Sheet (2) Magnetic circuits Solutioneee.guc.edu.eg/courses/electronics/elct708 electric...dimensions are as shown in the diagram, and the depth of the core is 7 cm. The air gaps on

Sheet (2)

Magnetic circuits

Solution

Problem (1):

A two-legged core is shown in the figure. The winding on the left leg (N1) has 600

turns, and the winding on the right (N2) has 200 turns. The coils are wound in the

directions shown in the figure. If the dimensions are as shown, then what flux will

be produced by currents i1 = 0.5 A and i2 =1.0 A? Assume μr =1000 and constant.

Solution:

Draw the equivalent electric circuit:

Hint: Use right hand rule to determine the direction of the flux and consequently

the polarity of the sources.

Calculate the circuit elements:

Using OHM’s law:

∑

Faculty of Information Engineering

& Technology

Electrical Machine (ELCT708)

Dr. Adel Ahmed Fouad

Winter 2019

Problem (2):

A ferromagnetic core with a relative permeability of 2000 is shown in Figure. The

dimensions are as shown in the diagram, and the depth of the core is 7 cm. The air

gaps on the left and right sides of the core are 0.050 and 0.070 cm, respectively. If

there are 300 turns in the coil wrapped around the center leg of the core and if the

current in the coil is 1.0 A, what is the flux in each of the left, center, and right legs

of the core? What is the flux density in each air gap?

Solution:

Draw the equivalent electric circuit:

Calculate the circuit elements:

The equivalent reluctance is:

The flux in the central leg is:

The flux in the right and left legs using flux divider rule are:

The flux densities are:

Problem (3):

Calculate the flux density in the air gap of the magnetic circuit shown in the figure.

Assume μr =2000 and constant.

Solution:

Draw the equivalent electric circuit:

14 cm

12 cm

Calculate the circuit elements:

The equivalent reluctance is:

The flux in the central leg is:

The flux density in the air gap is:

Problem (4):

A core with three legs is shown in the figure. Its depth is 5 cm, and there are 200

turns on the left most leg.100 turns on the right most leg. The relative permeability

of the core can be assumed 1500 and constant. What flux exists in each of the three

legs of the core? What is the flux density in each of the legs?

Solution:

Draw the equivalent electric circuit:

Calculate the circuit elements:

Applying Mesh Analysis in the two loops:

Loop 1:

( ) (

)

( ) ( )

Loop 2:

( ) (

)

( ) ( )

Solving the equations (1) and (2):

The flux densities are:

Problem (5):

The magnetic circuit shown in the figure has two windings and two air gaps. The

core can be assumed to be of infinite permeability. The core dimensions are

indicated in the figure. Find the self-inductances of windings 1 and 2 and the

mutual inductance between the windings.

Solution:

To find the self-inductance of coil 1 ( ) we have to activate source 1 only:

𝐿

𝑁 𝑔 𝑔

𝑔 𝑔

To find the self-inductance of coil 2 ( ) we have to activate source 2 only:

To find the mutual inductance between coil 1 and coil 2 ( ) we have to

activate source 1 and find flux linking coil 2:

𝐿 𝑁

𝑔

𝐿 𝑁 𝑁

𝑔

Problem (6):

A system of three coils on an ideal core is shown in the fig, where

N1=N3=2N2=500 turns,

g1=2g2=2g3=4mm, and A=1000mm2. Calculate:

a) The self-inductance of coil N1.

b) The mutual inductance between coils N2 and N3.

Solution:

a) The self-inductance of coil N1

b) The mutual inductance between coils N2 and N3.

To get the mutual inductance activates either coil 2 or coil 3, and find the

flux flowing in the other coil due to the triggered current.

Problem (7):

The symmetric magnetic circuit shown in the figure has three windings. Windings

A and B each have N turns and are wound on the two bottom legs of the core. The

core dimensions are indicated in the figure.

a) Find the self-inductances of each of the windings.

b) Find the mutual inductances between the three pairs of windings.

Solution:

a) The self-inductances:

For coil (1) activate source 1 only:

⁄

For coil (a) activate source a only:

For coil (b) activate source b only:

b) The mutual-inductances:

Between coil (1) and coil (a):

⁄

⁄

⁄

Due to symmetry:

Between coil (a) and coil (b):

( )

𝐿 𝑎

𝑁 𝑁𝑎 𝑔 𝑎 ⁄

𝐿 𝑎 𝐿 𝑏

( ) ( )

( ) ( )

( )

𝐿𝑎𝑏

𝑁 𝑔

𝑎 ( 𝑔 𝑎)