## 1 Introduction  
Pulsed eddy current (PEC) is a widely used technology for inspecting borehole casings due to its rapid and accurate acquisition of data over a broad frequency range and accessibility to targets [1; 2]. The basic PEC probe comprises transmitting coils to excite the magnetic field in the conductive tube being tested, as well as a receiving sensor that picks up the magnetic field from the eddy currents induced by the abrupt cutoff of emissions [3; 4]. Many researchers have focused on designing different PEC probes to improve the performance of inspections [5; 6; 7; 8]. On the receiver side, different receiving sensors have been utilized to improve the signal-to-noise ratio (SNR), including the coil sensor [5], the Hall sensor [6], and the magnetic resistance sensor [7]. On the transmitter side, research has focused on optimizing the shape and geometric parameters of the sensor design as performance is affected by the magnetic field distribution [8]. [PERSON] et al. [9] used a longitudinal probe and transverse probe to generate magnetic fields in different directions, and the experimental results showed that the orientation of the magnetic field distribution had a significant influence on the detection sensitivity for cracks in the casing. [PERSON] et al. [10] analyzed the field and impedance of a rotary coil in a conductive tube considering the induced current. [PERSON] et al. [11] analyzed the magnetic flux density distribution of different types of probes to optimize the performance of the transmitting coils. [PERSON] et al. [12] analyzed the effect of the probe size on the eddy current field at different times, and their results indicated that the longitudinal spatial resolution could be improved by reducing the longitudinal distribution range of the magnetic field. In recent years, magnetic focusing (MF) has attracted attention as an effective method of controlling the magnetic field distribution. A common approach to realizing MF is based on magnetic shielding theory [13; 14]. [PERSON] et al. [13] proposed a multilayer MF sensor framework with a driver coil, a pickup coil, and a three-layer shield with different shielding materials to enhance the detectability of ferromagnetic materials at greater depths. [PERSON] et al. [14] applied high-temperature superconducting plates together as a magnetic shield to concentrate the magnetic flux inside of the slit between the plates to improve the sensitivity and noise characteristics of a tunnel magnetoresistive sensor. Another approach to realizing MF is based on beamforming or spatial signal processing, which is similar to radar, sonar, and wireless communications [15; 16; 17; 18; 19]. This approach uses multiple transmitting coils (MTCs) to control the vector of the magnetic field. [PERSON] et al. [15; 16] proposed a method for calculating the current distribution, which enables active sidelobe suppression of MF while maintaining the resolution. However, while MF is suitable for the application of PEC at the surface, the high temperature, high pressure, and narrow space of downhole environments significantly limit the applicability of MF to PEC in metal pipelines [20; 21; 22]. With regard to magnetic shielding, the effect of MF largely depends on the geometric design of the shield [13]. To ensure trafficability [20], a downhole PEC instrument needs to be much smaller than the inner diameter of the casing. This results in some distance between the instrument and the casing, which can cause a poor focusing effect. Meanwhile, MF using MTCs requires different transmitting currents and corresponding electrical circuits, which both increase the hardware cost, and are difficult to fit in the narrow underground space [17; 21]. To solve this problem, we propose a synthesized transmitting coil (STC) for PEC downhole casing inspection that combines MTCs into a single coil with multiple connected sections using different winding directions and number of turns to adjust the magnetic field distribution.