## 1 Introduction
### Equivalent Circuit Considering the End Effect  
When designing a single LS-secondary DLIM for EM propulsion, the secondary is assumed to be the same length as the primary. If the cogging ratio is defined as the ratio of the slot width to the ladder width, the slot width and the ladder width can be expressed as follows:  
\[b_{2}=\frac{L_{1}}{N+(N-1)k},b_{1}=\frac{(k+1)L_{1}}{N+(N-1)k}. \tag{1}\]  
If Equation (1) is substituted into the equivalent circuit in Figure 2a, the model of the LS-secondary DLIM becomes a function of the cogging ratio. Referring to [PERSON]'s model and considering the discrete ladders, the eddy current per unit length induced in the LS-secondary is shown in Figure 2b, the envelope of which is consistent with that in [PERSON]'s model [14]. A series of ladder ends on a normalized time scale is proposed as a modification of the time scale's length. To simplify the formula, we define:  
\[Q=Q_{21}=\frac{b_{1}R_{21}}{(L_{\text{m}0}+L_{21})v}.\]  
Thus, the series of ladder ends on a normalized time scale can be represented as:  
\[Q_{l1}=\ (l-1)Q,\ Q_{l2}=Q_{l1}+\frac{b_{2}}{b_{1}}Q,\ l=1,2,...,N. \tag{2}\]  
Figure 1: Structure of the ladder-slot (LS)-secondary. Figure 2: Modeling of the LS-secondary DLIM: (**a**) Equivalent circuit of the LS-secondary; (**b**) Eddy current per unit length in the LS-secondary, combining [PERSON]’s model with discrete ladders. Then we express the distribution of eddy current induced in the secondary as a function of Equation (2):  
\[I_{2\text{e}}=\left\{ \begin{array}{l}I_{\text{m}0}e^{-x},x\in[Q_{1},Q_{l2}]\\ 0,x\
otin[Q_{l1},Q_{l2}]\end{array} ,l=1,2,...,N\right. \tag{3}\]  
By applying segmental integration, the modified magnetizing inductance and magnetizing resistance can be written as:  
\[L_{\text{m}}=L_{\text{m}0}\left[1-\frac{\left(1-e^{-NQ}\right)\left(1-e^{-Qb_{ 2}/b_{1}}\right)}{Q(1-e^{-Q})(N-1+b_{2}/b_{1})}\right],\ R_{\text{m}}=R_{21} \frac{\frac{1-e^{-2 NQ}}{1-e^{-2Q}}-2e^{-\frac{b_{2}}{b_{1}}Q}\frac{1-e^{-NQ}}{ 1-e^{-Q}}+N}{2(N-1+b_{2}/b_{1})Q}. \tag{4}\]  
It can be seen from Equation (4) that the longitudinal end effect influences the magnetizing inductance via the root-mean-square (RMS) value of the secondary eddy current, while it is the sole cause of the magnetizing and the secondary resistances. On one hand, the secondary resistance referred to as primary \(R_{21}\) (as in Equation (5)), differs from that of the flat secondary, and consists of the resistance of the ladders \(R_{\text{L}}\) and the resistance of end connections \(R_{\text{S}}\), both of which can be derived according to Kirchhoff's law. \[R_{21}=R_{L}+R_{\text{S}}\frac{N-1}{2N\sin^{2}\frac{\pi p}{N}},R_{L}=\frac{K_{ \text{w}}\rho l_{\delta 2}}{d_{2}b_{2}},R_{\text{S}}=\frac{K_{\text{w}}\rho 2(b_{1}-b_{ 2})}{d_{2}(l_{\delta 2}-l_{\delta 1})},K_{\text{w}}=\ \frac{4m_{1}\left(w_{1}K_{dp1} \right)^{2}}{N}. \tag{5}\]  
On the other hand, the magnetizing inductance without the end effect and the remaining inductances in the equivalent circuit can be derived from the inductances of the flat-secondary DLIM [7], as in Equation (6). The superscript \(f\) and coefficient \(K_{c2}\) indicate the parameters of the flat-secondary DLIM and Carter's coefficient, respectively. \[L_{\text{m}0}=\frac{L_{\text{m}0}^{f}}{K_{c2}},L_{1}=L_{1}^{f},L_{21}=1\times 1 0^{-7}\cdot\frac{m_{1}\left(w_{1}K_{dp1}\right)^{2}\tau}{p}. \tag{6}\]  
In addition, the primary resistance remains the same as that of the flat-secondary DLIM. By combining the above six equations, we arrive at the equivalent circuit of the LS-secondary DLIM, upon which the ensuing numerical analysis is founded. Here, we also discuss the distribution of the EM shear stress.