# High Frequency Transformers for Solid-State Transformer Applications
## 2 Modulation of the Converter System  
### Solid-State Transformer  
The SST can assume several topologies; some examples are given in Figure 1. The main difference between the various topologies is the number of constituent stages, the availability of direct current (DC) buses, the allowed functions, and the control complexity [1; 18]. The three-stage topology, Type D, has the greatest popularity due to its features such as reactive power compensation, voltage drop compensation, integration of renewable energy sources, and integration with energy storage systems, among others. It is designed to provide two DC buses, a low voltage (LV), and a high voltage (HV), which allow the integration of other components to improve the power quality or to supply direct current equipment. The main disadvantage of this topology is the large number of power semiconductors required, which generally translates into lower efficiency and lower reliability [18; 19]. In all the topologies depicted in Figure 1, power bidirectionality can be ensured by using power semiconductors that support operation in all four quadrants. The topology of the transformer is as important as the components in meeting the requirements of power density and efficiency. As SST applications, we highlight the solenoidal structures, namely the core type, shell type, and matrix type in Figure 2[2; 18]. \begin{table}
\begin{tabular}{c c c c} \hline \(A_{c}\) & Cross section area of the core (m\({}^{2}\)) & \(MLT\) & Average length of a coil (m) \\ \hline \(A_{p}\) & Product between \(W_{a}\) and \(A_{c}\) (m\({}^{4}\)) & \(P_{fe}\) & Power losses in iron (W) \\ \hline \(B_{opt}\) & Optimal density flux (T) & \(P_{cu}\) & Power losses in copper (W) \\ \hline \(B_{max}\) & Maximum density flux (T) & \(S\) & Apparent power (VA) \\ \hline \(B_{sat}\) & Saturation density flux (T) & \(T_{a}\) & Ambient temperature (K) \\ \hline \(f\) & Operation frequency (Hz) & \(V_{P}\) & Primary voltage (V) \\ \hline \(h_{c}\) & Convective heat-transfer coefficient (KW\({}^{2}\)/m); & \(V_{S}\) & Secondary voltage (V) \\ \hline \(J_{o}\) & Current density (A/m\({}^{2}\)) & \(V_{c}\) & Core volume (m\({}^{3}\)) \\ \hline \(k_{a}\) & & \(V_{w}\) & Window volume (m\({}^{3}\)) \\ \hline \(k_{c}\) & Dimensionless constants & \(W_{a}\) & Window area (m\({}^{2}\)) \\ \hline \(k_{w}\) & & \(\Delta T\) & Temperature variation (K) \\ \hline \(K_{c}\) & Magnetic material loss coefficient & \(\alpha\) & \\ \hline \(k_{f}\) & Stacking factor & \(\beta\) & Magnetic core material parameters \\ \hline \(k_{u}\) & Window utilization factor & \(K_{c}\) & \\ \hline \(K_{v}\) & Wave factor & \(\rho\) & Electrical resistance of the conductor \\ \hline \(l_{c}\) & Average length of the magnetic part of the core & \(\mu_{o}\) & Vacuum Permeability (H/m) \\ \hline \(L_{I}\) & Leakage inductance (H) & \(\mu_{r}\) & Relative permeability \\ \hline \end{tabular}
\end{table} 
Table 1: List of variables and parameters. The main advantage of the core type is that the windings are around a single magnetic core, which results in a weight reduction of the transformer [2]. However, a large part of the magnetic core is surrounded by windings, which, associated with the poor thermal conductivity of the magnetic core, become a disadvantage of the core type and consequently an advantage of the shell type from a thermal point of view. Apparently, the matrix type has the same advantages as the shell type, but it has two major disadvantages that restrict its use: first, the number of magnetic cores, which results in the increased volume and weight of the transformer and, consequently, increased losses and cost; second, the primary and secondary are separated at opposite ends of the magnetic core, leading to higher leakage inductance, an unfavorable quality in large power converters [2].