Induction
In 1830-1831, Joseph Henry & Michael Faraday discovered electromagnetic
induction. Induction requires time varying magnetic fields and is the subject of another
of Maxwell's Equations.
Modifying Ampere's Law to include the possibility of time varying electric fields gives
the fourth Maxwell's Equations.
Faraday's Law
Changing magnetic flux in a conducting loop produces an induced emf in the
conducting loop that depends on the time rate of change of the magnetic flux.
For the conducting loop, find the magnetic flux and its time rate of change:
Φ = B⊥ ⋅ A = BACos(θ )
SI unit of Φ is the Weber 1W = 1Tm2
Three ways by which the flux may vary in time are:
1) Amount of magnetic field changes
2) Loop cross-section area changes
3) Changes in the relative orientation of
B _ to _ nˆ
Method 3) is used in an electric generator with input mechanical energy.
Effectively the opposite of an electric motor, the changing orientation of the loop
relative to
B results in a generated emf via induction in the conducting loop.
For a changing magnetic flux, the induced emf in the conducting loop is:
Emf = −
ΔΦ
Δt
In an AC generator, if the loop rotates at a constant angular velocity
Emf = −
ω:
ΔΦ
Δ
= − BA (Cos(ωt ) ) = BAωSin(ωt )
Δt
Δt
Using a coil with
N tightly packed loops,
Emf = NBAωSin(ωt ) = E0 Sin(ωt )
n̂
Emf = −
Note the minus sign in the induced emf:
ΔΦ
Δt
The minus sign implies the direction of flow for the electric current resulting from the
induced emf. This is described by Lenz's Law.
Lenz's Law
Lenz's Law is: The direction of current corresponding to the induced emf is such that the
magnetic field resulting from this current opposes the change in magnetic flux that
induces the current.
A changing magnetic field flux
Induced current direction is such that its
B field direction opposes flux field changes.
Motional Emf
For a circuit or loop of wire moving within a magnetic field, changing magnetic flux can
induce an emf because of the conductor motion.
As the loop is removed from the region of magnetic field, a '_________' current is
induced. Moreover, since this is a current carrying wire in a magnetic field, a force
F = BiL is present on each of the segments within the field.
Along the top and bottom portions of the wire, these forces cancel. However, the external
agent must supply power
P = F ⋅ v to move the loop within the magnetic field
since the force on the vertical segment of wire in the field is directed opposite to
FExt
v.
E
BL ΔΦ BL Δ
B 2 L2 v
= BiL = B * * L =
*
=
* ( BXL) =
R
R Δt
R Δt
R
B 2 L2v 2
P = FvCos(0) =
R
Rate of doing work to move the loop at
v.
E 2 B 2 L2 v 2
P=i R=
=
R
R
2
The resistive heat in the loop is also
Fixing the loop in place, emf can be generated by moving a rail on a U-shaped conductor:
L
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V
Electric Potential and Time Varying Magnetic Fields
Within a time varying magnetic field, the possibility of induced emf subverts Kirchhoff's
loop rule:
ΔVClosed_Loop≠ 0
Emf= −
dΦ
ΔΦ
Emf = −
dt
Δt
Note electric potential has lost meaning when electric fields are the result of time varying
magnetic fields. Forces associated with changing magnetic fields are not conservative.
Counter Emf, Counter Torque, and Eddy Current
In the case of an electric motor, as the driving current turns coils within the magnetic
field there is an emf generated in the conducting loops that has current opposing the
supply current. This is the back emf and depends on the rotation rate of the coils.
For the electric generator, an attached load draws current through the generator coils,
which produces a counter torque at the axel.
Eddy currents form in general when a conductor moves through a region of magnetic
field such that forces exist on the free electrons in that material. Eddy currents require
energy that is extracted from the moving conductor thereby having a dissipating effect.
Transformers
Transformers are used to increase or decrease voltages to an appropriate value for a given
application.
Primary and secondary coils with differing numbers of turns are flux-linked such that
voltages may be either stepped-up or stepped-down:
V2 = N 2 *
ΔΦ B
V
= N2 * 1
Δt
N1
V2 N 2
=
V1 N1
For 100% efficient transformers, power input equals power output. If the voltage is
stepped-up the current in the secondary must decrease and vice-versa.
I1V1 = I 2V2
Inductors and Inductance
For devices and circuit elements that produce magnetic fields and emf inductance
processes in general:
A short ideal solenoid is the inductor prototype. Inductance is defined:
L≡
NΦ
i
The SI unit of inductance is the Henry.
N = # _ of _ turns
1H = 1Tm2A-1
NΦ is the flux linkage. Inductance is: flux linkage per unit current.
Increased inductance may be achieved by increasing the number of turns per unit length
on the inductor.
E.g., Ideal Solenoid:
B = µ 0 ni
L≡
n here is turns per unit solenoid length.
Nµ 0 niA
NΦ N
= * BA =
= nlµ 0 nA
i
i
i
L = µ 0 n 2lA = f (Geometry)
L = µ0n2 A
l
In units of henry/meter.
Note the units of permeability are, therefore, henry/meter and recall from capacitance
that the units of permitivity are farad/meter.
Self Induction
Δi
Given an inductor and given a changing current
current produces a changing
Δt within its coils, this changing
B in the inductor, and therefore a self-induced emf.
The induced current direction is such that the induced magnetic field produced from
this current counters whatever change is occurring in the inductor flux.
E = −N
ΔΦ
Δ
Δ
Δi
Δi
= − N ( BA) = − NA µ0 ni = − µ0 n2lA = − L
Δt
Δt
Δt
Δt
Δt
E = −L
Δi
Δt
For steady currents, the induced emf is zero.
For an ideal inductor, the resistance of the wire material is negligible, and the voltage
across the inductor is the self-induced emf.
If current is steady the inductor behaves as a short with zero voltage drop.
In practice, real inductors may be modeled by the series combination of an inductor
resistance outside the region of changing magnetic fields plus an ideal inductor.
Mutual Inductance
Consider interactions between two inductors in 'nearby' proximity. As current in inductor
#1 changes in time, this produces a changing magnetic field in inductor #1 and therefore
a changing flux in nearby inductor #2. Changing flux in inductor #2 produces an emf in
inductor #2. This is mutual inductance.
Similarly, any changes in inductor #2 current will induce an emf in inductor #1.
The mutual inductance of inductor 2 with respect to inductor 1 is:
M 21 =
M 21
N 2Φ 21
i1
As with self-inductance,
M
is characterized by geometry.
Δi1
ΔΦ12
= N2
= − E2
Δt
Δt
E2 = − M 21
Δi1
Δt
Induced emf in inductor #2 from a changing current in #1.
Similarly
E1 = − M 12
Δi2
Δt
Usually a small-unwanted effect in circuitry, mutual inductance is the way step-up / stepdown transformers operate.
Magnetic Field Energy Density
The energy stored in an inductor is magnetic energy.
UB 1 2
L i2
u=
= Li / lA =
Inductor magnetic field energy density is
V
2
l 2A
Which, given
Using
1
2 2
L = µ0 n 2 A
u
=
µ
n
i
0
, reduces to
l
2
B = µ 0 ni gives:
B2
u=
2µ 0
Energy density of magnetic field.
RL Circuits
In an approach analogous to what was done with charging/discharging in an RC circuit,
the current response in an RL circuit may be evaluated starting from Kirchhoff's rule:
E − iR −
Recall the RC circuit results as follows:
Charging
Q = CE (1 − e
Discharging
Q = Q0 e
−t
−t
τ
Q
=0
C
τ = RC
)
τ
In an RL circuit, the inductor initially appears as open and eventually t
E − iR − L
Δi
=0
Δt
E
−t
i(t ) = (1 − e τ )
R
Taking
Q→i
τ=
&
∞ as a short.
1
R
→
RC
L
L
= Inductive _ Time _ Const.
R
The voltage on the inductor is a maximum to start and exponentially tends to zero:
Δi LE − t τ
−t
e = Ee τ
VL = − L =
Δt Rτ
Using Kirchhoff's loop rule, the resistor voltage is therefore:
VR = E − VL = E − Ee
−t
τ
= E (1 − e
−t
τ
)
Zero to start and asymptotically tending to E as the inductor appears to short.
Removing the circuit emf,
i (t ) = i0 e
−t
τ
E − tτ
= e
R
LC Oscillators
Directly analogous to the mass-spring oscillating mechanical system, the electronic
counterpart is that of an inductor-capacitor oscillating circuit.
Oscillations are electromagnetic oscillations of capacitor electric fields and inductor
magnetic fields.
The circuit has all the properties of an oscillating system including a resonance driving
frequency, in this case
ω = 1 / LC
vs.
ω = k / m for the mechanical system.
A correspondence of
L→m
and
C → k −1 is made.
q (t )
Solutions to the differential equations describing the capacitor charge
may be
extracted from the mechanical system solutions by making the appropriate variable
substitutions into those results.
Simple Harmonic Oscillation
A capacitor charge that is periodic or repeats in regular time intervals, and is a sinusoidal
or co-sinusoidal function of time is referred to as simple harmonic in time.
q (t ) = QCos(ωt + φ )
Q is the amplitude and is the maximum +/- capacitor charge.
ω is the angular frequency of the oscillator and related to the frequency by ω=2πf
ω=2π
φ is a phase factor or phase angle in units of radians.
f is the frequency or number of oscillations per second. Units of f are Hertz, Hz.
LC Circuit
⇔ Mass-Spring Analogy
Newton's 2nd Law for the mass-spring oscillator is:
ma = −kx
x(t ) = xmCos(ωt )
a(t ) = −ω 2 xmCos(ωt )
Substituting:
− mω 2 xmCos(ωt ) + kxmCos(ωt ) = 0
This equation is true iff
ω = k/m
From the circuit energies, and conservation of energy without resistive damping,
electromagnetic oscillations take place as follows:
1 2 1 q2
E = Li +
= Total _ Energy = const.
2
2C
Δi q Δq
E& = 0 = Li +
Δt C Δt
Δ Δq
1
+
q=0
Δt Δt LC
Using
L→m
&
Compared To
Δ Δx k
+ x=0
Δt Δt m
C → k −1 , then q → x is also appropriate.
For the circuit oscillator, 'kinetic' and 'potential' energies are the energies within the
inductor and capacitor respectively. The capacitor is spring like in its electric potential
energy, inductance mass like, and current a 'velocity' term.
q (t ) = QCos (ωt + φ )
i (t ) =
ω = 1 / LC
Δq
= −ωQSin(ωt + φ ) = − ISin(ωt + φ )
Δt
I = ωQ = current _ amplitude
Let
φ = 0 and 'extend the mass', i.e., charge the capacitor to its full charge:
q(t = 0) = Q
q(t ) = QCos (ωt )
i (t ) = − ISin(ωt )
i (t = 0) = 0
The system oscillates as shown:
Initially:
2
1 2
1
Q
2
U E (t ) =
q =
[QCos (ωt )] =
2C
2C
2C
1 2 1
U B (t ) = Li = L * [− ISin(ωt )]2 = 0
2
2
T
t
=
At
4
=π
2ω
1 2
1
U E (t ) =
q =
[QCos (ωt )]2 = 0
2C
2C
U B (t ) =
1 2 1
1
Li = L * [− ISin(ωt )]2 = LI 2
2
2
2
Etc., the system evolves periodically transferring energy between the capacitor electric
field and the inductor magnetic field as shown.
Both 'kinetic' and 'potential' energy peaks occur twice over the course of 1 period [T].
Since
ω = 1 / LC the total energy at all times is:
E = U B (t ) + U E (t ) =
1 2
1 2
Q {Sin 2 (ωt + φ ) + Cos 2 (ωt + φ )} =
Q
2C
2C
The electromagnetic energy is a constant provided resistive damping is absent.
RLC Circuits, Damped Harmonic Oscillations
Most non-driven oscillating systems will come to rest after a finite amount of time due to
dissipative losses. In the RLC circuit, the resistor damps the LC oscillations causing
exponentially decaying oscillation amplitude.
The general solution is a linear combination of the two possibilities:
−γt
q(t ) = e { Ae
( γ 2 −ω 0 2 )*t
+ Be
−( γ 2 −ω 0 2 )*t
}
Three cases exist depending on whether
γ2 < ω02 , γ2 > ω02 , or γ2 = ω02
Case 1 Underdamped Solution:
2
γ <
ω02
R<2
−γt
q(t ) = e { Ae
L
C
i ( ω 0 2 −γ 2 )*t
+ Be
−i ( ω 0 2 −γ 2 )*t
}
q(t ) = e −γt {( A + B)Cos ( ω 0 − γ 2 )t + ( A − B)iSin( ω 0 − γ 2 )t}
2
2
q (t ) = e −γt { A' Cos (ω ' t ) + B ' Sin (ω ' t )}
With initial conditions q(0)
= Q and i(0) = 0,
A' = Q
B'ω
ω' - A'γγ = 0
B' = A'γγ/ω
ω' = Q {γ/ω
γ/ω'}
γ/ω
γ
q(t ) = Qe {Cos(ω ' t ) + Sin(ω ' t )}
ω'
−γt
Wishing to write something like:
q (t ) = Q
ω 0 −γt
e {Cos (ω ' t + φ )}
ω'
We find the phase angle φ must be:
φ = −Tan −1
γ
ω'
Proving this requires:
ω0
γ
{
Cos
(
ω
'
t
+
φ
)}
=
Cos
(
ω
'
t
)
+
Sin(ω ' t )
'
ω
ω'
From trigonometry the LHS is
ω0
{Cos(ω ' t )Cos(φ ) − Sin(ω ' t ) Sin(φ )}
'
ω
Cos (φ ) = Cos{Tan −1
The reference triangle is:
γ
}
ω'
γ 2 + ω '2
φ
γ
ω'
ω ' = ω02 − γ 2
Cos (φ ) = ω '
ω0
γ 2 + ω '2 = ω 0
Sin (φ ) = − γ
ω0
From which,
ω0
γ
{
Cos
(
ω
'
t
+
φ
)}
=
Cos
(
ω
'
t
)
+
Sin(ω ' t )
'
ω
ω'
q(t ) = Q
ω 0 −γt
e {Cos (ω ' t + φ )}
ω'
φ = −Tan −1
γ
ω'
The oscillation amplitude is decaying exponentially in time.
Case 2 Critically Damped Solution:
2
γ =
ω02 R=2
L
C
In this case ω' is identically zero and we can evaluate the limiting form of:
γ
q (t ) = Qe {Cos (ω ' t ) + Sin(ω ' t )}
ω'
−γt
As ω' 0.
γ
Lim{Qe [Cos (ω ' t ) + Sin(ω ' t )]}
ω ' − >0
ω'
−γt
The cosine term goes to 1 in the limit and the sine term is:
γ
Sin(ω ' t )
Sin(ω ' t )} = Lim{γt *
} = γt
ω
'
−
>
0
ω'
ω 't
Lim{
ω ' − >0
q (t ) = Qe −γt {1 + γt}
The circuit damps to equilibrium as quickly as is possible without oscillation about the
equilibrium point.
Case 3; Overdamping:
2
γ >
ω02 R>2
L
C
−γt
q
(
t
)
=
Qe
{Cos (ω ' t ) +
Starting with
γ
Sin(ω ' t )}
ω'
ω ' = ω02 − γ 2 = i γ 2 − ω02
Since
e i ( ix ) + e − i ( ix ) e x + e − x
Cos (ix ) =
=
= Cosh ( x)
2
2
And
e i ( ix ) − e − i ( ix ) e x − e − x
Sin (ix ) =
=
= iSinh ( x)
2i
2i
γi
2
q(t ) = Qe {Cosh( γ − ω 0 t ) + Sinh( γ 2 − ω 0 t )}
ω'
−γt
q(t ) = Q
2
2
ω 0 −γt
e {Cosh(ω " t + φ )}
ω"
ω" = γ 2 − ω0 2
Alternating Current and AC Circuits
For a sinusoidal oscillating voltage, the currents in a circuit will be alternating current.
Since any input signal may be Fourier analyzed into a series summation of sine and
cosine inputs, the response of a circuit to a general AC signal will be important.
Depending on the combination of circuit elements R, L, and/or
and circuit current will in general be out of phase with each other.
E = Em Sin (ω d t )
i = ISin (ω d t − φ )
C, the driving emf
Phasor projections along the vertical current / voltage at time =
t
Resistive Load
For the above emf / resistor only circuit:
E − vR = 0
iR =
vR = Em Sin (ω d t ) = VR Sin (ω d t )
VR
Sin(ω d t ) = I R Sin(ω d t )
R
VR = I R R
Current and voltage are in phase.
Capacitive Load
Using the AC emf with a capacitor:
vC = VC Sin (ω d t )
iC =
ΔqC
= CVCω d Cos (ω d t )
Δt
Defining the capacitive reactance
iC =
qC = CvC = CVC Sin (ω d t )
1
XC =
ωdC
VC
π
Sin(ω d t + )
XC
2
Current leads voltage by 90 degrees
Then since
And
iC = I C Sin (ω d t − φ )
φ = −π 2
VC = I C X C
Inductive Load
AC source and inductor only:
v L = VL Sin(ω d t ) = L
Defining the Inductive reactance as
iL =
Δi
Δt
X L = ωd L
π
VL
Sin (ω d t − )
XL
2
Then since i L = I L Sin (ω d t − φ )
And
VL = I L X L
Current lags voltage by 90 degrees
φ = +π 2
Series RLC Circuits
In general:
E = Em Sin (ω d t )
I
and
φ
i = ISin (ω d t − φ )
to be determined.
The Phasor algebra looks like:
From Kirchhoff's Rule,
The current and
2
Em − VR − VL − VC = 0
VR are in phase and the phase angle between current and emf is φ .
2
22
2
Em = VR + (VL − VC ) = I R + I 2 ( X L − X C ) 2
2
Em
I=
R2 + ( X L − X C )2
Z = R 2 + ( X L − X C )2
The impedance is:
I=
Em
Z
Em
=
From the figure, the phase angle
R 2 + (ω d L −
1 2
)
ωd C
φ is:
VL − VC
XL − XC
Tan(φ ) =
=
VR
R
Three cases are as follows:
1) Inductive circuit
φ > 0 Voltage leads current
2) Capacitive circuit
3) Resonance
φ < 0 Voltage lags current
φ = 0 Voltage and current are in phase.
Starting the emf at low frequency and scanning to higher values of
observable phase shift in
ω d will produce an
i relative to E as the circuit moves from a capacitive circuit
1
ω
=
d
to an inductive circuit across the resonance point at
LC
Resonance
Driving an RLC circuit at resonance condition produces output current and voltage
amplitudes that are maximal. The condition for resonance is:
ω Driving = ω 0
Gain = Av =
I rms =
Vout
=
Vin
Vrms
=
Z
Vrms
R 2 + (ωL −
1 2
)
ωC
R
R 2 + (ωL −
1 2
)
ωC
Notice as the damping R is reduced the resonant amplitude peaks become larger and
have narrower half-width. The series RLC is a useful bandwidth filter circuit near the
resonance frequency.
Power in AC Circuits: RMS Current and Voltage
Electronics meters measuring AC voltages and currents are reading a time-averaged
quantity known as the RMS value or 'root-mean-square'.
The RMS values of current and voltage are related to the peak values of these quantities:
I
I RMS =
VRMS =
2
Em
2
For the RLC circuit, the phase difference between voltage and current in both the
inductor and capacitor means that the average power transferred to these elements is zero
and the average power dissipated by the circuit is due solely to the resistor:
2
1
 I 
2
P = i 2 R = I 2 R Sin2 (ω d t − φ ) = I 2 R = 
 R = ( I RMS ) R
2
 2
P = ( I RMS ) 2 R
Using RMS current power is calculated just like DC.
ERMS
R
P = ( I RMS ) R =
I RMS R = ERMS I RMS
Z
Z
2
R
= Cos(φ ) = Power _ factor
Z
R=Z
⇒
Maximum power is transferred to
φ =0
i.e.,
resonance
R at resonance.
0
φ
=
±
90
For inductors or capacitors,
and no average power is dissipated.
Impedance Matching
If output from the left side of the above circuit is the input to a 'device' on the right, then
maximum power transfer takes place when R1=R2. When power transfer is
important any two circuit impedances will need to be matched Z1=Z2.
Large mismatches in impedance affect both the amount of signal transferred to the
second circuit and the amount of signal reflected back to the first.
Transformers may be used to impedance match two circuits:
VS N S
=
VP N P
I PVP = I SVS
2
Z P VP I S N P
=
⋅ =
Z S I P VS N S 2