Copyright © 2012 by Wayne Stegall
Updated May 25, 2013. See Document History at end for
details.
Feedback Distortion Analysis
Taming
the
capricious
transfer
curve
Introduction
The process of trying to create a desired distortion result in
a feedback amplifier baffled me. I presumed that the transfer
curve
would hold about the same shape under feedback only flatter, the
distortion as a whole reduced by the feedback factor 1 + A
_{OL}β.
Instead,
the
transfer
curve
behaved
unpredictably
under
different
openloop
gains
and
feedback
factors
even
paradoxically
flipping
the
curve
for
high feedback factors.
First Things
If transfer curve polynomials are to be used to evaluate distortion
characteristics, it is first necessary to determine how power terms
relate to specific harmonics. Since input signals are sinusoids
the power terms must be interpreted as powers of sinusoids. The
first few powers are well documented.
(1) 
cos^{2}(x) 
= ^{2} 
½
+ ½cos(2x) ^{2} 

sin^{2}(x) 
= ^{2} 
½ – ½cos(2x) ^{2} 





cos^{3}(x) 
= ^{2} 
¾cos(x)
+ ¼cos(3x) ^{2} 

sin^{3}(x) 
= ^{2} 
¾sin(x) –
¼sin(3x) ^{2} 





cos^{4}(x) 
= ^{2} 
⅜
+ ½cos(2x) + ⅛cos(4x) ^{2} 

sin^{4}(x) 
= ^{2} 
⅜ – ½cos(2x) +
⅛cos(4x) ^{2} 
In order to understand these results and extend them to higher powers
it is necessary to consider raising a sinusoid to a power to be an
intermodulation process. In this sense all distortion is
intermodulation distortion, the case of harmonic distortion being the
intermodulation of a single signal with itself. Since
intermodulation produces sum and difference frequencies from the inputs
the above power of sinusoidal trend is to have harmonics of some level
up to the n
^{th} harmonic for an x
^{n} term varying
according to the bias.
(2) 
cos^{n}(x
+
φ) 
= ^{2} 
a_{0} +
a_{1}cos(x + φ_{1}) + a_{2}cos(2x + φ_{2})
+
...
+
a_{n}cos(nx + φ_{n})^{2} 
where φ
_{i} represents any phase shift that may accrue.
Feedback Hacks Up the Transfer Curve
The gain factors in the negativefeedback equation are valid in the
both the time and frequency (sdomain) domains. In the time
domain they can represent the nonlinear transfer curves we want to
analyze.
Start with the standard negativefeedback equation.
(3)

A_{CL} =

A_{OL}
1 + A_{OL}β 
Then solve to separate the linear gain term from the remainder.
(4)

A_{CL} =

1/β + A_{OL}
1 + A_{OL}β 
– 
1/β
1 + A_{OL}β 
=

1
β

– 
1
β + β^{2}A_{OL}

If the undistorted closed loop gain is known to be:
What remains is the error or distortion term:
(6)

ε = – 
1
β + β^{2}A_{OL}

For a nonlinear openloop transfer curve polynomial
(7)

H(x) = h_{0}
+ h_{1}x + h_{2}x^{2} + ... + h_{n}x^{n}

its nonlinear gain A
_{OL} is defined as
^{1}
(8)

A_{OL}(x) = 
d
H(x)
dx

= a_{0}
+ a_{1}x + a_{2}x^{2} + ... + a_{n1}x^{n1}

The expanded error term now involves division by a polynomial the
general nature of
which the constants β and β
^{2} do not alter.
(9)

ε = 

1
β + β^{2}(a_{0}
+ a_{1}x + a_{2}x^{2} + ... + a_{n1}x^{n1})

Long division then synthesizes an infinite numerator polynomial of the
form:
(10)

ε =

b_{0} + b_{1}x +
b_{2}x^{2} + b_{3}x^{3} + ... 
Feedback will then take any transfer curve, even the simplest
second
order one producing only second harmonic distortion, and wring an
infinite number of harmonics out of it. It is left to the
designer to
juxtapose the various amplifier parameters affecting the final transfer
curve to reduce any undesirable higher harmonics to vanishing
levels or a profile that is suitable to him.
An Analogy
One plausible explanation of this result is the following. That
uncorrected distortion becomes a signal itself which input with the
correct signal gets distorted again. This process would occur
repeatedly until an infinite number of harmonics are produced.
This
infinite process would be conceived to occur instantly. This is
an
analogy only, one representing the long division that created the
closedloop transfer curve polynomial from the openloop one.
Conclusions
I deduct that obtaining a desired euphonic distortion outcome is an art
requiring experience and experimentation. Designing to an
objective
specification only is perhaps like throwing dice in comparison, a
process not guaranteeing a desirable result. Also, under certain
feedback conditions the BJT transistor with its already infinite
transfer polynomial may compare more favorably with
squarelaw devices than at least my design prejudices normally
judge. (I still prefer the squarelaw devices regardless!)
Example of a Long Division by Polynomial
Here a representative long division created by a first order gain
polynomial representing a second order transfer characteristic.
In each case the numerator is manipulated into the sum of a divisible
term and a remainder then the possible division completed. The
process proceeds infinitely.
(11)

1
10 + x

=

1 + 0.1x
10 + x

– 
0.1x
10 + x 
= 0.1 – 
0.1x
10 + x 
(12)

0.1 – 
0.1x
10 + x 
= 0.1 – 
0.1x + 0.01x^{2}
10 + x 
+

0.01x^{2}
10 + x 
= 0.1 – 0.01x +

0.01x^{2}
10 + x 
(13)

0.1  0.01x + 
0.01x^{2}
10 + x 
= 0.1 – 0.01x + 
0.01x^{2} + 0.001x^{3}
10 + x 
– 
0.001x^{3}
10 + x 
= 0.1 – 0.01x + 0.001x^{2}
– 
0.001x^{3}
10 + x 
(14)

0.1 – 0.01x + 0.001x^{2}
– 
0.001x^{3}
10 + x 
= 0.1 – 0.01x + 0.001x^{2}
– 
0.001x^{3} + 0.0001x^{4}
10 + x 
+

0.0001x^{4}
10 + x 
= 0.1 – 0.01x + 0.001x^{2}
– 0.0001x^{3} + 
0.0001x^{4}
10 + x 
What a mess! Division by a larger polynomial is even more
complicated.
Happy amplifiers ;)
^{1}The gain polynomials in this analysis
are derivatives of the transfer functions polynomials they represent
and are therefore one order lower than them. This does not affect
the validity of the analysis.
Document History
August 11, 2012 Created.
August 11, 2012 Added footnote about polynomial representation,
improved some wording, added derivation of gain polynomial from
transfer curve polynomial, and added an example of long division.
May 25, 2013 Corrected + to – in equation 4.