# Analysis and Design of a Transient-State Resonant Converter Used as a Frequency Multiplier
## 3 Design Methodology and Experimental Testing
### Implications on the Efficiency of the Resonant Converter Acting as a Frequency Multiplier  
The resonant converter met the performance specifications and succeeded in acting as a high frequency multiplier. As shown in Figure 12a-d, tests were performed with different odd values in the multiplier and for different decay factors \"\(df\)\". Using high values in the multiplier, or in the decay factor brings some implications in the circuit performance. As shown in Figure 13, increasing the value in the multiplier \"\(n\)\" leads to an increase in the supply voltage V\({}_{\text{DC}}\). This is because the higher the value in the multiplier \"\(n\)\", the higher the operating frequency \"\(f_{o}\)\" and the higher the R\({}_{\text{L}}\) load value as indicated in (8). The increase in the supply voltage brings with it an increase in the losses in the inverter, because the parasitic capacitances of the Mosfet are charged to the supply voltage V\({}_{\text{DC}}\), especially the output capacitance C\({}_{\text{oss}}\) which is normally the highest capacitance. Equation (16) defines the energy in joules at which the C\({}_{\text{oss}}\) capacitance is loaded, and (17) defines the power losses in watts. \[P_{M}=\bigg{(}\frac{1}{2}C_{\text{oss}}(2V_{\text{DC}})^{2}\bigg{)}f_{sw} \tag{16}\]  
\[E_{\text{cos}}=\frac{1}{2}C_{\text{oss}}(2V_{\text{DC}})^{2} \tag{17}\]  
Considering that in a class D inverter only two Mosfet switches switch in one cycle of the switching signal, the total losses of the switching stage are defined by (18). \[P_{\text{inver}}=\bigg{(}C_{\text{oss}}(2V_{\text{DC}})^{2}\bigg{)}f_{sw} \tag{18}\]  
As shown in (17), the losses increase due to the increase in the supply voltage V\({}_{\text{DC}}\) as the multiplier \"\(n\)\" increases, where the inverter switching frequency \(f_{sw}\) is a fixed value and does not increase the losses in the inverter as the operating frequency \(f_{o}\) in the resonant tank increases. In order to check the effect on the efficiency of the converter when operating as a frequency multiplier, power measurements were made in the circuit using the digital oscilloscope, more precisely in the inverter stage and in the resonant tank, which is where the efficiency of these stages is affected due to the transient operation. The measurements in the inverter stage were performed with the switching frequencies of 100 kHz, 300 kHz, and 500 kHz. Measurements were made of the power entering P\({}_{\text{in(inver)}}\) and the power leaving the inverter P\({}_{\text{o(inver)}}\), where the difference in power would be the losses, determined by (19). \[P_{loss(\text{inver})}=P_{in(\text{inver})}-P_{o(\text{inver})} \tag{19}\]  
Figure 14 shows the measurements made in the circuit to determine the losses in the inverter. Figure 15 shows the inverter efficiency percentages when using multiplier values with \(n=3\), 5,7. As shown in Figure 15, the lowest efficiency was obtained using the switching frequency \(f_{sw}=100\) kHz, because at this frequency, the predominant losses were conduction losses due to the RMS current flowing through the resonant tank and the R\({}_{\text{dson}}\) resistance of the Mosfet. With the switching frequencies of 200 kHz and 300 kHz, efficiencies were very close with the multiplier \(n=3\), and were moving away from the multiplier \(n=5\), where it is observed that the higher the frequency multiplication, the lower the efficiency, since the required V\({}_{\text{DC}}\) supply voltage is higher. Figure 13: Increase in V\({}_{\text{DC}}\) supply voltage due to the “\(n\)” multiplier. Regarding the efficiency of the resonant tank, it is mostly affected by the series inductance L\({}_{\rm s}\), which depends on the decay factor \(d_{f}\) and the multiplier \"\(n\)\". Figure 16 shows how the decay factor increases the value of the series inductance L\({}_{\rm s}\) needed in the circuit to operate underdamped, where using a decay factor of \(d_{f}=0.9\) (sinusoidal waveform) would increase almost forty-five times the value of its inductance.