Electromagnetic Oscillations and
Currents
March 23, 2014
Chapter 30
1
Driven RLC Circuit
!  The voltage V can be thought of as the projection of the
vertical axis of the phasor Vm representing the time-varying
emf in the circuit
!  The voltage phasors must sum as
vectors
!  Impedance
Z = R + ( X L − XC )
2
2
!  The impedance of a circuit depends on
the frequency of the time-varying emf
!  Impedance of an AC circuit has unit Ω
March 23, 2014
Chapter 30
2
Resistance and Reactance
!  Time-varying emf
Vemf = Vmax sin ω t
!  Time-varying emf VR with resistor
VR
iR =
= I R sin ω t
R
Resistance R
!  Time-varying emf VC with capacitor
1
XC =
ωC
VC
iC =
sin(ω t + 90°)
XC
!  Time-varying emf VL with inductor
XL =ωL
October 22, 2008
VL
iL =
sin ω t − 90°
XL
(
)
Capacitive
Reactance XC
Inductive
Reactance XL
Physics for Scientists & Engineers 2, Lecture 33
4
Driven RLC Circuit
!  Phase constant ϕ is the phase difference between the
voltage phasors VR and Vm
⎛ VL −VC ⎞
−1 ⎛ X L − X C ⎞
φ = tan ⎜
= tan ⎜
⎟
⎟
⎝
R ⎠
⎝ V ⎠
−1
R
!  The current is then
i = I m sin (ω t − φ )
!  Remember that the voltage
across all components is
V = Vemf (t ) = Vm sin (ω t )
March 23, 2014
Chapter 30
5
Driven RLC Circuit
!  Thus we have three conditions for an alternating
current circuit
•  For XL > XC, ϕ is positive, and the current in the circuit 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 π/2
•  For XL < XC, ϕ is negative, and the current in the circuit 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 -π/2
•  For XL = XC, ϕ is zero, and the current in the circuit will be 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
March 23, 2014
Chapter 30
6
Driven RLC Circuit
March 23, 2014
XL > XC
XC > XL
XL = XC
Inductive
Capacitive
Resonance
Chapter 30
7
Driven RLC Circuit
!  The current amplitude, Im, in the series RLC circuit depends
on the frequency of the time-varying emf, as well as on L
and C
!  The maximum current occurs when
ωL −
1
=0
ωC
!  This corresponds to ϕ = 0 and XL = XC
!  The angular frequency, ω0, at which the maximum current
occurs, is called the resonant angular frequency
1
ω0 =
LC
March 23, 2014
Chapter 30
8
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 Vm = 7.5 V, L = 8.2 mH, C = 100 μF, and three
resistances
!  As the resistance is lowered,
the maximum current at the
resonant angular frequency
increases and there is a more
pronounced resonant peak
March 23, 2014
Chapter 30
9
Energy and Power
!  When an RLC circuit operates, some of the energy is stored
in the electric field of the capacitor, some of the energy is
stored in the magnetic field of the inductor, and some
energy is dissipated in the form of heat in the resistor
!  The sum of energy stored in the capacitor and inductor does
not change in steady state operation
!  Therefore the energy transferred from the source of emf to
the circuit is transferred to the resistor
!  The rate at which energy is dissipated in the resistor is the
power P given by
P = i 2 R = ( I sin (ω t − φ )) R = I 2 R sin 2 (ω t − φ )
2
!  The average power is given by
2
1 2
⎛ I ⎞
R
P = I R =⎜
⎟
⎝ 2⎠
2
March 23, 2014
Chapter 30
10
Energy and Power
!  We define the root-mean-square (rms) current to be
I
I rms =
2
!  So we can write the average power as
2
P = I rms
R
!  We can make similar definitions for other quantities
Vrms =
Vm
2
!  The currents and voltages measured by a alternating current
ammeter or voltmeter are rms values
!  For example, we normally say that the voltage in the wall
socket is 110 V
!  This rms voltage would correspond to a maximum voltage
2 ⋅110 V ≈156 V
March 23, 2014
Chapter 30
11
Energy and Power
!  We can then re-write our formula for the current as
Vrms
=
I rms =
Z
Vrms
1 ⎞
⎛
2
R + ⎜ωL −
⎟
⎝
ωC ⎠
2
!  Which allows us to express the average power dissipated as
R
Vrms
I rms R = I rmsVrms
P =I R =
Z
Z
2
rms
!  We can relate the phase constant to the ratio of the
maximum value of the voltage across the resistor divided by
the maximum value of the time-varying emf
R
VR
IR
=
=
cosφ =
Z
Vm
IZ
March 23, 2014
Chapter 30
12
Energy and Power
!  We can then express the power in the circuit as
P = I rmsVrms cosφ
!  We can see that the maximum power is dissipated when
ϕ=0
!  We call cosϕ the power factor
!  The maximum power is dissipated in the circuit when the
frequency of the time-varying emf matches the resonant
frequency of the circuit
!  We define the quality factor of a series RLC circuit as
ω0L 1 L
Q=
=
R
R C
!  The quality factor is the ratio of the total energy stored in
the system divided by the energy per cycle
March 23, 2014
Chapter 30
13
Transformers
!  When using or generating electrical power, high currents
and low voltages are desirable for convenience and safety
!  When transmitting electric power, high voltages and low
currents are desirable, P=VI
•
•
•
•
The power loss in the transmission wires goes as P = I2R
Assume we have 500 MW of power to transmit
If we transmit at 750 kV, the current would be 667 A
If the resistance of the power lines is 200 Ω, the power dissipated in
the power lines is 89 MW
•  18% loss
•  Suppose we transmit at 375 kV instead
•  75% loss
!  The ability to raise and lower alternating
voltages would be very useful in everyday life
March 23, 2014
Chapter 30
16
Transformers
!  To transform alternating currents and voltages from
high to low one uses a transformer
!  A transformer consists of two sets of
coils wrapped around an iron core
!  Consider the primary windings with
NP turns connected to a source of emf
Vemf = Vmax sin ω t
!  We can assume that the primary windings act as an inductor
!  The current is out of phase with the voltage and no power is
delivered to the transformer
!  A transformer that takes voltages from lower to higher is a
step-up transformer and a transformer that takes voltages
from higher to lower is a step-down transformer
March 23, 2014
Chapter 30
17
Transformers
!  Now consider the second coil with NS turns
!  The time-varying emf in the primary
coil induces a time-varying magnetic
field in the iron core
!  This core passes through
the secondary coil
!  Thus a time-varying voltage is
induced in the secondary coil described by Faraday’s Law
Vemf = −N
dΦ B
dt
!  Because both the primary and secondary coils experience
the same changing magnetic field we can write
dΦ B
VS = −N S
dt
March 23, 2014
dΦ B
VP = −N P
dt
Chapter 30
18
Transformers
!  Dividing these two equations gives us
VP VS
=
NP NS
⇒ VS = VP
NS
NP
!  The transformer changes the voltage
of the primary circuit to a secondary
voltage, given by the ratio of the
number turns in the secondary coil
divided by the number of turns in the primary coil
!  If we now connect a resistor R across the secondary
windings, a current will begin to flow through the secondary
coil
!  The power in the secondary circuit is then PS = ISVS
March 23, 2014
Chapter 30
19
Transformers
!  This current will induce a time-varying magnetic field that
will induce an emf in the primary coil
!  The emf source then will produce enough current IP to
maintain the original emf
!  This induced current will not be 90° out of phase with the
emf thus power can be transmitted to the transformer
!  Energy conservation tells that the power produced by the
emf source in the primary coil will be transferred to the
secondary coil so we can write
PP = I PVP = PS = I SVS
!  The current in the secondary circuit is then
VP
NP
IS = IP
= IP
VS
NS
March 23, 2014
Chapter 30
20
Transformers
!  When the secondary circuit begins to draw current, then
current must be supplied to the primary circuit
!  We can define the current in the secondary circuit using
VS = ISR
!  We can then write the primary current as
2
NS
N S VS N S ⎛ N S ⎞ 1 ⎛ N S ⎞ VP
IP =
IS =
=
VP
=⎜
⎜
⎟
NP
N P R N P ⎝ N P ⎠ R ⎝ N P ⎟⎠ R
!  With an effective primary resistance of
2
2
⎛ NP ⎞ R ⎛ NP ⎞
VP
RP =
= VP ⎜
=⎜
R
⎟
⎟
IP
⎝ N S ⎠ VP ⎝ N S ⎠
!  Note that these equations assume no losses in the
transformers and that the load is purely resistive
March 23, 2014
Chapter 30
21