ECE 201 Lecture 15 Borja Peleato Capacitance and capacitors 1 Capacitors In its simplest form a capacitor consists of two parallel metal plates separated by a dielectric barrier that does not allow the charges to go through If a current pumps positive charges onto one plate these charges will push the positive charges from the other plate away The number of charges entering one plate is the same as the number of charges leaving the other so we treat it as if the current were going THROUGH the capacitor KCL holds In practice the positive charges entering one plate accumulate there while mostly negative charges remain on the other plate This creates a voltage difference between the plates which keeps increasing while charges keep arriving If the current stops no more charges arrive the voltage remains constant Capacitors are represented by or 2 Capacitance The ratio between the amount of charge in the plates and the voltage that appears between them is known as Capacitance Symbol C Units Farads F Coulombs Volt A capacitor of 1F would store 1C of charge on each plate positive on one negative on the other when subject to a voltage of 1V In general the charge stored in a capacitor at time t is Q t C V t The capacitance of a capacitor depends only on its shape and materials not on the voltage or current it is subject to For example in a parallel plate capacitor 3 Voltage Current equations So far we have studied the following laws KCL KVL hold for everything R L C Ohm s law holds for resistors only V R I Inductor equations Now we study the corresponding equations for a capacitor Passive sign convention Observe C L parallelism replace v t by i t and C by L 4 Example Now we are ready to analyze circuits 5 Important properties A capacitor can sustain a constant voltage with zero current so for DC it behaves as an OPEN circuit for inductors it was the other way around they behaved like short circuits The integral equation shows that the VOLTAGE in a capacitor has to be CONTINUOUS unless it gets an infinite current for inductors it was the current that had to be continuous 6 Energy stored Just like inductors capacitors do not dissipate energy they just store it A charged capacitor will return all its charges to the circuit if needed The power absorbed or returned depending on the signs and convention for v t and i t is the same as always P t v t i t The energy stored in a given interval is Does not depend on the waveform just on initial and final voltages The instantaneous stored energy is 7 Example 8 9 Example 2 Before the switch closes the 1F capacitor is discharged and the 2F one has 6V across it How much energy in total will the resistor dissipate after the switch closes Solution Eventually both capacitors will need to have the same voltage lets call it Vf Also by conservation of charge the total charge stored in both of them has to remain constant Hence 2 6 Vf 2 Vf 1 so Vf 4V Before the switch closed we had 0 5 2 6 2 36J stored in the capacitor Once the voltages stabilize we are left with 0 5 2 4 2 0 5 1 4 2 24J of energy stored between both capacitors Consequently the amount of energy that the resistor has dissipated is 12J 10
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