# A Wideband High-Voltage Power Amplifier Post-Linearizer for Medical Ultrasound Transducers
## 2 Materials and Methods
The output of the HVPA can be expressed by  
\[V_{out1}=-g_{m}\cdot V_{in1}\cdot\left\{\frac{1}{j\mathrm{wC}_{i}}\,//\,R_{L1} \,//\,\frac{1}{j\mathrm{wC}_{o}}\right\}=\,\frac{-g_{m}\cdot V_{in1}}{\frac{1}{ R_{L1}}+jw(C_{i}+C_{o})}. \tag{1}\]  
where \(g_{\mathrm{m}}\) is the transconductance of the transistor (M\({}_{\mathrm{T1}}\)), \(R_{L1}\) is the load resistance of the transistor (M\({}_{\mathrm{T1}}\)), and \(C_{i}\) is the combined input, parasitic gate-source, and gate-drain capacitances (C\({}_{\mathrm{input}}\), C\({}_{\mathrm{gs}}\) and C\({}_{\mathrm{gd}}\)), and C\({}_{\mathrm{o}}\) is the combined drain-source and output capacitances of the transistor (C\({}_{\mathrm{ds}}\) and C\({}_{\mathrm{out}}\)). As shown in Equation (1), the input and output capacitances of the transistor (C\({}_{i}\) and C\({}_{\mathrm{o}}\)) could deteriorate HVPA performances. Therefore, the capacitance variation generated from the post-linearizer circuit was designed to compensate for the parasitic capacitances of the transistor. This is achieved by placing the input and output capacitance, in addition to the capacitances in the post-linearizer, in opposite positions in the Volterra series expansion [19; 28; 30]. Accordingly, non-linear compensation between the HVPA and post-linearizer was enhanced. A DC voltage (V\({}_{1}\)) in the post-linearizer circuit was applied to offset the difference between the input capacitance variation of the transistor (M\({}_{\mathrm{T1}}\)) and the output impedance variation generated by the capacitors (C\({}_{1}\) and C\({}_{2}\)), diodes (D0-D5), and inductors (L\({}_{1}\), L\({}_{2}\), and L\({}_{3}\)). Figure 4 illustrates the equivalent HVPA with post-linearizer circuit model used to analyze the non-linearity components. The large values of the bias inductor and RF choke inductors and small values of the transmission lines were eliminated in order to simplify the analysis. The total output of the HVPA with post-linearizer circuit (V\({}_{\mathrm{out2}}\)) with ultrasound transducer load can be expressed by  
\[ \begin{array}{l}V_{out2}=-g_{m}\cdot V_{in1}\cdot\left\{\frac{1}{j\mathrm{wC }_{i}}\,//\,R_{L1}\,//\,R_{1}\,\right\}+\left(\frac{1}{j\mathrm{wC}_{2}}\,// \,j\mathrm{w}(L_{1}+L_{23})+\frac{1}{j\mathrm{wC}_{1}}\,//\,j\mathrm{w}L_{1} \right)+(R_{2}+\frac{1}{j\mathrm{wC}_{2}}\,//\,Z_{T})\right\}\\ \approx\frac{-g_{m}\cdot V_{in1}}{\frac{1}{j\mathrm{wC}_{1}}\,+\left\{\frac{1} {j\mathrm{wC}_{1}}\,+\frac{1}{j\mathrm{wC}_{2}}\,+\frac{1}{j\mathrm{wC}_{2}} \,+\frac{1}{j\mathrm{wC}_{2}}\,+\frac{1}{j\mathrm{wC}_{2}}\,+\frac{1}{j \mathrm{wC}_{2}}\,+\frac{1}{j\mathrm{wC}_{2}}\,+\frac{1}{j\mathrm{wC}_{2}}\, \right\}}Y_{L}=\frac{1}{j\mathrm{wC}_{2}+\frac{1}{j\mathrm{wC}_{2}}\,+j\mathrm{ wC}_{L}+\frac{1}{j\mathrm{wC}_{1}}\,\frac{1}{j\mathrm{wC}_{1}}\,\frac{1}{j \mathrm{wC}_{2}}\,\end{array}  \tag{2}\]  
Figure 4: Equivalent circuit model of the HVPA with post-linearizer. Figure 3: Equivalent circuit model of the HVPA. where \(C_{1}\) and \(C_{2}\) represent the equivalent capacitance of the capacitors C\({}_{1}\) and C\({}_{2}\), C\({}_{L}\) is the parasitic capacitance of the cross-coupled diode (D0 and D1), \(C_{E}\) is the total parasitic capacitances of the four switching diodes, and \(Z_{T}\) is the load impedance of the ultrasound transducer. Equation (2) indicates that the parasitic capacitance variation of the input and output capacitances (\(C_{i}\) and \(C_{o}\)) with switching diodes and transducers could be compensated by the capacitors (C\({}_{1}\) and C\({}_{2}\)) and diode equivalent capacitance (C\({}_{L}\)) because, as previously mentioned, they yield opposite variations in large signal operations. Furthermore, the inductances in the post-linearizer (L\({}_{1}\), L\({}_{2}\) and L\({}_{3}\)) also compensate for these parasitic capacitances; thus, impedances resonate at the operating frequencies of the transducers. Therefore, non-linear characteristics of the HVPA with post-linearizer could be appropriately controlled over a wide range of input powers and frequencies.