lesson 3: rlc circuits & resonance - niu - nicaddpiot/phys_375/lesson_3.pdf · lesson 3: rlc...

P. Piot, PHYS 375 – Spring 2008

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


• Start with Maxwell’s equation

• Integrate over a surface S (bounded by contour C) and use Stoke’s theorem:

• The voltage is thus

Inductor

temfVL ∂

Φ∂=−=

LiIVZ

dTdILV

L

L

ω==⇒

=

tBE∂∂

−=×∇r

rr

tAd

tBldEAdE

SCSS ∂Φ∂

−=∂∂

−==×∇ ∫∫∫∫∫∈

rr

rrrrr...

Magnetic flux in Weber

Wihelm Weber (1804-1891)


• 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:

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

• The voltage is thus

Inductor

IBrrlIdBd ∝⇒×= 2

0 ˆ4

rr

πµ

LI≡ΦInductance measuredin Henri (symbol H)

dtdIL

tVL =

∂Φ∂

=

Joseph Henri (1797-1878)


• Case of loop made with an infinitely thin wire

• If the inductor is composed of n loop per meter then total B-field is

• So inductance is

Inductor

IlB .4

δπµ

=

AnLAnIBAπµ

πµ

44=⇒=≡Φ

Increase number of wireper unit length increase L

nIBπµ4

=Increase magnetic

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

Area of the loop


Inductor in an AC Circuit

LiI

VZ

dTdILV

L

L

ω==⇒

= VT VL

VL

I P

IL VL

• Introduce reactance for an inductor:

LX L ω=


• Resistance = friction against motion of electrons

• Reactance = inertia that opposes motion of electrons

• Impedance is a generally complex number:

• Note also one introduces the Admittance:

Inductor , Capacitor, Resistor

iXRZ +=

CX

LX

C

L

ω

ω1

−=

= Z

RX

iBGZ

Y +==1

conductancesusceptance


CA

PAC

ITOR

IND

UC

TOR

Inductor versus Capacitor


RL series Circuits

LLRT iXRLiRZILiRdtdILRIVVV +=+=⇒+=+=+= ωω )(

°=Θ≈+=

Ω+=

02.37,262.67699.35||

)7699.35(22Z

iZ

VT VTVL

VR

• For the above circuit we can compute a numerical value for the impedance:


RL parallel Circuits

( ) 111)11(1 −−− +=⇒+=+=+= ∫ LiRZVLiR

VdtLR

VIII LR ωω

°=Θ≈Ω+=98.52,01.3||

)40.281.1(Z

iZ

V

• For the above circuit we can compute a numerical value for the impedance:


Inductor: Technical aspects

• Inductors are made a conductor wired around air or a ferromagnetic core

• Unit of inductance is Henri, symbol is H

• Real inductors also have a resistance (in series with inductance)


RLC series/parallel Circuits


• Compute impedance of the circuit below– Step 1: consider C2 in series with L Z1– Step 2: consider Z1 in parallel with R Z2– Step 3: consider Z2 in series with C

• Let’s do this:

• Current in the circuit is

• And then one can get the voltage across any components

RLC series/parallel Circuits: an example

iC

LiZ 34.152311 =

−=

ωω i

RZ

Z 41.13215.429111

1

2 −=+

=

iC

iZZ 79.62915.4291

23 −=−=ω

°=∠=⇒+== 371.58,64.146||86.12489.763

ImAIiZVI


LC circuit: An electrical pendulum


LC circuit: An electrical pendulum• Mechanical pendulum: oscillation between

potential and kinetic energy

• Electrical pendulum: oscillation between magnetic (1/2LI2) and electrostatic (1/2CV2)energy

• In practice, the LC circuit showed has some resistance, i.e. some energy is dissipated and therefore the oscillation amplitude is damped. The oscillation frequency keeps unchanged.

• LC circuit are sometime called tank circuit and oscillate (=resonate)


Example of a simple “tank” (LC) circuit

• ODE governing this circuit?

• Equation of a simple harmonic oscillator with pulsation:

• Or one can state that system oscillate if impedance associated to C and L are equal, i.e.:

∫+=+= VdtLdt

dVCIII LC1

LC1

=ω

dtdIV

LCdtVdV

LdtVdC

dtdI

=+⇔+=11

2

2

2

2

ωω

CL 1

=


Example of a simple “tank” (LC) circuit

• What is the total impedance of the circuits?

• So the tank circuit behaves as an open circuit at resonance!

• In a very similar way one can show that a series LC circuit behaves as a short circuit when driven on resonance i.e.,

∞=⇒=−=−=− ZLLiC

iiLZ 0)(1 ωωω

ω

0=Z


RLC series circuit• 2nd order ODE:

• Resonant frequency still

• Let’s define the parameter

• Then the ODE rewrites

)()()()(

;

2

2

tVtUdt

tdURCdt

tUdLC

CUQdtdQIwith

VCQ

dtdILRI

=++⇒

==

=++

VUdtdU

dtUd

=++ 202

2

2 ωζ

Voltage acrosscapacitor

U(t)


RLC series circuit: regimes of operation (1)

• Let’s consider V(t) to be a dirac-like impulsion (not physical…) at t=0. Then for t>0, V(t)=0 and the previous equation simplifies to

• With solutions

• Where the λ are solutions of the characteristics polynomial is

• The discriminant is

• And the solutions are

tt BeAetU −+ += λλ)(

02 202

2

=++ UdtdU

dtUd ωζ

LCCR 422 −=∆

( )∆±−=± ζλ 221



• If ∆<0 that is if Under damped

U(t) is of the form

A and B are found from initial conditions.

• If ∆=0 critical damping

+= −−−− ttt BeAeetU

20

220

2

)( ωδωδδ

20

2

2/12 122

ωδδλ −±−≡

−

±−=± LCL

RL

RCLR 2<

tAetU δ−=)(



• If ∆>0 that is if Strong damping

U(t) is of the form

A and B are found from initial conditions.

Which can be rewritten

+= −−−− titit BeAeetU

220

220)( δωδωδ

220

2/12

21

2δωδλ −±−≡

−±−=± i

LR

LCi

LR

CLR 2>

( )φδωδ +−= − tDetU t 220sin)(


Over-damped Critical damping

Under-damped


• For under-damped regime, the solutions are exponentially decaying sinusoidal signals. The time requires for these oscillation to die out is 1/Q where the quality factor is defined as:

CL

RQ 1≡


RLC series circuit: Impedance (1)• 2nd order ODE:

• Resonant frequency still

• Let’s define the parameter

• Then the ODE rewrites

dtdV

LI

LCdtdI

LR

dtId

dtICdt

dILRIVVVV CLR

11

.1

2

2

=++⇒

++=++= ∫

dtdV

LI

dtdI

dtId 12 2

02

2

=++ ωζ


• Take back (but could also jut compute the impedance of the system)

• Explicit I in its complex form and deduce the current:

• Introducing x=ω/ω0 we have

RLC series circuit: Impedance (2)

dtdV

LI

dtdI

dtId 12 2

02

2

=++ ωζ

22

220

20

2

1

1||2

12

−+

=⇒+−

=≡⇒

=++−

ωω

ζωωωω

ωωζωω

CLR

Yi

iVIY

VL

iIIiI

22 11

1||

−+

=

xxQR

Y


RLC series circuit: resonance

Values of Q


• The same formalism as before can be applied to parallel RLC circuits.

• The difference with serial circuit is: at resonance the impedance has a maximum (and not the admittance as in a serial circuit)

RLC parallel circuit : resonance

lesson 3: rlc circuits & resonance - niu - nicaddpiot/phys_375/lesson_3.pdf · lesson 3: rlc...

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