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Page 222 ï~~Simulation of Electron Tbe Audio Circuits Kai Lassfolk Department of Musicology P.O.Box 35 (Vironkatu 1) 00014 University of Helsinki Email: firstname.lastname@example.org Abstract This paper describes a signal processing software system that simulates vacuum tube audio electronics. The aim of this study was not to reconstruct tube circuits on component level but rather to simulate their effect on sound. The emphasis was put on tube distortion characteristics. The paper describes a simple tube circuit model. It uses non-linear processing to simulate the behavior of tube-based voltage amplifiers. Linear filtering is used to simulate coupling of contiguous tube stages. The paper is completed by the presentation of two example applications: a multi-stage amplifier and a compressor. 1 Introduction Amplifiers and peripheral equipment based on vacuum tubes are becoming increasingly popular in modem digital studios. Tube electronics are once again used widely in "re-issue" microphones, microphone preamplifiers, compressors, equalizers, distortion units, etc. Tubes are considered to produce a "warm" sound as opposed to the "cold" or "dead" sound of analog solid state and digital studio equipment. Our study addressed two questions: what makes vacuum tube equipment sound warm, and could tube hardware be substituted by software simulation. 2 The Software Implementation In simulating tube audio circuits we decided to concentrate on the transfer characteristics of grid-controlled tubes. Each type of tube has a distinctive anode-current/grid-voltage curve that can be accurately simulated by nonlinear processing. This technique, also called waveshaping, has been described e.g. by [Roads, 1985]. Nonlinear processing produces both harmonic and intermodulation distortion [Crowhurst, 1961]. Other types of distortion, such as phase and transient, as well as limitation of the frequency response caused by the coupling of consecutive tube stages, may be produced with conventional linear time-invariant filters. To simulate the transfer characteristics of a typical preamplifier circuit, the wave shaping transfer function should be somewhat asymmetrical,, even in the middle (i.e. D.C. level of the input signal). This generates the second harmonic overtone as a distortion component. Power amplifiers, on the other hand, often use a push-pull arrangement to reduce evenorder harmonic distortion. A push-pull tube pair may be simulated with a single symmetrical (but still nonlinear) transfer function which, in turn, emphasizes the third harmonic. The Sound Processing Kit, an object-oriented signal processing class library [Lassfolk, 1995], provided the software framework for our project. We expanded SPKit with a waveshaping module, which allows an external signal to control the input signal D.C. level. This allowed experimentation with timevariant biasing. Our waveshaping module uses a lookup table to store an approximation of the transfer function. Linear interpolation is used to reduce lookup distortion. We used hand-adjusted transfer tables to match the characteristics of different types of tubes. Transfer functions were normalized to yield output sample values between -1 and 1. 3 A Multi-stage Amplifier Our first example application is an amplifier with both a pre- and a power amplifier stage. The implementation follows a typical resistance-capacitance-coupled tube amplifier. Similar circuits may be found in many hi-fi power amplifiers as well as in line level studio equipment. The simplified block diagram of the equivalent analog circuit is shown in figure 1. preamp phase splitter power output am transformer output input Figure 1: An analog multi-stage amplifier Lassfolk 222 ICMC Proceedings 1996
Page 223 ï~~There, the input signal is fed to a preamplifier section. It is followed by a phase inversion stage, where the audio signal is converted to two signals which are in opposite phase. The signal is then fed to a push-pull output amplifier section. The circuit is completed by an output transformer, which matches the output impedance to drive either a loudspeaker or line level device. Figure 2 shows a block diagram of our software implementation. The input signal is first fed to an input amp 1 waveshaper 1 tgl hpfamp 2 tg2 waveshaper 2 lpf output Figure 2: A software multi-stage amplifier amplifier (amp 1), which controls the input gain. The first waveshaper and a high-pass filter (hpf) simulate the preamplifier stages. Waveshaper 1 should have an asymmetrical transfer function to follow the characteristics of input stage tubes (such as the popular ECC83/12AX7 double triode). The high-pass filter simulates a coupling capacitor often used in between tube stages. The filter removes the D.C. component produced by the asymmetrical transfer function of waveshaper 1. The second amplifier (amp 2) controls the amplitude of the signal fed to the "power amplifier" or output stage. To mimic the characteristics of power amplifier tubes operating in a push-pull arrangement, the transfer function of waveshaper 2 should be (nearly) symmetrical and smoothly curving. Finally the low-pass filter (lpf) simulates stray capacitances of an output transformer, which limits the high frequency response of the amplifier. The use of two waveshaper modules allowes to control preamplifier and power amplifier distortion levels separately. The transfer functions of both waveshapers bend synmnetrically on the edges to produce smooth clipping characteristic of an overdriven tube amplifier. 4 A Tube Compressor In tube electronics, amplitude compression may be implemented by using a triode tube with a smoothly curving control grid response curve (a.k.a. a remote cutoff grid tube [Reich, 1995 (1941)]) and by varying the negative bias voltage according to the amplitude of the input signal. On high input levels the negative bias voltage is increased so that the tube gives little amplification. Figure 3 shows the block diagram of our software implementation. Figure 3: A tube compressor The circuit includes two waveshaper modules, which use similar transfer functions, but receive their input signals in opposite phase. This arrangement, also used in tube hardware compressors, reduces even order harmonic distortion produced by the asymmetrical transfer functions used in the waveshaper modules. The amplitude estimator includes a signal rectifier and a low pass filter. It produces a signal used to control the bias (x-axis position of the input signal D.C. level) of the waveshaper transfer functions. Optional delay lines may be inserted in front of the waveshapers to reduce amplitude "bumping". 5 Conclusion Our simulations were indeed capable of producing a "warm" distorted sound similar to overdriven tube amplifiers. The evaluation process was, however, highly subjective. Systematic A-B tests against tube hardware are yet to be conducted. References [Crowhurst, 1961] Norman H. Crowhurst. High-fidelity sound Engineering. George Newness Ltd. London, pp.20-23, 1961. [Lassfolk, 1995] Kai Lassfolk. Sound Processing Kit. In: Proceedings of the 1995 International Computer Music Conference. The International Computer Music Association, San Francisco, California, pp. 503-504, 1995. [Reich, 1995 (1941)] Herbert J. Reich. Principles of Electron Tubes, First Reprint Edition. Audio Amateur Press, Peterborough, New Hampshire, pp. 62 -63, 1995. [Roads, 1985] Curtis Roads. A Tutorial on Nonlinear Distortion or Waveshaping Synthesis. In C. Roads and J. Strawn (Ed.): Foundations of Computer Music. The MIT Press, Cambridge, Massachusetts, pp.83-94, 1985. ICMC Proceedings 1996 223 Lassfolk