Vacuum Tube Amplifier Modeling with Dynamic Convolution

University of Florida | Journal of Undergraduate Research | Volume 17, Issue 2 | Spring 2016 1

Vacuum Tube Amplifier Modeling with Dynamic Convolution

Jason Traub

Department of Electrical and Computer Engineering, University of Florida

Though nearly obsolete within electronics, vacuum tubes remain in high demand for musical amplification [1]; and particularly for

electric guitar. This research discusses reasons for the observed preference and assesses its validity. Furthermore, amplifier electronic

circuits are studied and comparisons are made between vacuum tubes and bipolar junction and metal oxide field effect transistors; the

latter two have replaced vacuum tubes in nearly every application, and are responsible for the advancement of the digital computer.

Undoubtedly, there exists extreme cost effectiveness in obtaining a computer processing technique to mimic the sound that is applied

by a vacuum tube amplifier. Thus, many modeling techniques have been developed and used in commercially successful products. This

research discusses several of these methods and ultimately attempts to mimic a vacuum tube amplifier using the dynamic convolution

method, proposed by Kemp [2].

Introduction: Light Bulbs, Valves, and Transistors

acuum tubes, or “valves”, as the British call them,

were the first electronic component to be able to

function as an amplifier. To give a historical

perspective: Thomas Edison invented and patented the light

bulb in 1879 [3]; J.J. Thompson’s experiments led to the

discovery of the electron in 1897 [4]; and the work of John

Fleming and Lee de Forest ultimately led to the vacuum tube

triode in 1906 [5]. This three terminal device allows us to

amplify the current and/or voltage of an input signal.

A vacuum tube triode consists of two electrodes, the plate

(anode) and cathode, separated by a short distance in an

evacuated tube. Figure 1(a) shows a body diagram of a

vacuum tube triode. A third electrode (grid) is placed as a

wire mesh in between the plate and cathode. Figure 1(b)

shows the schematic representation.

As alluded to earlier, vacuum tubes act as an electronic

valve which controls the flow of current between the plate

and cathode by the voltage applied to the grid. The device

conducts current through thermionic emission [8]. The

cathode is heated, directly or indirectly, by a filament

connected to a high voltage. A potential difference of

hundreds of volts is applied between the plate and cathode

terminals. When the plate is more positive with respect to

the cathode, electrons will be emitted across the vacuum, i.e.

a current will flow. As we make the grid voltage more and

more negative, the more the current between the plate and

cathode is impeded. Figure 2 depicts the operation of a

12AX7 vacuum tube triode (taken directly from its’

datasheet). We can clearly see, as we make the grid voltage

more negative (lines moving to the right), the current gets

corralled.

From Tubes to BJTs and MOSFETs

This relatively crude device was relied upon in electronics

throughout the first half of the 20th century. Everything from

radio, television, radar, sound reproduction and process

control, used vacuum tubes. Today, however, vacuum tubes

are nearly extinct. By the early 1960s, solid-state transistors

were steadily replacing vacuum tubes [10]. The same

functionality that the vacuum tube pioneered, solid state

transistors did in an immensely more size, power, and cost

efficient manner [11].

The solid-state transistors most commonly used are

Bipolar Junction (BJT) and Metal Oxide Field Effect

(MOSFET) transistors. The physical operation of these

V

Figure 2. Vacuum Tube Diagrams

(a) Body diagram [6] (b) circuit symbol [7]

Figure 1. 12AX7 Current vs. Plate Voltage characteristic at several

grid voltages (Eg) [9]

JASON TRAUB


devices rely on charge transport between doped

semiconductors [12]. A BJT is a current controlled device,

while the MOSFET is a voltage controlled device. The

construction of these devices consists of no gaps, or

vacuums, and involves point to point contact of solid-state

doped semiconductors.

A rudimentary cross section of a MOSFET and a BJT is

shown in Figure 3. The fundamental physics of MOSFET

operation works like this: in the substrate, there exists a p-

doped region (hole dominant) between two n-doped regions

(electron dominant). An oxide insulator is placed above the

p-doped region and an electrode (gate) is placed on top. The

two n-doped regions are the drain and source electrodes.

When a positive voltage is applied at the gate, electrons in

the p-doped region are attracted to it. Thus, a channel for

current flow develops. The direction of the conventional

current in each device is given by the direction of the arrow.

A BJT works on similar fundamental. Here we have,

again, a p-doped region (base) between two n-doped regions

(collector and emitter). In this case, instead of a voltage

induced channel, we directly inject carriers into the doped

region, which allows for current flow. It is of note that, no

current flows through the gate of a MOSFET, due to the

insulator oxide.

It is essential to note that, the principle of operation of all

three devices (tubes, BJT, and MOSFET) remain the same:

the current between two nodes is controlled by the

current/voltage applied to the third.

To conclude this introduction, the development of the

vacuum tube and its associated electronic functions have

been monumental in the advancement of technology and

human achievement. The advent of the solid-state transistor

has progressed this technology, by implementing the same

electronic function, yet, much smaller, more power efficient,

and more reliable. It is the fact that we can fit billions of

transistors on a single computer chip [13] that is responsible

for the highly advanced functions computers perform today.

In spite of the many advantages of transistors, the major

question of this research remains: Why do vacuum tubes

remain relevant? Why do they remain in high demand for

guitar and other audio amplification purposes?

We will discuss the validation of this preference in the

following section. After an appreciation for the complex and

“imperfect” signal processing that vacuum tube circuits

inherently apply, we will discuss methods of modeling this

effect in the digital domain. The dynamic convolution

method has been chosen for detailed study and

experimentation. The final section documents the results and

methodology of this experiment and explains the

tribulations experienced. Finally, this research offers a

course of action for our future successful implementation of

dynamic convolution.

Perception and Psychoacoustics: Why Do Vacuum Tubes Remain Relevant?

It is known that a large amount of musicians and

audiophiles prefer the sound of vacuum tube amplifiers over

their solid-state counterparts [1]. Perhaps the sound of a

vacuum tube amplifier has become iconic due to their

extensive use on music from the 1960s. While this argument

is somewhat relevant, it does not reveal the objective and

subjective merits that tubes actually warrant.

Much of the desired sound of electric guitar stems from

how the signal distorts. In most practical engineering

situations, we do not want to distort the signal at all.

However, when musicality is our primary concern, this does

not necessarily apply, since the listening experience is

ultimately subjective. People continue to use tubes, because

people simply like the sound better! Let’s explore the, fairly

convincing, objective reasoning behind this.

According to research done by Russel O. Hamm [14],

there is an audible quality difference between tubes and

solid-state components that is perceivable and objectively

measureable. Vacuum tubes distort more gently than solid-

state transistors, particularly in the high frequency range.

Non-linearity causes distortion, and distortion generates

harmonics. Hamm’s research explains the difference in

terms of the harmonics generated when driven into

saturation. Tubes exhibit strong 2nd and 3rd harmonic

content, with the 4th and 5th harmonic’s power increasing as

the signal is driven more and more into saturation. The

author claims that even harmonics add body to the sound,

whereas, the 3rd harmonic contributes to softening the sound.

He adds that, the 5th harmonic adds a “metallic sound that

gets annoying in character as its amplitude increases” [14].

The author further notes that, higher order harmonics add

attack and bite to the sound. Thus, the perceived tonal

advantage of vacuum tubes can be explained by the fact that

solid-state components have strong, objectionable high

frequency components when only slightly driven into

saturation. Tubes, on the other hand, deliver pleasing, and

sought after, harmonic tones that rise in harmonic character

as the input signal increases.

According to the IEEE spectrum article, “The Cool Sound

of Tubes” [1], other characteristics of vacuum tube

amplifiers also have a substantial effect on their sound. For

Figure 3. Transistor cross sections

(a) MOSFET (b) BJT

VACUUM TUBE AMPLIFIER MODELING WITH DYNAMIC CONVOLUTION


example, it is noted that the high voltage output transformer,

used specifically in vacuum tube amplifiers, has a

tremendous effect. This effect is explained by the 2nd and 3rd

order harmonics generated with surprisingly low inter-

modulation distortion [1]. The author further notes that the

unique circuit components used can also affect the sound.

Finally, it is claimed that a natural compression of the audio

signal takes place when played through a tube amplifier, an

effect known as “infinite sustain” [1].

Modelling the sound in a computer

There have been plenty of attempts to recreate the desired

tube distortion, both in analog and digital (or hybrid)

implementations. It has been alluded to, and explicitly noted

in [15], that the shortcomings of tube amplifiers (large size,

weight, poor efficiency), offers an obvious motivation for

obtaining an emulation method. Cost and convenience, for

the consumer, as an alternative to buying a vacuum tube

amplifier, suggests a market. This research, in the following

paragraphs, focuses primarily on digital methods of

emulation.

To be able to model the sound applied by a vacuum tube

amplifier, we must think about how all factors affect the

output. Since the output sound is a function of how the

device physics alter the signal along its path, it seems

reasonable that, if we obtain a mathematical transfer

function, we can emulate our output. Advanced research,

focusing on the use of physical modeling for emulation of a

distortion overdrive pedal, has been done [16]. Through

advanced methods of solving linear and non-linear

differential equations, this method has delivered pleasing

results.

Another method of obtaining musical distortion is to

utilize a wave-shaping function, as performed by Fernandez

[17]. In this method, wave parameters can be chosen by the

musician to achieve “highly personal” sounding distortion.

The abstract notes that this should be regarded as a

“distortion synthesizer” of sorts, rather than an emulation

technique.

An example of a commercially successful technique in

tube digital emulation is given under a patent [18], and

known as Tube Tone Modeling (owned by Line 6). In this

method, an eight times oversampling block, in an embedded

processor, is used to handle the high frequency distortion

and obtain a vacuum tube-like sound.

Due to the complexity of the precise signal alteration, the

unpredictable nature of the underlying device physics, the

number of interrelated variables, and other factors, this

research chose to utilize a black-box approach and take

system measurement from input to output to derive a

transfer function. How to obtain this transfer function and

employ it to generate an output, is given by the dynamic

convolution method [2].

Dynamic Convolution

Convolution can be implemented to model a system, with

perfect accuracy, in theory, if the system is linear, and time-

invariant. However, the distortion we seek to emulate is, in

it of itself, non-linear, therefore violating the conditions of

the convolution theorem. However, dynamic convolution

has been proposed and implemented, by Kemp [2], to get

around the non-linearity. Kemp’s research gives specific

reference to its success in modeling vacuum tube amplifiers.

The discrete convolution formula used is:

(𝑥 ∗ ℎ)[𝑛] = 𝛴𝑖ℎ[𝑖]𝑥[𝑛 − 𝑖] (1)

For dynamic convolution, instead of inputting one unit

amplitude impulse into the system, we input a series of

impulses, at different amplitudes, and obtain the impulse

response from each. We normalize the response by the

impulse amplitude, as performed by Kemp.

Figure 4 shows a block diagram of the dynamic

convolution algorithm.

Each sample input, x[n], is compared to the test impulse

amplitudes, di. The hi, impulse response, corresponding to

the di that most closely matches x[n] is selected. The output

sample y[n] is computed directly using (1), using only the

current value of n. Then we increase n by 1, take the next

x[n], find the new hi corresponding to it, take the single-n

convolution again, and repeat the process until all points in

the input, x, have been processed, and our output, y, has been

generated.

Experimental Method

We obtained a range of impulse responses from our

system corresponding to impulse amplitude di. Specifically,

di = ± [1 – (i/128)]; i = 0,1,…,128 (2)

If Zi is the response of the system from di, we normalize Zi

by di to obtain hi.

In practice, a la Kemp, we input a series of step signals at

the amplitudes given in (2). Thus, we obtain the step

response. The impulse responses are calculated by

differentiation.

The method was performed as follows. First, we gather

test data (sampled at 4 times our sampling rate of 44.1 kHz,

Figure 4. Dynamic Convolution Algorithm Diagram

JASON TRAUB


i.e. 192 kHz). Processing of the data consists of down-

sampling, differentiation to get our impulse response, and

normalizing the response by the test amplitude. We

developed a ParseImpulseResponse.m function to first find

the peak in the impulse signal, and associate it to its

corresponding impulse response. The output of this function

is a matrix H and row vector D. The columns of H are hi,

corresponding to impulse amplitude, di, stored at the same

column index in D.

A dynamic convolution algorithm was constructed that

takes in an input vector, the impulse matrix, H, the

amplitude vector, D, and generates the output, y.

Experimental Results

The step input used is shown, by measurement with a

Digilent Analog Discovery oscilloscope, in Figure 5(a). Its

corresponding step response is given in Figure 5(b).

We were unable to satisfactorily implement the dynamic

convolution algorithm with this test data. As an illustration

of our source of error, we see our derived impulse and

impulse response in Figure 6 (a) and (b). The derived signal

did not meet expectation. First of all, we see a maximum

impulse of around 2, when our maximum step was at 1.

Also, the impulses are not consistently decreasing; some

impulses are drastically smaller or bigger than adjacent

impulses. This is a clear source of error.

We feel confident in the dynamic convolution algorithm

developed, since it collapses to normal convolution when

given a matrix of all of the same impulse response vectors.

Since it is difficult to input pure impulses into our system,

and we achieved poor results through the method attempted

thus far, we decided to attempt to characterize and obtain

our system data in the frequency domain. The Digilent

Analog Discovery’s Network Analyzer function was

employed in this development of the research. Here, the

network analyzer inputs the entire frequency range and

outputs a graphical display of the Bode plot. It is reassuring

to have a mild check of the accuracy of the data, by being

able to hear the signal and see the response match.

In this development, we input sinusoids of different

amplitudes. The Bode diagram, which plots magnitude and

phase, already takes care of the normalization for us. From

this experimental data we were able to analyze key factors

in the distortion we hear, namely, how the voltage output

compares to the voltage input as the amplitude and

frequency of our signal change. These results are depicted

in Figure 6, 7, and 8.

(a) Step Input

Figure 5. Step Input and Step Response

(b) Step Response

(a) Impulse Input

Response

Figure 6. Derived Impulses and Impulse Response

(b) Impulse Response



All three graphs see saturating distortion at high voltage

levels. The non-linear distortion obtained is much harsher

and abrupt at higher frequencies. This is a reassuring result

and coincides with the results from Hamm.

To generate these plots, we first observed the output Bode

plot from a range of amplitudes:

Ai = 100 mV · i; i = 1,2,3,…,10 (3)

For greater precision, we might want to use a 10 mV step.

These results are depicted in figure 9.

Here, we see a general high pass filter effect of the

amplifier, which is likely primarily due to the 6” speaker.

The cutoff frequency for this high pass effect is roughly 80

Hz. We see a common dip in the pass band, at around 450

Hz, whose presence diminishes as our test amplitude

increases. The high frequency response has an additional

gain boost. The gain gets steadily larger from 1 kHz to 20

kHz in what is ultimately more than a +10 dB gain than that

of lower frequencies in the pass band. Again, this effect also

diminishes as our test amplitude increases. As the test

amplitude approaches 1 V our response is fairly flat for 80 –

20,000 Hz. These results show substantial non-linearity, and

thus distortion, especially for the high frequencies.

To proceed with this test data, we need further

conditioning of our data. We seek to obtain the impulse

response from our Bode diagram. The discrete Fourier

transform (DFT) of the impulse response is essentially the

frequency sampled version of our Bode plot. Thus, we must

convert the magnitude and phase presented in the Bode

diagram to a polar coordinate. We need to be aware of

frequency resolution in a discrete Fourier transform and

sample the Bode plot accordingly. This requires a fairly

advanced interpolation algorithm. We could then apply the

inverse Fourier transform to obtain our impulse response

and be able to apply our dynamic convolution algorithm

(unless we can figure out how to apply dynamic convolution

in the frequency domain). This attempt to apply the

dynamic convolution method proposes its own challenges

and limitations and perhaps makes achieving our end goal

even more difficult. Further experimentation must be

explored and developed.

Figure 6. Vout vs. Vin for a 500 Hz sinusoid

Figure 7. Vout vs. Vin for a 1000 Hz sinusoid

Figure 8. Vout vs. Vin for a 10 k Hz sinusoid

From left to right:

Row1:100mV,200mV,300mV,400mV,500mV

Row2: 600mV,700mV,800mV,900,mV,1V

Figure 9. Magnitude plots from different test amplitude sweeps

JASON TRAUB


CONCLUSIONS

The chronological history surrounding the advent of the

vacuum tube was introduced. This component provided

remarkable advancement within electronics, allowing a

signal to be amplified for the first time, among other

functions. Over the next few decades, more efficient

technology came in the form of solid-state transistors, which

displaced vacuum tubes due to their vast improvement over

tubes in size, cost, and power efficiency. The digital

computer would not exist as we know it, if not for the

incredibly small size we can manufacture transistors.

Objective benefits aside, people still revere vacuum tubes

for their use in musical equipment. The subjective sound

quality of tubes is attributed to several reasons. One

particularly illuminating reason is that the harmonic content

of tubes in saturation is much more pleasing than that of

solid-state transistors. Others claim that the role of the

output transformer, circuit topology, and components used

in old vacuum tube amplifiers provide much of the desired

sound, as well.

Nonetheless, the tube sound is here to stay for subjective

and objective reasons. Moreover, this presents extreme

motivation for obtaining an accurate computer modeling

method to apply the sound of a vacuum tube amplifier

without experiencing the disadvantages of the technology

(poor power efficiency, frequent maintenance, bulky size

and weight).

Of the several modeling techniques employed in the

musical distortion realm, this research chose to investigate

the dynamic convolution method.

Dynamic convolution is a promising method. It utilizes

the power of convolution, and applies it to a non-linear

system. By choosing an appropriate impulse response for

each input sample, we can perform convolution by

dynamically changing the impulse response used.

Difficulties in data obtainment and accurate algorithm

construction prevented this research from achieving a

successful implementation. However, we are optimistic

about this method’s future success. Going forward, the

following inquiries will be addressed:

1. How can we rely on and obtain accurate impulse

response data?

2. Is 10 different impulse amplitudes enough? Is 128

required, as performed by Kemp?

If we find reliable data, perform accurate algorithms, and

take enough impulse amplitudes (this may correspond to the

resolution of our dynamic transfer function), we believe this

method will lead to a successful implementation for the

emulation of vacuum tube amplifiers. Moreover, we have

illustrated a very powerful tool of non-linear system

modeling.

REFERENCES

[1] Barbour, E. 1998. “The Cool Sound of Tubes.” IEEE Spectrum 35(8):24–35.

[2] Kemp, Michael J. Analysis and Simulation of Non-Linear Audio

Processes Using Finite Impulse Responses Derived at

Multiple Impulse Amplitudes. Audio Engineering Society 106.

[3] "The History of the Light Bulb." Energy.gov. Web. 04 Apr. 2016.

[4] "Joseph John Thomson | Chemical Heritage Foundation." Joseph

John Thomson | Chemical Heritage Foundation. Web. 04 Apr.

2016.

[5] "Sir Ambrose Fleming: Father of Modern Electronics." Sir Ambrose

Fleming: Father of Modern Electronics. Web. 04 Apr. 2016.

[6] By Svjo (Own work) [CC BY-SA 3.0

(http://creativecommons.org/licenses/by-sa/3.0)], via

Wikimedia Commons

[7] Bemoeial at nl.wikipedia [GFDL

(http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0

(http://creativecommons.org/licenses/by-sa/3.0/)], from

Wikipedia Commons

[8] "Thermionic Emission." Web. 4 Apr. 2016.

<http://www.physics.csbsju.edu/370/thermionic.pdf>. Lecture

Notes

[9] Radio Corporation of America, “Hi Mu Twin Triode”,

12AX7 datasheet. 30 July 1947.

[10] Bortz, Alfred B., PhD. Physics: Decade by Decade.

[11] "Vacuum Tubes and Transistors Compared." Effectrode. 2011.

Web. 04 Apr. 2016.

[12] Streetman, Ben G., and Sanjay Banerjee. Solid State Electronic

Devices. Upper Saddle River, NJ: Pearson/Prentice Hall, 2006.

[13] "Intel Microprocessor Quick Reference Guide - Product Family."

Intel Microprocessor Quick Reference Guide – Product

Family. Web. 04 Apr. 2016.

[14] Hamm, R. O. 1973. ”Tubes Versus Transistors—Is

There an Audible Difference?” Journal of the Audio

Engineering Society 21(4):267–273.

[15] Pakarinen, Jyri, and David T. Yeh. "A Review of Digital

Techniques for Modeling Vacuum-Tube Guitar

Amplifiers." Computer Music Journal33.2 (2009): 85-100.

Web.

[16] Yeh, David Te-Mao. Digital Implementation of Musical Distortion

Circuits by Analysis and Simulation. PhD diss., Stanford

University, 2009.

[17] Fernandez-Cid, P., and J. C. Quir ´ os. 2001. “Distortion of

Musical Signals by Means of Multiband Waveshaping.”

Journal of New Music Research 30(3):219–287.

[18] Diodic, Michel, Michael Mecca, Marcus Ryle, and Curtis Senffner.

Tube Modeling Programmable Digital Guitar Amplification

System. Diodic, assignee. Patent US 5789689 A. 4 Aug. 1998.

Print.



[19] Damelin, Steven B., and Willard Miller. The Mathematics of

Signal Processing. Cambridge: Cambridge UP, 2012. Print.

[20] ALLEY, Charles Loraine, and Kenneth Ward. ATWOOD.

Electronic Engineering. 3rd Ed. New York, London: Wiley,

1973.

[21] Yoon, Y.K. "MOSFET." Lecture.

[22] Yoon, Y.K. “BJT.” Lecture.

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Vacuum Tube Amplifier Modeling with Dynamic Convolution