Download Waves & Oscillations Physics 42200 Spring 2013 Semester

Physics 42200
Waves & Oscillations
Lecture 10 – French, Chapter 4
Spring 2013 Semester
Matthew Jones
Forced Oscillators and Resonance
+
+
=
• Natural oscillation frequency:
=
• Amplitude of steady-state oscillations:
⁄
=
−
cos
=
+
• Phase difference:
= tan
−
−
2
Resonance Phenomena
• Change of variables:
=
•
Natural oscillation frequency:
=
•
=
Amplitude of steady-state oscillations:
=
•
Strong damping force
Strong damping force
=
−
/
−
Phase difference:
= tan
2
+
1/
−
=
1
1−
/
2
1
large
small
Resonance Phenomena
• Steady state amplitude:
( ⁄ )
Q=5
Q=4
Q=3
Q=2
Q=1
Peak is near ⁄
/
≈ 1. The peak occurs at exactly ⁄
= 1.
Average Power
• The rate at which the oscillator absorbs energy is:
'(( ) =
'(( )
/2
2
−
1
+
1
= 10
Full-Width-at-Half-Max:
=5
,-. =
=3
=1
/
=
Examples
2
• Resonant RLC circuit:
0(1)
/
3
ℰ cos
67
5 +7
6
1 ;
8+ : 7
9
6
The transformer does not play a role in the analysis of the circuit. It is just
a convenient way to isolate the driving voltage source from the part of the
circuit that oscillates.
Resonant Circuit
67
1 ;
5 + 7 8 + : 7 6 = ℰ cos
6
9
• Differentiate once:
6 7
67 1
5
+8 + 7
= −ℰ sin
6
6
9
• Redefine what we man by “ = 0”:
6 7
67 1
5
+8 + 7
= ℰ cos
6
9
6
• Change of variables:
6 7
67
ℰ
+
+
7
=
cos
6
6
5
Resonant Circuit
6 7
67
ℰ
+
+
7
=
cos
6
6
5
• Amplitude of steady state current oscillations:
/
=ℰ 9
1
−
+
• Voltage across the capacitor:
=
=7
>? =
7
/
9
• Amplitude of voltage oscillations measured across C:
/
=
=ℰ
/
1
−
+
Actual Data
Voltage
• If we know that 5 = 235@-, can we estimate 8 and 9?
5 = 235@-
Peak voltage is at
AC DE = 290 -G
C DE = 2B AC DE
A = /2B
[Mhz]
Energy
• Energy stored on a capacitor is H = 9=
• The graph of the stored energy is proportional to the
square of the voltage graph.
• If we defined 2∆ as the FWHM on the graph of
power vs frequency, then it will correspond to 1/ 2
of the peak voltage.
Voltage
Resonant Circuit
2∆A = 110 -G
= 2B ∆A = 6.91 × 10L M
5 = 235@-
But
A=
/2B
Peak position is
[Mhz]
C DE
=
= 8/5 so we can find 8:
8 = 5 = 6.91 × 10L M
= 162Ω
O?
−
PQ
9 = 1.20nF
9=
O
235@-
Q XQ
RSTUV W Q
Lifetime of Oscillations
• Amplitude of a damped harmonic oscillator:
= Z P;/ cos
• Maximum potential energy:
1
[=
∝ Z P;
2
• After time = 1/ , the energy is reduced by the
factor 1/Z.
• We call ] = 1/ the “lifetime” of the oscillator.
Other Resonant Systems
Other Resonant Systems
Other Resonant Systems
Wire Bond Resonance
• Wire bonds in a magnetic field:
^
e
_
Lorentz force is ` = a b 6ℓ × d
The tiny wire is like a spring.
A periodic current produces
the driving force.
Wire Bond Resonance
Resonance in Nuclear Physics
• A proton accelerated through a potential
difference = gains kinetic energy f = Z=:
=
h
Phys. Rev. 75, 246 (1949).
Resonance in Nuclear Physics
• In quantum mechanics, energy and frequency
are proportional:
H=i
• A given energy corresponds to a driving force
with frequency .
• When a nucleus resonates at this frequency,
the proton energy is easily absorbed.
Nuclear Resonance
“Lifetime” is
defined in terms of
the width of the
resonance.
Resonance
• Resonances are the main way we observe fundamental
particles.