Physics 202, Lecture 20
Current And Voltages in a Series RLC Circuit
ΔvR=(ΔVR)max Sin(ωt)
ΔvL=(ΔVL)max Sin(ωt + π/2)
ΔvC=(ΔVC)max Sin(ωt - π/2)
Today’s Topics: Finishing up Chapt.33:
  Power in RLC
  Resonance in RLC
Moving on Chapt.34:
  Wave Motion (Review ch. 16)
  General Wave
  Transverse And Longitudinal Waves
  Wave Function
  Wave Speed
  Sinusoidal Waves
  Wave and Energy Transmission
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 )
 lowest impedance  largest current resonance
 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 
i= Imax Sin(ωt)
ΔVmax Sin(ωt +φ)
ΔVL - ΔVC
XL= ωL, XC=1/(ωC)
Also useful: cosφ= ImaxR/ΔVmax
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)
  i(t) = Imax sin(ωt - φ)
 P(t)= Imax sin(ωt - φ) *ΔVmax sin(ωt) = Imax ΔVmax sin(ωt - φ) sin(ωt)
 Paverage = ½ Imax * ΔVmax * cos(φ)
Power Factor
  For resistor: φ =0  Paverage = ½ Imax * Δvmax
  For inductor: φ=π/2  Paverage = ½ Imax * ΔVmax * cos(π/2) = 0 !
  For Capacitor: φ= - π/2  Paverage = ½ Imax * ΔVmax * cos(π/2) = 0 !
Ideal inductors and capacitors NEVER consume energy!
Only resistors do !
  For a general AC circuit:
Pave = ½ Imax * ΔVmax * cos(φ) = ½ Imax2R = ½ (ΔVmax)2 R/Z2
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Notes About rms Values
 Widely used “root-mean-squared” values:
 rms values are used when discussing alternating
currents and voltages because
  AC ammeters and voltmeters are designed to read
rms values.
  Many of the equations that will be used have the
same form as their DC counterparts.
Pave = ½ Imax2R = Irms2R
Section 33.2
General Waves (Review of Ch. 16)
  Wave:
Propagation of a physical quantity in space over time
q = f(x, t)
  Examples of waves:
Mechancical:
Water wave, wave on string, sound wave,
earthquake wave ….
Electromagnetic wave, “light”
 Some waves need medium to go (mechanical);
Some don’t (E & M)
 Waves can be
transverse: vibration perpendicular to the propagation
longitudinal: vibration along the propagation.
Wave On A Stretched Rope
Seismic Waves
Direction of Propagation
  It is a transverse wave
Longitudinal
  The wave speed is determined
by the tension and the linear
density of the rope:
Transverse
Transverse
Transverse
Direction of Propagation
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Electro-Magnetic Waves are Transverse
E
B
Wave Function
  Waves are described by wave functions in the form:
y
c
z
y: A certain physical quantity
e.g. displacement in y direction
f: Can be any form
x
y(x,t) = f(x-vt)
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
Two polarizations possible (will explain next week)
Identify Wave Speed in A Wave Function

If a wave function is in the form:
where t in seconds, x in m, what is the wave speed?
  Answer: 3.0 m/s to the right
  Illustrate wave form at t = 0s,1s,2s
t=0
t=1
Sinusoidal Wave
  A wave describe a function y=Asin(kx-ωt+φ)
is called sinusoidal wave (Harmonic wave).
  Check its wave speed (v=ω/k)
  At each fixed position x, the element is undergoing
a harmonic oscillation.
What are the amplitude, (angular) frequency,
and phase of this oscillation? (|A|, ω, -kx-φ)
  If we fix t and take a snap shot, the wave form is
periodic over space (x).
The spacing period λ is called
wavelength and λ=2π/k
t=2
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Parameters Summary:
 As motion in propagating in the form of wave in a
medium, energy is transmitted.

Snapshot:Fixed t
Go
a
 Note: power dependence on A,ω,v
od
co to
ur
se kno
re w
qu bu
ire t
me
nt
 It can be shown that the rate of energy transfer by a
sinusoidal wave on a rope is:
Snapshot:Fixed x
no
t
Snapshot with fixed t:
wave length λ=2π/k
  Snapshot with fixed x:
angular frequency =w
frequency f =ω/2π
Period T=1/f
Amplitude =A
  Wave Speed v=ω/k
 v=λf, or
 v=λ/T
  Phase angle difference
between two positions
Δφ =-kΔx
Waves Transfer Energy
Linear Wave Equation
 Linear wave equation
certain
physical quantity
 Sinusoidal wave
f: frequency
φ:Phase
Solution
A:Amplitude
Wave speed
λ:wavelength
v=λf
k=2π/λ
ω=2πf
General wave: superposition of sinusoidal waves
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