Inductors
A coil of wire can create a magnetic field if a
current is run through it. If that current
changes (as in the AC case), the magnetic
field created by the coil will change. Will
this changing magnetic field through the
coil cause a voltage to be created across the
coil? YES!
This is called self-inductance and is the basis
behind the circuit element called the
inductor.
Inductors
Since the voltage created depends in this case
on the changing magnetic field, and the
field depends on the changing current, we
have: Vinductor = -L dI /dt
where the L (called the inductance) depends
on the shape and material (just like
capacitance and resistance).
Inductors
Vinductor = -L dI /dt
Here the minus sign means that when the
current is increasing, the voltage across the
inductor will tend to oppose the increase,
and it also means when the current is
decreasing, the voltage across the inductor
will tend to oppose the decrease.
Units: Henry
From
Vinductor = -L dI /dt
L has units of Volt / [Amp/sec] which is
called a Henry:
1 Henry = Volt-sec / Amp .
This is the same unit we had for mutual
inductance before (recall the transformer).
Solenoid type inductor
For a capacitor, we started with a parallel
plate configuration since the parallel plates
provided a uniform electric field between
the plates.
In the same way, we will start with a device
that provides a uniform magnetic field
inside the device: a solenoid.
Solenoid inductor
Bsolenoid = mnI where n = N/Length .
Recall Faraday’s Law: V = d/dt [ BdA ] .
Since the I in the B is a constant with respect
to the variable of integration (dA), we have:
V = L dI/dt where L =  (mn) dA , and
since B is uniform over the area for the
solenoid (and in the same direction as area):
L = (mn)A. But the area is for each of N loops,
so: L = m N2A/Length where Aeach loop = pR2.
L = mpN2R2 / Length .
Size of a Henry
From the inductance of a solenoid:
L = mpN2R2 / Length
we see that with vacuum inside the solenoid, m
becomes mo which has a value of 4p x 10-7 T-m/A,
a rather small number. R will normally be less
than a meter, and so R2 will also make L small.
However, N can be large, and m can be a lot larger
than mo if we use a magnetic material.
Altogether, a henry is a rather large
inductance.
Inductors
Lsolenoid = mpN2R2 / Length .
As we indicated before, the value of the
inductance depends only on the shape (R,
Length, N) and materials (m).
The inductance relates the voltage across it to
the changing current through it:
V = - L dI/dt . We again use Lenz’s Law to
give us the direction (sign) of the voltage.
Energy Stored in an Inductor
We start from the definition of voltage: V =
PE/q (or PE = qV). But since the voltage
across an inductor is related to the current
change, we might express q in terms of I:
I = dq/dt, or dq = I dt. Therefore, we have:
Estored = S qi Vi =  V dq =  V I dt and now
we use VL = L dI/dt to get:
Estored =  (L dI/dt) I dt =  L I dI = (1/2)LI2.
Review of Energy in Circuits
There is energy stored in a capacitor (that has
Electric Field): Estored = (1/2)CV2 .
Recall the V is related to E.
There is energy stored in an inductor (that
has Magnetic Field): Estored = (1/2)LI2 .
Recall the I is related to B.
There is power dissipated (as heat) in a
resistor: Elost = RI2 .
Review of Circuit Elements
Resistor: VR = R I
where I = Dq/Dt
Capacitor: VC = (1/C)q
(from C = q/V)
Inductor: VL = -L DI/Dt
We can make an analogy with mechanics:
q is like x;
t is like t,
I = Dq/Dt is like v = Dx/Dt; DI/Dt is like a = Dv/Dt
V is like F;
C is like 1/k (spring);
R is like air resistance, L is like m.
RL Circuit
What happens when we have a resistor in
series with an inductor in a circuit?
From the mechanical analogy, this should
be like having a mass with air resistance. If
we have a constant force (like gravity), the
object will accelerate up to a terminal speed
(due to force of air resistance increasing up
to the point where it balances the gravity).
SF=ma  -bv - mg = ma, or
m dv/dt + bv = -mg
Mechanical Analogy: mass
falling with air resistance
20 0
15 0
10 0
ti m e in se c o n d s
6 4.0
5 6.0
4 8.0
4 0.0
3 2.0
2 4.0
1 6.0
0
8.0
50
0
sp e e d in m / s
M a s s fa l lin g w i th a ir re s i s ta n c e
RL Circuit (cont.)
If we connect the resistor and the inductor to a
battery and then turn the switch on, from the
mechanical analogy we would expect the
current (which is like velocity) to begin to
increase until it reaches a constant amount.
LR Circuit - qualitative look
From the circuit point of view, initially we
have zero current so there is no VR (voltage
drop across the resistor). Thus the full voltage
of the battery is trying to change the current,
hence VL = Vbattery, and so dI/dt = Vbattery /L.
However, as the current increases, there is
more voltage drop across the resistor, VR,
which reduces the voltage across the
inductor (VL = Vbattery - VR), and hence
reduces the rate of change of the current!
RL Circuit:
Differential Equation
To see this behavior quantitatively, we need to
get an equation.
We can get a differential equation by using
the Conservation of Energy (SVi = 0):
Vbattery - Vresistor - Vinductor = 0.
[This looks like an ordinary algebraic
equation. But all the V’s are not constants:
we have relations for Vresistor and Vinductor.]
RL Circuit:
Differential Equation
Vbattery - Vresistor - Vinductor = 0.
With Vbattery = constant, Vresistor = IR, and
Vinductor = L dI/dt, we have the differential
equation (for I(t)):
Vbattery - IR - L dI/dt = 0. This can be
rewritten as: IR + L dI/dt = Vbattery
which is an inhomogeneous first order differential
equation, just like we had for the mechanical
analogy: m dv/dt + bv = -mg .
RL Circuit (cont.)
IR + L dI/dt = Vbattery
The homogeneous equation is: LdI/dt + RI = 0.
This has a dying exponential solution:
IH(t) = Io e-(R/L) t .
The inhomogeneous equation is:
L dI/dt + RI = Vbattery .
This has the simple solution: II(t) = Vbattery / R.
RL Circuit (cont.)
I(t) = IH(t) + II(t) = Io e-(R/L) t + Vbattery / R .
To find the complete solution (that is, find Io ,
we apply the initial conditions:
I(t=0) = 0 = Io + Vbattery/R so Io = -Vbattery/R .
Therefore, we have:
I(t) = [Vbattery/R]*[1 - e-(R/L) t ] .
RL Circuit
I(t) = [Vbattery/R]*[1 - e-(R/L) t ] .
As time goes on, the current does increase and
finally reaches the value Vbattery /R which is
what it would be without the inductor
present.
The graph on the next slide shows this
function when Vbattery = 24 volts; R = 24 W,
and L = 1 H. Note that the max current is 1
Amp, and the time scale is in milliseconds.
RL Circuit: I(t) versus t
t in milliseconds
46
41
36
31
26
21
16
11
6
1.2
1
0.8
0.6
0.4
0.2
0
1
I in amps
I(t) vs t for RL Circuit
Mechanical Analogy: mass
falling with air resistance
20 0
15 0
10 0
ti m e in se c o n d s
6 4.0
5 6.0
4 8.0
4 0.0
3 2.0
2 4.0
1 6.0
0
8.0
50
0
sp e e d in m / s
M a s s fa l lin g w i th a ir re s i s ta n c e
RL Circuit
Note that this graph of I(t) versus t looks
qualitatively just like v(t) versus t for the
mass falling under constant gravity with air
resistance. According to the analogy, the
inductance acts like the mass, and the
resistance acts like the coefficient for air
resistance.
R, L and C in a circuit
We have already considered an RC circuit (in
Part 2) and an RL circuit (just now). We
looked at these cases for a circuit in which
we had a battery and then threw the switch.
However, the main reason these circuit
elements are important is in AC circuits.
We look next at the case of the three
elements in a series circuit with an AC
voltage applied.
LRC Circuit: Oscillations
Newton’s Second Law: S F = ma can be
written as: S F - ma = 0 . With a spring and
resistance, this becomes: -kx -bv -ma = 0
This is like the equation we get from
Conservation of Energy when we have a
capacitor, resistor and inductor: S V = 0 , or
(1/C)Q + RI + L dI/dt = 0 ,
where VC = (1/C)Q, VR = RI, and
VL = L dI/dt.
Resonance
If we put an inductor and a capacitor with an AC
voltage, we have the analogy with a mass
connected to a spring that has an oscillating
applied force.
In both of these cases (mechanical and
electrical), we get resonance.
We’ll demonstrate this in class with a mass and
spring.
This, it turns out, is the basis of tuning a radio!
LRC Circuit:
Differential Equation
Starting with Conservation of Energy
(SVi = 0) and using VR = IR, VC = Q/C,
VL = LdI/dt , and applying a sine wave
voltage (AC voltage), we get:
(1/C)Q + RI + LdI/dt = Vosin(wt)
Putting this in terms of Q (I = dQ/dt):
(1/C)Q + RdQ/dt + Ld2Q/dt2 = Vosin(wt)
we have a second order linear inhomogeneous differential equation for Q(t).
LRC Circuit:
Differential Equation
(1/C)Q + RdQ/dt + Ld2Q/dt2 = Vosin(wt)
We again look at the homogeneous solution
(that is, without the applied voltage), and
then try to find an inhomogeneous solution.
If the resistance is small, the homogeneous
equation becomes fairly simple
(1/C)Q + Ld2Q/dt2 = 0 and has the solution:
Q(t) = Qosin(wot) .
LC Circuit
When we substitute this expression
Q(t) = Qosin(wot)
into the diff. Eq.,
(1/C)Q + Ld2Q/dt2 = 0
we find the “natural frequency”: wo = [1/LC]
And just like the case of the mass on the
spring (mechanical analogy), when the
applied frequency approaches the “natural
frequency”, the amplitude of the resulting
oscillation gets very large (it resonates).
LRC Circuit
By placing a resistor in the circuit, the
differential equation becomes a little harder.
We need to consider either both sines and
cosines, or we need to consider exponentials
with imaginary numbers in the exponent.
This can be done, and reasonable solutions
can be found, but we will not pursue that
here. We will pursue an alternative way.
LRC Circuit:
Impedance
An alternative way of considering the LRC
Circuit is to use the concept of impedance.
The idea of impedance is that all three of the
major circuit elements impede the flow of
current.
A resistor obviously limits the current in a
circuit. But a capacitor and an inductor also
limit the current in an AC circuit.
LRC Circuit
The basic idea we will pursue is that in a
series circuit,
a) the same current flows through all of the
elements: I = Io sin(wt) ; and
b) the voltages at any instant add up to zero
around the circuit:
VR(t) + VC(t) + VL(t) = VAC(t) .
Resistance
Because of Ohm’s law (VR = IR), we see that
the current and the voltage due to the
current are in phase (that is, when the
oscillating voltage is at a maximum, the
oscillating current is also at a maximum.
VR = Vro sin(wt)
I = Io sin(wt)
Vro = IoR
Capacitive Reactance
For a capacitor: VC = (1/C)Q, and with
I = dQ/dt, or Q = I dt, if I = Iosin(wt), then
Q = (-Io/w) cos(wt), so VC = -(1/wC)Iocos(wt).
Note that cosine is 90o different than sine - we
say it is 90o out of phase. This means that VC
is 90o out of phase with the current!
Note that the constant (1/wC) acts just like R.
We call this the capacitive reactance,
XC = (1/wC) .
Voltage across the capacitor
VC = -VCo cos(wt)
I = Io sin(wt)
VCo = IoXC where XC = 1/wC
Inductive Reactance
For an inductor, VL = L dI/dt, if I =
Iosin(wt), then dI/dt = Iow cos(wt). Note
that cosine is 90o different than sine - we
say it is 90o out of phase. This means that
VL is 90o out of phase with the current.
Since VL = L dI/dt, VL = wLIo cos(wt) , and
we see that the constant wL acts just like R.
We call this the inductive reactance,
XL = wL .
Voltage across the inductor
VL = VLo cos(wt)
I = Io sin(wt)
VLo = IoXL where XL = wL
LRC in series
Note that in a series combination, the current
must be the same in all the elements, while
the voltage adds. However, the voltage
must add to zero across a complete circuit at
every instant of time. But since the voltages
are out of phase with each other, the
amplitudes of the voltages (and hence the
rms voltages) will not add up to zero!
LRC in series
For: I = Io sin(wt)
VR = IR = RIo sin(wt)
VC = (1/C)*Q = -(1/wC) Io cos(wt)
VL = L*dI/dt = (wL) Io cos(wt)
VR + VC + VL = VAC =
Io [R sin(wt) – (1/wC) cos(wt) + wL cos(wt)]
= Io Z sin(wt+θ) = Vo sin(wt+θ)
Impedance
In 2-D space, x and y are 90o apart. We
combine the total space separation by the
Pythagorean Theorem: r = [x2+y2]1/2 .
If we add up the V’s, this is equivalent to
adding up the reactances. But we must take
the phases into account. For the total
impedance, Z, we get : Vrms = Irms Z
where Z = [R2 + (wL - 1/wC)2]1/2 .
(Note that we had to subtract XL from XC because of
the signs involved.)
Resonance
Vrms = IrmsZ where Z = [R2 + (wL-1/wC)2]1/2
Note that when (wL - 1/wC) = 0, Z is smallest
and so I is biggest! This is the condition for
resonance. Thus when w = [1/LC]1/2, we have
resonance. This is the same result we would
get using the differential equation route.
Note that this is equivalent to the resonance of a
spring when w = [k/m]1/2.
V(t) versus Vrms
From Conservation of Energy, SVi = 0.
This holds true at every time. Thus:
VC(t) + VR(t) + VL(t) = Vosin(wt+qo) . But
due to the phase differences, VC will be
maximum at a different time than VR, etc.
However, when we deal with rms voltages,
we take into account the phase differences
by using the Pythagorean Theorem. Thus,
VC-rms + VR-rms + VL-rms > VAC-rms .
Net Power Delivered
What about energy delivered?
For a resistor, electrical energy is changed into heat,
so the resistor will remove power from the circuit.
For an inductor and a capacitor, the energy
is merely stored for use later. Hence, the
average power used is zero for both of
these. Note that Power = I*V, but I and V
are out of phase by 90o for both the inductor
and capacitor, so on average there is zero
power delivered.
Net Power Delivered
Thus, even though we have V=IZ as a
generalized Ohm’s Law, power is still:
Pavg = I2R
(not P=I2Z).
Computer Homework
There is a computer homework program on
Inductance on Vol. 4, #4, that gives you
practice on DC and AC behaviors of
inductors.
Magnetism in Matter
Just as materials affect the electric fields in
space, so do materials affect the magnetic
fields in space.
Recall that we described the effect of
materials on the electric fields with the
dielectric constant, K. This measured the
“stretchability” of the electric charges in the
materials. This stretching due to applied
electric fields caused electric fields itself.
Two main effects
Atoms have electrons that “orbit” the positive
nuclei. These “orbiting” electrons act like
little current loops and can create small
magnets.
Effect 1: A diamagnetic effect similar to the
dielectric effect.
Effect 2: An aligning effect.
1. Diamagnetic Effect
Effect 1: By Lenz’s law, when the applied
magnetic field changes, there is a tendency
in the circuit to resist the change. This
effect tends to create a magnetic field
opposing the change. In this effect, the
material acts to reduce any applied external
magnetic field. This is similar to the
dielectric effect that leads to the dielectric
constant for electric fields.
2. Aligning Effect
Effect 2: Since the normal “currents” due to
the “orbiting” electrons act like tiny
magnets, these magnets will tend to align
with an external magnetic field. This
tendency to align will tend to add to any
external applied magnetic field.
Heat tends to destroy this ordering tendency.
Net Result
When both of these effects (Lenz’s law and
aligning) are combined, we find three
different types of results:
1. Diamagnetic: Magnetic field is slightly
reduced in some materials
2. Paramagnetic: Magnetic field is slightly
increased in some materials
3. Ferromagnetic: Magnetic field is greatly
increased in a few materials
B, M and H
We are already familiar with B. It is called
the magnetic field or the magnetic flux
density (from its use in V = -d/dt [BdA] ).
This is the total field in space.
We have H, called the magnetic field strength
or magnetic intensity. This is the field due
to external currents only.
We have M, called the magnetization. It is
the field due to the material only.
Magnetic Susceptibility and
Magnetic permeability
To give a quantitative measure to the effects
of materials on magnetic fields, and to
relate B, M and H to one another, we define
two additional quantities:
m = magnetic permeability: B = mH , and
(mo is the value of m in vacuum)
 = magnetic susceptibility: M = H .
Further Relations
B = mH = mo(H + M) = mo(H + H)
= mo(1+)H so m = mo(1+) .
Diamagnetic materials:
<0, << 1
Paramagnetic materials:
>0, << 1
(in both cases,   10-4 so m  mo )
Ferromagnetic materials: >0, >> 1
at room Temp, only Fe, Ni and Co are ferromagnetic;
depends on how its made and past history [hysteresis];
 values can be between 100 and 100,000 !
Hysteresis
m is not constant - it depends on history
hard - permanent
soft magnet
transformer
B
B
H
H