Lossless Equations Research Articles

Shock waves micro‐damage induction in cortical bones: Comparison between experimental and simulations results.

Shock waves (SW) are considered a promising method to treat bone non unions. One potential mechanisms of action is the initiation of local micro‐fractures, which may in turn trigger the start of bone healing. In this study, a set of eight intact rat femurs have been subjected to SWs (peak positive pressure of 40 MPa and peak negative pressure of −8 MPa, PRF of 2 Hz). The number of SW was varied from 50 to 1500. Micro‐CT images of the specimens were acquired before and after treatment (16 microns resolution). In parallel, numerical simulations were used to quantify the stresses induced by SWs in cortical bone tissue. We used a 3‐D FDTD code to solve the linear lossless equations that describe wave propagation in solids and fluids. A 3‐D model of a fractured rat femur was obtained from micro‐CT data. Results demonstrate that damages were induced in the bone tissues from 150 SWs. Comparison of the location of the induced damages on micro‐CT images with predictions of maximum stresses by numerical simulations...

Differential equation solutions of MOS transmission line models generalized to lossy cases

Lumped transmission line equivalent circuit models offer a proven technique for calculating small signal admittances of MOS capacitors. Typically, this is done by means of 6 × 6 matrix multiplications. Recently, however, we derived for the lossless MOSC a set of three differential equations which are easier and more convenient to solve than the more general matrix method. Here we review simple circuit topology to explore the fundamental limitations of the three differential equation system and find that minority carrier scattering and bulk recombination losses can be included. Each of these losses contributes an additional term to each of the three lossless equations, resulting in a system of equations that is modular with respect to the inclusion of either of these two losses. The majority carrier scattering loss is more easily taken care of by the matrix method, but is only important at very high frequency (e.g. 10 7 Hz).