# Reactance Regulation Using Coils with Perpendicular Magnetic Field in the Tubular Core design
## 2 Method of Power Flow Control  
The schematic diagram of power flow control is shown in Figure 1a. The prosumer load is connected to the grid through controllable reactance. The auxiliary generating module is connected to the prosumer to provide load requirements. Generally, auxiliary generating modules have a slow response and a relatively narrow range of regulation. Therefore, the fast regulation of power flow is preferable using the controllable reactance.  
Figure 1: **(a)** The schematic of load power regulation through controllable reactance, **(b)** Non-linear magnetic curves of controllable reactance for different DC currents in the control winding [\(B\) is magnetic flux density, \(\varphi\) is magnetic flux, \(l\) is the average path of magnetic lines in the core, \(A\) is cross-section of core, \(H\) is field strength, \(I\) is current in the coil, \(WP_{1}\) is working point at \(I_{1}\), and \(WP_{2}\) is working point at \(I_{2}\).  
Let us consider that a coil is wound on the magnetic core. The magnetic characteristic of the core is explained in Figure 1b. The current (\(I\)) induced by AC voltage (\(V\)) is represented as:  
\[I=\frac{V}{X_{L}}=\frac{V}{\omega L}=\frac{VR_{m}}{\omega N^{2}} \tag{1}\]  
where \(\omega\) is frequency, \(X_{L}\) is the reactance, \(N\) is the number of coil's turns, and \(R_{m}\) is the magnetic reluctance. The magnetic permeability of core material (\(\mu=\mathrm{d}B/\mathrm{d}H\)) is varied with the location of different working points and becomes smaller as the voltage enhances. Therefore, an additional excitation coil on the same core can regulate the working point and current. The excitation current in the second coil controls the impedance of the first one.  
The impedance variations due to the non-linear magnetic curve are used in the traditional MAs (Figure 1b). The DC current of one of the magnetically independent winding shifts a working point of another winding from one position to another (WP\({}_{1}\) to WP\({}_{2}\)). The derivative at point WP\({}_{1}\) is much smaller than derivative at WP\({}_{2}\) due to the strong non-linearity of a function B = f(H). However, the use of the nonlinear behavior of the B-H curve leads to high harmonics, large total harmonic distortion (THD), and a current offset. These distortions are responsible for poor electricity quality and low service duration of AC motors and transformers. The method to modify the magnetization curve by a control current seems more efficient, compact, and cheaper. This method originated from the fact that the magnetic anisotropy depends on the magnetization orientation [34; 35; 36; 37]. The largest permeability is obtained in the longitudinal axis of magnetic strips, whereas the lowest permeability is observed in the perpendicular direction of the largest magnetization. This arrangement allows for impedance control by altering the B-H curve. In this method, the change of DC current in one winding (designated as a control coil) causes the shift of magnetization vector angle of another winding (designated as a working coil) in the direction from maximum to minimal permeability that provides the corresponding decrease of the working coil impedance. It is important to emphasize that the initial working point always remains at the beginning of the B-H axes. Therefore, the symmetry of a working coil current is ensured.