## 1 Introduction  
As a prerequisite for evaluating lightning-induced voltages in distribution networks, it is of great significance to calculate the electromagnetic field generated by a lightning return stroke over a lossy ground accurately and efficiently. At present, there are two kinds of methods for the calculation: Finite-difference time-domain method (FDTD) and Cooray-Rubinstein (CR) formula. The FDTD method [1; 2; 3] has great applicability when considering of the effect of lossy ground, such as the lightning electromagnetic field over layered ground. However, the space and time are required to be discretized in FDTD, which results in a huge amount of computation time and relatively complicated programming, especially for the absorbing boundary. In order to reduce the computation time, analytical formulae are an alternative choice. The Sommerfeld integral [4] can be used to rigorously evaluate the lightning electromagnetic field generated by a lightning return stroke over a lossy ground; however, the highly oscillatory and slow convergent integrand make it difficult to evaluate the integral efficiently. The Cooray-Rubinstein formula [5; 6; 7; 8] is a good choice to overcome this difficulty, and has become a widely used method. In the past years, most studies have mainly focused on efficient evaluation of the CR formula in the time domain [9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. Essentially, the main task of these methods is how to describe the integral kernel function of the CR formula in the time domain and accelerate the calculation of the convolution.  
In practice, the implementation of the CR formula to evaluate the lightning electromagnetic field over a lossy ground requires a two-step procedure. The first step involves calculating the lightning electromagnetic field over a perfectly conducting ground, and the second step is to evaluate the CR formula. Actually, compared with the second step, the calculation of the ideal lightning electromagnetic field (\(E_{r,ideal}\) and \(H_{q,ideal}\)) accounted for most of the computation time. Therefore, increasing the efficiency of the first step is more conducive to improving the overall computational efficiency. The relevant studies about this tend to be neglected. The formulae for evaluating \(E_{r,ideal}\) and \(H_{q,ideal}\) have been established based on the dipole method, and they are composed by integrals with respect to the lightning channel. The common method to evaluate these integrals are mainly based on the numerical integration, by means of a discretization of the lightning channel. However, only a sufficiently small discretization step is essential to get an accurate result, which leads to a relatively large number of calculations and results in a lengthy computation time. Besides, the programming is relatively complicated because the propagation of the lightning current along the channel must be considered.  
In order to increase the efficiency and simplify the programming, an improved method is proposed in this paper. Firstly, regarding the return stroke current in the lightning channel as a series of current sources distributed along the channel, the original time-varying system can be transformed into a time-invariant system. Then, the calculation formulae are rewritten in the frequency domain by means of the Fourier transform, which simplifies the integral and differential operations about time in the original formula, so that the formulae are reduced from having two variables (\(z^{\prime}\), \(t\)) in the time domain to having only one variable (\(z^{\prime}\)) in the frequency domain. Finally, a series of analytical formulae according to the integrals are derived, which effectively simplify the calculation procedure. Additionally, compared with the conventional method, the efficiency of the proposed method can be increased, which can be examined by the simulation example and discussion.