Closed-form representations for triangle impulse responses associated with single and coupled lossy transmission lines

A fast simulation method was proposed for single and coupled transmission lines that are connected to linear and non-linear circuit elements by Z. Chen et al. (1999) [1]. In that method, the time-domain voltages and currents at the ends of the lines are approximated by a series of triangular expansion functions. A time-stepping procedure can then be employed for the circuit simulation provided that the triangle impulse responses for the lines are known. In [1], a triangle impulse response database for the lossy transmission lines is employed. Other simulation tools are used to calculate the required lossy transmission line triangle impulse responses numerically. The numerical results for the triangle impulse responses are then used with a convolution algorithm to carry out the circuit simulation. In our work, analytic frequency-domain expressions for single and coupled transmission lines with triangular input waveforms are first developed. The inverse Laplace transform is then used to obtain an expression for the time-domain triangle impulse responses. The integral associated with inverse Laplace transform is solved analytically using a differential-equation-based technique. Closed-form expressions for the triangle impulse responses are given in the form of incomplete Lipschitz-Hankel integrals (ILHI's) of the first kind. The ILHIs can be efficiently calculated using algorithms developed by S.L. Dvorak and E.F. Kuester (1990). Combining these closed-form expressions for the triangle impulse responses with the method proposed in [1], provides an accurate and efficient simulation method for transmission lines embedded within linear and non-linear circuits.