## 2 The Lockhart Wavefolder
### Circuit Analysis
Therefore, we can simplify this result as:  
\[I_{\text{E,1}}-I_{\text{E,2}}=-\frac{2}{R}V_{\text{in}}. \tag{12}\]  
Substituting Equation (12) into Equation (10) produces an expression for the total output current \(I_{\text{out}}\) in terms of the input voltage and the currents through the collector diodes:  
\[I_{\text{out}}=-\frac{2}{R}V_{\text{in}}-I_{\text{CD,1}}+I_{\text{CD,2}}. \tag{13}\]  
The current-voltage (I-V) relationship of diodes can be modeled using [PERSON]'s ideal diode equation, defined as  
\[I_{\text{D}}=I_{\text{s}}\left(\exp\left(\frac{V_{\text{D}}}{\eta V_{\text{T} }}\right)-1\right), \tag{14}\]  
where \(I_{\text{D}}\) is the current through the diode, \(I_{\text{s}}\) is the reverse bias saturation current, \(V_{\text{D}}\) is the voltage across the diode, \(V_{\text{T}}\) is thermal voltage and \(\eta\) is the ideality factor of the diode [56]. For the p-n junctions inside transistors we can assume a reverse saturation current value \(I_{\text{s}}=10^{-17}\,\text{A}\) and an ideality factor \(\eta=1\). A thermal voltage value \(V_{\text{T}}=25.864\,\text{mV}\) is used throughout this study. Applying [PERSON]'s diode equation to the collector diodes and substituting into Equation (13) gives us:  
\[I_{\text{out}}=-\frac{2V_{\text{in}}}{R}-\underbrace{I_{\text{s}}\left(\exp \left(\frac{V_{\text{CD,1}}}{\eta V_{\text{T}}}\right)-1\right)}_{I_{\text{CD,1}}}+\underbrace{I_{\text{s}}\left(\exp\left(\frac{V_{\text{CD,2}}}{\eta V_{ \text{T}}}\right)-1\right)}_{I_{\text{CD,2}}}. \tag{15}\]  
Next, we use KVL to derive expressions for \(V_{\text{CD,1}}\) and \(V_{\text{CD,2}}\) in terms of \(V_{\text{in}}\) and \(V_{\text{out}}\):  
\[V_{\text{CD,1}} = V_{\text{out}}-V_{\text{in}} \tag{16}\] \[I_{\text{ED,1}} \approx I_{\text{E,1}} \tag{8}\]0  
Now, the collector diodes in the large-signal model are antiparallel. Therefore, we can make the further assumption that only one of them will conduct at a time depending on the polarity of \(V_{\text{in}}\). A similar treatment is presented in [57] for the case of diode pairs in guitar distortion circuits. This means that  
\[I_{\text{ED,1}} \approx I_{\text{E,1}} \tag{8}\]1  
By combining these new assumptions with Equations (15)-(17) we arrive at the piecewise expression  
\[I_{\text{ED,1}} \approx I_{\text{E,1}} \tag{8}\]2where \(\lambda=\text{sgn}\left(V_{\text{in}}\right)\) and \(\text{sgn}\left(\right)\) is the signum function  
\[\text{sgn}(x):= \begin{cases}-1&\text{if}\quad x<0\\ \phantom{-}0&\text{if}\quad x=0\\ \phantom{-}1&\text{if}\quad x>0.\end{cases}  \tag{19}\]  
Equation (18) can be further simplified if we consider that the independent constant factor \(\lambda I_{\text{s}}\) that results from its expansion will be very small (\(\pm 10^{-17}\) A) and can therefore be neglected. This gives us:  
\[I_{\text{out}}=-\frac{2}{R}V_{\text{in}}+\lambda I_{\text{s}}\exp\left(\frac{ \lambda\left(V_{\text{in}}-V_{\text{out}}\right)}{\eta V_{\text{T}}}\right). \tag{20}\]  
Finally, we multiply both sides of this expression by \(R_{\text{L}}\) to derive an input-output voltage relationship for the Lockhart wavefolder:  
\[V_{\text{out}}=-\frac{2R_{\text{L}}}{R}V_{\text{in}}+\lambda R_{\text{L}}I_{ \text{s}}\exp\left(\frac{\lambda\left(V_{\text{in}}-V_{\text{out}}\right)}{ \eta V_{\text{T}}}\right). \tag{21}\]