Emulation of Amplifier Overdrive Through the Use of the Nonlinear Output Response of Diodes
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Emulation of Amplifier Overdrive Through the Use of the Nonlinear Output Response of Diodes Dustin Lindley, Nishant Parulekar Department of Physics, University of Illinois at Urbana-Champaign 1110 West Green Street, Urbana, IL 61801-3080, USA Harmonic distortion is the most commonly used electric guitar effect. We have constructed a distortion pedal to emulate the sounds of overdriven tube amplifiers. The fuzz box created was designed for high versatility in controlling many parameters of an audio signal while at the same time retaining a simplicity that would allow it to be reproduced as a laboratory experiment. The output of the fuzz box is analyzed in terms of harmonic content and sound. I. INTRODUCTION within the device's manageability. The primary method of achieving this status was to increase Harmonic distortion of signals is a the levels of the signal to an amount that desired sound for many musicians. However, exceeded the amplifiers processing capabilities. there are many parameters that can be altered in This resulted in a clipping of the signal such that order to subtlety or drastically change the wave the top and bottom portions of the wave were cut form and hence sound. We have tried to design a off, which limits as a square wave. This is one laboratory experiment for the Physics of Musical of the simplest ways to describe distortion. Instruments class that would allow students to understand the physics, theory, and application The Demand for Distortion of distortion. Additionally, the lab is designed so that students can investigate the physics behind The sounds of overdriven amplifiers various stages of the signal processor in order became popular for many musicians. As such, create their own desired fuzz box. the demand for controlling the type of distortion After studying how various diode also surfaced. There were several problems with clamps produce current in response to an applied strictly using the amplifier for distortion. One voltage, the selected diodes were used in the box major problem was that a large volume was as primary sources of differing distortion. To necessary to get to the critical stage of quantify output signals, Fourier analysis was overdriving an amplifier. This was not practical performed to examine the harmonic content of a for performers who wanted a specific sound at signal. Though the fuzz box is intended for use lower levels. A consequence of intensely driving with an electric guitar signal, we used a an amplifier was that the electronics were often simplified signal of frequency f, produced by a subject to erosion of effectiveness. Another wave generator to produce plots of the frequency major drawback was that each amplifier tended response of the fuzz box. to have its own characteristic distortion sound, so After the circuitry was perfected for that the option of having a variety of distortions use, a design was laid out to house the was not offered. As such, these limitations electronics. Instructions were then created for created a need for a portable preamp that allowed the production of both the circuitry and the box. musicians a cheap, small, and practical control The entire process and results will be made over their sound. available as a laboratory experiment as well as a step-by-step instructional/educational process on III. THE PHYSICS OF THE DIODE the Internet. Physical Structure II. BACKGROUND One of the simplest types of Older Methods of Distortion semiconductor devices is the diode. A diode is made up of an N junction (cathode) and a P Before the popularization of fuzz boxes, junction (anode). Chemical composition ranges the frequently undesired sound of distortion was from germanium to silicon to gallium arsenide. produced by driving an amplifier at levels not All materials work essentially the same way. Unlike some other semiconductors, a where Ai is the amplitude of the input stimulus. diode cannot produce gain in signal (a higher Thus the output response also becomes time output than input). However, its nonlinear dependent: response to voltage is useful. The diode Ro(t)= Aocos(wt) (3) functions as a one-way valve for current. where Ao = K Ai is the amplitude of the output Current-Voltage Relationship response. To grasp the basic idea of harmonic When a voltage difference is applied distortion, we use a very simple example. Let us across an N-type crystal forming an electric first consider the mild nonlinearity of a quadratic field, the free electrons travel in the direction of response: the field and carry current. 2 The P-N junction, where the two types Ro(Si) = K (Si + eSi ) = K Si (1+eSi) (4) of semiconductor materials join, contains atoms are then fixed in the crystal so that a potential (úeSiú <<1) barrier on each side of the junction is formed within a depletion zone. This is the region where Here, the non-linearity parameter, e (with units all free electrons and holes recombine and all of 1/Si), is assumed to be small. So here, we carriers are depleted from the zone. Essentially, have in addition to the KSi linear response term, the depletion zone acts as an insulating barrier a small, non-linear quadratic response term, between the cathode and anode regions. As the 2 eKSi , which is the lowest-order deviation voltage is increased, the depletion zone becomes possible from a purely linear response, (1). thinner and thinner, and the current through the Because the output response is depletion zone grows exponentially as more and dependent on time, more electrons are able to tunnel across the barrier. The most commonly used model of a R (t)= K A cos(wt) + e K A 2 cos2 (wt) (5) diode claims that the diode is a switch, and at a o i i certain threshold voltage the switch is turned on, We can use the trigonometric identity and current flows. This works well for most applications, but the idea here is to take cos2?= ½ (1+cos (2?)) (6) advantage of that exponential relation by driving the diode around its threshold voltage. The most to get the new output response with harmonic common threshold voltages are from 0.5 V to 1.5 content information: V. Ro(t) = K A cos(wt) + ½ e K A 2 + IV. THEORY OF HARMONIC i i DISTORTION 2 ½ e K Ai cos(2wt) (7) In a “clean” signal, a linear relationship We now see that this quadratic non- is desired between the output response, R and o linear response to a pure input tone of f the input stimulus, frequency produces not only the fundamental Si : Ro(Si) = K Si (1) frequency (or first harmonic) f, but also a small second harmonic component with double the where K is the constant of proportionality. To fundamental frequency and with amplitude 2 simplify the examination of a guitar signal, we ½ e K Ai . This is due to the cos (2wt) term, that assume it to be a pure tone signal produced by a is, 2f. An additional product of this nonlinearity wave generator with a single frequency, f. We is a DC shift or offset, that is, a shift in the define angular frequency as w º 2pf average value of the signal, as a result of the ½ e 2 The input stimulus is a time-dependent K Ai term. function, Because this output waveform is more sharply peaked at top and flatter at the bottom Si(t)=Ai cos(wt) (2) than its input waveform, we consider it distorted. Here, the higher-order Taylor series terms Increasing Nonlinear Response correspond to the higher-order harmonics of the fundamental frequency: 2f, 3f, 4f, 5f, …. With increased nonlinearity, such as The result is a general clipping of the 4 5 quartic (e K Si ), quintic ( eK Si ), and so on, waveform. The degree of clipping/distortion on higher order harmonics are produced (4th, 5th, 6th) the signal depends on the specific exponential respectively in addition to the fundamental non-linearity of each diode loop’s current- frequency. voltage relationship, as well as the amplitude of The basic diode circuit we use is a the input signal. Several diode loop I-V plots are symmetric clamp: shown below. In general, the higher the exponential response to the voltage, or the higher the amplitude of the input, the more distorted the wave signal sounds-- because there are more odd harmonics produced. The waveform itself approaches that of a square wave with equation: m= ¥ sin[(2m -1)q] f (q) = å = m =1 (2m -1) 4 1 1 {sinq + sin3q + sin 5q +.....} p 3 5 Fig. 1 – Symmetric Diode Clamp (11) In this case, there is an exponentially This limiting case consists of only odd growing non-linear response, Ro(Si) which is harmonics. In most cases we find a mix of both modeled as: odd and even harmonics in varying degrees. Ro(Si)= K [exp(aúSiú) - 1] for Si ³ 0, a>0 (8) V. FUZZ BOX CIRCUITRY Ro(Si)=K [1 – exp(a÷Si÷)] for Si < 0, a>0 (9) The circuitry used to produce harmonic distortion is fairly straightforward in design. where K is a positive constant as before and the Operational amplifiers (op amps) are the active nonlinear parameter a is also constant. elements in the circuit, and we opted to use Through Taylor series expansion, we Texas Instruments TL-072 chips, which have find that very small values of input stimulus, two op-amps per 8-pin package. These chips 12 ÷Si÷, result in a fairly linear output response. have input impedance on the order of 10 ?, However, as the Si value increases, each of the with field-effect transistors on the inputs.