# Design of a Rotary Transformer for Installations on Large Shafts
## 2 Rotary Transformer: Model and Parameters
### Compensation Strategy  
It can be stated that the source sees an impedance which is, in general, complex, from Equations (6) and (7). Therefore, a significant amount of the input power is related to the nonzero imaginary part of \(\hat{Z}_{\text{in}}\). This reactive nature of the impedance does not cause any increase in the active power transfer, but it raises the absolute value of the absorbed current and therefore the ohmic losses (\(\propto l^{2}\)), leading to a drop in the efficiency of the system \(\eta\). To eliminate \(X_{\text{in}}\) and to increase \(\eta\), it is possible to design the transformer as:  
\[\omega L_{1}=\frac{\omega^{3}M^{2}L_{2}}{\left(R_{2}+R_{\text{L}}\right)^{2}+ \omega^{2}L_{2}^{2}} \tag{8}\]  
which is not so simple to obtain in practice, since this equation is frequency-dependent and it has three degrees of freedom (\(L_{1}\), \(L_{2}\) and \(M\)). Moreover, this equation may lead to mathematical solutions that correspond to unfeasible conditions (i.e., values of \(L_{1}\), \(L_{2}\) and \(M\) may be such that \(k>1\)) and they must be discarded. Alternatively, it can be thought to insert some capacitors which can resonate with the coils to provide a compensation of \(X_{\text{in}}\) for a specified frequency \(f\). In the literature, many solutions are available involving two capacitors connected in series and/or in parallel with respect to each winding, as can be seen in [21; 22; 23; 24; 25]. In this work, two solutions involving one or two series capacitors [26] were selected. They are represented in Figure 4a,b. These capacitors were tuned in such a way to cancel out the imaginary part of the corresponding input impedance. To this aim, the new expressions of \(\hat{Z}_{\text{in}}\) are considered here:  
Figure 3: Equivalent circuit of the rotary transformer with mutual inductor: phasor domain. \[\hat{Z}_{\text{in,IC}}=R_{1}+j\bigg{(}\omega L_{1}-\frac{1}{ \omega C_{1}}\bigg{)}+\frac{\omega^{2}M^{2}}{(R_{2}+R_{\text{L}})+j\omega L_{2}} \tag{9}\] \[\hat{Z}_{\text{in,2C}}=R_{1}+j\bigg{(}\omega L_{1}-\frac{1}{ \omega C_{1}}\bigg{)}+\frac{\omega^{2}M^{2}}{(R_{2}+R_{\text{L}})+j\bigg{(} \omega L_{2}-\frac{1}{\omega C_{2}}\bigg{)}} \tag{10}\]  
where \(\hat{Z}_{\text{in,IC}}\) and \(\hat{Z}_{\text{in,2C}}\) stand for the input impedance for the single- and double-capacitor configurations, respectively. Therefore, for the single-capacitor case, the strategy is to tune \(C_{1}\) so that the whole imaginary part simplifies:  
\[C_{1}=\frac{(R_{2}+R_{\text{L}})^{2}+\omega^{2}L_{2}^{2}}{\omega^{2}\Big{[}L_ {1}(R_{2}+R_{\text{L}})^{2}+\omega^{2}L_{1}L_{2}^{2}-\omega^{2}L_{2}M^{2} \Big{]}} \tag{11}\]  
and the input impedance of this configuration becomes purely real and equal to (6). Notably, the capacitor does not affect the real part of the input impedance (only the imaginary), so the active input power associated with the fundamental frequency is increased:  
\[P_{\text{in}}=V_{s}^{2}\frac{\hat{R}_{\text{in}}}{|\hat{Z}_{\text{in}}|^{2}} \tag{12}\]  
The single-capacitor configuration is thus beneficial both to the system efficiency and effectiveness. On the other hand, the tuning procedure for the double-capacitor configuration has, in principle, an infinite number of solutions, being one equation with two unknowns.