Drawing Linear Graphs
One of the methods used to find the transfer function of a system is to use linear graphs. This graph can then be processed into a normal tree and can be used to determine the elemental and state equations needed to simplify the problem.
Identify the through and across variables of the system shown below, then draw its linear graph.
In order to draw the linear graph, we first must identify the across and through variables of the system.
Through variable: I
Across variable: V
Because this problem is in the electrical domain its through variable is current (I) and its across variable is voltage (V). Current is measured through each element, and voltage is measured across each element.
Next we must identify how many distinct values of the across variable are present in the system. Each of these values will form a node on the linear graph.
This system has 3 distinct values of voltage in addition to ground. While it may be hard at first to determine the number of distinct values of the across variable, with practice it will grow easier.
To connect these nodes, first we will represent our source in the linear graph. For this problem the source will be modeled as an ideal voltage source, meaning that \(V_s\) will be constant regardless of drawn current.
An ideal across source is drawn as a vector in the direction of the decreasing across variable, hence here the voltage source is drawn as a vector with its arrow pointing in the direction of decreasing voltage. Had the source instead been an ideal through (current) source, the direction of this arrow would be reversed.
Finally, we must connect these nodes according to the provided system. Each element (capacitor, resistor, inductor etc…) is likewise represented as a vector between two nodes.
While the direction of these vectors is not as important, it is conventional to draw arrows in the direction of decreasing across variable (in this case in the direction of decreasing voltage).
That's it! Every element of the system has now been represented as a linear graph. This graph is now ready to be processed as a normal tree and then represented as a system of equations or transfer function. For more information, read up on how to draw normal trees, and how to develop elemental and constraint equations from a normal tree.
This process was conducted for an electrical problem, however the process of drawing linear graphs also applies to the mechanical (translational/rotational), thermal, and fluid domains.