EE301 – CAPACITORS AND INDUCTORS

EE301 – CAPACITORS AND INDUCTORSLearning Objectivesa.b.c.d.e.f.g.h.Define capacitance and state its symbol and unit of measurementPredict the capacity of a parallel plate capacitorAnalyze how a capacitor stores charge and energyExplain and Analyze capacitor DC characteristicsDefine inductance and state its symbol and unit of measurementPredict the inductance of a coil of wireExplain Inductor DC characteristicsAnalyze how an inductor stores charge and energyCapacitors and Inductors For resistive circuits, the voltage-current relationships are linear andalgebraic. Resistors can only dissipate energy; they cannot store energy and return it to a circuit at a latertime. This is not the case for capacitors and inductors. Capacitors and inductors are dynamic elements. The voltage-current relationships are non-linear and differential. They are dynamic because they store energy.The CapacitorA capacitor is passive element designed to store energy in its electric field.This energy can then be provided to a circuit at a later time. Capacitorsoften consist of parallel plates separated by an insulator (or dielectric).Capacitors accumulate electric charge. In the figure below, suppose the plates are uncharged and theswitch is open. When the switch is shut, electrons are pulled from the top plate (creating a net positivecharge on the plate). Electrons are deposited on the bottom plate (creating a net negative charge).When charging a capacitor, the voltage developed across the capacitor willincrease as charge is deposited. Current goes to zero once the voltagedeveloped across the capacitor is equal to the source voltage12/6/2017

EE301 – CAPACITORS AND INDUCTORSDefinition of Capacitance The capacitance of a capacitor is a measure of the capacitor’s ability tostore charge. The relationship between the capacitance, the charge and the voltage is given byThe capacitance of a capacitor is one farad if it stores one coulomb of charge when the voltage across itsterminals is one volt.Q CV C Q( Farads )V1 Example: Suppose that when we apply 50 V to a capacitor we find that eventually 8 1014 electrons aredeposited on the negative plate. What is the capacitance of this capacitor?Solution:Effect of Surface Area Capacitance is increased proportional to the area of the plates, A. Thus:Increased surface area increased capacitance.Effect of Separation Between the Plates Reduced separation distance increased capacitanceEffect of Dielectric Substituting a dielectric material for the air gap will increase the CapacitanceDielectric constant is calculated by using the relative dielectric constant and the absolute dielectricconstant for vacuum: r 0 (F/m) 0 8.854 x 10-12 F/m22/6/2017

EE301 – CAPACITORS AND INDUCTORSCapacitance of Parallel-Plate Capacitor Putting the factors together, capacitance of parallel-platecapacitor is given byAA(farads, F)C o rddwhere r is the relative dielectric constant 0 8.854 10 12 F/mA is aread is the distance between plates2 Example: A capacitor consists of 2” by 2” plates separated by 1/32 of an inch. No dielectric is used.Determine the capacitance and determine the charge on each plate if V 48Volts.Solution:Dielectric Voltage BreakdownHigh voltage will cause an electrical discharge between the parallel plates(lightning strike!) Above a critical voltage, the force on the electrons is sogreat that they become torn from their orbit within the dielectric.This damages the dielectric material, leaving carbonized pinholes whichshort the plates together.The working voltage is the maximum operating voltage of a capacitorbeyond which damage may occur.This voltage can be calculated using the material’s dielectric strength K(kV/mm):Dielectric Strength (K)MaterialkV/mmAir3Ceramic (high 60V KdExample: What is the maximum voltage that can be applied across a mica capacitor when the plates areseparated by 2mm?Solution:32/6/2017

EE301 – CAPACITORS AND INDUCTORSCapacitance and Steady State DC In steady state DC, the rate of change of voltage is zero, thereforethe current through a capacitor is zero. A capacitor looks like an open circuit with voltage vc in steadystate DC.Capacitors in Series. Capacitors in series are combined in the same manner as resistor sin parallel.In the figure to the right:11 11 CT C1 C2 C3Capacitors in Parallel. Capacitors in parallel are combined in the same manner as resistors in series.In the figure to the right:CT C1 C2 C33 Example: Determine the total capacitance of the circuit shown below.Solution:42/6/2017

EE301 – CAPACITORS AND INDUCTORS4 Example: The circuit shown below has been connected for a long time. Determine the voltage acrossand the charge on the capacitor.Solution:Power and workThe energy (or work) stored in a capacitor under steady-state conditions is given byw 1 2Cv25 Example: For the circuit of the previous example, determine the energy stored be each capacitor.Solution:Introduction to Inductors An inductor is a passive element designed to store energy in its magneticfield. Inductors consist of a coil of wire, often wound around a core of high magnetic permeability.Self-induced voltage Voltage is induced across coil when iis changing. When i is steady state, the voltage across coilreturns to zero. From these observations: A voltage isinduced in a circuit whenever the flux linking the circuit ischanging and the magnitude of the voltage is proportional tothe rate of change of the flux linkages.emf d dt52/6/2017

EE301 – CAPACITORS AND INDUCTORSEMF in an InductorCounter EMF The induced voltage tries to counter changes in current. As a result, the current in aninductor changes gradually or smoothly. An inductor RESISTS the change of current in a circuit.Inductor Calculations Induced emf in an inductor can be calculated:e Ldidt(volts, V)L is called the inductance of the coil and the unit of L is the henry (H). The inductance of a coil is onehenry if changing its current at 1 A/sec induces a potential difference of one volt across the coil. In termsof an inductor’s physical characteristics, we can determine the inductance from the formula:L N2AlIn the formula, 0 r where 0 is the permeability of air and is equal to 4 10 7 .6 Example: For the inductor shown below (where the core is just air), determine the inductance.Solution:62/6/2017

EE301 – CAPACITORS AND INDUCTORSInductors in Series and Parallel. Inductors in series are combined in the same way as resistors in series.7 Example: Simplify the network shown below.Solution:Inductance and Steady State DC In steady state DC, the rate of changeof current is zero, therefore the induced voltage across an inductor is alsozero. An inductor looks like a short circuit in steady state dc.Inductor Energy Storage An ideal inductor does not dissipate power.The power applied to an inductor is p v i, and previously we determined v L di/dt, thustt00W pdt L iidi1dt L idi Li 20dt2(J)8 Example: The circuit shown below has been in operations for a long time. Determine the current I1 .Solution:72/6/2017

EE301 – CAPACITORS AND INDUCTORS9 Example: The circuit shown below has been in operations for a long time. Determine the energy storedby the inductor.Solution:10 Example: Determine the current I L and the voltage VC for the circuit shown below.Solution:82/6/2017

Inductors in Series and Parallel. Inductors in series are combined in the same way as resistors in series. Example: Simplify the network shown below. Solution: Inductance and Steady State DC In steady state DC, the rate of change of current is zero,