Thought ExperimentCapacitor ChargeCapacitor VoltageDefinition of CapacitanceCapacitor AnalogyCapacitor EnergyField LinesParallel Plate CapacitorPractical CapacitorsCapacitor Current Capacitor Current-Voltage RelationshipCapacitor Voltage-Current RelationshipSinusoidal DriveCircuits with Capacitors: Shunt ConnectionCircuits with Capacitors: Series ConnectionSeries CapacitorsKCL with CapacitorsNon-Linear CapacitorsCapacitors Everywhere!Can Capacitors Replace Batteries?EE 42/100Lecture 12: CapacitanceELECTRONICSRev B 2/22/2012 (9:59 PM)Prof. Ali M. NiknejadUniversity of California, BerkeleyCopyrightc 2012 by Ali M. NiknejadA. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 1/21 – p.Thought Experiment•Imagine a current source connected to an ideal short circuit as shown. Thewaveform for the current is shown. It’s a constant current I0for a time T , so a totalcharge of Q = I0· T circulates around but no net work is done since v(t) = 0(short circuit).•Now imagine that we break the conductor in two and leave a gap between theconductors. Let’s repeat the experiment by applying the same current waveform.•What happens? The current flow is interrupted but the same amount of charge Qleaves the positive terminals of the current source. Where does it go?A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 2/21 – p.Capacitor Charge•In addition to the positive charge Q leaving the positive terminal, the same amountof charge enters the negative terminal. Equivalently, a charge of −Q leaves thenegative terminal and goes into the conductors.•We see that the charge cannot go anywhere but into the conductors, and thereforethe charge is stored there.•Since like charges repel, we have to do work to force the charges to accumulateon the conductors. In fact, the smaller the conductor, the more work that we haveto do.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 3/21 – p.Capacitor Voltage•By definition, the potential v across the capacitor represents the amount of workrequired to move a unit of charge onto the capacitor plates. This is the work doneby the current source.•For linear media, we observe that as we push more charge onto the capacitor witha fixed current, it’s voltage increases linearity because it’s more and more difficultto do it (like charges repel).V ∝ QA. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 4/21 – p.Definition of Capacitance•The symbol for a capacitor is shown above. Sometimes a + label indicates that acapacitor should only be charged in a given direction. Most capacitors, though, aresymmetric and positive or negative charge can be applied to either terminals.•The proportionality constant between the charge and the voltage is defined as thecapacitance of the two terminal elementq = Cv•The units of capacitance are given by charge over voltage, or Farads (in honor ofMichael Faraday)[C] =[q][v]=CV= F•We expect that a physically larger conductor should heave a larger capacitanceC1> C2, because it has more surface area for the charges to reside. The averagedistance between the like charges determines how much energy you have toprovide to push additional charges onto the capacitor.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 5/21 – p.Capacitor Analogy•Imagine a tank of water where we pump water into the tank from the bottom. Aswe initially pump water, there is no water and it takes virtually no work. But as thetank fills up, it takes more and more work since we the liquid obtains gravitationalpotential energy.•A smaller tank requires more work (it has less capacity) because the liquid columngets higher and higher.•Notice that we can always recover the work by emptying the tank.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 6/21 – p.Capacitor Energy•The incremental of amount of work done to move a charge dq onto the plates ofthe capacitor is given bydE = vdq•where v is the potential energy of the capacitor in a given state. Since q = Cv, wehave dq = Cdv, ordE = Cvdv•If we now integrate from zero potential (no charge) to some final voltageE =ZV00Cvdv = Cv22˛˛˛˛V00=12CV20•This is the energy stored in the capacitor. Just like the water tank, it’s stored aspotential energy that we can later recover.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 7/21 – p.Field Lines•Since we get charge separation in a capacitor, we expect that field lines emanatefrom the positive charges to the negative charges. The energy of the capacitor isin fact stored in these field lines.•So there are two competing charge mechanisms in a capacitor. Like charges areforced to reside on the same plate, which requires energy. On the other hand,unlike charges are placed in close proximity, which have attractive forces. So wecan see that the charges should bunch up as close as possible to the charges ofopposite sign.•The smaller the gap spacing, the more capacity we have in a capacitor, becausenow the “unhappy" feelings of being cramped up to similar charge is somewhatalleviated by the “happy” feeling due to close proximity of unlike charges.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 8/21 – p.Parallel Plate Capacitor•From basic physics, it’s easy to show that the capacitance of a parallel platestructure is given byC =ǫAd•where A is the plate area, ǫ is the permittivity of the dielectric (also called thedielectric constant), and d is the gap spacing.•The permittivity of free space is ǫ0= 8.854 × 10−12F/m. Most materials have ahigher permittivity which is captured by the unitless relative permittivity ǫr= ǫ/ǫ0,with typical materials ǫ ∼ 1 − 10. For instance, air is mostly empty space and soǫr≈ 1.•Some materials, such as water, have polar molecules that align when an electricfield is applied. Thus the dielectric constant is very large.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 12 p. 9/21 – p.Practical Capacitors•Real capacitors are made of large sheets of conductors (to maximize surface area)and thin dielectric layers (to minimize the gap). A multi-layer sandwich structurecan then be wrapped together to form a large capacitor.•In integrated circuits, thin dielectrics and/or multiple metal fingers in close proximityform a high density
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