## 1 Introduction  
Trains are the fastest means of transportation on land, contributing greatly to industrial development. Railway tracks are required for trains to move, and rails are composed of supports, sleepers, and roadbeds to distribute the train's load. Rails are constantly subjected to fatigue load due to friction with the wheels of the train, and defects in the rail occur because of corrosion caused by the external environment (climate, etc.) [1]. Various surface defects in rails occur most frequently during rolling stock operation, and internal defects may develop, which lead to accidents if maintenance, such as grinding, is insufficient or not performed in advance.  
In general, a visual inspection method is used for inspecting rail defects. As a method in which inspectors walk around and check whether a defect has occurred in the rail, visual inspection makes it possible to easily perform defect inspection, even without proficiency. However, in land transport inspection, it is difficult to determine defects occurring inside the rail, and there is a limit to the area that can be inspected. Another non-destructive method of detecting defects is ultrasonic testing.  
Ultrasonic flaw detection is a method of inspecting the inside of a rail with a wave generated by vibrating a specific material [2, 3]. Inspection using ultrasound has developed to the stage where it is possible to check the inside of a rail with an image using phased arrays, but it has a disadvantage in that it is difficult to detect defects on the surface of the rail because of the flaw detection structure. In particular, it is impossible to detect squat-type defects that occur on the rail surface because the ultrasonic wave does not propagate because of the instantaneous void between the rail surface and the ultrasonic probe surface.  
Eddy current flaw detection is a technique proposed to compensate for the shortcomings of visual inspection and ultrasonic inspection. The eddy current inspection technique is a contactless method in which a high-frequency alternating current is applied to generate a magnetic field to detect changes in impedance between a coil and the specimen or the changes in voltage induced in a coil [4; 5; 6].  
Therefore, eddy current inspection can be used to inspect the inside of the rail, which gives it an advantage over visual inspection, and it can be used very usefully because it can inspect the surface area, which gives it an advantage over ultrasonic inspection [7; 8]. In the eddy current inspection method, a Lissajous figure can be drawn using X data (resistance) and Y data (reactance), which are signal outputs from the sensor, and the size of a defect can be obtained using amplitude and phase [9; 10].  
Previously developed eddy current flaw detection equipment is designed in a structure that cannot inspect the entire head of the rail using a small number of sensors. In order to inspect the entire head of the rail, technology that can minimize the dead zone area using multiple eddy current sensors is required. Along with the minimization of the dead band region, the performance of the eddy current sensor also plays an important role. In general, an eddy current sensor consists of a coil, and since a coil that responds according to the direction of a defect is determined, a technology capable of inspecting all defects using one sensor is required.  
In this study, the number of sensors was optimized by manufacturing a Plus-Point coil so that flaw detection could be performed regardless of the direction of the defect, and a total of 16 eddy current sensors were used to inspect the entire head of the rail. In addition, two-dimensional and three-dimensional rail surface shapes were obtainable through separate software, and a relational expression for analyzing the depth of the defect was derived by manufacturing artificial defects on the rail surface.