Lesson 3: RLC circuits & resonance

[Pages:25]Lesson 3: RLC circuits & resonance

? Inductor, Inductance ? Comparison of Inductance and Capacitance ? Inductance in an AC signals ? RL circuits ? LC circuits: the electric "pendulum" ? RLC series & parallel circuits ? Resonance

P. Piot, PHYS 375 ? Spring 2008

Inductor

VL=LddTI

?

Start with Maxwell's equation

r

?

r E

=

-

Br

t

Z=VL=iL

I

? Integrate over a surface S (bounded by contour C) and

use Stoke's theorem: r ? Er.dAr =

Er.dlr = -

r B

.dAr

=

-

S

SC

S t

t

? The voltage is thus

VL

= -emf

=

t

Magnetic flux in Weber

P. Piot, PHYS 375 ? Spring 2008

Wihelm Weber (1804-1891)

Inductor

? Now need to find a relation between magnetic field

generated by a loop and current flowing through the

loop's wire. Used Biot and Savart's law:

r dB

=

?0 4

r Idl

?

r^ r2

B

I

? Integrate over a surface S the magnetic flux is going to be of the form

LI

? The voltage is thus

Inductance measured in Henri (symbol H)

VL

=

t

=

L

dI dt

Joseph Henri (1797-1878)

P. Piot, PHYS 375 ? Spring 2008

Inductor

? Case of loop made with an infinitely thin wire

B

=

? 4

l.I

? If the inductor is composed of n loop per meter then

total B-field is

B

=

? 4

nI

? So inductance is

BA =

? 4

AnI

L

=

? 4

An

Increase magnetic permeability (e.g. use metallic core instead of air)

Increase number of wire

per unit length increase L

Area of the loop

P. Piot, PHYS 375 ? Spring 2008

Inductor in an AC Circuit

VL

=

L

dI dT

VT

VL

Z = VL = iL

I

IL

VL

? Introduce reactance for an inductor:

X L = L

VL I P

P. Piot, PHYS 375 ? Spring 2008

Inductor , Capacitor, Resistor

? Resistance = friction against motion of electrons

? Reactance = inertia that opposes motion of electrons

X L = L

1

XC

=

-

C

X

? Impedance is a generally complex number:

Z R

Z = R + iX

? Note also one introduces the Admittance:

1 Y = = G + iB

Z

susceptance conductance

P. Piot, PHYS 375 ? Spring 2008

CAPACITOR

Inductor versus Capacitor

P. Piot, PHYS 375 ? Spring 2008

INDUCTOR

RL series Circuits

VT

=V R+VL

=

RI

+ L dI dt

=

(R + iL)I

Z

=

R + iL

=

R + iX L

VT

VL

VT

VR ? For the above circuit we can compute a numerical

value for the impedance:

Z = (5 + 3.7699i) | Z |= 52 + 3.76992 6.262, = 37.02?

P. Piot, PHYS 375 ? Spring 2008

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