In previous sections, we introduced the electrical properties of capacitance, resistance and inductance that characterize the elementary elements of an electrical circuit, capacitors, resistors and inductors respectively. These elements have an impact on the electrical behavior of the circuit and influence its response, power dissipation, reliability and overall performance.
In order to study those effects, we need to introduce electrical models that estimate and approximate the real behavior of an electrical circuit as a function of those parameters. These models vary from very simple to very complex ones, depending upon the effects that are being studied and the required accuracy.
There are two ways to represent an electrical system by its components, the lumped and distributed system. The elements building a lumped system are considered concentrated at singular points in space. In contrast, the elements in distributed systems are considered distributed in space, so that physical quantities depend on both time and space. The more you distribute a system, the more you approach reality, however, this adds a lot of complexity.
For systems like touchscreens, that usually consist of cells, it makes sense to use distributed systems. Therefore, a touchscreen with 20×30 electrodes that consists of 600 cells and can be represented by a distributed system consisting of 600 concentrated (lumped) cells. This is also very consistent with the fact that the circuit parasitics are usually distributed along its length and are not lumped into a single position.
There are two common methods of showing an electrical network with its distributed components, the τ- and π- equivalent circuits, shown in figure 11.
Figure 11. π- and τ- equivalent circuits.
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In electrical engineering, layout, schematic diagram, equivalent electrical circuit and netlist are the most common ways to represent the different stages of the product development. Each serves its own role and can be used to communicate information or validate the product concept.
A schematic, or schematic diagram, represents the elements of a system with abstract symbols instead of realistic pictures. A schematic focuses more on comprehending and spreading information rather than doing physical operations. For this reason, a schematic usually omits all details that are not relevant to the information it intends to convey.
For example, a touchscreen consists of inductances, resistances and capacitances, but the inductances are omitted because they are negligible in the operation frequencies of the system.
In electronics, having a schematic diagram on hand may help a user design an entire circuit before building it, or troubleshoot an electronic that has stopped working. Schematic diagrams may also be used to explain the general way that an electronic functions without detailing the hardware or software used in the actual electronic.
Often a netlist of connections is produced from a schematic.
In comparison to a Schematic Diagram, in electrical engineering and science, an equivalent circuit refers to a realistic representation of the components of an electrical circuit and its aim is to represent the signal flow.
Figure 13. Example of equivalent circuit diagram.
In electronic design, a netlist is a description of the connectivity of an electronic circuit. In its simplest form, a netlist is a text file including a list of the electronic components in a circuit and a list of the nodes they are connected to. A network (net) is a collection of two or more interconnected components.
The structure, complexity and representation of netlists can vary considerably, but the fundamental purpose of every netlist is to convey connectivity information.
The physical design is a geometric description, that is through volumes, surfaces, lines. This geometric description is governed by layout design rules that specify the geometry. Design rules represent a tolerance which insures very high probability of correct fabrication. A drawing meant to depict the physical arrangement of the electrical components called layout and it is 2D (top view).
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In principle, a touch sensor consists of electrodes and traces inside layers of dielectrics. The electrodes and traces form the conductive parts of the sensor (conductors). Every electrode has a self capacitance towards infinite earth and a mutual capacitance towards other conductive parts, such as neighboring traces, electrodes, shielding entities, a metal casing and a finger approaching. The traces add to that capacitance as well, since every electrode is connected to one. Furthermore, every conductor has a total resistance, which is resulting from the individual resistance of the trace and the electrode.
Understanding that every conductor of a sensor has a self capacitance, a resistance and a mutual capacitance towards other conductors can enable a designer to prepare the schematic diagram of any touch sensor.
Let’s start the representation of a touch sensor by using a simple capacitive touch button. The touch button is conductive and is placed on top or inside dielectric layers. Depending on the material the conductor is made off, its shape and the general shape of the physical layout, the simple touch button will have a resistance and a self capacitance. If we consider it isolated, it means that no other conductive parts are in close vicinity to influence the capacitance value, and we can use a lumped model representation and the touch sensor will look like:
Figure 14. Simple button and its lumped model representation.
This of course is a simplified case, as normally the button also includes traces, maybe shielding entities and of course more buttons close by.
We can now try to expand the very simple model to a more complex one: the case of two buttons. For this new case, we have each button described with their resistance and capacitance, however, now there is capacitive coupling between the two buttons. This means that since the conductive parts are close to one another, they form a capacitor with a capacitance dependent on the shape, size and materials of the physical layout. This capacitance is called mutual capacitance, or because it is unwanted can sometimes be mentioned as parasitic capacitance. Again, we can use the lumped model representation for each button and add a mutual capacitance between them. Depending on many parameters, such as geometry, frequency, materials, position of buttons etc., the lumped schematic diagram might not be an accurate depiction of the physical design and the components might need to be distributed more.
Figure 15. Two buttons and their lumped model representation.
The case of three buttons is becoming even more complex, because there are more mutual capacitances introduced. Each button now has its self capacitance and its resistance, but also has a mutual capacitance with each other button:
Figure 16. Three buttons and their lumped model representation.
Naturally, Cself, Cmutual and R depend on the properties of the materials and the geometry and do not have an equal value from one button to the other.
It is worth mentioning here that if the geometry/shape of the conductive part changes, for example by simply adding the routing towards a microcontroller, this will add to its resistance and the capacitance of the equivalent circuit. It will also add a change in the mutual capacitive coupling to all other conductive parts.
Adding trace electrodes to the buttons will also change the schematic representation. Using the distributed and lumped systems a trace itself has its resistance and capacitance, distributed along its length and can therefore be represented in a similar manner as the single button. For simple geometries, using a lumped model to represent the trace and a lumped model to represent the button should work fine (figure 17). For more complex geometries, where parasitics are distributed across the sensor or the frequencies get higher, it makes more sense to use even more distributed models.
Figure 17. Simple button and its trace (lumped representation).
We can follow a similar process to model a touchscreen. If we isolate a single cell of a capacitive touchscreen, we have the self capacitances and resistances of the two electrodes of the grid, but we also have the mutual capacitance between them. This is the most common case, as capacitive touchscreens work with the mutual capacitive sensing method and they consist of transmitting and receiving electrodes.
Figure 18. Simple cell of a touchscreen.
We are going to use the distributed system for the whole screen and lumped components connected with the τ-equivalent for the representation of the cell. In order to do so, we split the resistance in half and use two resistors with half the original resistance of the cell on the left and right. In the middle of the circuit, we introduce the self capacitance. The two electrodes of the cell eventually look like Chapter 1, Figure 8 and have a capacitive coupling according to their geometrical properties. This capacitive coupling is their mutual capacitance. We introduce this capacitance exactly in the middle of the τ-equivalent. This way we remain very consistent with how touchscreen geometries look like, where transmitting and receiving electrodes overlap in the center of the cell they define. One touchscreen cell is represented by 2 τ-equivalent circuits.
Following the same principles, we can expand this model to larger screen sizes. For example, if we want to build a 2×2 screen, we need now to take into account the mutual capacitances of each electrode to every other electrode. A 2×2 screen consists of 4 cells and a total of 8 τ-equivalent parts (figure 19).
Figure 19. Using distributed cells for a touchscreen.
Each of those electrode parts will be represented by their resistances and self-capacitances and also by their mutual capacitance to all other electrodes. For example, in figure 21, we can see that X11 has a mutual capacitance with Y11, Y12, Y21, Y22 and X21, X22. Of course, there is no mutual capacitance to X12, since they are parts of the same electrode, X1. Depicting the mutual capacitances of the other nodes would complicate the drawing, so they are left to the reader’s imagination.
Figure 20. Lumped and distributed cell combinations to represent a touchscreen.
It is worth noticing that a full screen model consists of a very high number of Cmutual per node. Usually an experienced designer can simplify the Schematic and discard some of those values that have no real influence to the performance, because they are too low. A good example is electrodes that are far from each other. This is easily understandable if we read the equation of the capacitance introduced in Chapter 1 once again, where by increasing distance, the capacitance is reduced.
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