Lesson 3: RLC circuits & resonance - NIU - NICADDpiot/phys_375/Lesson_3.pdf · Lesson 3: RLC...
Transcript of 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
P. Piot, PHYS 375 – Spring 2008
• 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)
P. Piot, PHYS 375 – Spring 2008
• 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)
P. Piot, PHYS 375 – Spring 2008
• 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
P. Piot, PHYS 375 – Spring 2008
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=
P. Piot, PHYS 375 – Spring 2008
• 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
P. Piot, PHYS 375 – Spring 2008
CA
PAC
ITOR
IND
UC
TOR
Inductor versus Capacitor
P. Piot, PHYS 375 – Spring 2008
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:
P. Piot, PHYS 375 – Spring 2008
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:
P. Piot, PHYS 375 – Spring 2008
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)
P. Piot, PHYS 375 – Spring 2008
RLC series/parallel Circuits
P. Piot, PHYS 375 – Spring 2008
• 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
P. Piot, PHYS 375 – Spring 2008
LC circuit: An electrical pendulum
P. Piot, PHYS 375 – Spring 2008
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)
P. Piot, PHYS 375 – Spring 2008
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
=
P. Piot, PHYS 375 – Spring 2008
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
P. Piot, PHYS 375 – Spring 2008
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)
P. Piot, PHYS 375 – Spring 2008
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
P. Piot, PHYS 375 – Spring 2008
RLC series circuit: regimes of operation (2)
• 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−=)(
P. Piot, PHYS 375 – Spring 2008
RLC series circuit: regimes of operation (3)
• 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)(
P. Piot, PHYS 375 – Spring 2008
Over-damped Critical damping
Under-damped
RLC series circuit: regimes of operation (4)
• 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≡
P. Piot, PHYS 375 – Spring 2008
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
P. Piot, PHYS 375 – Spring 2008
• 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
P. Piot, PHYS 375 – Spring 2008
RLC series circuit: resonance
Values of Q
P. Piot, PHYS 375 – Spring 2008
• 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