## 2 The Lockhart Wavefolder
### Circuit Analysis  
The [PERSON] wavefolder consists of an NPN and a PNP bipolar junction transistors connected at their base and collector terminals. In order to model the large-signal behavior of the circuit, we replace transistors \(Q_{1}\) and \(Q_{2}\) with their corresponding Ebers-Moll large-signal models [23; 24]. Figure 1b shows the large-signal equivalent circuit of the [PERSON] wavefolder. We use a double subscript notation to distinguish the voltages and currents in transistor \(Q_{1}\) from those in \(Q_{2}\). For example, \(I_{\mathrm{ED,2}}\) denotes the current through the base-emitter diode in \(Q_{2}\). Component values for the circuit have been compiled in Table 1.  
We begin the analysis of the circuit by assuming that the supply voltages are equal but opposite in sign (i.e., \(V_{\mathrm{CC}}=-V_{\mathrm{EE}}\)), and that \(|V_{\mathrm{in}}|\ll V_{\mathrm{CC}}\). This assumption means that the base-emitter junctions of both \(Q_{1}\) and \(Q_{2}\) will always be forward-biased and their voltage drops will remain approximately constant for all expected values of \(V_{\mathrm{in}}\). In [PERSON]'s version of the circuit, \(V_{\mathrm{in}}\) is assumed to be bounded between approximately \(\pm 1.2\) V [53]. Applying KVL around both input-emitter loops gives us  
\[V_{\mathrm{in}} = V_{\mathrm{CC}}-RI_{\mathrm{E,1}}-V_{\mathrm{BE,1}} \tag{1}\] \[V_{\mathrm{in}} = V_{\mathrm{BE,2}}+RI_{\mathrm{E,2}}+V_{\mathrm{EE}}, \tag{2}\]  
where \(I_{\mathrm{E,1}}\) and \(I_{\mathrm{E,2}}\) are the emitter currents, and \(V_{\mathrm{BE,1}}\) and \(V_{\mathrm{BE,2}}\) are the voltages across the base-emitter junctions of \(Q_{1}\) and \(Q_{2}\), respectively. Solving Equations (1) and (2) for \(I_{\mathrm{E,1}}\) and \(I_{\mathrm{E,2}}\) gives us:  
\[I_{\mathrm{E,1}} = \frac{V_{\mathrm{CC}}-V_{\mathrm{BE,1}}-V_{\mathrm{in}}}{R} \tag{3}\] \[I_{\mathrm{E,2}} = \frac{V_{\mathrm{in}}-V_{\mathrm{BE,2}}-V_{\mathrm{EE}}}{R}. \tag{4}\]  
Next, we apply KCL at the collector node, which gives us  
\[I_{\mathrm{out}}=I_{\mathrm{C,1}}-I_{\mathrm{C,2}}, \tag{5}\]  
where  
\[I_{\mathrm{C,1}} = \alpha_{\mathrm{F}}I_{\mathrm{ED,1}}-I_{\mathrm{CD,1}} \tag{6}\] \[I_{\mathrm{C,2}} = \alpha_{\mathrm{F}}I_{\mathrm{ED,2}}-I_{\mathrm{CD,2}}. \tag{7}\]  
If we then assume that the contributions of the reverse currents \(\alpha_{R}\,I_{\mathrm{CD_{1}}}\) and \(\alpha_{R}\,I_{\mathrm{CD_{2}}}\) to the total currents associated with the emitter nodes are negligible (i.e., \(\alpha_{\mathrm{R}}\approx 0\)), we can establish that \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Value (k\(\Omega\))** & **Component** & **Value (V)** \\ \hline \(R\) & 15 & \(V_{\mathrm{CC}}\) & 15 \\ \(R_{\mathrm{L}}\) & 1–50 & \(V_{\mathrm{EE}}\) & \(-15\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Component values for the [PERSON] wavefolder circuit. \[I_{\text{ED,1}} \approx I_{\text{E,1}} \tag{8}\] \[I_{\text{ED,2}} \approx I_{\text{E,2}}. \tag{9}\]  
Assuming \(\alpha_{\text{F}}=1\), as suggested in [23], and inserting Equations (8) and (9) into Equations (6) and (7), respectively, yields a new expression for the total output current of the circuit:  
\[I_{\text{out}}=I_{\text{E,1}}-I_{\text{E,2}}-I_{\text{CD,1}}+I_{\text{CD,2}}. \tag{10}\]  
We then combine Equations (3) and (4) to derive an expression for the difference between emitter currents:  
\[I_{\text{E,1}}-I_{\text{E,2}}=\frac{V_{\text{CC}}+V_{\text{EE}}-2V_{\text{in} }+V_{\text{BE,2}}-V_{\text{BE,1}}}{R}. \tag{11}\]  
Since \(V_{\text{CC}}+V_{\text{EE}}=0\), and voltage drops \(V_{\text{BE,1}}\) and \(V_{\text{BE,2}}\) are assumed to be constant and equal, their contribution to this expression disappears.