e.p. 155.3: electric and magnetic circuits...

E.P. 155.3: Electric and Magnetic Circuits I

Lecture 24

April 5th, 2005

Magnetic Circuits &

Inductance Reading:

Boylestad’s Circuit Analysis, 3rd Canadian Edition

Chapter 11.1-11.5, Pages 331 - 338 Chapter 12.1-12.4, Pages 341 - 349 Chapter 12.7-12.9, Pages 363 – 367

Note: We do NOT cover RL Transients (12.5-6) Assignment: Assignment #11 Due: N/A


April 5th, 2005 Magnetic Circuits & Inductance 2

Magnetic Circuits &

Inductance

Electric Circuit Magnetic Circuit

Name Equation Units Units Equation Name Electric Current

RV

I = A ⇔ Wb

ℜℑ=Φ

Magnetic Flux

Electromotive Force (emf)

V V ⇔ At NI=ℑ Magneto-motive force (mmf)

Resistance

Al

R ρ= Ω ⇔ At/Wb

Al

Rµ

= Reluctance

Permittivity ε0 F/m ⇔ Wb/Atm µ0 Permeability Capacitance C Farads ⇔ Henries L Inductance τ RC seconds ⇔ seconds R/L τ Electric flux density

AD

Ψ= ⇔ WB/m2

AΦ=Β

Magnetic Flux density

Electric Field Intensity

dV

E = V/m At/m

lH

ℑ= Magnetic Field Intensity

Aside: The following 3 constants are exact:

Permittivity of free space: ε0 = 8.85418781762 x 10-12 F/m Permeability of free space: µ0 = 4π x 10-7 H/m Speed of light: c = 299 792 458 m/s Now

ε0µ0=1/c2

Coincidence???



Magnetic Circuits & Inductance: In what follows a new circuit element for use in E.P.

155.3 is going to be introduced. It is called an inductor. You will notice during the development that inductance is in some ways similar to previous circuit elements while at the same time having some “unusual” characteristics! Magnetic circuit analysis is difficult and many assumptions are made. Magnetism:

You probably have noticed at some point in your life that opposite poles of magnets attract (and that similar poles of magnets repel). This implies that there exists a magnetic field, analogous to an electric field, responsible for the force.



It is noted that magnetic field lines always go from the north pole of a magnet to the south pole of the magnet. Unlike electric field lines, which start on one charge and terminate on another charge, they do not start and end on each pole but are closed loops.

Some materials, called ferromagnetic materials, have

natural magnetic properties; the classic example is iron. They can affect neighbouring magnetic fields by “capturing” the magnetic field and causing it to bend. Of course there are other materials, such as plastic, that possess no magnetic properties and have no effect on the magnetic field lines.



Electromagnetism:

It was noticed in past history that moving charge generates a magnetic field and therefore currents generate a magnetic field. This is the concept that electric motors operate on. The input electrical energy generates a current that creates a magnetic field that interacts with permanent magnets to turn the motor. The Permanent Magnet Moving Coil (PMMC) discussed as part of your E.P. 155.3 meter lectures works on this concept (although on a much smaller scale than an electric motor).

The magnetic field strength is given the symbol B and

has units of Teslas (T). B is also called the magnetic flux density (recall the previous information on electric flux density). Thus 1 Tesla = (1 Weber/m2) = (1 Wb/m2).

Like the electric field E, the B field is a vector field so at all points in space it has a magnitude and direction. Unlike the electric field which is set up due to stationary charge, the magnetic field is set up due to moving charge.



Current Carrying Wire:

Consider a current carrying wire (current is moving charge).

p

r

I At point p, a perpendicular distance r from a long

current carrying wire, the magnetic field strength (magnetic flux density) is

rI

Bπµ2

=

where µ is the permeability of the medium between the wire and the point p, and I is the current in the wire. Note that while we would like µ to be constant is it not, especially when we talk about ferromagnetic materials (in part due to an effect called saturation). In the examples and problems that are given as part of E.P. 155.3, the µ values used for the various materials were obtained from the Normal Magnetization Curves (I & II) on Page 37.



Example #1: A long straight conductor carries a current of 100A.

At what distance from the conductor is the magnetic field caused by the current equal in magnitude to the earth’s magnetic field in Saskatoon (about 0.5 x 10-5 T)?



Now we have to determine the direction. This is where the concept of a magnetic field starts to diverge from that for an electric field (if it hasn’t already).

The direction of the magnetic field is given by the

“Right Hand Rule” (RHR). Point your thumb of your right hand in the direction of the current and your fingers will curl in the direction of the magnetic field. In order for us to visualize this we also use arrows to help. If an arrow was coming straight at you would see the tip (designated •) while if it was going away from you would see notch end (designated ××). An example diagram indicating the arrow convention is shown below.

Note that the above diagram indicates something

other than B (i.e., Φ). More on this very soon.



Thus for the current carrying wire you would see

p

r

I An end view would show the following

Note that the farther we move away from the wire the

smaller the magnetic field strength (and magnetic flux density).



Current Loop: If we loop the wire with the current in it

The direction of the magnetic field (RHR still applies)

is as shown. Since B is a magnetic flux density, define Φ as the

total flux with units Webers (Wb). A good way to visualize this is that the total flux is the number of magnetic field lines through a surface.

AB

Φ= .

If the current loop is coiled to form more loops (for a total of N loops) as below

What happens to the total flux?



Note that for an air coil, not all of the flux passes through all of the loops of the coil (since some of it “escapes” between the loops). If iron is used in the center of the coil (the core) then much more flux will pass through all of the loops since iron tends to “capture” the flux. The flux that passes through all of the loops is called the “flux linkage”. It is the flux linkage of a magnetic circuit that gives it its property of inductance.

Magnetic Circuits:

The flux producing ability of a coil is called its magneto-motive force (mmf). This is simply the current through the wire multiplied by the number of turns in the coil

NI=ℑ The units are Ampere-turns (At). The reluctance of a magnetic circuit is its ability to

oppose the magnetic flux (this is analogous to resistance which opposes current)

Al

µ=ℜ

where l is the length of the magnetic circuit, A is the cross sectional area of the magnetic circuit and µ is the permeability of the core in the magnetic circuit. The units are Ampere-turns/Wb (At/Wb).

Therefore

lANIµ

=ℜℑ

=Φ



where Φ is the total flux generated by the magnetic circuit. There is also a “Right Hand Rule” (RHR) for Φ (in fact we have already seen it). If you place the fingers of your right hand along the current carrying conductors in the direction of the current in a coil, your thumb will point in the direction of the Φ.

Note that permeability is analogous to permittivity.

Therefore

0µµµ r= where

70 104 −×= πµ Henries/m

is the permeability of a vacuum (free space) and

rµ is the relative permeability of the material in the core

of the magnetic circuit. The relative permeability for ferromagnetic materials can be quite large (> 1000).

Here is a summary of some classes of materials:

Class Example Paramagnetic 1≤≅rµ Silver, Copper

Diamagnetic 1≅≥rµ Platinum, Aluminium

Ferromagnetic 1>>rµ Iron, Nickel,AlNiCo

Ferrite 1>>>>rµ Ceramics



Magnetizing force: While the use of reluctance, ℜ, is comfortable to use,

since it looks like resistance, it turns out to be hard to use in reality. The magneto-motive force per unit length is called the magnetic field intensity and is much more commonly used

lH

ℑ=

and since NI=ℑ

then

lNI

H = .

The units of H are At/m. Using some of our previous definitions it can now be

shown that HB µ= .

The graphs shown on page 37 are examples of the variation of B with respect to H and show the non-linearity and changing value of µ for some common magnetic circuit materials.



Some Magnetic Circuits: There are two coil type magnetic circuits that we are

interested in. Solenoid:

Note that even with a magnetic core (like iron), the

flux has to leave the core and pass through the air to get to the other end of the coil (remember the magnetic field is a continuous loop leaving one end and coming in at the other). This causes a problem. What is it???

Toroid:

If we attach the ends of the solenoid to each other

there is no (or very little) flux that leaves the core (especially if it is magnetic). Much more useful.



Example #2: Consider the copper core toroid with circular cross section shown below. The inside diameter is 8cm and the outside diameter is 12cm. The current through the 2000 turn coil is 1A. What is the total flux in the core?



If a conductor is moved through a magnetic field so that it cuts magnetic lines of flux, a voltage will be induced across the conductor, as shown below. The greater the number of flux lines cut per unit time (by increasing the speed with which the conductor passes through the field), or the stronger the magnetic field strength (for the same traversing speed), the greater will be the induced voltage across the conductor. If the conductor is held fixed and the magnetic field is moved so that its flux lines cut the conductor, the same effect will be produced.

Faraday’s Law:

Voltage is induced in a magnetic circuit whenever the flux linkage is changing. The magnitude of the induced voltage is proportional to the rate of change of the flux linkages.

Mathematically (note use of lower case)

dtd

eind

φ∝



where eind is the time dependent induced voltage. As more turns are used the induced voltage is seen to increase. This leads to

dtd

Neind

φ=

Since Φ is proportional to the current, i, through the magnetic circuit

dtdi

did

Ndtd

Neφφ

==

The proportionality constant is denoted L, the self inductance of the magnetic circuit

dtdi

lAN

dtdi

did

Ndtdi

Leind

=

==

2µφ

The units of inductance are Henries. Note that

ondAmpere

VoltAmpere

ondVoltHenry

sec

sec =−=.

The denominator is a rate of change of current.



Lenz’s Law:

The polarity of the induced voltage is such that it opposes the cause of the changing flux.

Therefore (from Faraday’s Law)

dtd

Neind

Φ−= .

The importance of this is that an inductance opposes changes in its flux. Since the flux is set up by current, the inductor opposes changes in its current (analogous to a capacitor not allowing the voltage across it to change instantaneously).



Example #3: Consider the air core solenoid shown below. What is

the inductance?



Example #4: Consider the iron core solenoid shown below. The

relative permeability of iron is µr=2000µc. What is the inductance?



So far we have looked at simple coil type magnetic circuits. In order to consider more complex circuits we need another law. Ampere’s Circuital Law for Magnetic Circuits

Just like there is an Ohm’s Law for electric circuits there is an equivalent law for magnetic circuits, Ampere’s Circuital Law.

The nature of the law is essentially the same. Using the following table

Electric Circuit Magnetic Circuit V ℑ I Φ R ℜ

we can see that V can be replaced by ℑ, I can be replaced by Φ, and R can be replaced by reluctance. We can have rises and drops around a magnetic circuit just like an electric circuit.

If the “elements” are in series ℑT = ℑ1 + ℑ2 + ℑ3 + ℑ4 + … ℜT = ℜ1 + ℜ2 + ℜ3 + ℜ4 + …

Note that some of the ℑs will be negative since the sum has to be 0 (like KVL).

What about total flux (as shown below)?



Example #5: Consider the following magnetic circuit. Apply

Ampere’s Circuital Law. Use ℑ and H.



Example #6: A 1000 turn coil is wound on the center leg of the

magnetic circuit shown below. What is the maximum current through it if the total flux in the core must not exceed 2x10-3 Wb? The core is made of laminated sheet steel 4 cm in total thickness. The width of the outer legs is 2.5cm while the inner leg is 5 cm wide. The average path length is 25cm. Use µsheet steel = 2.283x10-3 H/m. Hint: Symmetry may help you here.



Example #7: What is the current, I, needed to get a total flux of

2.4x10-4 Wb? µsheet steel=1.0x10-3H/m.



Electric Circuits with Inductors: Series inductors:

Consider the following diagram. Note the presence of

the resistor, R. More on R later.

E

L1

L2

R

The current through both inductors is the same,

therefore

21 LL vvE +=

dtdi

Ldtdi

LE 21 +=

Rearranging

( )dtdi

LLE 21 += .

Thus

21 LLLtotal +=

Series inductors are similar to series resistances.



Parallel inductors:

Consider the following diagram.

E L1 L2

R

i

i1 i2

b

a

Note (from KCL)

21 iii += . and

21 LLab vvv == . So

dtdi

Ldtdi

Ldtdi

L total== 22

11 .

Noting that

dtdi

dtdi

dtdi 21 +=

+=+=

2121

1121

LLv

L

v

L

v

dtdi

abLL

.

Thus

21

111LLLtotal

+= or 21

21

LLLL

Ltotal += .

Parallel inductances are similar to parallel

resistances.



Example #8: Find the voltage V1 and the current through each

inductor.



Some interesting items:

Energy stored in an inductor The energy stored in an inductor is contained within

the magnetic field that is created by the current through the inductor

( ) ∫=t

pdttW0

( ) ∫=t

vidttW0

( ) ∫=t

idtdtdiLtW

0

( )( )

∫=ti

idiLtW0

( ) ( )2

2 tLitW =

Note that there is no direct relationship to the

voltage across the inductor only the current through the inductor.

While a capacitor can store energy in its electric field when disconnected from a circuit, an inductor cannot store energy in its magnetic field when disconnected from a circuit (since i = 0).



Example #9: Find the energy stored in each inductor in the circuit

shown below.



Voltage across an inductor:

Recall that

dtdiLvL = .

• If the current through the inductor is constant

0=dtdi

.

Therefore is no voltage across the inductor. The significance of this is that for a circuit in which the

currents are not changing (i.e., a dc circuit), no voltage exists across an inductor under steady state. As a result of this, under steady state conditions, an inductor is equivalent to a short circuit.

Recall in the discussion of series and parallel inductances there was a resistor, R, in the circuit. This is to prevent the inductor(s) from shorting out the battery under steady state conditions.



What happens if current is flowing in the inductor and a switch in series with the inductor is opened?

=↓dtdi

The inductor doesn’t like having the current through it change. Therefore it will induce a voltage (whatever it takes) to try and keep the current flowing. This induced voltage can be quite large and will cause a breakdown of the gap between the switch contacts (switch is opening). This can cause a flashover arc.

This can be quite spectacular in the dark.



Real World Effects: Note:

While we discuss the following effects, in E.P. 155.3 we always assume (unless told otherwise) that all inductors are ideal coil inductors.

Air Gaps: When an air gap in introduced into a magnetic circuit,

some of the flux bulges outside of the area of the core. This is called fringing (just like in a capacitor). See diagram below. The left side shows the air gap with the bulge while the right side shows an “ideal” air gap.

In E.P. 155.3 we ignore the fact that it bulges out. If we didn’t, a common adjustment is to increase the length (L) and width (W) of the air gap area by the length of the air gap (l) thus creating a larger area:

Area ignoring bulge: A = L x W Area with bulge: A = (L + l) x (W + l)



Laminated sheet steel cores: A large number of magnetic circuit cores are made up

of laminated sheet steel cores.

Laminations

Varnish

This is because they are cheap and they reduce the power lost through “eddy currents”. Eddy currents are internal currents within the steel cores due to changes in the flux and as a result of the induced voltage creating a current that circulates around the periphery of a solid core. The use of laminations reduces the power to almost negligible values. The downside to this is that the laminations have to be insulated from one another. This is done with a thin coat of varnish between the laminations. As a result, for any given area of core, the effective area has to be reduced by the total thickness of the varnish (since it is not magnetic). A common adjustment to use is a 10% reduction in area due to the varnish.



Actual inductor: Since an inductor is made up of coiled wire, it must

have some resistance associated with it (due to the wire in the coil).

As well, the closely spaced coils give rise to a “stray” capacitance between the wires of the coil.

Therefore an actual inductor is modeled as below:

L

R

C1

The value of the resistance for a “good” inductor is on

the order of ohms (Ω) and obviously depends on the wire size used in the coils which in turn depend on the current carrying capability of the inductor.

The capacitance is usually ignored.



Variation of µ The following graphs are representative of the

variation of µ with B and H. They are a) Variation of permeability with respect to H b) Hysteresis and Saturation curve c) Normal magnetization curve (I) d) Normal magnetization curve (II)

We will not directly use these graphs in E.P 155.3.

a) Variation of permeability with respect to H

From graph a) you can see that µ is anything but constant.



b) Hysteresis and Saturation curve

From graph b) you can see that as H increases the B produced is non-linear and saturates near the ends. As H is then reduced there is a residual BR (top and bottom on B axis) indicating that there is a magnetic field present even if H is 0. This is the case for permanent magnets. The size of the grey area is indicative of the power lost (which causes heating) during cycling around the loop.



c) Normal Magnetisation Curve (I)

Graph c) represents the initial magnetization (from o-a-b) shown in graph b).

d) Normal Magnetization Curve (II)

Knowing the value of B you can find the corresponding H using curve c) or d) above. This graph is an expanded version of graph c).

e.p. 155.3: electric and magnetic circuits...

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