Physics 202, Lecture 20

Today’s Topics   Power in RLC   Resonance in RLC   Wave Motion (Review ch. 15)

  General Wave   Transverse And Longitudinal Waves   Wave Function   Wave Speed   Sinusoidal Waves   Wave and Energy Transmission

!

"Vmax = "VR2 + ("VL #"VC )2

= Imax R2 + (XL # XC )2

)(tan 1

RXX CL != !"

Note: XL= ωL, XC=1/(ωC)

!

"V = "Vmax sin(#t + $)

ΔvR=(ΔVR)max Sin(ωt) ΔvL=(ΔVL)max Sin(ωt + π/2) ΔvC=(ΔVC)max Sin(ωt - π/2)

RCL Series Circuit

I=Imaxsinωt, the same for all three elements

Impedance  For general circuit configuration:   ΔV=ΔVmaxsin(wt+φ) , ΔVmax=Imax|Z|   Z: is called Impedance.

  e.g. RLC circuit :

  In general impedance is a complex number. Z=Zeiφ. (the above result is specific to a RLC series circuit) The impedance in series and parallel circuits follows

the same rule as resistors. Z=Z1+Z2+Z3+… (in series) 1/Z = 1/Z1+ 1/Z2+1/Z3+… (in parallel) (All impedances here can be complex numbers)

ΔV

i

22 )( CL XXRZ −+=

Z

Summary of Impedances and Phases of series circuts

Comparison Between Impedance and Resistance

Resistance Impedance Symbol R Z

Application Circuits with only R Circuits with R, L, C Value Type Real Complex: Z=|Z|eiφ

I - ΔV Relationship ΔV=IR ΔV=IZ, ΔVmax=Imax|Z| In Series: R=R1+R2+R3+… Z=Z1+Z2+Z3+… In Parallel: 1/R=1/R1+1/R2+1/R3+… 1/Z=1/Z1+1/Z2+1/Z3+…

Resonances In Series RLC Circuit  The impedance of an AC circuit is a function of ω.

  e.g Series RLC:

•  when ω=ω0 = (i.e. XL=XC ) à lowest impedance à largest currentè resonance à Same as the phase of LC circuit in harmonic

oscillation  For a general AC circuit, at resonance:

  Impedance is at lowest   Phase angle is zero. (I is “in phase” with ΔV)   Imax is at highest   Power consumption is at highest

2222 )1()(C

LRXXRZ CL !! "+="+=

LC1

Power in AC Circuit   Power in a circuit: P(t)= i(t)ΔV(t) true for any circuit, AC or DC   In an AC circuit, current and voltage on any component can be written

in general:   ΔV(t)= ΔVmax sin(ωt + φ)   i(t) = Imax sin(ωt )  P(t)= Imax sin(ωt) *ΔVmax sin(ωt + φ) = Imax ΔVmax sin2(ωt ) cos(φ)  Paverage = ½ Imax * ΔVmax * cos(φ)

 For resistor: φ =0 à Paverage = ½ Imax * Δvmax

 For inductor: φ=π/2 à Paverage = ½ Imax * ΔVmax * cos(π/2) = 0 !  For Capacitor: φ= - π/2 à Paverage = ½ Imax * ΔVmax * cos(π/2) = 0 ! (Ideal inductors and capacitors NEVER consume energy!)  For an AC circuit at resonance: Pave = ½ Imax * Δvmax = ½ Imax

2R = ½ (ΔVmax)2/R =Irms2R =ΔVrms

2/R

  Widely used definitions:

2

ΔVΔV , 2

II maxrms

maxrms ==

Power Factor, max for R only or at resonance for RLC

General Waves (Review of Ch. 15)   Wave: Propagation of a physical quantity in space over time q = q(x, t)   Examples of waves: Water wave, wave on string, sound wave, earthquake

wave, electromagnetic wave, “light”, quantum wave….

 Waves can be transverse or longitudinal.

Example wave: Stretched Rope

  It is a transverse wave

  The wave speed is determined by the tension and the linear density of the rope:

lmTv!!

"= µµ

;

Seismic Waves

Longitudinal

Transverse

Transverse

Transverse

Dire

ctio

n of

Pro

paga

tion

Direction of Propagation

Electro-Magnetic Waves are Transverse

x

y

z

E

B c

A changing magnetic field can cause an electric field (this electric field was the source of the Induced EMF)

A changing electric field also cause an magnetic field (next chapter)

Wave Function   Waves are described by wave functions in the form:

y(x,t) = f(x-vt)

y: A certain physical quantity e.g. displacement in y direction (or electric and magnetic fields)

f: Can be any form

x: space position. Coefficient arranged to be 1

t: time. Its coefficient v is the wave speed v>0 moving right v<0 moving left

Practical Technique: Identify Wave Speed in A Wave Function

  A wave function is in the form:   The wave speed:

  3.0 m/s to the right

  Illustrate wave form at t = 0s,1s,2s

1)0.3(2),( 2 +−

=tx

txy

t=0 t=1 t=2

Sinusoidal Wave: Fixed X

  A wave describe a function y=Asin(kx-ωt+φ) is called sinusoidal wave. (Harmonic wave)

  The wave speed: v=ω/k   At each fixed position x,

  Amplitude: |A|

  Period: T = 2π/ω   Frequency: f =ω/2π   Angular frequency: ω

  Phase constant: -kx-φ

Sinusoidal Wave: Fixed T  Wave: y=Asin(kx-ωt+φ)  Snapshot with fixed t:

  Amplitude: |A|   Wave length: λ=2π/k

 Wave Speed v=ω/k à v=λf, or à v=λ/T

Waves Transfer Energy  As motion in propagating in the form of wave in a

medium, energy is transmitted.

  It can be shown that the rate of energy transfer by a sinusoidal wave on a rope is:

 Note: power dependence on A,ω,v  We’ll find that EM waves can transfer energy also!

vAP 22

21

ωµ=

Linear Wave Equation  Linear wave equation

2

2

22

2 1ty

vxy

!!

=!!

certain physical quantity

Wave speed

 Sinusoidal wave

)22sin( !"#"

+$= ftxAy

A:Amplitude

f: frequency φ:Phase

General wave: superposition of sinusoidal waves

v=λf k=2π/λ ω=2πf

λ:wavelength

Solution

EM waves will obey such a wave equation. Time changing current à Time changing Magnetic field à

Time changing Electric field (dB/dt) à Time changing Magnetic Field (dE/dt or d2B/d2t)