## 2 Materials and Methods
### The Output Pulse Shaping Unit  
The most important part of the presented high-voltage pulsed converter design is the output stage based on the Tesla transformer concept. This allows for shaping of the output voltage pulse through the resonance of the circuit's L and C components, as schematically depicted in Figure 4. To obtain the highest amplitudes of output voltage pulse without mutual oscillations between pulses, the magnetic coupling coefficient k should be equal to approx. 0.6, and the rules of Equation (1) must be fulfilled [12; 13]. \[\text{L}_{\text{L1}}\text{-C}_{2}=\text{L}_{\text{L2}}\text{-C}_{3}\left( \omega_{1}=\omega_{2}\right) \tag{1}\]  
Most Tesla-based transformers are coreless (air coupling) but there are also constructions with magnetic cores [14]. Low permeability magnetic cores (e.g., distributed gap cores) were used in the presented design, instead of air coupling, which allows for significant reduction in the size of the transformer, although there is still a problem with the external magnetic field (electromagnetic interference, i.e., EMI) [15]. A low value in the transformer's magnetic coupling factor results in large values of leakage inductance for the transformer windings. Magnetic field lines of these two leakage inductances are still closed through air, which results in distortions of the magnetic field in transformer surroundings. The external magnetic field of the transformer means that in addition to electromagnetic compatibility problem, there will also be additional power losses since external magnetic fields will interact with magnetic elements, e.g., transformer or converter enclosures, etc. [1; 16]. In order to keep the magnetic field of the Tesla coil-like transformer inside its magnetic construction, a core-based Tesla transformer design is proposed, as shown in Figure 5.  
Proper selection of the number of cores and magnetic properties of magnetic cores led to a defined magnetic coupling. In the constructed prototype, seven cores were used in a 2-3-2 arrangement, which led to the desired k = 0.6 coupling coefficient. The term 2-3-2 means that two cores were used for primary side leakage inductance L\({}_{\text{L1}}\), three cores were used for main part of the transformer T\({}_{1}\), and two cores were implemented for secondary side leakage inductance L\({}_{\text{L2}}\) of the design transformer. The different numbers and properties of the cores and their proper arrangement allows for reaching any defined magnetic coupling ratios, such as k = 0.6(6), k = 0.75, k = 0.8, k = 0.9, etc. This design idea could be easily implemented in LLC DC/DC converter design to simplify the magnetics. An important aspect of the design was to use the same magnetic core material for leakage inductances and the main transformer. This means that the system will be using the same part of the magnetic hysteresis loop of the material to maintain a constant coupling ratio during transformer operation. For the same reason, the number of primary windings, N1 (windings of the primary leakage  
Figure 4: Output pulse shaping circuit with Tesla coil-like cored transformer. inductance and primary side of the transformer), should be equal. Similarly, N2 (for secondary leakage inductance and secondary side of transformer) should be equal. The prototype was built with seven MS-250026-2 micrometal cores with a reduced magnetic permeability of \(\mu_{\mathrm{r}}\) = 60. The primary winding is made of 64 coils connected in parallel. Each of these coils were made as a single turn of \(\sim\)1 mm\({}^{2}\) litz wire. The secondary winding is made of 50 turns of the same litz wire that was used in the primary side. The cores and windings of the primary leakage inductance and primary side of the transformer were insulated by epoxy resin impregnation. Figure 6 shows the transformer components before the final assembly and the main transformer part mounted with the primary leakage inductance and the primary windings.