class 13 - impedance in ac crcuitsclass 13 physics 106 learning outcomes inductors and...

Class 13

Physics 106

LearningOutcomes

Inductors andSelf-Inductance

AC CircuitTerminology

Resistors,Capacitors, andInductors in ACCircuits

The Series RLCCircuit

Phasors

Phasors in theSeries RLCCircuitImpedance

Power

Class 13Impedance in AC Crcuits

Physics 106

Winter 2019

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Class 13

Physics 106

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Phasors


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Learning Outcomes

Last time weI learned about generatorsI looked at the details of a motional emf problemI reviewed for Exam #1

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Learning Outcomes

Today we will discuss:I Inductors and self-inductanceI RL Circuit ProblemsI AC Circuit TerminologyI R, L, and C in AC Circuits

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Self-inductance

I A coil of wire in a circuit is called an inductor. Itproduces a magnetic field inside itself.

I If the current is changing, it produces an EMFbecause its own B field is changing in time.

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Self-inductance

I If current is increasing, the inductor produces aninduced current that opposes the current in thecircuit.

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Self-inductance

I If current is decreasing, the inductor produces aninduced current in the direction of the current in thecircuit.

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Self-inductance

I The faster the current changes, the larger the is theinduced EMF

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Self-inductance

I We define the inductance L of a coil by therelationship:

� = −L∆I∆t

I The negative sign indicates that a changing currentinduces an emf in opposition to that change

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Self-inductance

I The SI unit of self-inductance is the Henry (H)

I A useful expression for L is:

L = NΦBI

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RL Circuits

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Inductors in a Circuit

I If an inductor, a resistor, a battery, and a switch areplaced in in a series circuit:I Just after the switch is closed, the inductor

produces an emf that completely stops the flowof current− but only instantaneously

I After a long time, the inductor acts like a wire.

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RL Circuit

I As the currentgets larger, itchanges less, andthe inductor’svoltagedecreases.

I The timeconstant, τ , for anRL circuit is thetime required forthe current in thecircuit to reach63.2% of its finalvalue

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RL Circuit

I The time constant depends on R and L

τ =LR

I The current at any time can be found by

I =�

R

(1− e−t/τ

)

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Energy Stored in a Magnetic Field

I An inductor stores energy in its magnetic field.

UL =12

LI2

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RL Circuit Example

I A 3.0 Ω resistor, a 12 mH inductor, and a 12 V batteryare connected in series. A switch is closed at t = 0.

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AC Circuit Terminology

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AC Circuit

I The output of an AC generator is sinusoidal andvaries with time according to the following equation

I V (t) = Vmax sin2πftI V (t) is the instantaneous voltageI Vmax is the maximum voltage of the generatorI f is the frequency at which the voltage changes,

in HzI The period is T = 1/f

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Average Voltage

I V (t) = Vmax sin2πftI The average value of sine is 0I But that’s not useful

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Average Voltage

I We could try Vmax | sin 2πft |I But absolute values are awkward functions.

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Average Voltage

I Try V 2max sin2 2πft

I The average value of sin2 is 1/2

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Average Voltage

I Try V2max2

I But this is the average value of V 2

I So we take the square root of thisI This is called the root-mean-square or "rms" voltage.

Vrms =Vmax√

2

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rms Values

I rms values are the most practical effective values forAC circuits

I We can have rms voltages or currentsI Our outlets have 110-120 V rms.

Vrms =Vmax√

2

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AC Circuit

I rms values will be used when discussing AC currentsand voltagesI AC ammeters and voltmeters are designed to

read rms valuesI Many of the equations will be in the same form

as in DC circuits

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Resistors, Capacitors, andInductors in AC Circuits

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Resistor in an AC Circuit

I Consider a circuitconsisting of an ACsource and a resistor

I The graph shows thecurrent through andthe voltage acrossthe resistor

I The current and thevoltage reach theirmaximum values atthe same time

I The current and thevoltage are said tobe in phase

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Ohm’s Law in an AC Circuit

I Ohm’s Law for a resistor, R, in an AC circuit

VR,rms = IrmsR

or

VR,max = ImaxR

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Capacitors and Inductors in AC Circuits

I Ohm’s Law for a capacitor or inductor in an AC

circuit:

VC,rms = IrmsXCVL,rms = IrmsXL

I The effective resistance is called "reactance"I Remember ELI the ICE man

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Capacitors in an AC Circuit

I The voltageacross thecapacitor lagsbehind thecurrent by 90◦

I I in a capacitor,C, reaches itsmaximumbefore the EMF

I Positive voltageopposescurrent flow, asa resistor.

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Capacitive Reactance and Ohm’s Law

I The effective resistance in a capacitor is called

capacitive reactance,

XC =1

2πfCI The SI unit of XC is ohmsI Ohm’s law for a capacitor in an AC circuit is

VC,rms = IrmsXC

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Capacitive Reactance and Ohm’s Law

I A capacitor has a large effective resistance when itbuilds up a large voltage to oppose current flow.

I Large capacitances have small voltages, V = Q/C.I If the frequency is large, the capacitor doesn’t have a

chance to build up large voltages.

XC =1

2πfC

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Inductors in an AC Circuit

I The current isaffected by theemf of theinductor

I The voltageacross theinductor alwaysleads thecurrent by 90◦

I EMF in aninductor, L,reaches itsmaximumbefore I

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Inductive Reactance and Ohm’s Law

I The effective resistance in an inductor is called

inductive reactance,

XL = 2πfL

I The SI unit of XC is ohmsI Ohm’s law for an inductor in an AC circuit is

VL,rms = IrmsXL

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Inductive Reactance and Ohm’s Law

I An inductor has a large effective resistance when ithas a large emf.

I Large inductances have large emf’s.I If the frequency is large, the inductor produces a

large emf

XL = 2πfL

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The Series RLC Circuit

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The RLC Series Circuit

I A resistor,inductor, andcapacitor can becombined with anAC power supplyin a series circuit

I The current in thecircuit is the sameat any time andvaries sinusoidallywith time

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Current and Voltage Relationships in anRLC Circuit

I The instantaneousvoltage across theresistor is in phasewith the current

I The instantaneousvoltage across theinductor leads thecurrent by 90◦

I The instantaneousvoltage across thecapacitor lags thecurrent by 90◦

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I The voltages acrossthe R, L, and C addto equal the voltageof the power supplyat all times

I Currents are thesame everywhere

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Summary of Circuit Elements, Impedanceand Phase Angles

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Phasors and the SeriesRLC Circuit

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I When we add currentand voltages in ACcircuits, we have touse the values at onespecific time.

I The hard way is towrite sine and cosinefunctions and usetrig identities.

I The easy way is touse "phasors."

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Representing an AC Current

I If a current varies sinusoidally, we usually represent itgraphically.

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Representing an AC CurrentI But we can also represent it with an arrow that goes

up and down in time.

Click on the image to start the animation. Click on the forward arrow to move to the next slide as the down key may not be functional.

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Representing an AC CurrentI An arrow that goes up and down sinusoidally is just

the y component of a vector that rotates at aconstant angular frequency.

I The length of the arrow is i0 and the angle of thevector is θ = 2πft .


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Phasors

I A phasor is a vector that represents a sinusoidalfunction. Its magnitude is the amplitude of the sinewave (it’s maximum value) and its angle is the phaseangle (the argument of the sine function).

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Phasor Diagrams

I We represent thecurrent or voltageacross eachelement as aphasor

I The diagram iscalled a phasordiagram

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Phasor Diagram for RLC Series Circuit

I LetI = Imax sin(2πft).

I At time t = 0, thecurrent is 0.

I Since the currentis they -component ofthe phasor, attime t = 0, thecurrent phasor isin the x direction.

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Phasor Diagram for RLC Series Circuit

I The voltage acrossthe resistor is on the+x axis since it is inphase with thecurrent

I The voltage acrossthe inductor is on the+y axis since it leadsthe current by 90◦

I The voltage acrossthe capacitor is onthe −y axis since itlags behind thecurrent by 90◦

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Resistor Phasors in Time


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Inductor Phasors in Time


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Capacitor Phasors in Time


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Phasors in the SeriesRLC Circuit

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Circuit Rules for AC Circuits

If two circuit elements are in series:

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1)their current phasors are the same.

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1) their current phasors are the same.2) their voltage phasors add to give the total voltage.

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Phasor Diagram

I The phasors areadded as vectors toaccount for thephase differences inthe voltages

I VL and VC are inopposite directionsline and so the net ycomponent isVL − VC

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Phasor Diagram

I VR is on the x axis,so it is at right anglesto VL − VC

I Vmax is the vectorsum of VL − VC andVR

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Impedance of a Circuit

I Divide eachside of thevoltage triangleby the current

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I Divide eachside of thevoltage triangleby the currentVRI

= R

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I Divide eachside of thevoltage triangleby the current

VL − VCI

= XL−XC

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I Divide eachside of thevoltage triangleby the currentVmax

I= Z

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I The anglebetween thepower supplyvoltage and thecurrent is

φ = tan−1XL − XC

R

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Impedance and Ohm’s Law

I Ohm’s Law can be applied to the impedanceI Vmax = ImaxZI This can be regarded as a generalized form of Ohm’s

Law applied to a series AC circuit

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Summary of Circuit Elements, Impedanceand Phase Angles

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Power in an AC Circuit

I No power losses are associated with pure capacitorsand pure inductors in an AC circuit

I In a capacitor, during one-half of a cycle energy isstored and during the other half the energy isreturned to the circuit

I In an inductor, the source does work against theback emf of the inductor and energy is stored in theinductor, but when the current begins to decrease inthe circuit, the energy is returned to the circuit

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Power Dissipated by a Resistor

I The average power dissipated in resistor in an AC

circuit carrying a current I is

Pave = I2rmsR

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Power in an AC Circuit

I The average powerdelivered by thegenerator isconverted to internalenergy in the resistor

I Pav = IrmsVR =IrmsVsource cosφ

I cosφ is called thepower factor of thecircuit

Learning OutcomesInductors and Self-InductanceAC Circuit TerminologyResistors, Capacitors, and Inductors in AC CircuitsThe Series RLC CircuitPhasorsPhasors in the Series RLC CircuitImpedancePower

class 13 - impedance in ac crcuitsclass 13 physics 106 learning outcomes inductors and...

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