strongly blur these traditional categories
Indeed, there are a number of op-amps that feature the combination of excellent DC performance, decent slew rate and GBW characteristics, low noise, and low non-linear distortion…examples include the OPA1611/12, OPA1641/42, OPA827, and LT1468
My apologies if your own favorite op-amp is not in this list…
A much more useful op-amp distinction is the input device technology: Bipolar vs. JFET
Both can offer input voltage offset performance to <200 μV
However, JFET op-amps have near-zero input bias current
Bipolar op-amps still have the lowest input noise voltage performance, although some current JFET designs rival older classic bipolar designs

32. Op-Amp Distortion Mechanisms
Input stage trans-conductance non-linearity
ΔIinput = Ccomp * d/dt(Vout) in a dominant pole design
Input stage common mode impedance non-linearity
Junction capacitance variation with signal voltage, JFETs are worse
Input stage offset voltage modulation
Unequal thermal coupling from the output stage to the input device pair
Output stage non-linearity, crossover non-linearity
Non-linear power supply currents (often overlooked!)
Mutual inductance between power supply connections and critical circuit loops can be a significant distortion contributor (Faraday’s Law)
Imperfect power supply rejection (PSRR) of non-linear ripple voltages 32

33. 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 pF between the input and pin 8 33

34. Two-Pole Compensation

35. “Feed-Forward”

36. More Tips to Minimize Op-Amp Distortion
Use inverting topologies whenever possible
Input capacitance modulation effects can be very significant with common mode signals in a non-inverting configuration
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. 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
Slight noise penalty due to extra resistive elements
Distortion reduction can be surprisingly significant! 36

37. Common-Mode Distortion Cancellation

38. Which Op-Amp is Best for Ultra-Low THD?
NONE! Although some can deliver -120 dB THD at 1 kHz, none can maintain that level of performance up through 20 kHz or higher frequencies
The solution is to think “outside of the box” and not be limited to just a single op-amp!
Consider a 2 op-amp cascade topology: the first op-amp selected for its input characteristics, the second selected for speed and high slew rate
Stability is an obvious concern, but it can be done… Indeed, this circuit topology is used extensively in the state-of-the-art Audio Precision APx555 audio test system (introduced in late 2014)
Unfortunately our discussion of the 2 op-amp cascade topology and some of its design considerations must end at this point due to IP issues…

39. In Conclusion…
Today we have examined some of the design factors responsible for distortion in analog circuits
As you have seen, the analog design engineer must contend with real-world components having many forms of imperfection and non-linearity that can significantly limit THD
Hopefully I have also shown you that analog design engineering can be both an exciting and challenging career…especially at the cutting edge of performance
Questions, comments, slide set:

40. Designing Analog Circuits with Ultra-Low THD (<-120 dB)
Bruce E. Hofer Ex-Chairman & Co-Founder Audio Precision, Inc. (now retired)