Power Electronics

# Mathematical Spreadsheets and Simulators: Both Belong in Your Toolbox

Since SPICE was first developed in the Electronics Research Laboratory at the University of California, Berkeley, it has been used by power-supply designers and engineers to investigate power-circuit behavior under all sorts of conditions. Spice gives engineers feedback on their designs before they build their first prototypes. Furthermore, it enables them to solve problems before they encounter them in the lab and without losing a single component. I have used SPICE-class simulators for the last 27 years to understand the reasons for problems in my power circuits by probing deep into the circuit to an extent usually not possible with the actual circuit.

That said, I must note that the accuracy of a simulation depends to a very large extent on the accuracy of the component models. As a result, simulation of power circuits with very complex power semiconductor and transformers/inductors models gives the user a good idea of the circuit performance with reasonable accuracy. So, if you want to put your circuit “under the microscope,” make sure that your models are at least as accurate as you need your simulation results to be.

In my experience, if you have a modern complex power circuit, you will have convergence problems and issues unless you set the simulation relative accuracy of voltages and current to a minimum of 1% to 2% in most cases. That level of accuracy is quite enough in most practical applications. One caution, though: the increased dependency on circuit simulation may, at times, make you lose sight of the actual underlying laws of physics and some of the not-so-obvious dependencies of the solution on different component parameters within a given circuit. That’s an inherent potential of simulation, which by definition, finds a solution for a single set of parameters, components and conditions at a given time.

I have discovered that using mathematical spreadsheet software like Maple gives me a totally different experience in understanding how different individual parameters affect the final performance of the circuit. One added benefit is that the mathematical approach will require you to write the equations from scratch, an exercise that we have almost forgotten with our dependency of circuit simulations where only a circuit schematic is needed.

Mathematical software packages like Maple are capable of finding closed-form solutions to almost any equation or a set of equations including differential equations. A thorough understanding of ordinary differential equations (ODE) is a mandatory prerequisite before attempting to use the software, which is not difficult since ODE is part of any college mathematics course.

Obtaining a closed-form solution to a complex set of differential equations using mathematical software packages may not always be possible. The greatest advantage of the mathematical solution to a problem is once you get the closed-form solution for a set of equations, you can now do several probing activities that will help you understand the scope of your application and give you a very good understanding of the above mentioned dependencies of the solution on different parameters.

Using math software, you can generate a family of curves that relates any two parameters, helping you to understand the effect of a given perturbation of a given parameter on the solution. Better yet, by using the same mathematical package you can generate a 3-D graph that relates any two independent parameters and time. This type of result gives you an unparalleled visual understanding of the relationship of these two parameters over time.

One more thing that can easily be done is to derive equations for dependent parameters by solving one or more of the independent parameter equations for a specific condition. You can also perform optimization calculations by finding the maxima or minima of a certain parameter and design for either, if needed. One last use of the same set of solutions is performing sensitivity analysis using the closed-form solution with obvious advantages for numerical sensitivity analysis. You can use the same math software to do numerical solutions (used in simulation software), too, if you so desire.

Of course, mathematical software is not going to replace simulation, which plays a great role in power engineering design today and will continue to do so in the future. Instead, it offers engineers another tool for understanding and optimizing their power circuit designs. So, for any power designers who have yet to experience the benefits of math software, its time to take a closer look at this very powerful design tool and discover just how well it complements simulation.

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