electronics design engineers

Abstract: Inductor- and transformer-coupled balanced mixers are usually analyzed with switching approximations, conversion matrices, harmonic balance, or circuit simulation. This paper instead derives their loaded quasistatic equations from the device laws, keeping the transformer ratios, reflected load, bias, and the connections that set the IF common mode. The functions needed to solve these equations are surprisingly varied. A Shockley single-balanced pair is solved by the principal Lambert function $W_0$ under a quantified one-diode-dominance condition. The double-balanced ring has a unique loaded solution that reduces to an exact inverse-hyperbolic-tangent form only for an open IF port. Vacuum diodes and triodes lead to Cardano cubics and structured sextics, whereas depletion JFET/MESFET crossbars need only square roots. Bilateral MOS channels give piecewise quadratics in strong inversion, or a Ferrari quartic when the IF winding floats. In weak inversion they give an exact $W_0$, the real $r$-Lambert function, or the inverse of $\sinh x+ρx$, depending on topology and IF connection. In matched BJT and HBT Gilbert cells, equal emitter resistors turn the hyperbolic-tangent current steering into an exact $r$-Lambert function. Fourier projection brings in Beta/Gamma, modified Bessel, and complete elliptic functions. We checked the equations against 37 LTspice simulations using seven independent Python programs. In the 12 primary comparisons, which implement the assumed device laws exactly, every RMS waveform error was below 0.017% of the analytical RMS value; built-in and vendor device models were examined separately. Two design examples show how the closed forms size a thermionic IF load and a Gilbert-cell tail current without parameter sweeps.

Abstract: Process-induced warpage, ceramic stress and solder strain limit the reliability of insulated-gate bipolar transistor (IGBT) modules on direct-bonded copper (DBC) substrates. Surrogate-assisted design studies train regression models on finite-element analysis (FEA) databases, but rarely check the optimized designs against new FEA or report how surrogate error interacts with the optimizer. This paper builds and evaluates an open pipeline: an open-source process finite-element model, surrogates tuned by Simplified Swarm Optimization (SSO), multi-objective design search, FEA confirmation of selected designs and confirmation-driven infill. The model starts at the second reflow and reproduces measured warpage within 18.2%, 33.6% and 15.3% at the reflow, housing and molding stages without fitted parameters. On a balanced 60-design database, all stochastic tuners reach the same test accuracy, outperform the published grid on cross-validated performance for every output but generalize better only for warpage; the cross-validated ranking of tuners does not transfer to the test set. With equal result reporting, multi-objective SSO and the non-dominated sorting genetic algorithm II give comparable Pareto fronts; a corrected multi-objective particle swarm optimizer trails both. FEA confirmation shows that warpage predictions hold (mean absolute error 0.5%), whereas at the design-space bounds reached by the optimizers the ceramic-stress surrogate is optimistic by up to 26%. Two confirmation-driven infill rounds reduce this error to 1-10% and halve the out-of-sample error; no confirmed design improves on the database in ceramic stress. Ceramic-stress results are indicative, as the metric is mesh-sensitive at production resolution. The model, database, scripts and pre-registered and post-registration results are released.