Inductors

Energy Storage

Current passing through a coil causes a magnetic field Energy is stored in the field Similar to the energy stored by capacitors

We saw a charging time for a capacitor An inductor takes time to store energy also

Simple RL Circuit

t

ILIRV

L must have units of Ohmsseconds

From the Construction

Inductance

AN

L2

N = number of turns on the coil

= permeability of the core (henrys/m)

A = cross sectional area (m2)

l = length of core (m)

L = inductance in henrys

Relative Permeability

Many texts and handbooks publish Km, where = Km o

o = permeability of free space = 4 X 10-7 Wb/A

Ex: Compute L for the following coil: N = 100 turns A = 1.3 X 10-4 m2

l = 25 X 10-3 m Km = 400 (steel)

mH1.26 H0261.0 10X25

)10X3.1)(10X4)(400((100)

AKNL

3

472

om2

Time Dependence

)e1(R

EI /t = L/R

This is the same way that voltage varied in the capacitor

Try it!

Notes

The final current (E/R) doesn’t depend on L There is no voltage drop across the inductor after the full

current has been established The coil then acts as a short circuit (as if it weren’t there)

The inductance depends on the change of current (once I is established, I/t → 0 and V=IR)

At first I = 0, so V = IR = 0 As current rises the voltage drop across the resistor (IR)

gets greater, leaving less voltage to be dropped through the coil.

Voltage/tEeV

Inductors in Series

Kirchhoff’s Voltage Law

T

321

321

LLL

Lt

I

)LLL(t

I

t

I L

t

I L

t

I L

VVVE321

LT = L1 + L2 + L3

Inductors in Parallel

321 LLL VVVE

The analysis is difficult in a dc circuit since the voltage drains to zero, but the result is…

321T L

1

L

1

L

1

L

1

Real Inductors

Inductors have… Internal Resistance Internal Capacitance between windings

So a real inductor in a circuit looks like…

Example

The equivalent circuit is

k38R

k8

1

k4

1

R

1

T

T

Continued

)e1(mA76.3)e1(R

EI

s5.710X66.2

H10X20

R

L

s5.7/t/t

T

3

3

T

T

Comparing inductors to capacitors

After about 5, the current has reached a maximum for the coil and zero for a capacitor. The coil acts as a short, while the capacitor acts like an open circuit.

Sample RLC Circuit

After about 5 , the equivalent circuit is

No current flows through C1

and L1 acts as if it’s not there

Solve Circuit

R1 and R2 are in series, so…

AV

RR

EI 2

5

10

21

For the path ABCD IR1 + IR2 = E

EVV RR 21

6V)(2A)(2-V 10

10 112

IRVVEV RR

Notice that R2 and C1 are in parallel, so VR2 is the voltage drop across the capacitor also.

Stored Energy

Capacitor WC = ½CV2

Inductor WL = ½LI2