Today in Physics 122: inductance

Leftovers: electric motors and back EMF

Mutual inductance

Self inductance

Inductors and resistors in circuits

4 November 2019 Physics 122, Fall 2019 1

Commercial ferrite-core inductors (FIEK-Kompjuterike)

The concept works in reverse, too. Run a DC current I through the loop, and magnetic torque rotates the loop. This time t = 0 when the torque is maximum.

The DC motor

4 November 2019 Physics 122, Fall 2019 2

xz

y yL

BB

Ltω

1F

2F

3F

4F

2F

4F

I

DC motor (continued)

The loop gains angular momentum, so it will overshoot the position where τ = 0:perfect alignment of µ with B.

4 November 2019 Physics 122, Fall 2019 3

tω

1F

2F

3F

4F

2F

4F

xz

y y

L

BB

LI

DC motor (continued)

If the current direction stays the same, the loop’s position will now oscillate about the τ = 0 position.

4 November 2019 Physics 122, Fall 2019 4

tω

1F

2F

3F

4F

2F

4F

xz

y y

BB

I

L

L

DC motor (continued)

But if, just after it passes that point, the loop current is reversed, then the forces and torque reverse too,

4 November 2019 Physics 122, Fall 2019 5

tω

1F

2F

3F

4F

2F

4F

xz

y y

BB

I

DC motor (continued)

and the loop keeps rotating in the same direction. So the trick is to switch current polarity just after passing through τ = 0.

4 November 2019 Physics 122, Fall 2019 6

tω

2F

4F

2F

4F

xz

y y

BB

Then every pair of positions 180 ° apart are the same except for segments 2 and 4 being swapped.

DC motor (continued)

4 November 2019 Physics 122, Fall 2019 7

tω

1F

2F

3F

4F

2F

4F

xz

y y

BB

I

DC motor (continued)

Swapping the direction of I is generally done by having the loop connected to its power source with

metal-bristle brushes (as in the generator), in contact with

a split circular ring (“commutator”) which is fixed to the loop and rotates with it about its axis.

As rotation swaps the brushes from one half to the other of the commutator, the polarity of the voltage on the loop switches.

4 November 2019 Physics 122, Fall 2019 8

Back EMF (continued)

That’s not all there is to it, though:

Since the motor’s loop rotates about its diameter in a magnetic field, there is an additionalcomponent of current and EMF for which we haven’t yet accounted.

… the one that is there when the same components are functioning as an AC generator.

This additional EMF opposes the EMF which drives the rotation. Thus the name back EMF.

4 November 2019 Physics 122, Fall 2019 9

DC motor and back EMF (continued)

For example: compare pages 15 (motor) and 12 (generator): the drive and generated currents always flow oppositely.

4 November 2019 Physics 122, Fall 2019 10

I’

tω

1F

2F

3F

4F

2F

4F

xz

y yL

BB

LI

DC motor and back EMF (continued)

Suppose that the resulting motor turns at constant angular speed ω. Then, as we saw in the case of the generator,

When starting up (ω near zero), the total current I – I’ is larger (I’ near zero). This is why the lights dim when powerful motors on the same circuit start up, but return to their brightness when the motor is up to speed.

4 November 2019 Physics 122, Fall 2019 11

R

back

I I ′−2

back2

sin

sin

BL t

BLI tR

ω ω

ω ω

=

′ =

Mutual inductance

Consider again a transformer which couples magnetic flux perfectly between two coils.

Suppose the transformer coils are long solenoids, with the same cross-section area A and the same length .

Apply a sinusoidally-varying (i.e. AC) voltage to the “primary” side.

For the secondary coil,

4 November 2019 Physics 122, Fall 2019 12

PP P BS

P SS BS P

NB A I A

N N AN I

µ

µ

= = Φ

Φ =

per turn of the coil

(from Giancoli)

Mutual inductance (continued)

If instead we impose a current on the secondary side, the flux on the primary side is

Evidently the two coils have an symmetrical relation between the current in one and the flux in the other. Define:

4 November 2019 Physics 122, Fall 2019 13

SS S BP

P SP BP S

NB A I A

N N AN I

µ

µ

= = Φ

Φ =

S BS P SP BP

S PM

N N N ANI I

µΦΦ= = ≡

(from Giancoli)

per turn of the coil

Mutual inductance

Mutual inductance (continued)

The mutual inductance M for the transformer depends only upon the geometrical factors of the coils and the permeability of the magnetic material which “conducts” the flux.

This result turns out generally to be obtained for circuits which can induce currents in each other, since B generated by a current I depends linearly on I in most cases.

4 November 2019 Physics 122, Fall 2019 14

(from Giancoli)

Mutual inductance (continued)

In general, for two loops with number of turns and currents

,

where the mutual inductances depend only on loop geometry and permeability of the medium between them.

Furthermore, it can be proven that, as we just saw for the transformer,

4 November 2019 Physics 122, Fall 2019 15

1N

2N

2I

1I

2 2 21 1

1 1 12 2

B

B

N M IN M I

Φ =Φ =

1 2 and N N1 2 and I I

21 12M M M= =

Mutual inductance (continued)

The main implication of mutual inductance is that the EMF induced in onecircuit due to a time dependent flux from another circuit is proportional to the time derivative of the current in that othercircuit:

since M depends only on geometry and permeability, not upon time.

4 November 2019 Physics 122, Fall 2019 16

2 12 2

1 21 1

B

B

d dIN Mdt dt

d dIN Mdt dt

Φ= − = −

Φ= − = −

1N

2N

2I

1I

Self inductance

It’s easiest to imagine this sort of inductive circuit coupling between two circuits, but of course it happens with just one:

External voltage applied to circuit drives current,

current generates B which has flux through the circuit,

induced EMF appears in the circuit.

For a long solenoid (N turns, length , loop area A, current I),

4 November 2019 Physics 122, Fall 2019 17

2

B

B

B

NBA IA

N N AI

d dIN Ldt dt

L

µ

µ

Φ = =

Φ=

Φ= − =

≡

−

IN

ASelf inductance

Self inductance (continued)

Other sorts of loops will also have self inductance, though it isn’t usually as easy to calculate as it is for the solenoid.

Main implication: we have another “lumped” circuit element, along with voltage sources, resistors, and capacitors: the inductor.

In honor of the noble Solenoid, inductors are represented as coils in circuit diagrams, whether solenoidal or not.

In honor of Lenz, self inductance is given the symbol L.

4 November 2019 Physics 122, Fall 2019 18

2

2

1C

R

L

QV IdtC C

dQV IR Rdt

dI d QV L Ldt dt

= =

= =

= =

∫I

+

+

+

-

-

-

C

R

L

Inductors (self-inductances) in circuits: R-L

Simplest example: an R-L circuit.

Example 1. In the circuit at right, switch S is closed at t = 0. What is the current as a function of time?

As usual we use Kirchhoff’s rules. No nodes and only one loop:

4 November 2019 Physics 122, Fall 2019 19

I

V

R

L

S +

++

-

--

( )

0 0

0 , or

, or

I t t

dIV IR Ldt

dI R VIdt L R

dI R dtI V R L

− − =

= − −

′′= −

′ −∫ ∫

R-L (continued)

I expect you’re tired of doing this integral so often:

4 November 2019 Physics 122, Fall 2019 20

( )( )

( )ln

I t V R

V R

Vq I dq dIR

V Vq I tR R

dq R tq L

I t V RV R

−

−

′ ′= − =

= − → −

= −

−= −

∫I

V

R

L

S +

++

-

--

R-L (continued)

Exponentiate both sides, as usual:

Like the RC circuit, this one has a time constant:

4 November 2019 Physics 122, Fall 2019 21

( )

( ) ( )1

Rt L

Rt L

I t V Re

V RVI t eR

−

−

−=

−

= −

( ) ( )max

max

1 tI t I e

I V RL R

τ

τ

−= −

==

0 2 4 6 8 10

t (L/R)

I(t) Imax 1

1e

−

⋅

LR

R-L (continued)

Note the strong resemblance of current in the RL circuit to charge in the RCcircuit.

4 November 2019 Physics 122, Fall 2019 22

0 2 4 6 8 10

t (L/R)

I(t) Imax 1

1e

−

⋅

LR

0 2 4 6 8 10

t (RC)

Q(t) Qmax 1

1e

−

⋅

R C⋅

0 2 4 6 8 10

t (RC)

I(t)

1e

Imax⋅

R C⋅