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Review
! Consider a circuit consisting of an
inductor L and a capacitor C
! The charge on the capacitor as a
function of time is given by
Physics for Scientists &
Engineers 2
q = qmax cos (! 0t + " )
! The current in the inductor as a
function of time is given by
Spring Semester 2005
i = !imax sin(" 0t + # )
Lecture 29
! where ! is the phase and "0 is the angular frequency
!0 =
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Review (2)
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qmax
cos 2 (! 0t + " )
2C
! If there is a resistance in the circuit, the current flow in
the circuit will produce ohmic losses to heat
! The total energy stored in the circuit is given by
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2
! We observed that the energy of a circuit
with a capacitor and and an inductor
remains constant and that the energy translated from
electric to magnetic and back gain with no losses
L 2
imax sin 2 (! 0t + " )
2
U = UE + UB =
Physics for Scientists&Engineers 2
! Now let’s consider a single loop circuit
that has a capacitor C and an
inductance L with an added resistance R
! The energy stored in the magnetic field of the inductor L as a function
of time is
UB =
1
LC
RLC Circuit
! The energy stored in the electric
field of the capacitor C as a function
of time is
UE =
1
=
LC
2
qmax
2C
! Thus the energy of the circuit will decrease because of
these losses
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RLC Circuit (2)
RLC Circuit (3)
! The rate of energy loss is given by
! We can then write the differential equation
d 2 q dq
q
L 2 +
R+ = 0
dt
dt
C
dU
= !i 2 R
dt
! We can rewrite the change in energy of the circuit as a
function time as
dU d
q dq
di
= (U E + U B ) =
+ Li = !i 2 R
dt dt
C dt
dt
! The solution of this differential equation is
q = qmax e
! Remembering that i = dq/dt and di/dt =
we can
write
2
q dq
di
q dq
dq d 2 q ! dq $
+ Li + i 2 R =
+L
+
# & R=0
C dt
dt
C dt
dt dt 2 " dt %
Physics for Scientists&Engineers 2
Rt
2L
cos (" t + # )
! where
d2q/dt2
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!
# R&
! = ! 02 " % (
$ 2L '
5
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2
!0 =
1
LC
Physics for Scientists&Engineers 2
RLC Circuit (4)
6
RLC Circuit (5)
! Now consider a single loop circuit that contains a capacitor, an inductor
and a resistor
! The charge varies sinusoidally with but the amplitude is
damped out with time
! If we charge the capacitor then hook it up to the circuit, we will
observe a charge in the circuit that varies sinusoidally with time and
while at the same time decreasing in amplitude
! After some time, no charge remains in the circuit
! We can study the energy in the circuit as a function of
time by calculating the energy stored in the electric field
of the capacitor
! This behavior with time is illustrated below
UE =
Rt
!
$
'
2L
q
e
cos (" t + # ))
max
&
(
1%
1 q2
=
2C 2
C
2
=
Rt
2
!
qmax
e L cos 2 (" t + # )
2C
! We can see that the energy stored in the capacitor
decreases exponentially and oscillates in time
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Alternating Current
Alternating Current (2)
! Now we consider a single loop circuit
containing a capacitor, an inductor,
a resistor, and a source of emf
! The current induced in the circuit will also vary sinusoidally with
time
! This source of emf is capable producing
a time varying voltage as opposed the
sources of emf we have studied in previous chapters
! However, this current may not always remain in phase with the
time-varying emf
! This time-varying current is called alternating current
! We can express the induced current as
i = I sin (! t " # )
! We will assume that this source of emf provides a
sinusoidal voltage as a function of time given by
! where the angular frequency of the time-varying current is the
same as the driving emf but the phase ! is not zero
Vemf = Vmax sin ! t
! Note that traditionally the phase enters here with a negative sign
! where " is that angular frequency of the emf and Vmax is
the amplitude or maximum value of the emf
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Physics for Scientists&Engineers 2
! Thus the voltage and the current in the circuit are not
necessarily in phase
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Circuit with Resistor
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Circuit with Resistor (2)
! To begin our analysis of RLC circuits, let’s start
with a circuit containing only a resistor and a
source of time-varying emf as shown to the right
! Thus we can relate the current amplitude and the voltage amplitude by
VR = I R R
! We can represent the time varying current by a phasor IR and the timevarying voltage by a phasor VR as shown below
! Applying Kirchhof’s loop rule to this circuit we get
Vemf ! vR = 0
Physics for Scientists&Engineers 2
" Vemf = vR
! where vR is the voltage drop across the resistor
! Substituting into our expression for the emf as a function of
time we get
vR = VR sin ! t
! Remembering Ohm’s Law, V = iR, we get
! The current flowing through the resistor and the voltage across the
resistor are in phase, which means that the phase difference between
the current and the voltage is zero
v
iR = R = I R sin ! t
R
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Circuit with Capacitor
Circuit with Capacitor (2)
! Now let’s address a circuit that contains a capacitor
and a time varying emf as shown to the right
! We can rewrite the last equation by defining a quantity that is
similar to resistance and is called the capacitive reactance
! The voltage across the capacitor is given by
Kirchhof’s loop rule
XC =
vC = VC sin ! t
! Which allows us to write
! Remembering that q = CV for a capacitor we can write
iC =
q = CvC = CVC sin ! t
! We would like to know the current as a function of time rather
than the charge so we can write
Physics for Scientists&Engineers 2
VC
cos ! t
XC
! We can now express the current in the circuit as
iC = I C cos ! t = I C sin (! t + 90° )
dq d ( CVC sin ! t )
iC =
=
= ! CVC cos ! t
dt
dt
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!C
! We can see that the current and the time varying emf are out of
phase by 90°
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Circuit with Capacitor (3)
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Circuit with Capacitor (4)
! We can represent the time varying current by a phasor IC
and the time-varying voltage by a phasor VC as shown below
! We can also see that the amplitude of voltage across the
capacitor and the amplitude of current in the capacitor are
related by
VC = I C XC
! This equation resembles Ohm’s Law with the capacitive
reactance replacing the resistance
! One major difference between the capacitive reactance
and the resistance is that the capacitive reactance depends
on the angular frequency of the time-varying emf
! The current flowing this circuit with only a capacitor is
similar to the expression for the current flowing in a
circuit with only a resistor except that the current is out
of phase with the emf by 90°
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Circuit with Inductor
Circuit with Inductor (2)
! We are interested in the current rather than its time derivative
so we integrate
! Now let’s consider a circuit with a source of
time-varying emf and an inductor as shown
to the right
iL = !
! We can again apply Kirchhof’s Loop Rule to
this circuit to obtain the voltage across the inductor as
! We define inductive reactance as
XL = ! L
vL = VL sin ! t
! A changing current in an inductor will induce an emf given by
di
vL = L L
dt
! So we can write
di
L L = VL sin ! t
dt
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"
diL VL
= sin ! t
dt
L
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Circuit with Inductor (3)
! The current in the inductor can then be written as
iL = !
vL
cos " t = !I L cos " t = I L sin (" t ! 90° )
XL
! Thus the current flowing in a circuit with an inductor and a source timevarying emf will be -90° out of phase with the emf
! We can write the relationship between the amplitude of the current
and the amplitude of the voltage as
VL = I L X L
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diL
V
V
dt = ! L sin " tdt = # L cos " t
dt
L
"L
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! which, like the capacitive reactance, is similar to a resistance
! We can then write
v L = iL X L
! which again resembles Ohm’s Law except that the inductive
reactance depends on the angular frequency of the time-varying
emf
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