This is a particular case implying only three frequencies, but the solution, implying the inversion of a square matrix, is easily scalable.
You can move the frequencies (blue dots on the x-axis) and the charge values (blue dots Y1, Y2 or Y3 placed vertically on them). The app renders the propose3d solution (black dots joined by black segmnents). Observe that there is always at least one black dot restin on the x-plane (implying a dispensable branch) and the rest are all positive.
This means an acceptable solution.
The app begins with the resonant frequencies (red dots r1 and r2 on the x-axis) placed in the middle of the operating frequencies (blue dots w1, w2 and w3). If you move the blue dots only, the red dots keep in the midst of them. This is the initial proposal.
But you can move also the red dots in order to see how their position influences the solution. In some cases you will be able to place an additional black dot on the x-axis (this would mean that it is possible to eliminate not just one but two branches from the compensator.
To return to the initial middle-placed resonancies you can refresh the page.
As explained in the reference text, the elimination of a certain branch implies a condition which lies on a hyperplane in the compensation space (whose coordinates are the values of the compensator capacities).
In this app, this space is three-dimensional, and its hyperplanes are two-dimensional (sides). Their intersections define regions where not one, but two blanches are eliminated. These are lines (edges) whose intersections correspond to points (vertexes) where three branches are eliminated, leaving the compoensator with only one branch.
These sides, edges and vertexes are visible as planes, lines and points at the right side of the app, where the simplex appears in the form of a tetrahedron.
The selection of the charges is done in this case moving only one point. It is the blue point at the right side of the screen. It scrolls around the surface of a sphere (which can be seen by pressing a check button)..
You can observe that when a certain branch is eliminated, the compensation happens on the corresponding side of the simplex (the colors help to identify frequencies and sides). When you move the blue point until the compensatin line passes over a line, you can notice that two black points rest on the x-axis, and in some cases you will be able to find a solution where only a branch is needed (compensation line on a vertex of the simplex).
This allows to understand the underlying geometric features better
The simplex is now only a triangle. The sides of this triangle correspond to the solutions whetre one branch is absent. The vertexes arew the points where two branches could be eliminated, and the compensation would be possible using only one branch.
Move the blue point around the triangle to show all the possible charge configurations and their compensation.
You can click on the check buttons to see the whole figure, namely the "compensating ray" linking the origin and the blue point.
Reducing one more dimension we keep with only one frequency. This is a trivial case for compensation purposes, but we rely on it to show the basics of the linear programming procedure.
Move firrst the blue point to change the vector column of the matrix Then move the green point on the green projection line, and the red line (rejection hyperplane) moves accordingly.
The intersection with any of the axes from the compensation space shows the desired solution.