UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
LECTURE NOTES 22
Inductance: Mutual Inductance and Self-Inductance
Inductance is the magnetic analog of capacitance in electric phenomena. Like capacitance,
inductance has to do with the geometry of a magnetic device and the magnetic properties of the
materials making up the magnetic device. The capacitance C of an electric device is associated
with the ability to store energy in the electric field of that device. The inductance L of a
magnetic device is associated with the ability to store energy in the magnetic field of that device.
Mutual Inductance:
Suppose we have two arbitrary shaped loops of wire (both at rest) in proximity to each other,
as shown in the figure below. Suppose Loop # 1 carries a current I1(t).
B1 ( r , t )
Loop # 2
Loop # 1
I1 ( t )
The current I1 ( t ) flowing in Loop # 1 produces a magnetic field B1 ( r , t ) , and some of these
magnetic field lines will pass through Loop # 2, linking it.
The magnetic flux from Loop #
1 linking Loop # 2 is:
Φ12
m ( t ) = ∫ B1 ( r , t )i da⊥ 2
S2
Where:
⎛μ
B1 ( r , t ) = ⎜ o
⎝ 4π
d 1′ × rˆ
⎞
⎟ I1 ( t ) ∫ C1
r2
⎠
Note that B1 ( r , t ) is linearly
proportional to I1(t).
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
1
UIUC Physics 435 EM Fields & Sources I
Then:
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
⎛
d 1′ × rˆ ⎞
⎛ μo ⎞
I1 ( t ) ∫ ⎜ ∫
Φ12
⎟ida⊥2
m ( t ) = ∫ B1 ( r , t )i da⊥ 2 = ⎜
⎟
2
S2
S2
C
⎝ 4π ⎠
⎝ 1 r
⎠
Now irrespective of the details of doing the double integral in the above formula, we know that
that B1 ( r , t ) is linearly proportional to I1(t) and thus to is Φ12
m ( t ) , i.e.:
Φ12
m ( t ) ≡ M 12 I1 ( t )
The magnetic flux from Loop 1 linking Loop 2 is:
d 1 × rˆ ⎞
⎛μ ⎞ ⎛
Where the constant of proportionality: M 12 = ⎜ o ⎟ ∫ ⎜ ∫
⎟ida⊥2 =
2
⎝ 4π ⎠ S2 ⎝ C1 r
⎠
Mutual Inductance of
Loop 1 and Loop 2
what is this physically?
Equivalently, note that we can also obtain M 12 using: Φ12
m (t ) =
∫
C2
A1 ( r , t )id
2
{obtained previously, using B = ∇ × A }
⎛μ
Where: A1 ( r , t ) = ⎜ o
⎝ 4π
Then:
Φ12
m (t ) =
∫
C2
d 1
⎞
← Note A1 ( r , t ) is also linearly proportional to I1 ( t )
⎟ I1 ( t ) ∫ C1
r
⎠
A1 ( r , t )id
Then if: Φ12
m ( t ) = M 12 I1 ( t )
2
2
⎛μ
=⎜ o
⎝ 4π
⎞
⎟ I1 ( t ) ∫ C2
⎠
∫
d 1 id
⎛μ ⎞
Then: M 12 = ⎜ o ⎟ ∫
⎝ 4π ⎠ C2
2
r
C1
∫
d 1 id
C1
r
2
⇐
Known as
Neumann’s Formula
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
⎛μ ⎞
Thus we see that: M 12 = ⎜ o ⎟
⎝ 4π ⎠
∫ ∫
C2
Fall Semester, 2007
d 1 id
2
r
C1
∫ ∫
C2
d 1 id
C1
r
2
Prof. Steven Errede
d 1 × rˆ ⎞
⎛μ ⎞ ⎛
= ⎜ o ⎟∫ ⎜ ∫
⎟ida⊥2
2
⎝ 4π ⎠ S2 ⎝ C1 r
⎠
Depends only on geometry;
Has dimensions of length.
We also see that:
Lecture Notes 22
Depends only on geometry;
Has dimensions of length.
⎛
d 1 × rˆ ⎞
=∫ ⎜∫
⎟ida⊥2
2
S2
C
⎝ 1 r
⎠
⎛ Webers ⎞
The SI units of mutual inductance: M 12 = Henrys = ⎜
⎟
⎝ Ampere ⎠
Φ12
Flux
Webers Tesla − m 2
m (t )
=
=
= Henrys
Φ ( t ) = M 12 I1 ( t ) →
= M 12 =
Current Ampere
Ampere
I1 ( t )
12
m
Note also that: μo = 4π × 10−7
N
= 4π × 10−7 Henrys
⇒ M 12 = Henrys
meter
A2
Why is M 12 called the mutual inductance of a two-circuit system?
Let’s reverse the roles of Loop 1 and Loop 2 – i.e. Loop 2 carries current I2
(I2 not necessarily = I1 in original situation).
The magnetic flux from Loop 2 linking Loop 1: Φ 21
m (t ) =
⎛μ
With: B2 ( r , t ) = ⎜ o
⎝ 4π
d 2 × rˆ
⎞
⎟ I 2 ( t ) ∫ C2
r2
⎠
and
∫
C1
A2 ( r , t )id
⎛μ
A2 ( r , t ) = ⎜ o
⎝ 4π
1
= ∫ B2 ( r , t )ida⊥1
S1
d 2
⎞
⎟ I 2 ( t ) ∫ C2
r
⎠
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
3
UIUC Physics 435 EM Fields & Sources I
⎛μ
Then: Φ m21 ( t ) = ⎜ o
⎝ 4π
Fall Semester, 2007
⎡ ⎛
⎤ ⎛ μo
d 2 × rˆ ⎞
⎞
I
t
da
i
(
)
⎢
⎜
⎟
⎟ 2
∫S ∫C 2 ⎠ ⊥1 ⎥⎥ = ⎜⎝ 4π
⎠
⎣⎢ 1 ⎝ 2 r
⎦
Lecture Notes 22
⎡
⎞
⎟ I 2 ( t ) ⎢ ∫ C1
⎠
⎣
∫
C2
d 2 id
r
Prof. Steven Errede
1
⎤
⎥
⎦
Again, define Φ 21
m ( t ) ≡ M 21 I 2 ( t ) with constant of proportionality = mutual inductance M 21
⎛μ
M 21 = ⎜ o
Φ 21
m ( t ) Loop 2→ Loop 1:
⎝ 4π
But:
d 2 id
⎞⎡
⎟ ⎢ ∫ C1 ∫ C2
r
⎠⎣
⎛μ
M 12 = ⎜ o
Φ12
m ( t ) Loop 1→ Loop 2:
⎝ 4π
⎞⎡
⎟ ⎢ ∫ C2
⎠⎣
∫
d 1 id
C1
r
1
2
⎤ ⎛ μo
⎥=⎜
⎦ ⎝ 4π
⎤
d 2 × rˆ ⎞
⎞⎡ ⎛
da
i
⎥
⎟
⊥
⎟ ⎢ ∫S1 ⎜ ∫ C2
1
r2 ⎠
⎠ ⎣⎢ ⎝
⎦⎥
⎤ ⎛ μo
⎥=⎜
⎦ ⎝ 4π
⎤
d 1 × rˆ ⎞
⎞⎡ ⎛
⎟ida⊥2 ⎥
⎟ ⎢ ∫S2 ⎜ ∫ C1
2
r
⎠ ⎢⎣ ⎝
⎥⎦
⎠
Thus we see that M 12 ≡ M 21 !!! Hence why it is known as the mutual inductance of Loop # 1
with Loop # 2 (or vice versa)!! Thus, we don’t need subscripts M 12 = M 21 = M
If one thinks about it, the fact that M 12 ≡ M 21 is not a trivial consequence.
If I1 = I 2 = I , independent of the geometrical shapes / configurations of the two loop circuits,
it is not immediately obvious that:
[magnetic flux Φ12
m ( t ) (due to current I1 = I (t) in Loop 1) linking Loop 2]
= [magnetic flux Φ 21
m ( t ) (due to current I2 = I (t) in Loop 2) linking Loop 1].
If I1 = I 2 = I then:
Φ12
m ( t ) = M 12 I1 ( t ) = M 12 I ( t )
Φ 21
m ( t ) = M 21 I 2 ( t ) = M 21 I ( t )
And since M 12 = M 21 = M
21
then Φ12
m (t ) ≡ Φm (t ) .
This is a consequence of the Reciprocity Theorem.
The underlying physics of the Reciprocity
Theorem has to do with the intrinsic /
fundamental properties of the
electromagnetic interaction at the
microscopic / particle physics level – i.e.
exchange of virtual photons between
electrically-charged particles, as well as
the fundamental symmetry principles
obeyed by the EM-interaction:
Charge Conjugation
(C)
Parity (Space Inversion) (P)
Time Reversal
(T)
4
The EM interaction is invariant under C, P and T.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
Also, the intrinsic properties / nature of our 3-space dimensional and 1-time dimensional
universe is also important e.g. 3-D space is isotropic – not anisotropic.
The Reciprocity Theorem has many consequences in all branches of physics / chemistry /
science. It is also relevant e.g. in optics (i.e. visible light/real photons). Nearly everything in the
“everyday world” deals with the EM interaction at the microscopic scale…
For two magnetically-coupled / flux linked circuits, the Reciprocity Theorem also predicts that:
∂Φ12
∂I1 ( t )
∂Φ m21 ( t )
∂I ( t )
m (t )
ε mf ε 2 ( t ) = −
= − M 12
= − M 21 2
≡ ε mf ε1 ( t ) = −
∂t
∂t
∂t
∂t
If I1 ( t ) = I 2 ( t ) = I ( t ) then:
∂I1 ( t ) ∂I 2 ( t ) ∂I ( t )
=
=
, then since: M 12 = M 21 = M
∂t
∂t
∂t
∂I ( t )
∂t
∂I ( t )
= ε mf ε1 ( t ) = − M
∂t
Then: ε mf ε 2 ( t ) = − M
by the
Reciprocity
Theorem
in Loop 1
in Loop 2
∂I ( t )
= 1 Amp/sec change in the current flowing in Loop 1 with M = 1 Henry of mutual
∂t
inductance between Loop 1 and Loop 2 will produce an EMF ε 2 = 1 Volt in Loop 2, which is
i.e. a
∂I ( t )
= 1 Amp/sec change in the current flowing in Loop 2 with M = 1 Henry of mutual
∂t
inductance between Loop 1 and Loop 2 will produce an EMF, ε1 = 1 Volt in Loop 1.
= to a
Thus, the EMF induced in a loop b due to a time-varying current flowing in loop a producing a
time-varying magnetic field linking both loops via their mutual inductance, M is:
ε mf ε b ( t ) = −
∂Φ m ( t )
∂I ( t )
= −M a
∂t
∂t
This phenomenon then provides a convenient way for us to measure / determine the mutual
inductance, M of two circuits – i.e. we can compute the mutual inductance from:
M=
εb (t )
∂I a ( t )
∂t
measure in loop b
put in known (or measured)
∂I a ( t )
into loop a
∂t
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
5
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
As one might realize, e.g. from our previous experience(s) with dielectric and magnetic
media, if loops / circuits 1 and 2 are both uniformly embedded inside a magnetically permeable
material (with magnetic permeability μ = μo (1 + χ m ) = K m μo ) then for linear magnetic materials
with μ = K m μo we would expect:
⎛ μ
Φ bmμ ( t ) = ⎜
⎝ 4π
⎛ μ
Mμ = ⎜
⎝ 4π
where:
⎡
⎞
⎟ I a ( t ) ⎢ ∫ Cb
⎠
⎣
⎞⎡
⎟ ⎢ ∫ Cb
⎠⎣
∫
d
∫
d
a
id
b
r
Ca
b
r
Ca
a
id
⎤
⎥ = M μ Ia (t )
⎦
⎤
⎥
⎦
vs. that for non-magnetic materials:
⎛μ
Φ bmμ ( t ) = ⎜ o
o
⎝ 4π
⎛μ
M μo = ⎜ o
⎝ 4π
where:
⎡
⎞
⎟ I a ( t ) ⎢ ∫ Cb
⎠
⎣
⎞⎡
⎟ ⎢ ∫ Cb
⎠⎣
∫
∫
d
a
r
Ca
d a id
b
r
Ca
id
b
⎤
⎥ = M μo I a ( t )
⎦
⎤
⎥
⎦
Thus, we see that: M μ = K m M μo .
For example, for soft/annealed iron K mFe ≡ μ Fe μo 1000 . Then for two circuits embedded in
soft/annealed iron, their mutual inductance is also correspondingly increased by this same factor:
M μ = K m M μo
1000M μo .
This also implies that magnetic fluxes are also correspondingly increased: Φ bmμ ( t ) = K m Φ mμo ( t )
and the induced EMF’s in the loops are also increased by the same factor, since:
ε μ (t ) = −
b
∂Φ bmμ ( t )
∂t
= K mε μo ( t ) = − K m
b
∂Φ bmμ ( t )
o
∂t
⇒
ε μb ( t ) = K mε μb ( t ) = 1000ε μb ( t )
o
o
Of course, the above results implicitly assume that the flux-linking geometries of the two circuits
are identical – with and without the presence of the magnetically permeable material.
Thus, we see that the use of highly magnetically permeable materials ( μ μo ) can dramatically
improve the magnetic coupling between two circuits, over and above the free-space μo value!!
For the long solenoid, this also means that a high magnetic permeability flux return (external to
solenoid) from one end to other must also be provided to “complete” the magnetic circuit.
If only a magnetic core is used inside the solenoid, then:
6
1.7
Mμ
M μo
core
only
⎛L⎞
1.3 × ⎜ ⎟
⎝D⎠
where D = 2R
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
The Mutual Inductance Between Two Long Coaxial Solenoids
Take a long air-core solenoid of length L and radius R wound with n1 = NTOT1 L turns/meter.
Then wind a second winding over the first winding of the solenoid with n2 = NTOT2 L turns/meter.
If we put a steady current I through the 2nd (i.e. outer) winding of the long solenoid, the magnetic
field inside the bore of the outer solenoid is:
enclosed
B2inside = μo n2 Izˆ (Using Ampere’s Circuital Law ∫ Bid = μo ITot
)
The magnetic flux through the bore of the outer solenoid is:
Φ 2 = B2 i A2⊥ = B2π R 2 = μo n2π R 2 I
where A2⊥ = A2⊥ zˆ = π R 2 zˆ
However, this same magnetic flux links each and every one of the NTOT1 turns of the inner
solenoid winding (if the two windings are close-packed / carefully wound).
Thus, the magnetic flux (arising from I flowing in the outer solenoid winding) that links one turn
of the inner solenoid winding is:
Φ1one turn = B2 i A⊥1 = μo n2π R 2 I = Φ 2
But the inner solenoid has NTOT1 total number of turns, thus the total magnetic flux (arising from
current I flowing in the outer solenoid winding) linking NTOT1 total number of turns of the inner
solenoid is:
Φ1TOT = NTOT1 Φ1one turn = μo NTOT1 n2π R 2 I
∴ Φ
TOT
1
NTOT1
L
= μo ⎡⎣ n1n2π R L ⎤⎦ I
2
Current I flowing in
outer solenoid winding
Total magnetic flux linking
inner solenoid winding
But Φ1TOT = MI
but n1 =
⇒ mutual inductance of two long coaxial solenoids: M = μo ⎡⎣ n1n2π R 2 L ⎤⎦
Notice that:
1
⎡⎣ n1n2π R 2 L ⎤⎦ has dimensions of length, and =
4π
∫ ∫
C2
d 1 id
C1
r
2
=
1
4π
⎛
∫ ⎜⎝ ∫
S2
C1
d 1 × rˆ ⎞
⎟ida !!!
r 2 ⎠ ⊥2
μo = 4π ×10−7 Henrys/meter → M in Henrys
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
7
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
In the real world, it is very difficult to build any two-coil device (e.g. a transformer!!) with
perfect, 100.0000% magnetic flux linking / coupling between the two windings. It is always
some fraction of 100%.
Can define an efficiency of magnetic coupling of the two windings to each other, ∈ : 0 ≤∈≤ 1
= M actual I =∈ M ideal I =∈ μo ⎡⎣ n1n2π R 2 L ⎤⎦ I
Then: M actual =∈ M ideal =∈ μo ⎡⎣ n1n2π R 2 L ⎤⎦ and Φ1TOT
actual
Then:
(1− ∈) = inefficiency of magnetic coupling / flux linkage between the two windings
Define: Φ leakage ≡ (1− ∈) μo ⎡⎣ n1n2π R 2 L ⎤⎦ I = Leakage flux not coupled from one winding to the
other.
The value of ∈ in an actual device depends on the details of the design (and who made it)...
Generally speaking, everyone wants ∈≡ 100.000% . Manufacturers try to achieve this, but the
old adage: “Yous gits what yous payz for” is true . . .
Significant leakage flux in a transformer (a 2-winding
magnetically-coupled circuit) will adversely affect the
high-frequency response of the transformer output
vs. input for a real transformer.
The mutual inductance for two coils tightly wound together on a long solenoid is:
M =∈ μo n1n2π R 2 L (Henrys)
Note that M > 0 i.e. the mutual inductance M is always a positive quantity.
Note that how the two separate coils are wired up into a larger circuit does matter – on the sign
(i.e. polarity) of the voltage output from one coil relative to the voltage input of the other – i.e.
the relative phase polarity of the two coils.
Schematically, this phase polarity is indicated by black dots on the circuit diagram for two
magnetically coupled circuits (i.e. transformer) as follows:
The arrows indicate direction of positive current flow.
Coil
winding
#1
Current I1 enters coil winding #1 through lead #1
(with black dot), leaves through lead #2
Coil
Winding
#2
8
The direction of induced current flow in coil winding
#2 is shown.
When the voltage at point 1 on coil # 1 goes positive,
the voltage at point 3 on coil 2 also goes positive.
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
By the Reciprocity Theorem, either coil #1 or coil #2 could be viewed as the “primary”
winding, the other coil would then be the “secondary” winding (or vice versa).
If two long solenoid coils are coaxially wound together on a magnetically permeable core
( μ = μo K m = μo (1 + χ m ) ) , then:
M μ =∈ μ n1n2π R 2 L = K m M μo
M μo if μ
μo
n.b. adding μ also tends to improve ∈ 100% !!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
9
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
Self-Inductance – a.k.a. “Inductance”:
As we have seen in two-coil magnetically coupled circuits, a time-varying current with
∂I ( t ) ∂t in one coil induces an EMF ε = − M ( ∂I free ( t ) ∂t ) in the other coil due to their mutual
inductance, M.
Similarly, a time-varying current with ∂I free ( t ) ∂t in an isolated / single coil also induces a
so-called BACK EMF (i.e. a reverse EMF) in itself, due to Lenz’s Law (i.e. the coil tries to
maintain constant magnetic flux – maintain the flux status quo).
The Self-Inductance of a Long Solenoid
For a long solenoid the magnetic flux in the bore (cross-sectional area) of the long solenoid
(of length , radius R and n = NTOT turns/meter) is Φ m = μo nπ R 2 I free where A⊥solenoid = π R 2 .
However, this magnetic flux links each and every turn of the solenoid (ideally):
Total magnetic flux linking all NTOT turns
Magnetic flux linking one turn
= NTOT Φ m = μo NTOT nπ R 2 I free with n = NTOT
Then: ΦTOT
m
= ⎡⎣ μo n 2π R 2 ⎤⎦ I free
Thus: ΦTOT
m
is linearly proportional to I free , i.e.:
We see here again, for a single solenoid coil, that ΦTOT
m
ΦTOT
≡ LI free where the constant of proportionality (here) is: L = μo n 2π R 2 (Henrys)
m
The quantity L is known as the self-inductance of the long solenoid. SI units are (also) Henrys.
∂ΦTOT
( t ) = − L ∂I free ( t )
m
BACK EMF in a coil due to its self-inductance: ε ( t ) = −
∂t
∂t
Due to
Lenz’s Law
A two-lead device with many turns of wire – having much self-inductance is called an inductor.
{Analogous to a two-lead device that can store much charge – called a capacitor.}
For the long solenoid, unless a good / high-permeability external magnetic flux return is
provided, then using only a magnetic core:
1.7
Lμ
Lμo
10
core
only
⎛ ⎞
1.3 ⎜ ⎟
⎝D⎠
where D = 2R (Diameter)
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
The Self-Inductance of a Toroidal Coil
Dimensions of
Rectangular Toroid:
- Inner radius a
- Outer radius b
- Height h
- Toroid has NTOT turns
3-D View:
Direction of
Bm and
closed contour
path of
Integration, C
Cross-Sectional View:
From Ampere’s Circuital Law
(∫
C
)
enclosed
ITOT
( t ) = NTOT I free ( t )
enclosed
B ( r , t )id = μo ITOT
(t ) :
⎛μ
The magnetic field inside the bore of the toroid: Bin ( ρ , t ) = ⎜ o
⎝ 2π
⎞ NTOT I free ( t ) ˆ
ϕ , ρ = x2 + y 2
⎟
ρ
⎠
(in cylindrical coordinates)
The magnetic flux through the bore of the toroid is:
Φ m ( t ) = ∫ B ( ρ , t )ida⊥ with da⊥ = d ρ dzϕˆ
S⊥
⎛μ
=⎜ o
⎝ 2π
⎛μ
=⎜ o
⎝ 2π
ρ =b 1
z =+ h
⎞
2
⎟ NTOT I free ( t ) ∫z =− h ∫ρ = a d ρ dz
ρ
2
⎠
h
z =+
⎞
2
⎟ NTOT I free ( t ) ∫z =− h ⎡⎣ln ( b ) − ln ( a ) ⎤⎦ dz
2
⎠
( a)
= ln b
( )
⎛μ
=⎜ o
⎝ 2π
⎞
b
h
h
⎟ NTOT I free ( t ) ln a ⎡⎣ 2 + 2 ⎤⎦
⎠
magnetic flux linking
⎛μ ⎞
Thus: Φ m ( t ) = ⎜ o ⎟ NTOT I free ( t ) h ln b a = Φ1m ( t ) =
one turn of the toroid.
⎝ 2π ⎠
( )
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
11
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
But (here again) the magnetic flux inside the bore of the toroid links each and every turn of the
toroid (ideally).
μ
Thus: ΦTOT
( t ) = NTOT Φ1m ( t ) = ⎛⎜ o
m
⎝ 2π
( )
⎞ 2
b
⎟ NTOT I free ( t ) h ln a
⎠
=
total magnetic flux
linking all NTOT turns
( )
⎛μ ⎞ 2
h ln b
⇒ Self-inductance of rectangular toroid: L = ⎜ o ⎟ NTOT
a
⎝ 2π ⎠
∂ΦTOT
( t ) = − L ∂I free ( t ) = − ⎛ μo ⎞ N 2 h ln b ∂I free ( t )
m
Back EMF in Toroid: ε mf ε ( t ) = −
⎜
⎟ TOT
a
∂t
∂t
∂t
⎝ 2π ⎠
But: ΦTOT
( t ) = LI free ( t )
m
( )
Due to the nature of the toroid’s excellent magnetic self-coupling (toroid = solenoid bent back on
itself), if the toroid coil is wound on magnetically permeable core (of magnetic permeability
μ = K m μo = μo (1 + χ m ) ), then:
Air core toroid
( )
⎛ μ ⎞ 2
TOT
b
ΦTOT
mμ ( t ) = ⎜
⎟ NTOT h ln a I free ( t ) = K m Φ mμo ( t )
⎝ 2π ⎠
⎛ μ ⎞ 2
b
Lμ = ⎜
Magnetic core toroid
⎟ NTOT h ln a = K m Lμo
⎝ 2π ⎠
∂I ( t )
∂I free ( t )
ε mf ε μ ( t ) = − Lμ free
= − K m Lμo
= K m ε μo ( t )
∂t
∂t
( )
For inductors with magnetically permeable cores, the magnetization Μ and hence μ , χ m are
often (not all) frequency dependent!! Frequency dependence can be significant (100 % or more)
(over wide frequency range) and depends on the microscopic (and (eddy currents) macroscopic)
details of the magnetically permeable material(s) being used.
Magnetic materials have various magnetic dissipation mechanism(s) which can be/are frequency
dependent…
Furthermore, magnetic dissipation mechanisms can be/are also dependent on the strength of Bin
(Since Μ =
χm
B is a non-linear relationship, e.g. for soft iron) thus magnetic dissipation is
μ
frequently a non-linear function of the magnetization, Μ .
12
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Lecture Notes 22
Prof. Steven Errede
The Ideal Transformer
An ideal transformer is a crude representation of a real transformer; nevertheless it is a very
useful approximation to a real one. A simple (and very common version) of a real transformer is
one which has two windings that are wound on a magnetically-permeable core (such as iron),
such as is shown in the figure below:
Winding # 1
N1 total turns
Winding # 2
N2 total turns
For an idealized version of this transformer, we make the following simplifying assumptions:
(1) Each of the two coil windings have no resistance.
(2) There are no Eddy-current and/or hysteresis Joule-heating losses in the magnetic core of the
transformer. The hysteresis loop for the core is then a straight line through the origin, then B
is linearly proportional to H, which is in turn proportional to I, thus B is then proportional to I
(3) All of the magnetic flux is confined to the magnetic core – i.e. there is no leakage flux, and
thus the magnetic flux through one winding is the same as that through the other winding.
Since the two windings are perfectly magnetically coupled to each other in the ideal transformer,
then Φ1m1 ( t ) = Φ1m2 ( t ) ≡ Φ1m ( t ) = magnetic flux passing through one loop of winding # 1 (or # 2).
The mutual inductance of the two windings of the ideal transformer is M, and if e.g. a function
generator is connected to winding # 1, then a potential difference (i.e. a voltage) ΔV1 ( t ) is
present and thus a free current I1 ( t ) will flow in winding # 1; thus a magnetic field B1 ( t ) will be
present in the magnetic core of the ideal transformer. If the magnetic core of the ideal
transformer has a cross sectional area Acore , then a magnetic flux of Φ1m1 ( t ) = ⎡⎣ B1 ( t ) Acore ⎤⎦ is
present in the core of the ideal transformer.
1
Then the total magnetic flux through winding # 1 is: ΦTot
m1 ( t ) = N1Φ m1 ( t ) = N1 ⎡
⎣ B1 ( t ) Acore ⎤⎦
But: ΔV1 ( t ) = −
∂ΦTot
m1 ( t )
∂t
= N1
∂Φ1m1 ( t )
∂t
= ε1 ( t )
However since the two windings of the ideal transformer are perfectly magnetically coupled, i.e.:
Φ
1
m1
(t ) = Φ (t ) ≡ Φ (t )
1
m2
1
m
then:
∂Φ1m1 ( t )
∂t
=
∂Φ1m2 ( t )
∂t
∂Φ1m ( t )
=
.
∂t
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.
13
UIUC Physics 435 EM Fields & Sources I
Fall Semester, 2007
Then the resulting EMF induced in winding # 2 is: ε 2 ( t ) = −
Since:
Thus:
∂Φ1m1 ( t )
∂t
∂Φ1m1 ( t )
∂t
=
=
∂Φ1m2 ( t )
∂t
∂Φ1m2 ( t )
∂t
=
Lecture Notes 22
∂ΦTot
m2 ( t )
∂t
= N2
Prof. Steven Errede
∂Φ1m2 ( t )
∂t
= ΔV2 ( t )
∂Φ1m2 ( t ) ΔV2 ( t )
∂Φ1m1 ( t ) ΔV1 ( t )
∂Φ1m ( t )
=
=
and
and
∂t
∂t
∂t
N2
N1
∂Φ1m ( t ) ΔV2 ( t ) ΔV1 ( t )
ΔV2 ( t ) ΔV1 ( t )
=
=
=
=
or:
∂t
N2
N1
N2
N1
ε 2 ( t ) ΔV2 ( t ) N 2
=
=
= Turns Ratio of ideal transformer = Constant
ε1 ( t ) ΔV1 ( t ) N1
Or:
Each winding of the ideal transformer has its own associated self-inductance, L and for each
∂ΦTOT
∂I ( t )
mi ( t )
TOT
= − Li i
= ΔVi ( t ) .
winding, i = 1,2: Φ mi ( t ) = Li I i ( t ) and ε mf ε i ( t ) = −
∂t
∂t
As we saw in the case of the rectangular toroid (with soft-iron core), the self-inductances
associated with each of the two windings of the ideal transformer are proportional to the square
of the number of turns of their windings, i.e. L1 ∼ N12 and L2 ∼ N 22 . Then we can also see that:
ε 2 ( t ) ΔV2 ( t ) N 2
L2
=
=
=
L1
ε1 ( t ) ΔV1 ( t ) N1
If the ideal transformer is lossless, then electrical power in winding # 1 can be transferred
with 100% efficiency to winding # 1 (and vice-versa) (n.b. microscopically, all energy/power is
transferred from one winding to the other via virtual photons), and thus:
P1 ( t ) = ΔV1 ( t ) I1 ( t ) = ΔV2 ( t ) I 2 ( t ) = P2 ( t ) or:
Then we see that:
ΔV2 ( t ) I1 ( t )
=
ΔV1 ( t ) I 2 ( t )
ε 2 ( t ) ΔV2 ( t ) I1 ( t ) N 2
L2
=
=
=
=
and that N1 I1 ( t ) = N 2 I 2 ( t )
L1
ε1 ( t ) ΔV1 ( t ) I 2 ( t ) N1
Thus for an ideal transformer, e.g. if winding # 1 has many turns and winding # 2 has few turns,
the above formulae tell us that for N1 N 2 (i.e. a so-called “step-down” transformer):
ΔV1 ( t )
ΔV2 ( t ) and I1 ( t )
I 2 ( t ) with P1 ( t ) = ΔV1 ( t ) I1 ( t ) = ΔV2 ( t ) I 2 ( t ) = P2 ( t ) ;
whereas if winding # 1 has few turns and winding # 2 has many turns, then for N1
(i.e. a so-called “step-up” transformer):
ΔV1 ( t )
14
ΔV2 ( t ) and I1 ( t )
N2
I 2 ( t ) with P1 ( t ) = ΔV1 ( t ) I1 ( t ) = ΔV2 ( t ) I 2 ( t ) = P2 ( t ) .
n.b. By the
Reciprocity
Theorem:
A step-up
transformer
run backwards
= a step down
transformer!!!
© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2005-2008. All Rights Reserved.