Copyright © 2011 by Wayne Stegall
Updated January 6, 2011. See Document History at end for
details.
Computer verified.
Transfer Curve Shape and Distortion
Introduction
Investigated here is the common belief that the more bends a
transfer function has the worse the quality of its distortion.
Mathematical Analysis
To evaluate the assertion at hand it is necessary to choose suitable
mathematical models and methods. I chose to use sin/cos functions
as error functions added to a consistant linear function base (i.e. y = x  k(sin or cos)(m·x) ) in
order to have the same sort of distortion characteristic regardless
of the number of bends. I also decided to iterate for a value of
k in each case to give a total distortion of 0.01% to add another
element
of consistency. The transfer curve graphs are exaggerated as a
result because the actual transfer curves would still look very
flat. The methodology is to apply the transfer functions to one
cycle of a unity magnitude sine function then analyze the result with a
Fast Fourier
Transform.
The resulting Fourier series analysis gives the distortion breakdown by
harmonic.
Figure
1:
Transfer
Curve
1
—
y = x  k·cos(0.5π·x)


k = 0.000189781
Total distortion = 0.0001002042 0.01002042%
79.9823dB
Breakdown by harmonic
harmonic # 
value 
decibels 
1 
1.0000000000 
0.0000dB 
2 
0.0000947773 
80.4659dB 
4 
0.0000053124 
105.4942dB 
6 
0.0000001132 
138.9199dB 
8 
0.0000000013 
177.9077dB 

Figure 2: Transfer Curve 2 —
y = x  k·sin(1.0π·x) 

k = 0.000128746
Total distortion = 0.0001002025 0.01002025%
79.9824dB
Breakdown by harmonic
harmonic # 
value 
decibels 
1 
0.9999267138 
0.0006dB 
3 
0.0000858629 
81.3239dB 
5 
0.0000134259 
97.4411dB 
7 
0.0000008807 
121.1034dB 
9 
0.0000000322 
149.8463dB 
11 
0.0000000008 
182.4618dB 

Figure 3: Transfer Curve 3 —
y = x  k·cos(1.5π·x) 

k = 7.9155e005
Total distortion = 0.0001001989 0.01001989%
79.9827dB
Breakdown by harmonic
harmonic # 
value 
decibels 
1 
1.0000000000 
0.0000dB 
2 
0.0000231636 
92.7039dB 
4 
0.0000586435 
84.6356dB 
6 
0.0000162764 
95.7689dB 
8 
0.0000019717 
114.1033dB 
10 
0.0000001373 
137.2439dB 
12 
0.0000000063 
164.0541dB 
14 
0.0000000002 
193.8519dB 

Figure 4: Transfer Curve 4 — y = x  k·sin(2.0π·x) 

k = 8.4877e005
Total distortion = 0.0001004889 0.01004889%
79.9576dB
Breakdown by harmonic
harmonic # 
value 
decibels 
1 
1.0000360528 
0.0003dB 
3 
0.0000049419 
106.1221dB 
5 
0.0000632885 
83.9735dB 
7 
0.0000267398 
91.4568dB 
9 
0.0000049439 
106.1186dB 
11 
0.0000005344 
125.4434dB 
13 
0.0000000384 
148.3066dB 
15 
0.0000000020 
174.0485dB 

Figure 5: Transfer Curve 5 — y = x  k·cos(2.5π·x) 

k = 6.00815e005
Total distortion = 0.0001006654 0.01006654%
79.9424dB
Breakdown by harmonic
harmonic# 
value 
decibels 
1 
1.0000000000 
0.0000dB 
2 
0.0000180809 
94.8556dB 
4 
0.0000083474 
101.5690dB 
6 
0.0000415688 
87.6247dB 
8 
0.0000251344 
91.9946dB 
10 
0.0000064539 
103.8036dB 
12 
0.0000009738 
120.2309dB 
14 
0.0000000986 
140.1187dB 
16 
0.0000000072 
162.8095dB 
18 
0.0000000004 
187.8707dB 

You should not read too much into the fact the the harmonic results are
entirely even or odd. This is a function of the perfect
symmetry of the trigonometric error functions used. However there
is a trend here. The more bends in the transfer function, the
higher the order of the highest harmonics that result.
Topology Considerations
Care must be taken if successive stages of a system are
not to create multiple bends in the transfer curve. It is
commonly thought that bending the transfer function in an opposite
directions alternatively will reduce distortion. And it
may. However the nature of most transfer functions prevents
distortion cancellation. Instead the flatter curve will have more
bends. This is because the transfer curves of known amplifying
devices have a slope that increases with magnitude, resulting in the
greatest bend near cutoff. Alternate bends under varying gain and
bias contexts will create random bends to a less musical end. A
common topology creating an opportunity for multiple bends is one where
the same polarity of device is used in successsive inverting stages
(See
figure 6 below). Vacuum tube circuits force this type of topology
in
some cases because tubes have only ntype transfer curves.
Perhaps
this is why SET fans scorn multiple amplifiying stages in their signal
paths, sometimes only putting a passive volume control in front of
their
singlestage amp. Transistors have an advantage here because
ntype and ptype devices are available enabling an alternating
topology to match the bends in the transfer curves in the same
direction relative to the signal (See figure 7 below).
Figure 6: cascaded
nchannel topology can create as many bends in the transfer curve as
inverting stages.


Figure 7: Alternating
topology bends the transfer curve in the same direction relative to the
signal in each device.


Dominant Bend Masking
Perhaps one might think to overcome transfer function anomalies by
having one characteristic or bend in a transfer function dominate the
others. Will the dominant bend solely determine the distortion
outcome? First I added the error term of transfer
curve 5 reduced by 10 to transfer curve 1 then determined this an
improper model. The proper model below nests the transfer
functions instead, a more proper representation. The result is
somewhat inconclusive. Relative to multibend transfer curve, the
second harmonic is raised, the fourth lowered, odds are added at low
levels, the remainder lower. The result appears mostly additive
with some subtractions due to phase. It appears the intended
masking that results would be less than ideal. This might infer
that using a euphonic stage to mask the distortion of a less than
euphonic signal chain would require that chain to already have
inaudible distortion.
Figure 8:
Transfer
Curve 6 —
y = x 
0.1k·cos(2.5π·x)  k·cos(0.5π·(x 
0.1k·cos(2.5π·x)))


k = 0.000164986
Total distortion = 0.0001002453 0.01002453%
79.9787dB
Breakdown by harmonic
harmonic# 
value 
decibels

dominant bend (from figure
1)

full multibend (from
figure 5)

1 
0.9999999983 
0.0000dB 
0.0000dB 
0.0000dB 
2 
0.0000774293 
82.2219dB 
80.4659dB 
94.8556dB 
3 
0.0000000005 
186.3172dB 


4 
0.0000023261 
112.6675dB 
105.4942dB 
101.5690dB 
5 
0.0000000022 
173.0612dB 


6 
0.0000115134 
98.7760dB 
138.9199dB 
87.6247dB 
7 
0.0000000006 
184.6238dB 


8 
0.0000069031 
103.2191dB 
177.9077dB 
91.9946dB 
9 
0.0000000010 
180.4310dB 


10 
0.0000017723 
115.0295dB 

103.8036dB 
11 
0.0000000004 
189.1018dB 


12 
0.0000002674 
131.4568dB 

120.2309dB 
14 
0.0000000271 
151.3446dB 

140.1187dB 
16 
0.0000000020 
174.0354dB 

162.8095dB 
18 
0.0000000001 
199.0969dB 

187.8707dB 

Final Thoughts
Investigating the use of distortion cancellation in some topologies may
add additional benefit to this pursuit. Other than a matched
current mirror, I cannot immediately think of any circuit for which
this would be an easy task. The expensive OPA627 operational
amplifier derived its reputation for low distortion and quality sound
from a proprietary distortion cancelling circuit.
Document History
January 1, 2011 Created.
January 1, 2011 Made minor grammatical corrections.
January 6, 2011 Added investigation into distortion masking and
made an immediate model correction to same.