## Mathematics

Knowledge of Mathematics is essential in most fields of Engineering. It is interesting that knowledge of one field of engineering may be applied to another field of engineering. This is called the *Principle of Duality*. This is a principle we have found useful in more than one project.

Even apparently innocent problems may lead to unexpected complex equations. Consider the RC circuit below charging a capcaitor C through a reistor R and a __fixed voltage__ source Vin. The equation for the voltage over the capacitor C is well known: Vc = Vin*(1 - exp(-t / (R*C))).

However, if the input voltage V is a function of time e.g. V(t) = t, then the equation for Vc(t) is quite complex. One way to find Vin is by doing Laplace Tranformation of the circuit equations, and then solving the resulting rational functions for Vc(S). This gives: Vc[S] -> 1/(S^2 (1 + R*C*S)). The time function can now be found by doing an Inverse Laplace tranform of Vc(S). The solution is: Vc(t) = R*C*exp(-(t/(R*C))) + t -R*C.

Performing Laplace Tranforms requires that you remember your math. To save time we use Mathematica when we are faced with problems like this.

Since the field of Mathematics is a vast one, it is not possible for an engineer to know or remember all formulas needed to solve all kinds of problems. Engineers are expected to deliver on short deadlines and knowledge of efficient tools to solve mathematical and simulation problems becomes crucial!

Objective Technology has good knowledge of the mathematical tool *Mathematica*. (See the Foundation page). Mathematica has been used extensively to perform calculations in several projects, but especially in development of our HPC-09 system.

In the area of simulation, we have experience with Modelica based tools such as Maplesim and Mathmodelica.