wdgl eagle 98.1

Physics 2102
Jonathan Dowling
Lecture 29: WED 25 MAR 09
Ch. 31.1–4: Electrical Oscillations, LC
Circuits, Alternating Current
EXAM 03: 6PM THU 02 APR 2009
The exam will cover:
Ch.28 (second half) through
Ch.32.1-3 (displacement current,
and Maxwell's equations).
The exam will be based on:
HW08 – HW11
Final Day to Drop Course: FRI 27 MAR
What are we going to learn?
A road map
• Electric charge
 Electric force on other electric charges
 Electric field, and electric potential
• Moving electric charges : current
• Electronic circuit components: batteries, resistors, capacitors
• Electric currents  Magnetic field
 Magnetic force on moving charges
• Time-varying magnetic field  Electric Field
• More circuit components: inductors.
• Electromagnetic waves  light waves
• Geometrical Optics (light rays).
• Physical optics (light waves)
Oscillators in Physics
Oscillators are very useful in practical
applications, for instance, to keep time, or
to focus energy in a system.
All oscillators can store energy in
more than one way and exchange
it back and forth between the
different storage possibilities. For
instance, in pendulums (and swings)
one exchanges energy between
kinetic and potential form.
We have studied that inductors and capacitors are devices
that can store electromagnetic energy.
energy In the inductor it is
stored in a B field, in the capacitor in an E field.
PHYS2101: A Mechanical Oscillator
U tot = U kin + U pot = const
U tot
1
1 2
2
= mv + k x
2
2
dU tot
1 ! dv $ 1 ! dx $
= 0 = m # 2v & + k # 2x &
dt
2 " dt % 2 " dt %
dv
! m +kx = 0
dt
Solution :
v = x !(t)
a = v!(t) = x !!(t)
Newton’s law
d 2x
m 2 +k x =0
F=ma!
dt
k
!=
x (t ) = x0 cos(" t + !0 )
m
x0 :
amplitude
" :
!0 :
frequency
phase
PHYS2101 An Electromagnetic LC Oscillator
Capacitor initially charged. Initially, current is zero,
energy is all stored in the capacitor.
Energy!Conservation:!U tot = U B + U E
A current gets going, energy gets split between the
capacitor and the inductor.
2
1 2
1q
U B = L i !!!!!U E =
2
2C
Capacitor discharges completely, yet current keeps going.
Energy is all in the inductor.
The magnetic field on the coil starts to collapse,
which will start to recharge the capacitor.
U tot
1 2 1 q2
= Li +
2
2C
Finally, we reach the same state we started with (with
opposite polarity) and the cycle restarts.
Electric Oscillators: the Math
U tot = U B + U E
U tot
1 2 1 q2
= Li +
2
2C
dU tot
1 ! di $
1 ! dq $
= 0 = L # 2i & +
#" 2q &%
"
%
dt
2
dt
dt
2C
Energy Cons.
! di $ 1
VL + VC = 0 = L # & + ( q )
" dt % C
Or loop rule!
Both give Diffy-Q:
Solution to Diffy-Q:
d 2q q
0=L 2 +
dt
C
q = q0 cos(! t + " 0 )
!"
1
LC
i = q!(t)
i !(t) = q!!(t)
LC Frequency
In Radians/Sec
i = q!(t) = "q0# sin(# t + $ 0 )
i !(t) = q!!(t) = "# 2 q0 cos(# t + $ 0 )
Electric Oscillators: the Math
q = q0 cos(! t + " 0 )
i = q!(t) = "q0# sin(# t + $ 0 )
i !(t) = q!!(t) = "# 2 q0 cos(# t + $ 0 )
Energy as Function of Time
1
1
2
2
U B = L [ i ] = L [ q0! cos(! t + " 0 )]
2
2
1 [q]
1
2
UE =
=
q
cos(
!
t
+
"
)
[0
0 ]
2 C
2C
2
Voltage as Function of Time
VL = Li !(t) = $%" q0 sin(" t + # 0 ) &'
2
2
1
1
VC = [ q(t)] = [ q0 cos(! t + " 0 )]
C
C
Analogy Between Electrical
And Mechanical Oscillations
d 2q q
0=L 2 +
dt
C
1
!=
LC
d 2x
m 2 +k x =0
dt
k
!=
m
q = q0 cos(! t + " 0 )
x(t ) = x0 cos(" t + !0 )
i = q!(t) = "q0# sin(# t + $ 0 )
v = x !(t) = "x0# sin(# t + $ 0 )
i !(t) = q!!(t) = "# q0 cos(# t + $ 0 )
a = x !!(t) = "# 2 x0 cos(# t + $ 0 )
2
q! x
i!v
1/C ! k
L!m
Charqe q -> Position x
Current i=q’ -> Velocity v=x’
D-Current i’=q’’-> Acceleration a=v’=x’’
LC Circuit: Conservation of Energy
q = q0 cos(" t + !0 )
1.5
1
0.5
0
Time
-0.5
Charge
Current
-1
dq
i=
= #" q0 sin(" t + !0 )
dt
1 2 1
U B = Li = L! 2 q02 sin 2 (! t + " 0 )
2
2
1 q2
1 2
UE =
=
q0 cos 2 (! t + " 0 )
2C
2C
-1.5
1.2
And remembering that,
1
0.8
0.6
0.4
0.2
0
Time
Energy in capacitor
Energy in coil
1
cos x + sin x = 1, and ! =
LC
2
U tot
2
1 2
= UB + UE =
q0
2C
The energy is constant and equal to what we started with.
Example 1 : Tuning a Radio Receiver
The inductor and capacitor
in my car radio are usually
set at L = 1 mH & C = 3.18
pF. Which is my favorite FM
station?
(a) KLSU 91.1
(b) WRKF 89.3
(c) Eagle 98.1 WDGL
FM radio stations: frequency
is in MHz.
!=
=
1
LC
1
#6
1 " 10 " 3.18 " 10
= 5.61 " 10 8 rad/s
!
f =
2"
= 8.93 # 10 7 Hz
= 89.3!MHz
#12
rad/s
Example 2
• In an LC circuit,
L = 40 mH; C = 4 µF
• At t = 0, the current is
a maximum;
• When will the capacitor
be fully charged for the
first time?
1.5
1
0.5
0
Time
-0.5
-1
-1.5
Charge
Current
!=
1
1
=
rad/s
LC
16x10 "8
• ω = 2500 rad/s
• T = period of one
complete cycle
•T = 2π/ω = 2.5 ms
• Capacitor will be
charged after T=1/4
cycle i.e at
• t = T/4 = 0.6 ms
Example 3
• In the circuit shown, the
switch is in position “a” for a
long time. It is then thrown
to position “b.”
• Calculate the amplitude ωq0
of the resulting oscillating
current.
1 mH
1 µF
b
E=10 V
a
i = #" q0 sin(" t + !0 )
• Switch in position “a”: q=CV = (1 µF)(10 V) = 10 µC
• Switch in position “b”: maximum charge on C = q0 = 10 µC
• So, amplitude of oscillating current =
1
! q0 =
(10µC) = 0.316 A
(1mH)(1µF)
Example 4
In an LC circuit, the maximum current is 1.0 A.
If L = 1mH, C = 10 µF what is the maximum charge q0 on
the capacitor during a cycle of oscillation?
q = q0 cos(" t + !0 )
dq
i=
= #" q0 sin(" t + !0 )
dt
Maximum current is
i0=ωq0 → Maximum charge: q0=i0/ω
Angular frequency ω=1/√LC=(1mH 10 µF)–1/2 = (10-8)–1/2 = 104 rad/s
Maximum charge is q0=i0/ω = 1A/104 rad/s = 10–4 C