**1. Introduction**

Wireless power transfer (WPT) has become popular recently and is expected to be used in plenty of technological devices. The main reason for the recent intense interest in WPT is the sharp increase in the use of electrical devices with various sizes and types of batteries. The main purpose of WPT research was the transfer of as much power as possible, with the goal of high system efficiency in spite of low mutual inductance between coils.

In the literature, one of the popular publications that take range variations into consideration is [1]. Critical coupling, frequency splitting, and impedance matching issues were analyzed in [1] using single-turn axillary loops to amplify the magnetic coupling added to resonators. In [1], the authors revealed 50% efficiency for their system, which is suitable for various positions of the receiver.

It has become widely accepted to use a series resonating circuit as the equivalent circuit of a resonator to conduct analysis for resonant frequency [2]. In this context, different attempts have been made to develop a sufficiently precise model [3,4]. There have been many methods employed such as modification circuits and magnetic design of the core properties [5]. In those studies, the scattering parameters were calculated by two port network theory using a network analyzer.

The state space model was created to obtain an accurate mathematical model of wireless power transfer (WPT) in [6,7], in which the transfer function of the bidirectional IPT system was obtained. As the Ziegler–Nichols method has high overshoot, it was optimized the PID parameters with a phase margin of 60° [8]. The PID parameters were optimized using the genetic algorithm in order to achieve better transient performance in [9]. The dynamic behavior ought to be as fast as possible in [10], in which the H-bridge circuit structure was inserted into the dynamic model. In [11], authors proposed a dynamic model for a multi-pickup system. To overcome the higher order problem and make a straightforward real dynamic model from the energy point of view, it was built a dynamic model of the WPT including a nonlinear inverter and rectifier in [12].

In [13], authors proposed a design methodology using a series-series (S-S) contactless power transfer system for an EV battery charger with two fixed operating frequencies. The converter operates at one of the fixed frequencies for a load-independent current output and at the other operating frequency for a load-independent voltage output. It was proposed hybrid topologies using either SS and PS compensation or SP and PP compensation for battery charging [14]. Controlling output current or voltage is achieved with two different methods. One is PWM (Pulse width modulation) control at a constant frequency at which huge changes in the duty cycle will not allow zero voltage switching (ZVS) [15]. Cost analysis was performed in [16], and the results showed that coils are the most expensive part of the system.

System analysis in terms of electrical circuit is necessary since magnetic resonance coupling involves creating LC resonance. To derive the analytical equations of the input impedance and transferred power, a simplified circuit model of the WPT systems can be used. In order to calculate parameters such as current and voltage as a function of frequency at various air gap values and load conditions, mutual inductance between two self-resonators can be obtained from magnetic circuit analysis. And the parameters that are obtained by magnetic analysis can be used to solve the equivalent circuit which makes it possible to calculate the efficiency using electrical relations.
