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)