Digital Modeling/Implementation of Valve Amplifiers Senior Project Proposal By: Cody Frye & Mitchell Gould Advisor: Dr. Lu 1 TABLE OF CONTENT I. Introduction………………………………………………………………....pg. 3 II. Motivation and Objective…………………………………………………...pg. 3 III. System Description………………………………………………………….pg. 3 IV. Methods……………………………………………………………………..pg. 4 V. Preliminary Results………………………………………………………....pg. 5 VI. Bill of Material……………………………………………………………...pg. 8 VII. Schedule and Labor………………………………………………………....pg. 9 VIII. Summary and Conclusion…………………………………………………...pg. 9 IX. References…………………………………………………………………..pg. 10 2 Introduction It is estimated that seventy five percent of the market for vacuum tubes is taken up by electric guitar amplifiers. Since tubes were first used in guitar amplifiers in the 50’s and 60’s, their signature sound has become a staple for rock guitar players. Valve amplifiers come in a broad variety of shapes and sizes. Every model has its own signature tone that musicians specifically seek out. When it plays loudly, the sound of vacuum tube amplifier arguably presents much better than the sound from a solid state amplifier because of the more musical and smoother clipping. Also, since there are much greater operating voltages for valves, they have a much greater dynamic range. Unfortunately vacuum tubes are large components that overheat often. In addition, tubes have a short life span and they can burn out at any time. Sometimes this even can cause damage to the transformer and other components. Once they burn out, they often need to be replaced quickly. This is caused by the higher power consumption that results in a lot of heat being generated within the valves. The downside of using a tube amplifier is not only the overheating and aging problems. These pieces of equipment can be too heavy to be moved with ease. This problem is compounded by the fact that most bands have multiple guitar and bass players. Most of them need their own amplifier. The cost of transporting and maintaining these tube amplifiers can be a burden on the bands that use them. Musicians are constantly in need of gear that not only is lightweight and cost effective, but gives them the ability to have any tube amplifier at their disposal for the wide range of gigs and sessions. Motivation and Objectives Due to the inconvenient nature of the valves themselves and the need to preserve their highly sought after tone, digital implementation is a viable alternative. Affordable, convenient, and versatile equipment is what many musicians are looking for in their gear. Modeling vacuum tubes and their wide dynamic range digitally can eliminate the problems that valve amplifier owners face while maintaining all of the subtleties that valve amplifiers offer. This technology gives the musician the option of using multiple amplifiers. Since processing power follows Moore’s law, digital implementation will always be more powerful and cheaper than previous years. Not only will the musician have the option to incorporate more amplifiers into their arsenal, they will also have the benefits of lightweight equipment that doesn’t overheat. System Description Inputs and Outputs The system input will be audio signal (primarily guitar/bass). The input signal will be fed into an audio codec which will perform the A/D conversion. The codec will then send the data to the process using the I2S communication protocol. The processor will perform the signal processing. The processed data will be fed back to the audio codec which will then perform the D/A 3 conversions. The output of the codec will be the output of the system. The block diagram for the valve amplifier being modeled is described below in Figure 1. Figure 1. Block Diagram for preamp, gain, output, and power stages. ​ Methods Wave Digital Filtering Wave Digital Filtering is a way of modeling physical systems or analog circuits in such a way that filter coefficients or components can be easily adjusted in order to change the behavior of the model. As opposed to Kirchhoff circuits whose parameters are current and voltage, Wave Digital Elements have parameters called wave quantities. There are many different sub classes of Wave Digital Filters that are used to describe different models. The different components are connected using ports. The mapping of the individual components is shown in Figure 2 as referenced by “Wave Digital Filter Based Analog Circuit Emulation on FPGA”. Serial and parallel adapters are used to place components accordingly. Wave Digital Filters are modular, so they can easily be altered which is good for implementation of valve amplifiers. Theses filters also have a very good dynamic range performance and have guaranteed stability under linear conditions. The nonlinear portion of the valve amplifier will be modeled through look up tables in order to match the behavior of the original model. Figure 2. WDF Elements ​ ​ 4 The signal flow diagram for WDF adapters are also shown in Figure 3 as described by “Design and Implementation of Wave Digital Filters” by M. Anotsova and V. Davidek. The final product will use a combination of multi-port serial and parallel adapters. Two port adapters will be implemented similarly, however, their structures will be slightly more simple. Once the most bit-optimal solution for the WDF binary tree is found, the structure is then implemented on an embedded system. The system is then be compared with the original digital equivalent. Modularity of the structure as well as the ability to change the filter coefficients are both important criterion as well for modeling any given tube amplifier. Simulation results have been found for the triode valve’s nonlinear parameters using the Leach model as well as the Norman Koren Model. Figure 3 WDF adaptor Signal Flow Diagram: a) parallel reflection free adaptor, b) parallel adaptor, c) series reflection free adaptor, d) series adaptor. Preliminary Results Tube characteristics are modeled according to Leach’s model as well as Norman Koren’s model. Leach’s model gives the current as a function of Vgk (voltage from grid to cathode) and 5 Vpk (voltage from plate to cathode) for a common cathode triode tube. The equations for the leach model are described in equations 1 and 2 respectively. The simulation results for the Leach model is shown by Figures 3 and 4. For the Norman Koren model, a different set of equations are used. They are referred to as equations 4 and 5 respectively. The Norman Koren model is derived from the Leach model. The constants in these equations as well as the equations themselves can be found in the paper “Simulation of a guitar amplifier stage for several triode models” by Cohen and Helie. Simulation results for the Koren model can be found in Figures 6 and 7. Equations 1, 2 Equations 3,4 Figure 4. Plate current as a function Figure 5. Plate current as a function of Vgk ​ ​ ​ of Vpk for Leach’s model for Leach’s model 6 Figure 6. Plate Current as function of Vgk Figure 7. Plate Current as function of Vpk Norman ​ ​ ​ Norman Koren model. Koren model. The valve amplifier as a whole will be modeled using Wave Digital Filtering techniques. A simple high pass filter is designed using a two port parallel adapter and a three port parallel adapter scheme. The overall schematic is for a capacitor and resistor connected in series. The input source is then put in parallel with the three port adapter. The simulink circuit model is described Figure 8. Time and frequency domain representations are shown in figures 9 and 10 respectively. Figure 8. Wave Digital Filter Representation of Simple High Pass Filter ​ 7 Figure 9. Time Domain Response of Logarithmic Chirp Input ​ Figure 10. Frequency Response of Random Gaussian Noise ​ Bill of Materials Mikroe-506 Audio Codec Board $19.38 Digilent PYNQ Board $229.00 ¼” female adaptors (2) $0.86 per piece Crate Blue Voodoo 120 Valve Amplifier $239.00 used 8 Schedule and Labor Week of... Descriptions 1/26 Implementation of tone stack 2/2 Simulation of tone stack 2/9 Obtain characteristics of triode valve 2/16 Simulate of triode valve WDF model 2/23 Obtain characteristics of Pentode valve 3/2 Simulate of Pentode valve WDF model 3/9 Implementation of Gain Stages 3/16 Simulation of Gain stages 3/23 Combine and simulate all stages together 3/30 Fpga i2s controller implementation 4/6 Convert Simulink Model to FPGA IP model 4/13 Implement Design on hardware 4/20 Extra Time allocation 4/27 Student Expo 5/4 Final Presentation preparations 5/11 Finals Summary and Conclusion This project aims to design a digital emulation of an analog valve amplifier with the non-linear characteristics being reconfigurable. Emulating the overall response of the valve amplifier will be performed using a Wave Digital Filtering technique. The previous method, utilized the use of filter banks, where each bank was fed into a lookup table to generate the nonlinear response on a given input signal. As of now, the hardware to emulate the valve amplifier will be performed on an arm cortex m7 processor. The processor will communicate with an audio codec using the Inter-IC sound (I2S) protocol. 9 References Several papers have been published on Wave Digital Filtering and Implementation for non-linear amplifiers and other various models. There also exists a paper that describe the common cathode triode amplifier with the Norman Koren model, Leach model, and dynamic state space methods. ● Simulation of a guitar amplifier stage for several triode models: examination of some relevant phenomena and choice of adapted numerical schemes - aims to describe the high gain stage of a triode amplifier through simulation - Modeled through a nonlinear differential algebraic system ● WAVE DIGITAL SIMULATION OF A VACUUM-TUBE AMPLIFIER - Uses WDFs for efficient real time simulation of valve amplifiers ● Wave Digital Filters: Theory and Practice - Gives state flow diagrams for each individual WDF adapter - Describes the many methods of Wave Digital Filtering ● Wave Digital Filter based Analog Circuit Emulation on FPGA - Implementation of various R, L, and C circuits on an FPGA - Results for first study in FPGA based WDF emulation Work cited [1] Ivan Cohen, Thomas Hélie.