RCL Series Circuit
Physics 202, Lecture 20
"V = "Vmax sin(#t + $ )
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
Impedance
"Vmax = "VR2 + ("VL # "VC ) 2
" = tan !1 (
I=Imaxsinωt, the same for all
three elements
X L ! XC
)
R
!
ΔvR=(ΔVR)max Sin(ωt)
ΔvL=(ΔVL)max Sin(ωt + π/2)
ΔvC=(ΔVC)max Sin(ωt - π/2)
ΔV
Z
 For general circuit configuration:
ΔV=ΔVmaxsin(wt+φ) , ΔVmax=Imax|Z|
Z: is called Impedance.
e.g. RLC circuit : Z =
= Imax R 2 + (X L # X C ) 2
!
Note: XL= ωL, XC=1/(ωC)
Summary of Impedances and Phases of
series circuts
i
R2 + ( X L − X C )2
 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)
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.
1
  e.g Series RLC:
2
2
2
Z = R + ( X L " X C ) = R + (!L "
!C
)2
1
•  when ω=ω0 =
(i.e. XL=XC )
LC
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
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Power in AC Circuit
General Waves (Review of Ch. 15)
  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)
Power Factor, max for R only
or at resonance for RLC
  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 Imax2R = ½ (ΔVmax)2/R =Irms2R =ΔVrms2/R
  Widely used definitions:
I rms =
  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.
I max
ΔVmax
, ΔVrms =
2
2
Example wave: 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:
v=
T
;
µ
µ"
Transverse
!m
!l
Transverse
Transverse
Direction of Propagation
Electro-Magnetic Waves are Transverse
E
B
z
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)
  Waves are described by wave functions in the form:
y
c
Wave Function
x
y: A certain physical quantity
e.g. displacement in y direction
(or electric and magnetic fields)
f: Can be any form
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
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Practical Technique:
Identify Wave Speed in A Wave Function

A wave function is in the form:
2
y ( x, t ) =
( x − 3.0t )2 + 1
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|
  The wave speed:
  3.0 m/s to the right
  Illustrate wave form at t = 0s,1s,2s
  Period: T = 2π/ω
  Frequency: f =ω/2π
  Angular frequency: ω
  Phase constant: -kx-φ
t=1
t=0
t=2
Sinusoidal Wave: Fixed T
Waves Transfer Energy
 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
 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:
P=
1 2 2
µA ω v
2
 Note: power dependence on A,ω,v
 We’ll find that EM waves can transfer energy also!
Linear Wave Equation
 Linear wave equation
certain
physical quantity
!2 y 1 !2 y
=
!x 2 v 2 !t 2
 Sinusoidal wave
f: frequency
Solution
y = A sin(
2"
x $ 2"ft + ! )
#
v=λf
k=2π/λ
ω=2πf
A:Amplitude
Wave speed
φ:Phase
λ:wavelength
General wave: superposition of sinusoidal waves
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)
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