## 3 The Serge Middle Wave Multiplier  
The second circuit considered in this study is the middle section of the Serge Wave Multipliers (often abbreviated as the Serge VCM). The Serge VCM is a synthesizer module designed in 1977 by West Coast designer [PERSON], founder of Serge Modular Music Systems. It offered three separate and independent analog sound processors, namely the \"top\", \"middle\" and \"bottom\" sections. As described in an original 1980 Serge product catalog, \"The middle section generates a sweep of the odd harmonics (1, 3, 5, 7, 9, 11 and 13 th) when a triangle wave is applied to its input... This module can be used to explore timphal areas beyond the range of ring modulation because there are more varied harmonics than the sum and difference tones\" [62]. Figure 4: Time-domain view of the proposed Lockhart wavefolder model plotted against its SPICE simulation for a 500-Hz sinusoidal input (peak amplitude 1 V) with load resistance: (**a**) \(R_{\mathrm{L}}=10\,\mathrm{k}\Omega\); and (**b**) \(R_{\mathrm{L}}=50\,\mathrm{k}\Omega\). Figure 3: Absolute value of the difference between a SPICE simulation of the Lockhart wavefolder and its proposed VA model. Figure 2: Transfer function of the Lockhart wavefolder simulated using: (**a**) SPICE (Simulation Program with Integrated Circuit Emphasis); and (**b**) the proposed virtual analog (VA) model. Different colors indicate different values of \(R_{\mathrm{L}}\). The middle Serge VCM is essentially a waveshaping circuit consisting of six identical wavefolding stages arranged in series. An amplifier at the input of the circuit is used to modulate the gain of the input waveform and control the amount of folds introduced [63]. In this section we focus on the analysis of a single folding stage. The transfer function and frequency-domain behavior of the complete system are presented in Section 6. Figure 5 shows the schematic of a single wavefolding stage in the circuit. Component information is given in Table 3.  
To derive the transfer function for the Serge wavefolding circuit, we first assume ideal op-amp behavior and derive an expression for \(V_{\text{out}}\) in terms of \(V_{\text{in}}\) and \(V_{\text{x}}\), the voltage at the non-inverting input of the amplifier. This gives us:  
\[V_{\text{out}}=V_{\text{x}}-\frac{R_{3}}{R_{2}}\left(V_{\text{in}}-V_{\text{x} }\right). \tag{30}\]  
Since in this case \(R_{3}=R_{2}\), we can further simplify this result as:  
\[V_{\text{out}}=2V_{\text{x}}-V_{\text{in}}. \tag{31}\]  
Next, we derive an expression for \(V_{\text{x}}\) by considering the subcircuit shown in Figure 6, which is essentially a diode pair similar to those found in guitar distortion circuits [13; 57; 58]. Applying KVL around the outer loop of the circuit yields the relation  
\[V_{\text{in}}=R_{1}I+V_{\text{x}}, \tag{32}\]  
Figure 5: Schematic of a single folding cell in the middle section of the Serge Wave Multipliers (VCM). Figure adapted from [63]. Figure 6: Equivalent view of the diode saturator at the non-inverting input of the op-amp in Figure 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Value (k\(\Omega\))** & **Component** & **Description** \\ \hline \(R_{1}\) & 33 & Diodes & 1M4148 or similar \\ \(R_{2}\) & 100 & Op-Amp & TL072 or similar \\ \(R_{3}\) & 100 & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Component information for the Serge wavefolder circuit shown in Figure 5.  
where \(I\) is the current through resistor \(R_{1}\).