Monday, March 31, 2014
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 In an electric circuit the electromagnetic energy is
dissipated in
 A: Resistor
 B: Inductor
 C: Capacitor
 D: Electromagnetic energy is conserved
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Reminder: time-dependent currents
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RC Circuits
 Going around the circuit in a counterclockwise direction we can write
 We can rewrite this equation
remembering that i = dq/dt
 The solution is
 where q0 = CVemf and τ = RC
The term Vc is negative since
the top plate of the capacitor is
connected to the positive higher potential - terminal of
the battery. Thus analyzing
counter-clockwise leads to a
drop in voltage across the
capacitor!
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Monday, March 31, 2014
RL Circuits
 Thus we can write the sum of the potential drops around the circuit
as
 The solution to this differential equation is
 We can see that the time constant of this circuit is τL = L/R
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Monday, March 31, 2014
Summary: LC Circuit (1)
 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
 The current in the inductor as a
function of time is given by
 where φ is the phase and ω0 is the angular frequency
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Intrinsic EM oscillations!
 Frequency determined by the parameters of the
circuit, not external driving
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Monday, March 31, 2014
RLC Circuit (1)
 Now let’s consider a single loop circuit
that has a capacitor C and an
inductance L with an added resistance R
 We observed that the energy of a circuit
with a capacitor and an inductor
remains constant and that the energy
translated from electric to magnetic and
back gain with no losses
 If there is a resistance in the circuit, the current flow in the
circuit will produce ohmic losses to heat
 Thus the energy of the circuit will decrease because of these
losses
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Monday, March 31, 2014
RLC Circuit (2)
 The rate of energy loss is given by
 We can rewrite the change in energy of the circuit as a
function time as
 Remembering that i = dq/dt and di/dt = d 2q/dt2 we can write
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Monday, March 31, 2014
RLC Circuit (3)
 We can then write the differential equation for charge on
the capacitor
 The solution of this differential equation is
(damped harmonic oscillation!), where
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RLC Circuit (4)
 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
 This behavior with time is illustrated below
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Monday, March 31, 2014
RLC Circuit (3)
 We can then write the differential equation for charge on
the capacitor
 The solution of this differential equation is
(damped harmonic oscillation!), where
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Monday, March 31, 2014
RLC Circuit (5)
Observations:
•The charge varies sinusoidally with but the amplitude is damped
out with time
•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
 We can see that the energy stored in the capacitor
decreases exponentially and oscillates in time
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Alternating currents
(driven RLC circuits)
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DC and AC
Motors and
Generators
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Direct and Alternating Current Generators
 In a direct current generator the rotating coil is connected to an
external circuit using a split commutator ring
 As the coil turns, the connection is reversed
such that the induced voltage always has the
same sign
 In alternating current generator, each end of
the loop is connected to the external circuit
through a slip ring
• Thus this generator produces an induced voltage that varies
from positive to negative and back, and is called an alternator
 The voltages and currents produced by these
generators are illustrated below
Direct voltage/
current
Alternating voltage/
current
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Alternating Current (1)
 Now we consider a single loop circuit
containing a capacitor, an inductor,
a resistor, and a source of emf
 This source of emf is capable of producing
a time varying voltage as opposed to the
sources of emf we have studied in previous chapters
 We will assume that this source of emf provides a sinusoidal voltage as a
function of time given by
 where ω is the angular frequency
of the emf and Vmax is the
amplitude or maximum value
of the emf
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Alternating Current (2)
 The current induced in the circuit will also vary sinusoidally
with time
 This time-varying current is called alternating current
 However, this current may not always remain in phase with the timevarying emf
• The sinusoidal wave may crest earlier or later than that for emf
 We can express the induced current as
where the angular frequency of the time-varying current is the same as
the driving emf but the phase φ is not zero
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Monday, March 31, 2014
Alternating Current (3)
 Note that traditionally the phase enters here with a negative
sign
 Thus the voltage and the current in the circuit are not
necessarily in phase
 Notation: instantaneous values are denoted by small letters
(v, i ), amplitudes by capital letters (V, I)
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Circuit with Resistor (1)
 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
 Applying Kirchhoff’s loop rule to this circuit we get
 where vR is the voltage drop across the resistor
 Substituting into our expression for the emf as a function of
time we get
 Remembering Ohm’s Law, V = iR, we get
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Monday, March 31, 2014
Circuit with Resistor (2)
 Thus we can relate the current amplitude and the voltage
amplitude by

 Phase difference is 0
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Circuit with Resistor (2)
 We can represent the time varying current by a phasor IR and the time-varying voltage
by a phasor VR as shown below
 Phase difference is 0
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Monday, March 31, 2014
Circuit with Resistor (2)
 We can represent the time varying current by a phasor IR and the time-varying voltage
by a phasor VR as shown below
 Phase difference is 0
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Circuit with Resistor (2)
 We can represent the time varying current by a phasor IR and the time-varying voltage
by a phasor VR as shown below
 Phase difference is 0
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Monday, March 31, 2014
Circuit with Capacitor (1)
 Now let’s address a circuit that contains a capacitor
and a time varying emf as shown to the right
 The voltage across the capacitor is given by
Kirchhoff’s loop rule
 Remembering that q = CV for a capacitor we can write
 We would like to know the current as a function of time
rather than the charge so we can write
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Monday, March 31, 2014
Circuit with Capacitor (2): Capacitive
Reactance
 We can rewrite the last equation by defining a quantity that
is similar to resistance and is called the capacitive reactance
 Which allows us to write
1
Effective “resistivity” of a capacitor
 We can now express the current inωC
the circuit as
 We can see that the current and the time varying emf are
out of phase by 90°
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Monday, March 31, 2014
Circuit with Capacitor (2): Capacitive
Reactance
 We can rewrite the last equation by defining a quantity that
is similar to resistance and is called the capacitive reactance
 Which allows us to write
compare with
V
i=
R
1
Effective “resistivity” of a capacitor
 We can now express the current inωC
the circuit as
 We can see that the current and the time varying emf are
out of phase by 90°
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Monday, March 31, 2014
Circuit with Capacitor (3): Phasor
 We can represent the time varying current by a
phasor IC and the time-varying voltage by a phasor VC as shown below
 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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Monday, March 31, 2014
Circuit with Capacitor (4)
 We can also see that the amplitude of voltage across the capacitor and
the amplitude of current in the capacitor are related by
 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
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Monday, March 31, 2014
Circuit with Inductor (1)
 Now let’s consider a circuit with a source of
time-varying emf and an inductor as shown
to the right
 We can again apply Kirchhoff’s Loop Rule to
this circuit to obtain the voltage across the inductor as
 A changing current in an inductor will induce an emf given by
 So we can write
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Monday, March 31, 2014
Circuit with Inductor (2)
 We are interested in the current rather than its time
derivative so we integrate
 We define inductive reactance as
which, like the capacitive reactance, is similar to a resistance
 We can then write
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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Monday, March 31, 2014
Circuit with Inductor (3)
 The current in the inductor can then be written as
 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
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Monday, March 31, 2014
Summary: RLC Circuit
 If we have a single loop RLC circuit, the charge in the
circuit as a function of time is given by
where
 The energy stored in the capacitor as a function of time is given
by
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Monday, March 31, 2014
Summary: Resistance and Reactance
 Time-varying emf
 Time-varying emf VR with resistor
Resistance R
 Time-varying emf VC with capacitor
Capacitive
Reactance XC
 Time-varying emf VL with inductor
Inductive
Reactance XL
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Summary: Phase and Phasors
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Average powers
IV
< P >= IV < sin ωt sin ωt >=
2
< P >= IV < sin ωt cos ωt >= 0
< P >= IV < sin ωt cos ωt >= 0
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Monday, March 31, 2014
Series RLC Circuit (1)
 Consider a single loop circuit that has
a resistor, a capacitor, an inductor, and
a source of time-varying emf
 We can describe the time-varying currents
in these circuit elements using a phasor I
V
 The projection of I on the vertical axis
represents the current flowing in the
circuit as a function of time
• The angle of the phasor is given by ωt - φ
 We can also describe the voltage in terms
of a phasor V
 The time-varying currents and voltages in the circuit can have
different phases
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Monday, March 31, 2014
Series RLC Circuit (2)
 We can describe the current flowing in the circuit and the
voltage across the various components
• Resistor
• The voltage vR and current iR are in phase with
each other and the voltage phasor vR is in phase
with the current phasor I
iR
VR
• Capacitor
• The current iC leads the voltage vC by 90° so
that the voltage phasor vC will have an angle 90°
less than I and vR
• Inductor
• The current iL lags behind the voltage vL by 90°
so that voltage phasor vL will have an angle 90°
greater than I and vR
iC
VC
VL
iL
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Monday, March 31, 2014
Series RLC Circuit (3)
 The voltage phasors for an RLC circuit are shown below
 The instantaneous voltages across each of the components
are represented by the projections of the respective phasors
on the vertical axis
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Monday, March 31, 2014
Series RLC Circuit (4)
 Kirchhoff’s loop rules tells that the voltage drops across all
the devices at any given time in the circuit must sum to zero,
which gives us
 The voltage can be thought of
as the projection of the
vertical axis of the phasor Vmax
representing the time-varying
emf in the circuit as shown
below
 In this figure we have replaced the sum of the two
phasors VL and VC with the phasor VL - VC
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Monday, March 31, 2014
Series RLC Circuit (5): Impedance
 The sum of the two phasors VL - VC and VR must equal Vmax so
 Now we can put in our expression for the voltage across the components in
terms of the current and resistance or reactance
 We can then solve for the current in the circuit
 The denominator in the equation is called the impedance
 The impedance of a circuit depends on the frequency of the time-varying
emf
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Monday, March 31, 2014
Series RLC Circuit
impedance
Z=
�
�
1
R2 + ωL −
ωC
�2
active
reactive
resistance resistance (reactance)
Only active resistance
determines losses!
Reactive resistance can be 0, at resonance
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Monday, March 31, 2014
Series RLC Circuit (6): Phase
 The current flowing in an alternating current circuit depends
on the difference between the inductive reactance and the
capacitive reactance
 We can express the difference between the inductive
reactance and the capacitive reactance in terms of the phase
constant φ
 This phase constant is defined as the phase difference
between voltage phasors VR and VL - VC
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Monday, March 31, 2014
Series RLC Circuit (7)
 Thus we have three conditions for an alternating
current circuit
• For XL > XC, φ is positive, and the current will lag behind the voltage in the
circuit
• This circuit will be similar to a circuit with only an inductor, except that the
phase constant is not necessarily 90°
• For XL < XC, φ is negative, and the current will lead the voltage in the circuit
• This circuit will be similar to a circuit with only a capacitor, except that the
phase constant is not necessarily -90°
• For XL = XC, φ is zero, and the current in phase with the voltage in the circuit
• This circuit is similar to a circuit with only a resistance
• When φ = 0 we say that the circuit is in resonance
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Monday, March 31, 2014
Series RLC Circuit (8)
XL > XC
XL < XC
XL = XC
 For XL = XC and φ = 0 we get the maximum current in the circuit and we
can define a resonant frequency
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Monday, March 31, 2014
Resonances in RLC circuit
 If R=0
Vmax
Vmax
I=
=
2
2
|ωL − 1/(ωC)|
ωL |1 − ω0 /ω |
1
2
ω0 =
LC
 Infinite current for
ω = ω0
 Resonance!
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Monday, March 31, 2014
Near resonance
 The smallest R will make the resonance finite
 If XL=XC, I =Vmax/R
 At resonance, maximal current for given voltage
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Monday, March 31, 2014
Resonant Behavior of RLC Circuit
 The resonant behavior of an RLC circuit resembles the
response of a damped oscillator
 Here we show the calculated maximum current as a function of
the ratio of the angular frequency of the time varying emf
divided by the resonant angular frequency, for a circuit with Vmax
= 7.5 V, L = 8.2 mH, C = 100 µF, and three resistances
 One can see that as the
resistance is lowered, the
maximum current at the
resonant angular frequency
increases and there is a more
pronounced resonant peak
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Monday, March 31, 2014
Complex impedance
�
�
1
Z = j ωL −
+R
ωC
�
�
�2
1
2
|Z| = R + ωL −
ωC
 Imaginary unit: phase
shift by 90o
v = V0 e
i = I0 e
jωt
Imaginary = j
-1
sin φ
φ
cos φ
1
Real
jωt+∆φ
v = iZ
e
jφ
= cos φ + j sin φ
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Monday, March 31, 2014