For high speed digital integrated circuits, the interconnections of semi-conductor devices behave as transmission lines terminated with nonlinear loads. Nonlinearities become important when a device is changing its state or when it is excited by a large-amplitude signal. Signal delays and rising time along or at the terminals of these lines are investigated either by the direct time-domain approach or by the transform of the frequency domain data into the time-domain.
Since the transmission lines are analyzed and measured in the frequency domain and the nonlinear load boundary conditions may be analyzed only in the time domain, one may take the frequency domain data of the transmission line with the source and transform it into the time domain response and solve the non-linear load boundary condition numerically in the time domain by using the Newton-Raphson method.
It is known that the forward and backward propagating waves represent the total voltage and current along the lossless transmission line in the time domain. If we defines the reflection coefficients at the source and load boundaries in the time domain, one may use the forward and backward wave representations in the transmissions line.
A successive time stepping method may be utilized to linearize the given non-linear load voltage-current relation in the incremental time stepping period. For a nonlinear load having linearized conductance and capacitance in the incremental time interval, one may approximate the time derivative by the finite difference in the incremental time interval δt which gives the equivalent resistance of the capacitance, δt/$C_q$, where $C_q$ is the linearized capacitance at this time interval.
One may extend this method to the lossy, non-uniform and time-dependent lines terminated with an arbitrary non-linear load. Finite difference method is applied to the non-uniform transmission lines in the spatial domain, one may obtain the similar representation in the time domain. For a lossy and non-uniform transmission line, additional terms due to the continuous reflections along the transmission line appear.
Numerical calculations show that this direct time domain calculation is efficient in the computation time since it does not use the inverse Fourier or (Laplace) transformation and convolution integral for the frequency domain data of the transmission line. This method does not use the Newton-Raphson algorithm to deal with the non-linearity of the load, it gives the convergent results even when the voltage-current characteristics of the non-linear load are not monotonically increasing or multivalued, where the Newton-Raphson method may give more than one solution. The convergence of this method requires the incremental time period δ to be sufficiently small compared with the rising and falling time of the source, the time variation of the non-linear voltage-current characteristics, and L/R and C/G for the lossy transmission line, where L, R, C, G are respectively, inductance, resistance, capacitance, and conductance of the transmission line per unit length.