EES 6A Designing Information Devices an Systems I Spring 27 Official Lecture Notes Note 3 Touchscreen Revisite We ve seen how a resistive touchscreen works by using the concept of voltage iviers. Essentially, for a resistive touchscreen, we map the value of analog voltag to the position touche. Another way to esign a touchscreen is to break the screen own into a bunch of pixels. At each one of the pixels, we can etect whether the finger is touching it or not. This allows the touchscreen to sense multiple touch points at a time. In orer to o this efficiently, we nee a new element: capacitors. The beauty of the capacitor is that you on t have to ben anything like we i for the resistive touchscreen the presence or absence of the finger can moify the capacitance irectly! apacitor A capacitor is two pieces of metal that are separate by some other material that is not in general conuctive. If there is voltage across the two pieces of conuctors, charges will buil up on the surface of the capacitor. As epicte below, when we apply voltage V across the two plates, positive charges buil up on the (bottom) surface of the plate connecte to the positive terminal an negative charges buil up on the (top) surface of the plate connecte to the negative terminal. + V + + + apacitance represents how much charge can be store for a given amount of voltage. We usually enote capacitance with. Let s represent voltage across the capacitor as V an the charge store on the capacitor EES 6A, Spring 27, Note 3
as Q. In mathematical form, we have the following efinition Q V, () This relation is often written as Q = V. (2) When we raw circuits, we usually use the two short parallel lines to represent a capacitor: The unit of capacitance is Fara (F). A capacitor with capacitance F is charge with of charge when V of voltage is applie across it. Let s try to get some intuition on capacitors by rawing an analogy between capacitors an water buckets simply put, we can think of a capacitor as a bucket. The more water (charge) you put in the bucket, the higher the water level (voltage) woul be. The water level in the bucket is etermine by its imensions. Similarly, the capacitance is set by the imensions of the conuctors an some properties of the material separating them. In particular, the capacitance of a capacitor is: = ε A (3) where A is the area of the surface of the plates facing each other, is the separation between the two plates (illustrate below), an ε is the "permitivity" of the material between the plates. We can euce that ε has unit F/m. For example, if the material between the capacitors is air (vacuum), then ε = 8.85 2 F/m. A Let s see what happens when we connect a capacitor with a voltage source. I + V S + V EES 6A, Spring 27, Note 3 2
We know that initially, charges are going to buil up on the two plates positive charges on the top plate an negative charges on the bottom plate. Once enough charge has been store, the voltage across the capacitor becomes V = V S ; as we will see next, in this next, the current I flowing through the circuit becomes zero. To unerstan why I = in this case, let s reexamine the efining relationship of a capacitor: Differentiating both sies with respect to t, we have Q = V. (4) Q t = V t = V t. (5) However, we know that current I = Q t. Hence, I = V t. (6) Thus, the current through a capacitor is just the prouct of the capacitance an the rate of change of the voltage across the capacitor. An important implication of this is in orer for current to flow through the capacitor, the voltage of that capacitor can change with time. When a capacitor is charge, there is voltage across it. What oes this mean? This means that there is energy store in the capacitor. Why is there energy store? Recall that like charges repel each other. So for example if we take a positive charge an move it closer to another positive charge, we nee to exert force an thus supply energy to o so. Let s now ask the question: when the capacitor is fully charge, how much energy is store in it? We know from previous lectures that the energy require to store an aitional q amount of charge when the voltage across the capacitor is V is We also know that q = V. With this in min, we have Integrating both sies, we have E E = V q. (7) E = V (V ) = V V. (8) VS VS E = V V = V V (9) E = 2 V 2 S. () Hence, the energy store in the capacitor after it s fully charge is 2 V 2 S. apacitors in series an capacitors in parallel Just like resistors, we can connect capacitors in series an in parallel. Let s take a look at what happens when we capacitors are connecte in series an in parallel. EES 6A, Spring 27, Note 3 3
apacitors in parallel Suppose we combine two capacitors in parallel as follows A A 2 2 The capacitor on the left has capacitance = ε A an the capacitor on the right has capacitance 2 = ε A 2 Intuitively, the equivalent capacitance for capacitors in parallel woul just be the sum of the capacitance of each of the capacitors since connecting them in parallel is analogous to summing the surface area of the two plates. If we look at the combine capacitor above, its capacitance is equal to which is equal to the sum of the capacitance of the two capacitors eq = ε A + A 2 eq = ε A + A 2, () = ε A + ε A 2 = + 2, (2) In general, the equivalent capacitance of capacitors in parallel is the sum of their capacitance. Thus, the following circuit. 2 can be reuce to: + 2 apacitors in series Now let s look at the case where capacitors are connecte in series. If we take the following capacitor an raw a horizontal line in between, then essentially we can view it as two capacitors connecte in series EES 6A, Spring 27, Note 3 4
2 2 Assume that the surface area is A. Intuitively, we know that the equivalent capacitance shoul be smaller than both an 2 since the plates are farther from each other. We know that = ε A an 2 = ε A 2. Let the capacitance of the combine capacitor be eq. It satisfies Hence, the equivalent capacitance is equal to In general, circuits with capacitors in series = + 2 = eq ε A ε A + 2 ε A = +. (3) 2 eq = + = 2. (4) + 2 2 2 can thus be reuce to 2 + 2 EES 6A, Spring 27, Note 3 5