Lecture 2: Capacitors And Inductors

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Lecture 2: Capacitors and InductorsCapacitance:Capacitance (C) is defined as the ratio of charge (Q) to voltage (V) on an object: C Q/V Coulombs/Volt Farad Capacitance of an object depends on geometry and its dielectric constant. Symbol(s) for capacitors: A capacitor is a device that stores electric charge (memory devices).A capacitor is a device that stores energyQ2 CV 2E 2C2Capacitors are easy to fabricate in small sizes (µm), use in chipsHow to combine capacitance: capacitors in parallel adds like resistors in series:Ctot C1 C2 CiTotal capacitance is more than individual capacitance!K.K. Gan L2: Resistors and Capacitors1

capacitors in series add like resistors in parallel:11 11 Ctot C1 C2CiTotal capacitance is less than individual capacitance! Energy and Power in Capacitors How much energy can a "typical" capacitor store? Pick a 4 µF Cap (it would read 4 mF) rated at 3 kVE 12 CV 2 12 4 10 6 3000 2 18 J This is the same as dropping a 2 kg weight (about 4 pounds) 1 meter How much power is dissipated in a capacitor?dEPower dtd " CV 2 % 'dt # 2 &dVP CVdt dV/dt must be finite otherwise we source (or sink) an infinite amount of power!THIS WOULD BE UNPHYSICAL. K.K. GanL2: Resistors and Capacitors2

the voltage across a capacitor cannot change instantaneously a useful fact when trying to guess the transient (short term) behavior of a circuitthe voltage across a resistor can change instantaneously the power dissipated in a resistor does not depend on dV/dt:P I2R or V2/RWhy do capacitors come in such small values? Example: Calculate the size of a 1 Farad parallel capacitor with air between the plates.kε AC 0dk dielectric constant ( 1 for air)ε0 8.85 10 12 N 1m 2d distance between plates (assumed 1 mm) A area of plates 1.1 10 8 m 2 !!!square plate of 6.5 miles per sidebreakthroughs in capacitor technologies (driven by laptop/cell phone industries) 10 µF capacitor in 0402 package (1.0 mm 0.5 mm 0.5 mm) costs only 3 pennies How small can we make capacitors? A wire near a ground plane has C 0.1 pf 10-13 F.“Stray capacitance” slow down signal orinduces cross talk by capacitive coupling!K.K. GanL2: Resistors and Capacitors3

Some words to the wise on capacitors and their labeling. Typical capacitors are multiples of micro Farads (10-6 F) or picoFarads (10-12 F). Whenever you see mF it almost always is micro, not milli F and never mega F. picoFarad (10-12 F) is sometimes written as pf and pronounced puff. no single convention for labeling capacitors Many manufacturers have their own labeling scheme (See Horowitz and Hill lab manual).Resistors and Capacitors Examine voltage and current vs. time for a circuit with one R and one C. Assume that at t 0 all voltages are zero, VR VC 0. At t 0 the switch is closed and the battery (V0) is connected. Apply Kirchhoff's voltage rule:V0 VR VCQ IR CdQ Q R dt C Solve the differential equation by differentiating both sides of above equation:dV0 1 dQd 2Q R 2dt C dtdtWhy we care about the DC behavior? IdIWant to know how a circuit (characterize0 Ras R & C) responses to a pulse!CdtdII dtRCK.K. Gan L2: Resistors and Capacitors4

This is just an exponential decay equation with time constant RC (sec).The current as a function of time through the resistor and capacitor is:I(t) I0 e t /RCWhat's VR(t)? By Ohm's law:VR (t) I R R I0 R e t /RC V0 e t /RCAt t 0 all the voltage appears across the resistor, VR(0) V0.At t , VR( ) 0. K.K. GanL2: Resistors and Capacitors5

What's VC(t)? Easiest way to answer is to use the fact that V0 VR VC is valid for all t.VC V0 VR (VC V0 1 e t /RC )At t 0 all the voltage appears across the resistor so VC(0) 0.At t , VC( ) V0 . Suppose we wait until I 0 and then short out the battery. 0 VR VCVR VCdQQR dtCdQQ dtRCSolving the exponential equation yields,Q(t) Q0 e t /RCWe can find VC using V Q/C,VC (t) V0 e t /RCFinally we can find the voltage across the resistor using VR -VC,VR (t) V0 e t /RC K.K. GanL2: Resistors and Capacitors6

Suppose VC(t) V0sinωt instead of DC What happens to VC and IC?Q(t) CV (t) CV0 sin ωtIC dQ / dt ωCV0 cos ωt ωCV0 sin(ωt π /2) K.K. Gan current in capacitor varies like a sine wave too, but 900 out of phase with voltage.L2: Resistors and Capacitors7

We can write an equation that looks like Ohm's law by defining V*:V* V0 sin(ωt π /2) the relationship between the voltage and current in C looks like:V* IC / ωC IC R * 1/ωC can be identified as a kind of resistance, capacitive reactance:XC 1/ωC (Ohms) XC 0 for ω high frequencies: a capacitor looks like a short circuit “bypass capacitor” on power supply “short circuit” ripple (noise) to ground XC for ω 0 low frequencies: a capacitor looks like an open circuit (high resistance). look at AC output of a circuit via a capacitor (use AC coupling on scope)Inductance: Define inductance by: V LdI/dt Unit: Henry Symbol: Inductors are usually made from a coil of wire tend to be bulky and are hard to fabricate in small sizes (µm), seldom used in chips.Two inductors next to each other (transformer) can step up or down a voltage no change in the frequency of the voltage provide isolation from the rest of the circuitK.K. GanL2: Resistors and Capacitors8

10 Gb/s VCSEL (Laser) Array Driver 4-channel test chip (65 nm CMOS)2 mm240 µm Capacitance and inductance onlong transmission lines degrade signalL2: Resistors and Capacitors

How much energy is stored in an inductor?dE VdQdQI dtdE VIdtdIV LdtdE LIdIE L 0I IdIE 12 LI 2 How much power is dissipated in an inductor?dEPower dtd " LI 2 % 'dt # 2 &dI P LIdt dI/dt must be finite as we can’t source (or sink) an infinite amount of power in an inductor!THIS WOULD BE UNPHYSICAL. the current across an inductor cannot change instantaneously.K.K. GanL2: Resistors and Capacitors10

Two inductors in series: Apply Kirchhoff's Laws,V V1 V2dIdI L1 L2dtdtdI LtotdtLtot L1 L2 Li Inductors in series add like resistors in series.The total inductance is greater than the individual inductances. K.K. GanL2: Resistors and Capacitors11

Two inductors in parallel: Since the inductors are in parallel,V1 V2 VThe total current in the circuit isI I1 I2dI dI1 dI2 dt dt dtV V L1 L2V Ltot11 1 Ltot L1 L2LLLtot 1 2L1 L2 If we have more than 2 inductors in parallel, they combine like:11 LtotLi Inductors in parallel add like resistors in parallel.The total inductance is less than the individual inductances.K.K. GanL2: Resistors and Capacitors12

Resistors and Inductors Examine voltage and current versus time for a circuit with one R and one L. Assume that at t 0 all voltages are zero, VR VL 0. At t 0 the switch is closed and the battery (V0) is connected.Like the capacitor case, apply Kirchhoff's voltage rule:V0 VR VLdI IR Ldt Solving the differential equation, assuming at t 0, I 0:VI(t) 0 1 e tR /LR This is just an exponential decay equation with time constant L/R (seconds). What's VR(t)? By Ohm's law VR IRR at any time: VR I(t)R V0 1 e tR /L ()( At t 0, none of the voltage appears across the resistor, VR(0) 0.At t , VR( ) V0.K.K. Gan )L2: Resistors and Capacitors13

What's VL(t)? Easiest way to answer is to use the fact that V0 VR VL is valid for all t.VL V0 VR VL (t) V0 e tR /LAt t 0, all the voltage appears across the inductor so VL(0) V0.At t , VL ( ) 0. Pick L/R 1 msK.K. GanL2: Resistors and Capacitors14

Suppose VL(t) V0sinωt instead of DC, what happens to VL and IL?dIV L Ldt1I L 0t VdtLV 0 cos ωtωLVI L (t) 0 sin(ωt π /2)ωL The current in an inductor varies like a sine wave too, but 900 out of phase with the voltage. We can write an equation that looks like Ohm's law by defining V*:V* V0sin(ωt - π/2) V* ILωL ILR* ωL can be identified as a kind of resistance, inductive reactance:XL ωL (Ohms) XL 0 if ω 0 low frequencies: an inductor looks like a short circuit (low resistance). XL if ω high frequencies: an inductor looks like an open circuit. connect a circuit to the power supply via an inductor to filter out ripple (noise).Choke on power cordK.K. GanL2: Resistors and Capacitors15

Some things to remember about R, L, and C's. For DC circuits, after many time constants (L/R or RC): Inductor acts like a wire (0 Ω). Capacitor acts like an open circuit ( Ω). For circuits where the voltage changes very rapidly or transient behavior: Capacitor acts like a wire (0 Ω). Inductor acts like an open circuit ( Ω).Example, RLC circuit with DC supply: At t 0, voltages on R, C are zero and VL V0. At t , voltages on R, L are zero and VC V0.K.K. GanL2: Resistors and Capacitors16

VinputVInductorVcapacitorL2: Resistors and Capacitors

Two inductors in series: Apply Kirchhoff's Laws, Inductors in series add like resistors in series. V V 1 V 2 L 1 dI dt L 2 dI dt L tot dI dt L tot L 1 L 2 L i L2: Resistors and Capacitors The tot

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