2D GraphsCassiopeia includes a sophisticated 2D graph module for plotting functions. We will discuss a damped LC tank in this document and then present the result of the derivation in a 2D graph. If you are not interested in the concrete physical problem skip the math and scroll down to the graphs. :-) We consider the following electrical circuit consisting of a capacitance, a resistor and an inductance.
This circuit is described mathematically by the following two differential 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.
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
We now use Eq. 2 to get an expression for the current in the circuit.
We Alt-drag the expression for the angular frequency onto this last equation and replace Q with CU. This gets us the following two functions.
Now choose SDM - FunctionGraph2D from the menu to insert a 2DGraph and drag the two equations onto the textview of the FunctionGraph2D inspector.
The graph appears as follows In our document.
We change the From: and To: values for the abscissa 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 too bad. However, the current line is too flat to be easily examined. We therefore specify a scale factor of ten for the current function.
Click back into your document. The graph is updated accordingly.
We are not done yet. 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 now double-click onto the 2D graph to (re)open its inspector and simply drag this last equation onto the textview. We also drag the expression for onto the textview and 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 functions can be dragged as they are onto a 2D graph to be plotted.
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