## 3 Synthesized Transmitting Coil
### Magnetic Focusing  
We used the model of the PEC system presented in Section 2 to design the proposed STC for MF in downhole environments. Figure 2 compares the structures of the traditional MTCs and the proposed STC. Both are able to focus a synthesized magnetic field more narrowly on a certain area. As shown in Figure 2a, when MTCs send the same current \(I_{1}\) simultaneously, a non-focused magnetic field is excited. MTCs can control the vector of the magnetic field (or transmitting currents with \(I_{1}\), \(I_{2}\), \(I_{3}\), \(I_{4}\), and \(I_{5}\)) to achieve MF. However, MTCs not only require more transmitting electrical circuits, but also require the H-bridges and control signals to be almost the same to ensure consistency when the magnetic field is turned on and off; otherwise, large errors can be introduced. Figure 2b shows the proposed STC, which combines the MTCs into a single coil with multiple sections having different winding directions and number of turns to adjust the magnetic field distribution. Specifically, the STC comprises multiple transmitters, each defined as a section. The five sections are connected together, and each section has a different winding direction and number of turns. The sections of the STC can be divided into two types according to the winding direction: the P-sections have the same winding direction as the middle section, and the N-sections have the opposite winding direction of the middle section. If the total number of turns of the STC is \(M\), then the magnetic field can be expressed as the sum of the fields of each transmitting coil turn:  
\[\overline{H}(\mathbf{z}_{t},t,r,z)=\sum_{m=1}^{M}\alpha_{m}H(z_{t_{m}},t,r,z) \tag{4}\]  
Figure 2: Structural comparison: (**a**) MTCs and (**b**) the proposed STC. where \(z_{tm}\) is the position of the \(m\)th turn. \(\alpha_{m}\) is a coefficient related to the winding direction of the \(m\)th turn coil and can be expressed as:  
\[\alpha_{m}= \begin{cases}1&m\text{th turn coil}\in\text{P-Section}\\ -1&m\text{th turn coil}\in\text{N-Section}\end{cases}  \tag{5}\]  
By stacking \(\alpha_{m}\) to form a vector, we obtain:  
\[\alpha= \begin{bmatrix}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{M}\end{bmatrix}  \tag{6}\]  
The winding parameters of the STC can be expressed by \(\alpha\). Different magnetic field distributions can be achieved by designing the vector \(\alpha\). Let the number of STC sections be \(K\), and the number of turns for each section be \(n_{k}\). The two adjacent sections have opposite winding directions. Therefore, \(\alpha\) of the five-section STC shown in Figure 2b can be written as:  
\[\alpha= \begin{bmatrix}1&\cdots&1\\ &n_{1}\end{bmatrix} \ \ \ \underbrace{-1&\cdots&-1}_{n_{2}}\ \ \ \underbrace{1&\cdots&1}_{n_{3}}\ \ \ \underbrace{-1&\cdots&-1}_{n_{4}}\ \ \ \underbrace{1&\cdots&1}_{n_{5}}\end{bmatrix} \tag{7}\]  
Suppose that \(S_{k}\) represents the \(k\)th section for a five-section STC, where \(S_{1}\), \(S_{3}\), and \(S_{5}\) are P-sections, and \(S_{2}\) and \(S_{4}\) are N-sections. To focus the magnetic field on the longitudinal center of the probe, the proposed STC adopts a central-symmetry structure. If the number of sections \(K\) is odd, and symmetrical sections have the same number of turns (e.g., \(n_{1}=n_{5}\) and \(n_{2}=n_{4}\) for a five-section STC), then the number of turns of section \(k\) is equal to the number of turns of section \(K+1-k\):  
\[n_{k}=n_{K+1-k} \tag{8}\]  
Because \(K\) is an odd number, the central section is the \((K+1)/2\)-th section. When \(K\) changes, the sequence number for P-sections also changes. For example, \(S_{2}\) is a P-section for a three-section STC while \(S_{1}\), \(S_{3}\), and \(S_{5}\) are P-sections for a five-section STC. The P-sections of STCs with different sequence numbers can be expressed as follows:  
\[\text{P-sections}= \begin{cases}S_{2},\,S_{4},\,\ldots,\,S_{(K+1)/2},\,\ldots,\, S_{K-1}&K=3,7,11,\cdots\\ S_{1},\,S_{3},\,\ldots,\,S_{(K+1)/2},\,\ldots,\,S_{K}&K=5,9,13,\cdots\end{cases}