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        <h1>CPSC 490: Simulating a Triode Tube Guitar Amplifier Using a Non-Linear State-Space Representation</h1>
        <h3>Arthur Befumo - arthur.befumo@yale.edu</h3><h3>Advisor: Scott Petersen - scott.peterson@yale.edu</h3>
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        <h2>Abstract</h2>
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                A triode vacuum tube is a simple electrical device which can be used to amplify an electrical signal, 
                and consists of three electrodes—termed the ‘plate’ ‘cathode’ and ‘grid’—enclosed in a glass vacuum tube. 
                The plate is heated by a power source, acting as a pump which moves electrons from the plate to the cathode. 
                When current is applied to the grid, this flow is disrupted, allowing a relatively weak input signal to have 
                an amplified effect on the plate to cathode current. Although transistors have replaced vacuum tubes in many 
                modern applications, vacuum tube guitar amplifiers are still manufactured and sought after due to their distinct, 
                non-linear distortion. This non-linearity makes exact simulations of triode tube amplifier circuits computationally 
                prohibitive, and many commercial products for emulating tube distortion take a “black-box” approach which completely 
                ignores the underlying physics. For this project, an analytical approach was taken. Several equations describing 
                the behavior of a triode tube circuit, some phenomenological and some derived from fundamental physics, are used 
                to produce an extended non-linear state-space representation of the circuit. Numerical methods for approximation 
                can then be used to solve for the state of the circuit given its current state and the next input. 
                The tube amp simulation is implemented in C for efficiency. Listening tests and a visual analysis of processed signals 
                show the circuit simulation produces a reasonably close approximation of tube distortion, and previous work has shown 
                that this numerical method is efficient enough for real-time signal processing.
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        <h2>Links</h2>
        <ul>
            <li><a href="./code.html">Code</a></li>
            <li><a href="./proposal.pdf">Project Proposal</a></li>
            <li><a href="./finalreport.pdf">Final Report</a></li>
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