# High Frequency Transformers for Solid-State Transformer Applications
## 2 Modulation of the Converter System
### Transformer Modeling  
Over the last decades, the trend has been to increase the power and frequency (typically from some kHz to dozens of MHz) of magnetic circuits. Nevertheless, this leads to a challenging design of magnetic circuits where parasitic effects play a key role. According to [23; 24], most of the proposed designs are based on optimization routines, depending on the temperature rise and assuming uniform heat dissipation. Some authors consider the desired efficiency as a design input for the HFT [24]; otherwise, the optimization is focused on the \(A_{p}\) area [25; 26]. In [27], the magnetizing inductance is considered the first step for the design process of HFT. In [28], The leakage inductance is first investigated, and then the remaining parameters of HFT are given. This paper proposes a step-by-step methodology for designing medium and high-frequency transformers and their optimum electrical and magnetic specifications to reduce volume and increase efficiency. The proposed design methodology for the HFT first finds the optimum magnetic- flux density and then arrives at the other parameters according to the desired criteria such as leakage inductance and losses, as shown in Figure 4.  
1. System and Topology Specifications: Definition of the main system specifications, namely: apparent power (\(S\)), primary voltage (\(V_{P}\)), secondary voltage (\(V_{S}\)), operating frequency (\(f\)), ambient temperature (\(T_{a}\)), admissible temperature variation (\(\Delta T\)), wave factor (\(K_{\sigma}\)), window utilization factor (\(k_{u}\)) and stacking factor (\(k_{f}\));
2. Magnetic Core Properties: Selection of the material to be used for the magnetic core to find the saturation flux density (\(B_{sat}\)) and parameters of the constants \(\alpha\), \(\beta\) and \(Kc\) of the respective material, essential parameters for the following steps; most common cores, such as amorphous, ferrite, and nanocrystalline magnetic materials are compared in [29] for core losses at 10, 20, 50, and 100 kHz, respectively. 3. Calculate Optimum Flux Density: The optimal flux density corresponds to the flux density value that minimizes the sum of copper losses with iron losses. Knowing the expression that describes the losses in copper and iron, as a function of the frequency and flux density, and after numerical simplification, it is possible to obtain the expression of the optimal flux density by Equation (1), as laid out in [30]. Figure 3: Examples of conductor topologies. \[B_{opt}=\frac{(h_{c}\ k_{a}\ \Delta T)^{\frac{2}{3}}}{2^{\frac{2}{3}}(\rho_{w}\ k_{w} \ k_{u})^{\frac{1}{12}}(k_{c}\ K_{c}\ f^{\alpha})^{\frac{1}{12}}}\left(\frac{K_{ V}\ f\ k_{f}\ k_{u}}{\ \sum VA}\right)^{\frac{1}{6}} \tag{1}\]  
where \(k_{a}\), \(k_{c}\), \(k_{w}\) are dimensionless constants, which usually assume the values 40, 5.6 and 10 respectively [30]; \(\sum VA\) represents the apparent power of the transformer, \(h_{c}\) is the heat transfer coefficient, and \(\rho_{w}\) is the electrical resistance of the conductor that composes the windings. 4. Magnetic Core Sizing: For the optimal flux density, determined in the previous point, the dimensions of the magnetic core of the transformer are selected based on the product of the window area (\(W_{a}\)) by the cross-sectional area (\(A_{c}\)); this product is given the nomenclature \(A_{p}\), usually expressed in cm\({}^{4}\). For the analytical determination of this parameter, it is necessary to perform a comparison between the value of the optimal flux density (\(B_{opt}\)) and the saturation flux density value (\(B_{sat}\)). Based on the exposition in [30]:
* If \(B_{opt}<B_{sat}\), the value of the parameter \(A_{p}\) is given by Equation (2). \[A_{p}=\left(\frac{\sqrt{2}\sum VA}{K_{V}\ f\ B_{opt}\ k_{f}\ K_{t}\ \sqrt{k_{u}\ \Delta T}}\right)^{\frac{8}{7}}\] (2)
* If \(B_{opt}>B_{sat}\), the value of the parameter \(A_{p}\) is given by Equation (3).