PHY 202 (Blum)1 Capacitance

PHY 202 (Blum)2 What is a capacitor? A capacitor is an electronic component that stores electric charge (and electric energy) A simple version of a capacitor is the parallel-plate capacitor; it consists of two conducting plates separated by an insulating material –The insulating material is called a dielectric in this context.

PHY 202 (Blum)3 Parallel-plate capacitor dielectric I  conducting plates 2. Positive charges accumulates on first plate; dielectric does not let them through 1. Battery (not shown) pushes charges out 3. Neutral atoms on other plate are made up positive and negative charges 4. Negative charges are attracted and stay behind, while positive charges are repelled and move out giving rise to the current on other side

PHY 202 (Blum)4 We can’t go on like this dielectric I  conducting plates 2. There are positive charges on the first plate. Like charges repel. 1. There are positive charges coming from the battery 3. When the push from the battery is equal to the push back from the plate, the capacitor stops charging 4. The bigger the push from the battery (i.e. voltage), the more charge goes onto the plate. The voltage and charge are proportional.

PHY 202 (Blum)5 Q = VC When we talk about the charge on a capacitor, we mean the charge on one of the plates. The charge is proportional to the voltage: Q  V. The proportionality constant C is called the capacitance. Solving for V, the equation becomes V= Q C

PHY 202 (Blum)6 Farad The standard unit of capacitance is the farad (F). A farad is quite large, usually you see –millifarad mF (1 mF = F) –microfarad  F (1  F = F) –nanofarad nF (1 nF = F) –picofarad pF (1 pF = F)

PHY 202 (Blum)7 Capacitance Q=VC The capacitance is a measure of how easy it is to put charge on the plates, it is –directly proportional to the surface area of the plates, bigger plates can hold more charge –inversely proportional to the distance between the plates, the interaction between the positive and negative charges is greater when they are closer –dependent on the material (dielectric) separating the plates, having a good insulator between them is like their being further apart

PHY 202 (Blum)8 Capacitive keyboards Capacitive keyboards detect which key was pressed by looking at changes in capacitance. Under each key are two oppositely charged plates ( a capacitor). Pressing a key moves its upper plate closer to its lower plate, changing its capacitance, and hence changes the amount of charge the plates can hold for a given voltage. The keyboard circuitry detects this change and sends the appropriate information (interrupt request and ASCII code) to the CPU.

PHY 202 (Blum)9 DRAM Capacitors can be put on integrated circuit (IC) chips. Together with transistors, they are used in dynamic random access memory (D-RAM). The charge or lack thereof of the capacitor corresponds to a stored bit. Since these capacitors are small, their capacitance is pretty low. DRAM has to be recharged (refreshed) thousands of times per second or it loses its data.

PHY 202 (Blum)10 Power Supply Large capacitors are used in the power supplies of computers and peripherals. The capacitors (along with diodes) are used in rectifying: turning the alternating current (AC) into smoothed out direct current (DC). Capacitors do the smoothing part.

PHY 202 (Blum)11 Monitors Monitors (CRT’s) are another place that large capacitors are found. Since capacitors store charge, monitors and power supplies can be dangerous even when the power is off. –If you don’t know what you’re doing, don’t even open them up.

PHY 202 (Blum)12 Equivalent Capacitance When a combination of capacitors can be replaced by a single capacitor, which has the same effect as the combination, the capacitance of the single capacitor is called the equivalent capacitance. Having the same effect means that same voltage results in the same amount of charge being stored.

PHY 202 (Blum)13 Capacitors in parallel Recall that things in parallel have the same voltage And the charge is split between the capacitors. The charge has a choice, some will go onto one capacitor, the rest on the other. V C1C1 C2C2

PHY 202 (Blum)14 Capacitors in parallel Individual capacitors obey basic equation Q 1 = C 1 V and Q 2 = C 2 V Equivalent capacitor obeys basic equation Q total = C eq V The total charge is the sum of the individual charges Q total = Q 1 + Q 2 Solve basic equations for Q’s and substitute C eq V = C 1 V + C 2 V Divide by common factor V C eq = C 1 + C 2

PHY 202 (Blum)15 Capacitors in series Capacitors in series have the same charge –When the capacitors are uncharged, the region including the lower plate of C 1 and the upper plate of C 2 is electrically neutral. This region is isolated and so this remains true; the negative charge on the lower plate of C 1 when added to the positive charge on the upper plate of C 2 would give zero The voltage is split between them V C1C1 C2C2

PHY 202 (Blum)16 Capacitors in series Individual capacitors obey basic equation V 1 = Q 1 / C 1 and V 2 = Q 2 / C 2 Equivalent capacitor obeys basic equation V total = Q total / C eq The total voltage is the sum of the individual voltages V total = V 1 + V 2 Substitute basic equations into voltage equation Q total / C eq = Q 1 / C 1 + Q 2 / C 2 All of the charges are the same (Q total = Q 1 = Q 2 = Q), divide out the common factor 1 / C eq = 1 / C / C 2

PHY 202 (Blum)17 Example

PHY 202 (Blum)18 Example (Cont.) Something must take the brunt of the voltage when the battery is first connected, that’s why the 0.2-k  resistor is there. The 2.5-  F and 1.5-  F are in parallel, so they can be replaced with one 4.0-  F capacitor.

PHY 202 (Blum)19 Example (Cont.) The 3.5-  F and 4.0-  F capacitors are in series, so they can be replaced with one  F capacitor. The charge on it Q=CV would be 9.33  C

PHY 202 (Blum)20 Example (Cont.) Going back to the 3.5-  F and 4.0-  F capacitors are in series, capacitors in series have the same charge, so Q 3.5 = 9.33  C Then V 3.5 = Q 3.5 / C 3.5 or V 3.5 = 2.67 V And V 4.0 = Q 4.0 / C 4.0 or V 4.0 = 2.33 V (Note that V V 4.0 = 5)

PHY 202 (Blum)21 Example (Cont.) The 4.0-  F capacitor was really a 1.5-  F and a 2.5-  F in parallel Things in parallel have the same voltage, so V 1.5 = V 2.5 = 2.33 V Since Q 1.5 = V 1.5 C 1.5, Q 1.5 = 3.50  C Similarly, Q 2.5 = 5.83  C (Note Q Q 2.5 = 9.33  C)

PHY 202 (Blum)22 Testing in Electronics Workbench

PHY 202 (Blum)23 Two Cautions When the switches are closed, there is an easy path from one side of the capacitor to another, this makes sure that the capacitors are discharged. Even when the switches are open there is a path from one side to another for the upper capacitors, it is through the voltmeter. The voltmeter must be made extremely ideal (very high resistance) to get agreement with theory.

PHY 202 (Blum)24 RC circuits: Charging

PHY 202 (Blum)25 What happens When you connect an uncharged capacitor and a resistor in series to a battery, the voltage drop is initially all across the resistor. –Because the voltage drop across a capacitor is proportional to the charge on it and there is not charge on it at the beginning. But charge starts to build up on the capacitor, so some voltage is dropped across the capacitor now. –Capacitors have a gap and while current gets all the way around the circuit, individual charges are trapped on one side of the capacitor.

PHY 202 (Blum)26 What happens (Cont.) Since some voltage is dropped across the capacitor, less voltage (than before) is dropped across resistor. With less voltage being dropped across the resistor, the current drops off. –V = IR (smaller V  smaller I) With less current, the rate at which charge goes onto the capacitor decreases. The charge continues to build up, but the rate of the build up continues to decrease. In mathematical language, the charge as a function of time Q(t) increases but its slope decreases. Theory says the charge obeys Q(t) = C V (1 - e - t /  )

PHY 202 (Blum)27 Time Constant Q(t) = C V (1 - e - t /  )  in that equation is known as the “time constant” and is given by  = RC Note that since R = V / I, resistance has units volt/ampere Since C = Q / V, capacitance has units coulomb/volt RC = (V / I)  (Q / V) = Q / I Then RC has units coulomb/ampere but an ampere is coulomb/second RC = Q / (Q / T) = T So RC has units of second

PHY 202 (Blum)28 Charging Capacitor 11 63% 33 95%

PHY 202 (Blum)29 Voltage is similar If the charge on the capacitor varies according to the expression Q(t) = C V (1 - e - t /  ) then since the voltage across a capacitor is V=Q/C the voltage is V(t) = V (1 - e - t /  ) the voltage approaches its “saturation” value, which in a simple RC circuit in the battery’s voltage.

PHY 202 (Blum)30 Time Constant The time constant (  = RC) is the time required for a certain percentage (  63%) of the saturation charge (the charge after a very long time) to be put on the capacitor. If the resistance is large, the currents are small, even from the start and it requires more time to charge up the capacitor (   R) If the capacitance is large, the capacitor can take a lot of charge and again it requires more time to charge up the capacitor (   C)

PHY 202 (Blum)31 RC circuits: Discharging

PHY 202 (Blum)32 What happens This time we imagine that there are already charges on the capacitor, but there is no battery. With no battery to "push" the charges around, the opposite charges on the two capacitor plates would prefer to be together. They must pass through the resistor before they can reunite. With all those like charges on one plate, there is a strong incentive for charges to leave the plate.

PHY 202 (Blum)33 What happens (Cont.) However, as charges leave the plate, the voltage across the capacitor decreases (V = Q C) and the incentive for charges to leave the capacitor decreases, thus the rate at which charges leave decreases as well. In mathematical language, this time the charge as a function of time Q(t) decreases and its slope decreases. Theory says the charge obeys Q(t) = Q 0 e - t / . Same time constant as before.

PHY 202 (Blum)34 Discharging Capacitor 11 33 37% 5%