12AX7 Tube
The most commonly used tube in this amplifier is the 12AX7 tube. In order to model the tubes in this project, we used LTSpice to model the tubes reponse and get an input output relationship to design a transfer function for each. Figure 1 is an LTSpice schematic with input, output, and our tube model component in the center. We used 1/4 of a 1kHz sin wave to get the input/output response as shown in fig. 2.
Figure 1. LTSpice Circuit Model
At each stage in the amplifier, the tube acts a little differently because of the signal level it is being driven by. In order to take this into account, we adjusted the grid voltage, relative load, and amplitude of the incoming sin wave when modeling each instance of the tube. Then, for the simulation we mapped the input voltage to the output voltage and normalized the output around -1V to 1V. Figure 2 shows the results of this for the third of the 12AX7 (highest gain preamp).
Figure 2. Normalized Simulation Input and Output.
We then graphed the input as a function of output by eliminating time as a variable as shown in Figure 3. We used a cubic spline fitting calculator to approximate a function to fit to the curve in Figure 3 that is used in the Matlab function implementation of the tube.
Figure 3. Input Output Relationship of Tube.
Figure 4 includes images of all three Matlab scripts; one for each preamp stage in the block diagram pictured on the home page.
Figure 4. Scripts for 12AX7 stages 1-3.
Below are examples of a recording running through each stage of the 12AX7 preamp series.
Stage 1
Stage 2
Stage 3
The first two stages of the 12AX7 preamp distortion are fairly subtle. On particularly loud parts of the signal, we can hear some slight harmonic distortion. The third stage (the most drastic pre-amp stage) we can hear a very pronounced "squaring-off" of wave forms. There is also an inherent shift in phase because the transfer function is asymmetric. Because of this asymmetry, the distortion imparted on the signal is fairly unique and would have been hard to get sonically correct without the circuit analysis and modeling in LTSpice.
Video. With and without Effect