2D Graphs

Cassiopeia has built-in a sophisticated 2D graph module for plotting functions. We will discuss a real physical problem in this document and then present the result of the derivation in a 2D graph. Consider the following electrical circuit consisting of a capacitance, a resistor (wire resistance) and an inductance.

This circuit is described mathematically by the following two equations.


The first equation can be rewritten as follows:


We use the following approach to solve this linear differential equation:


The last equation is true for all t only if we assume


This gets us

We consider the first solution and insert the expression into our solution apporach.


We set


and use euler to rewrite

The differential equation is linear. Therefore both summands must be a solution for the differential equations.

A constant factor shouldn't harm a solution and the sum of two solutions is again a solution. We can thererefore write


This can be transformed to
with


We now use the second equation from above to get an expression for the current in the circuit.


We Alt-drag the expression for the angular frequency onto this last equation and rewrite the equation for Q. This gets us the following two functions.


Now choose SDM - FunctionGraph2D from the menu to insert a 2DGraph into the document. This should get us


in the document. Moreover the graph inspector should be opened in a separate window.


Now drag the two derived functions for voltage and current onto the textview in the middle left of this inspector.


Change the from to values for x (actually t) to 0 - 100 ms. After modifying the values press Return in one of the fields to trigger redraw of the graph.



Now change the coefficients C, L and R to more realistic values. Also check the Fine box to get a more accurate rendering of the graph.


Choose Tools - Colors from the menu and drag red color onto the current function in the tablevie win the lower left and blue color onto the function for the voltage.


This does not look bad. However, the current line is too flat to be easily examined. We therefore specify a scalation factor of ten for the current function.


Click back into your document. The graph is updated accordingly.

We are not done yet with the discussion of the problem. Let's assume we are interested in the heat loss in the wire resistance. This loss is given by


We Alt-drag the expression for the current onto this equation and get


We Alt-drag our expression for the angular frequency onto this equation and get


We now double-click onto the 2D graph to (re)open its inspector and simply drag this last equation onto the textview, then set a scaler of 20 and a color for this additional physical property.


Clicking back into the document updates the 2D graph.


The functions in a 2D graph can be interdependent. Assume the result of your creative work is the following set of functions (see Example Paper):


Note that the last function depends on all the other. These fucntions can be dragged as is onto a 2D graph to be plotted.

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