Magnetic Circuits
and Transformers
Discussion D10.1
Chapter 6
1
Hans Christian Oersted (1777 – 1851)
X
1822
In 1820 he showed that a current
produces a magnetic field.
Ref: http://chem.ch.huji.ac.il/~eugeniik/history/oersted.htm
2
André-Marie Ampère (1775 – 1836)
French mathematics professor who only
a week after learning of Oersted’s
discoveries in Sept. 1820 demonstrated
that parallel wires carrying currents
attract and repel each other.
attract
A moving charge of 1 coulomb
per second is a current of
1 ampere (amp).
repel
3
Michael Faraday (1791 – 1867)
Self-taught English chemist and physicist
discovered electromagnetic induction in
1831 by which a changing magnetic field
induces an electric field.
A capacitance of 1 coulomb per volt
is called a farad (F)
Faraday’s electromagnetic
induction ring 4
Joseph Henry (1797 – 1878)
American scientist, Princeton University
professor, and first Secretary of the
Smithsonian Institution.
Discovered selfinduction
Built the largest
electromagnets of
his day
Unit of inductance, L, is the “Henry”
5
Magnetic Fields and Circuits
A current i through a coil produces a
magnetic flux, f, in webers, Wb.
f   B dA
i
f  BA
+
A
v
B = magnetic flux density in Wb/m2.
N
-
B  H
H = magnetic field intensity in A/m.
 = magnetic permeability
Ampere's Law:
Hl  Ni
 H dl   i
Magnetomotive force F  Ni
reluctance
F Rf
6
Magnetic Flux
Magnetic flux, f, in webers, Wb.
i2
i1
+
v1
-
Current entering
"dots" produce
fluxes that add.
+
N1
N2
v2
-
f11  flux in coil 1 produced by current in coil 1
f12  flux in coil 1 produced by current in coil 2
f21  flux in coil 2 produced by current in coil 1
f22  flux in coil 2 produced by current in coil 2
f1  total flux in coil 1  f11  f12
f2  total flux in coil 2  f21  f22
7
Faraday's Law
i2
i1
+
i
v1
+
N1
N2
-
v2
-
Total flux linking coil 1: 1  N1f1
d 1
df1
 N1
Faraday's Law: induced voltage in coil 1 is v1 (t ) 
dt
dt
Sign of induced voltage v1 is such that the current i through
an external resistor would be opposite to the current i1 that
produces the flux f1.
Example of Lenz's law
8
Symbol L of inductance from Lenz
Mutual Inductance
i2
i1
+
+
v1
N1
N2
-
v2
-
Faraday's Law
v1 (t )  N1
df1
df
df
 N1 11  N1 12
dt
dt
dt
In linear range, flux is proportional to current
di1
di2
v1 (t )  L11
 L12
dt
dt
self-inductance
mutual inductance
9
Mutual Inductance
i2
i1
+
+
v1
N1
N2
-
-
di1
di2
v1 (t )  L11
 L12
dt
dt
di
di
v2 (t )  L21 1  L22 2
dt
dt
L2  L22
v1 (t )  L1
di1
di
M 2
dt
dt
di1
di2
v2 (t )  M
 L2
dt
dt
Linear media
L12  L21  M
Let
v2
L1  L11
10
Ideal Transformer - Voltage
AC
i1
i2
+
+
v1
N1
N2
-
df
v1 (t )  N1
dt
v2
Load
-
The input AC voltage, v1,
produces a flux
1
f
v1 (t )dt

N1
f
This changing flux through
coil 2 induces a voltage, v2
across coil 2
df
v1 N1 dt
N1


v2 N 2 df N 2
dt
v2 (t )  N 2
df
dt
N2
v2 
v1
N1
11
Ideal Transformer - Current
AC
i1
i2
+
+
v1
N1
N2
-
Magnetomotive force, mmf
v2
Load
F  Ni
-
f
The total mmf applied to core is
F  N1i1  N 2i2  R f
For ideal transformer, the reluctance R is zero.
N1i1  N 2i2
N1
i2 
i1
N2
12
Ideal Transformer - Impedance
AC
i1
i2
+
+
v1
N1
v2
N2
-
-
Load
V2
ZL 
I2
N1
V1 
V2
N2
Input impedance
V1
Zi 
I1
Load impedance
2
 N1 
Zi  
 ZL
 N2 
ZL
Zi  2
n
Turns ratio
N2
I1 
I2
N1
N2
n
N1
13
Ideal Transformer - Power
AC
i1
i2
+
+
v1
N1
N2
-
Load
P  vi
-
Power delivered to primary
Power delivered to load
P2  v2i2
P1  v1i1
N2
v2 
v1
N1
v2
N1
i2 
i1
N2
P2  v2i2  v1i1  P1
Power delivered to an ideal transformer by the source
is transferred to the load.
14
L.V.D.T.
Linear Variable Differential Transformer
Position transducer
http://www.rdpelectronics.com/displacement/lvdt/lvdt-principles.htm
http://www.efunda.com/DesignStandards/sensors/lvdt/lvdt_theory.cfm
15
LVDT's are often used on clutch actuation
and for monitoring brake disc wear
LVDT's are also used for
sensors in an automotive
active suspension system
16