## 5 Simulation Results
### Optimization Method  
In order to find the optimal value of source inductance to be inserted in the power loop, the method discussed hereafter is proposed. By performing parametric simulations on the \(L_{\text{s}}\) value of the circuit shown in Figure 8, and according to [23], it resulted that the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Name** & **Value** & **Name** & **Value** & **Name** & **Value** & **Name** & **Value** & **Name** & **Value** \\ \hline \(C_{1}\) & \(560\,\text{\mu F}\) & \(I_{\text{LOAD}}\) & \(3\,\text{A}\) & \(C_{2}\) & \(22\,\text{\mu F}\) & \(R_{\text{g,on}}\) & \(0\,\Omega\) & \(R_{\text{g,off}}\) & \(10\,\Omega\) \\ \(L_{\text{in}}\) & \(100\,\text{\mu H}\) & \(L_{\text{c1}}\) & \(30\,\text{nH}\) & \(R_{\text{c1}}\) & \(80\,\text{m}\Omega\) & \(L_{\text{c2}}\) & \(4.6\,\text{nH}\) & \(R_{\text{c2}}\) & \(50\,\text{m}\Omega\) \\ \(L_{\text{t1}}\) & \(2.1\,\text{nH}\) & \(C_{3}\) & \(10\,\text{\mu F}\) & \(R_{\text{c3}}\) & \(10\,\text{m}\Omega\) & \(R_{\text{drv}}\) & \(10\,\Omega\) & \(C_{\text{t1}}\) & \(135\,\text{pF}\) \\ \(C_{\text{ind}}\) & \(40\,\text{pF}\) & \(L_{\text{t2}}\) & \(700\,\text{pH}\) & \(L_{\text{t3}}\) & \(15.4\,\text{nH}\) & \(L_{\text{t4}}\) & \(9.6\,\text{nH}\) & \(R_{\text{g,int}}\) & \(3.3\,\Omega\) \\ \(T_{\text{HS}}\) & [29] & \(D_{\text{LS}}\) & [30] & \(V_{\text{DD,DRV}}\) & \(10\,\text{V}\) & \(L_{\text{d,pack}}\) & \(1.7\,\text{nH}\) & \(L_{\text{s,0}}\) & \(5.5\,\text{nH}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values of the simulated converter. Figure 8: Switching waveforms at the \(T_{\text{HS}}\) turn-on in the oscillating case (plus markers), and with a \(4\,\text{nH}\) (circle), \(12\,\text{nH}\) (square) and \(20\,\text{nH}\) (diamond markers) \(L_{\text{s}}\) inductances obtained from a time-domain simulation of the circuit shown in Figure 7.  
\(T_{\text{HS}}\) switching energy at the turn-on (\(E_{\text{sw}}^{\text{on}}\)) linearly increase with \(L_{\text{s}}\), as shown in Figure 9 by the dashed line. Conversely, the \(v_{\text{sw}}\) frequency spectrum should be compared against some target to determine whether the turn-on resonance is undamped or overly damped. In what follows, the frequency spectrum of an ideal trapezoidal waveform, with the same pulse-width, rise and fall time of the \(v_{\text{sw}}\) obtained from the simulation, is chosen as a target. The envelopes of the \(v_{\text{sw}}\) frequency spectrum (\(|V_{\text{sw}}(i\omega)|_{dB,sim}^{env}\)), obtained for the different \(L_{\text{s}}\) values, were evaluated and each of them compared against the corresponding envelope of the ideal trapezoidal frequency spectrum (\(|Y(j\omega)|_{dB,trapz}^{env}|\)), which was obtained from [6]. Then, the ratio (\(\alpha\)) between the envelope obtained from the simulation and the target one was evaluated at the turn-on resonant frequency as,  
\[\alpha_{\text{max}}=\max_{f_{\text{end}}\pm\Delta f}(|V_{\text{sw}}(j\omega)| _{dB,sim}^{env}-|Y(j\omega)|_{dB,trapz}^{env}|) \tag{6}\]  
where \(\Delta f=4\,\text{MHz}\) is to account for uncertainties and the resonant frequency was evaluated as  
\[f_{\text{end}}=\frac{1}{2\pi\sqrt{(L_{\text{d}}+L_{\text{s}}+L_{\text{s},0})( C_{\text{s}}+C_{\text{j}})}}, \tag{7}\]  
since it decreases with \(L_{\text{s}}\). For the EMI viewpoint, the optimal value of \(L_{\text{s}}\) is the one with \(R_{\text{max}}=0\), as the corresponding switching waveform is the fastest and non-oscillating. The maximum ratio between such spectra was evaluated for \(L_{\text{s}}\) in the (0,34) nH range and it is shown in Figure 9 by solid line. In the case of no source inductance added (\(L_{\text{s}}=0\)), the frequency peak is approximately 15 dB higher than the expected one, and two point with \(\alpha_{\text{max}}=0\) exist, i.e., \(L_{\text{s}}=6\,\text{nH}\) and \(L_{\text{s}}=22\,\text{nH}\).