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Why Negative Feedback Is the Audiophile’s Enemy

Technical data is just an improve that item was tested carefully before sending to customer. While our products consistently achieve good to excellent technical figures compared to commercial alternatives, those numbers serve only to confirm that the design contains no engineering errors.
We could, in most cases, apply active techniques to make the specifications look even more impressive — but if doing so strips away originality or musical character, we will not do it. Our aim is to build equipment that sounds beautiful, not equipment that measures beautifully.
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Abstract. Negative feedback (NFB) is a fundamental technique for reducing distortion and noise and for stabilising the gain of an amplifier. Yet many listeners feel that amplifiers with heavy feedback sound dry, lack dynamics and seem unnatural. This article analyses mathematically the mechanism by which negative feedback acts on the signal: the depth of feedback falls with frequency, so the higher-order harmonics of distortion are suppressed far less than the lower-order ones. As a result, the harmonic structure of the amplifier is altered, and the deeper the feedback, the greater the alteration. From this we derive design principles for amplifiers that use little or no feedback.

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Why Negative Feedback Is the Audiophile’s Enemy

Name: Why Negative Feedback Is the Audiophile’s Enemy
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A mathematical analysis of how negative feedback affects the harmonic spectrum of audio signals

Uesugi Ken, CTO, OTOMON LAB

Abstract. Negative feedback (NFB) is a fundamental technique for reducing distortion and noise and for stabilising the gain of an amplifier. Yet many listeners feel that amplifiers with heavy feedback sound dry, lack dynamics and seem unnatural. This article analyses mathematically the mechanism by which negative feedback acts on the signal: the depth of feedback falls with frequency, so the higher-order harmonics of distortion are suppressed far less than the lower-order ones.

Readers who are not interested in mathematics may skip the derivations and go straight to the Remarks, Section 5 and Section 6.

1. Introduction

It is often said that negative feedback makes an amplifier sound slow, dry, unnatural and lacking in dynamics. These, however, are subjective impressions of listeners. This article seeks a quantitative explanation: how does negative feedback act on an audio signal, and which of its effects can rob the sound of its natural character?

The article is organised as follows:

Benefits of negative feedback.
Mathematical model of a negative feedback system.
How negative feedback affects the sound.
Conclusions and practical application.

2. Benefits of Negative Feedback

The greatest benefit of negative feedback is that it corrects errors in the amplifier’s output signal, making specifications such as hum, background noise and total harmonic distortion (THD) look better. With a loop gain $T = A\beta$ (defined in Section 3), negative feedback provides three effects established since the work of H. S. Black (1934):

Distortion and noise generated inside the feedback loop are reduced by the factor $1 + T$ : $D_{\text{out}} = D/(1 + T)$ .
Output impedance is reduced by the same factor: $Z_{\text{out}} = Z_{0}/(1 + T)$ , giving better control of the loudspeaker (a higher damping factor).
The closed-loop gain becomes less sensitive to component variations: when $T \gg 1$ , $H \approx 1/\beta$ .

However, negative feedback is a double-edged sword when engineers overuse it to polish an amplifier’s specifications. Amplifier design is not merely a matter of measurements at the output. It is a whole process of harmonising the gain stages and minimising hum, noise and harmonic distortion every time the signal passes through a component such as a resistor, a capacitor or a transformer. Negative feedback should not be used to correct design flaws; it should only be the ribbon tied on a gift before it is handed to its owner.

3. Mathematical Model of a Negative Feedback System

3.1. Basic quantities

In a negative feedback system (Black, 1934; Bode, 1945), the quantity that determines the amount of feedback is the loop gain:

T(s) = A(s)\,\beta(s)

where $A(s)$ is the open-loop gain of the amplifier and $\beta(s)$ is the transfer ratio of the feedback network. The amount of feedback actually applied to the signal is determined by $T(s)$ , not by $A$ or $\beta$ alone.

The feedback factor is defined as:

F(s) = 1 + T(s) = 1 + A(s)\,\beta(s)

It is the denominator of the closed-loop transfer function:

H(s) = \frac{A(s)}{1 + A(s)\,\beta(s)} = \frac{A(s)}{F(s)}

The magnitude $|F(j\omega)|$ is called the depth of feedback at frequency $\omega$ and is usually expressed in dB as $20\log_{10}|F(j\omega)|$ .

3.2. Frequency dependence of $A$ and $\beta$

Every physical amplifier has a frequency response. The general model is:

A(j\omega) = A_{0}\,\frac{\prod_{i}\left(1 + j\omega/\omega_{zi}\right)}{\prod_{k}\left(1 + j\omega/\omega_{pk}\right)}

where $A_{0}$ is the DC gain, $\omega_{zi}$ are the zeros and $\omega_{pk}$ are the poles. Hence $A(j\omega)$ always varies with $\omega$ .

The feedback network $\beta$ is frequency-dependent as well. Even when designed to be purely resistive, in practice:

\beta(j\omega) = \beta_{0}\,\frac{\prod_{m}\left(1 + j\omega/\omega_{zm}\right)}{\prod_{n}\left(1 + j\omega/\omega_{pn}\right)}

owing to stray capacitance, wiring inductance and the frequency response of the components in the feedback network. In a tube amplifier whose feedback loop encloses the output transformer, the transformer’s frequency response is the most important source of poles.

Since both $A(j\omega)$ and $\beta(j\omega)$ depend on frequency, the loop gain $T(j\omega)$ , and therefore the depth of feedback $|F(j\omega)|$ , are frequency-dependent.

3.3. Single-pole amplifier model

Consider an amplifier with a single pole and a purely resistive feedback network:

A(j\omega) = \frac{A_{0}}{1 + j\omega/\omega_{p}}, \qquad \beta = \text{const}

Loop gain and feedback factor:

T(j\omega) = \frac{A_{0}\beta}{1 + j\omega/\omega_{p}}, \qquad F(j\omega) = 1 + \frac{A_{0}\beta}{1 + j\omega/\omega_{p}} = \frac{1 + A_{0}\beta + j\omega/\omega_{p}}{1 + j\omega/\omega_{p}}

Depth of feedback at frequency $\omega$ :

|F(j\omega)| = \sqrt{\frac{\left(1 + A_{0}\beta\right)^{2} + \left(\omega/\omega_{p}\right)^{2}}{1 + \left(\omega/\omega_{p}\right)^{2}}}

Numerical example. Take $A_{0}\beta = 9$ (DC loop gain of 9, so that the depth of feedback at DC is $1 + A_{0}\beta = 10$ , i.e. 20 dB) and a pole frequency $f_{p} = 10$ kHz ( $\omega_{p} = 2\pi\cdot 10^{4}$ rad/s):

Table 1. Depth of feedback versus frequency ( $A_{0}\beta = 9$ , $f_{p} = 10$ kHz)
Frequency	$\omega/\omega_{p}$	$\left\vert F(j\omega)\right\vert$	Depth of feedback
20 Hz	0.002	10.0	20.0 dB
1 kHz	0.1	9.95	20.0 dB
10 kHz	1	7.11	17.0 dB
60 kHz	6	1.92	5.7 dB
100 kHz	10	1.41	3.0 dB

Remark. The depth of feedback falls from 20 dB at 20 Hz to only about 3 dB at 100 kHz. In other words, high-frequency components receive far less feedback than low-frequency components.

3.4. Proof of monotonicity by differentiation

For the single-pole model, the squared depth of feedback is:

|F(j\omega)|^{2} = \frac{\left(1 + A_{0}\beta\right)^{2} + \left(\omega/\omega_{p}\right)^{2}}{1 + \left(\omega/\omega_{p}\right)^{2}}

Let $u = \left(\omega/\omega_{p}\right)^{2}$ and $a = \left(1 + A_{0}\beta\right)^{2}$ , so that $|F|^{2} = \dfrac{a + u}{1 + u}$ . Differentiating with respect to $u$ :

\frac{d|F|^{2}}{du} = \frac{(1 + u) - (a + u)}{(1 + u)^{2}} = \frac{1 - \left(1 + A_{0}\beta\right)^{2}}{(1 + u)^{2}} = -\,\frac{\left(A_{0}\beta\right)^{2} + 2A_{0}\beta}{(1 + u)^{2}}

Since $A_{0}\beta > 0$ , this derivative is always negative. Moreover $u$ increases with $\omega$ , with $\dfrac{du}{d\omega} = \dfrac{2\omega}{\omega_{p}^{2}}$ . By the chain rule:

\frac{d|F|^{2}}{d\omega} = -\,\frac{\left[\left(A_{0}\beta\right)^{2} + 2A_{0}\beta\right]\cdot 2\omega/\omega_{p}^{2}}{\left(1 + \omega^{2}/\omega_{p}^{2}\right)^{2}} < 0 \quad \text{for all } \omega > 0

Conclusion. For the single-pole model, the depth of feedback $|F(j\omega)|$ decreases monotonically as frequency rises. The rate of decrease is not constant but depends on $\omega$ : very slow when $\omega \ll \omega_{p}$ , fastest around $\omega_{p}$ and above.

3.5. General case: multi-pole amplifier

For an amplifier with $M$ poles:

T(j\omega) = \frac{A_{0}\beta}{\prod_{k=1}^{M}\left(1 + j\omega/\omega_{pk}\right)}, \qquad F(j\omega) = 1 + T(j\omega)

At low frequencies ( $\omega \ll \omega_{p1}$ ): $|F| \approx 1 + A_{0}\beta$ , the deepest feedback.
At very high frequencies ( $\omega \gg \omega_{pM}$ ): $|F| \approx 1$ , feedback practically vanishes.
Where $|T| \gg 1$ , $|F| \approx |T|$ . Each pole makes $|A|$ fall by a further 20 dB/decade, so the depth of feedback falls at the same rate.

4. How Negative Feedback Affects the Sound

Every sound, whether a voice or a musical instrument, consists of a fundamental and a series of harmonics. It is this harmonic structure that gives a singer’s voice and an instrument’s timbre their character. Two kinds of harmonics passing through an amplifier must be clearly distinguished:

Harmonics of the original signal: part of the sound to be reproduced.
Harmonics generated by the amplifier itself (distortion): foreign components that are not in the recording.

Negative feedback acts on these two kinds of harmonics through two different mechanisms.

4.1. Effect on the harmonics of the original signal

The original signal contains many harmonics:

x(t) = \sum_{n=1}^{N} X_{n}\cos\left(n\omega_{0}t + \phi_{n}\right)

Considering only the linear part of the amplifier, this signal passes through the closed-loop transfer function $H(j\omega)$ :

y(t) = \sum_{n=1}^{N} X_{n}\,\left|H(jn\omega_{0})\right|\cos\left(n\omega_{0}t + \phi_{n} + \angle H(jn\omega_{0})\right), \qquad \left|H(jn\omega_{0})\right| = \frac{\left|A(jn\omega_{0})\right|}{\left|F(jn\omega_{0})\right|}

Note that the amplitude of the $n$ -th harmonic is multiplied by $|H| = |A|/|F|$ , not by $1/|F|$ . For the single-pole model of Section 3.3:

H(j\omega) = \frac{A_{0}}{1 + A_{0}\beta + j\omega/\omega_{p}} = \frac{A_{0}/\left(1 + A_{0}\beta\right)}{1 + j\omega/\left[\omega_{p}\left(1 + A_{0}\beta\right)\right]}

Feedback moves the pole of the system from $f_{p} = 10$ kHz up to $f_{p}\left(1 + A_{0}\beta\right) = 100$ kHz. Comparing the 20th harmonic (20 kHz) with the 1 kHz fundamental:

Table 2. Linear response to the original signal
	Without feedback	With feedback ( $A_{0}\beta = 9$ )
Amplitude of 20 kHz relative to 1 kHz	−6.95 dB	−0.17 dB
Phase shift at 20 kHz	−63.4°	−11.3°

Remark. In linear terms, negative feedback does not distort the harmonic structure of the original signal; on the contrary, it flattens the amplitude and phase response. The cause of the loss of naturalness must therefore be sought in the non-linear part, i.e. in the distortion generated by the amplifier, analysed in Section 4.2.

4.2. Effect on distortion generated by the amplifier

Distortion generated inside the feedback loop at frequency $\omega$ is suppressed by the factor $1/|F(j\omega)|$ . If the fundamental has frequency $\omega_{0}$ , the $n$ -th harmonic distortion lies at $n\omega_{0}$ and is suppressed by the factor:

G_{n} = \frac{1}{\left|F(jn\omega_{0})\right|} = \sqrt{\frac{1 + \left(n\omega_{0}/\omega_{p}\right)^{2}}{\left(1 + A_{0}\beta\right)^{2} + \left(n\omega_{0}/\omega_{p}\right)^{2}}}

With $f_{0} = 1$ kHz, $f_{p} = 10$ kHz, $A_{0}\beta = 9$ :

Table 3. Suppression of harmonic distortion by harmonic order
Harmonic $n$	Frequency	$\left\vert F(jn\omega_{0})\right\vert$	Suppression factor $G_{n}$	Suppression
1	1 kHz	9.95	0.101	−20.0 dB
2	2 kHz	9.81	0.102	−19.8 dB
3	3 kHz	9.58	0.104	−19.6 dB
5	5 kHz	8.96	0.112	−19.0 dB
10	10 kHz	7.11	0.141	−17.0 dB
20	20 kHz	4.56	0.219	−13.2 dB

Remark. Second-harmonic distortion is suppressed by almost 20 dB, whereas 20th-harmonic distortion is suppressed by only about 13 dB, a difference of about 7 dB. The residual distortion spectrum after feedback is therefore shifted towards the higher-order harmonics. THD may be very low, but the remaining distortion is concentrated in the higher-order harmonics, which many authors consider more objectionable to the ear than the lower-order ones.

Influence of feedback depth. This difference depends directly on the depth of feedback. Table 4 shows by how many dB the 2nd harmonic is suppressed more than the 6th, the 20th and the 50th harmonics, for various depths of feedback (same $f_{0} = 1$ kHz, $f_{p} = 10$ kHz). Without feedback the difference is zero, meaning the amplifier’s harmonic structure is preserved.

Table 4. Change in harmonic structure versus depth of feedback
Depth of feedback (DC)	2nd vs 6th harmonic	2nd vs 20th harmonic	2nd vs 50th harmonic
3 dB	0.5 dB	2.1 dB	2.8 dB
5 dB	0.8 dB	3.3 dB	4.5 dB
10 dB	1.0 dB	5.4 dB	8.6 dB
20 dB	1.2 dB	6.7 dB	13.0 dB
40 dB	1.2 dB	6.8 dB	14.0 dB

Remark. With light feedback (3–5 dB), the ratios between harmonics change by only about 2–4.5 dB. With heavy feedback (20–40 dB), they change by as much as 7–14 dB: THD falls sharply, but the amplifier’s inherent harmonic structure is substantially altered. Between the 2nd and 6th harmonics, both of which lie well below the pole at 10 kHz, the change is small (0.5–1.2 dB); it grows rapidly for harmonics near and above the pole frequency.

5. Mathematical Conclusions

In a practical negative feedback system, the distortion suppression factor at frequency $\omega$ is:

G(\omega) = \frac{1}{\left|1 + A(j\omega)\,\beta(j\omega)\right|}

This quantity is a function of $\omega$ and, in the single-pole model, decreases monotonically with frequency. The ratio of the distortion suppression at the $n$ -th harmonic to that at the fundamental is:

\frac{G_{n}}{G_{1}} = \frac{\left|F(j\omega_{0})\right|}{\left|F(jn\omega_{0})\right|}, \qquad \left|F(j\omega)\right|^{2} = 1 + 2\,\mathrm{Re}\,T(j\omega) + \left|T(j\omega)\right|^{2}

When $|T| \gg 1$ , this ratio is approximately $\left|T(j\omega_{0})\right| / \left|T(jn\omega_{0})\right| > 1$ for all $n > 1$ : the higher the harmonic order, the less the distortion is suppressed.

From the above analysis we conclude:

Harmonics of the original signal pass through the closed-loop system without additional linear distortion; feedback even flattens the frequency and phase response.
Distortion generated by the amplifier is suppressed unevenly: low-order harmonics are strongly suppressed, high-order harmonics weakly. The residual distortion spectrum shifts towards the higher-order harmonics.
The deeper the feedback, the greater the alteration of the harmonic structure: from about 2–4.5 dB at 3–5 dB of feedback up to 7–14 dB at 20–40 dB (Table 4).
The extent of these effects depends on the depth of feedback $A_{0}\beta$ , the pole positions $\omega_{pk}$ , the fundamental frequency $\omega_{0}$ of the signal and the non-linear characteristics of each gain stage.

Thus negative feedback can make THD figures look excellent while at the same time changing the character of the distortion: the ratios between harmonics are altered, the residual distortion shifts to higher orders, and the deeper the feedback, the greater the change. This offers a plausible basis for the common observation that amplifiers with heavy feedback sound dry and less natural, despite better measurements.

6. Practical Application

If you use negative feedback to correct design flaws, you will need very deep feedback. THD can then be extremely low, but as shown in Sections 4 and 5, the residual distortion shifts to the higher-order harmonics and the amplifier’s inherent harmonic structure is substantially altered.

This does not mean that negative feedback is pure poison. It is like an antibiotic: used correctly, in the right dose and in the right place, it is an excellent weapon. When negative feedback is used to improve output specifications, around 10 dB or more is usually needed to make a clear difference to THD.

Therefore, to obtain beautiful sound before any feedback is applied, the first step is to use good components and to reduce non-linear distortion as far as possible. In a tube amplifier in particular, the heart and most important component is always the interstage transformer.

Next, the design must be uncompromising. An amplifier almost never has just one gain stage. In a tube amplifier, the gain of each stage is low, so one or two gain stages are usually needed before the power stage. Each gain stage produces its own harmonic distortion and hum. If the design is not good enough, the distortion of the stages accumulates and the total distortion at the output stage soars. Conversely, if the design is good enough, the harmonic distortion of the stages can cancel each other; the better the design, the greater the cancellation, and the amplifier’s specifications will be excellent even without feedback.

In the OUDDC circuit design, the distinctive coupling between the gain stages meets exactly this criterion. That is why, in almost all our amplifiers, since the specifications are already good enough without it, we keep the use of feedback to an absolute minimum in order to preserve the original character of the sound.

Most OTOMON amplifiers use no feedback at all. In certain special cases, a very small amount of negative feedback (typically 3–5 dB) is applied to lower the output impedance and help control the bass, the range most prone to losing control. According to Table 4, at this level the harmonic structure changes by only about 2–4.5 dB, far less than the 7–14 dB of deep feedback. Some amplifiers also offer a switch to select feedback on or off, so that listeners can experience for themselves the difference between with and without feedback, and how the sound changes as the amount of feedback changes.

References

H. S. Black, “Stabilized Feedback Amplifiers”, Bell System Technical Journal, vol. 13, 1934.
H. W. Bode, Network Analysis and Feedback Amplifier Design, Van Nostrand, 1945.

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