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 dI V=L L dT

VL r ⇒Z= =iωL r r ∂B • Start with Maxwell’s equation ∇× E = − I ∂t • Integrate over a surface S (bounded by contour C) and Magnetic use Stoke’s theorem: flux in Weber r r r r r r ∂B r ∂Φ ∫∫∇× E.dA = ∫ E.dl = − ∫∫ .dA = − S S∈C S ∂t ∂t • The voltage is thus ∂Φ V = −emf = L ∂t

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 µ r rˆ dB = 0 Idl × ⇒ B ∝ I 4π r 2 • Integrate over a surface S the magnetic flux is going to be of the form Φ ≡ LI Inductance measured • The voltage is thus in Henri (symbol H)

∂Φ dI V = = L L ∂t dt Joseph Henri (1797-1878)

P. Piot, PHYS 375 – Spring 2008 Inductor

• Case of loop made with an infinitely thin wire µ B = δl.I 4 π

• If the inductor is composed of n loop per meter then total B-field is µ B = nI 4π Increase magnetic • So inductance is permeability (e.g. use metallic core instead of air) µ µ Φ ≡ BA = AnI ⇒ L = An 4π 4π Increase number of wire per unit length increase L Area of the loop

P. Piot, PHYS 375 – Spring 2008 Inductor in an AC Circuit

dI VL = L VT dT VL

VL I ⇒ Z = = iωL L VL I • Introduce reactance for an inductor:

X L = ωL

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 = ωL Z 1 X = − C ωC X R • Impedance is a generally complex number: Z = R + iX

• Note also one introduces the Admittance: 1 Y = = G + iB Z susceptance conductance

P. Piot, PHYS 375 – Spring 2008 Inductor versus Capacitor CAPACITOR INDUCTOR

P. Piot, PHYS 375 – Spring 2008 RL series Circuits ω dI V =V +V = RI + L = (R + i L)I ⇒ Z = R + iωL = R + iX T R L dt L

V T VT VL

VR • For the above circuit we can compute a numerical value for the impedance: Z = (5 + 3.7699i) Ω | Z |= 52 + 3.76992 ≈ 6.262, Θ = 37.02°

P. Piot, PHYS 375 – Spring 2008 RL parallel Circuits

ω V 1 1 1 I =I +I = + Vdt = ( + )V ⇒ Z −1 = R−1 + ()iωL −1 R L R L ∫ R i L V

• For the above circuit we can compute a numerical value for the impedance: Z = (1.81+ 2.40i) Ω | Z |≈ 3.01, Θ = 52.98°

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 RLC series/parallel Circuits: an example

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

 ω 1  1 Z = i L −  =1523.34i Z = = 429.15 −132.41i 1 2 1 1  Cω  + i Z1 R Z3 = Z2 − = 429.15 − 629.79i C1ω • Current in the circuit is V I = = 76.89 +124.86i ⇒| I |=146.64 mA, ∠I = 58.371° Z3 • And then one can get the voltage across any components

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? dV 1 I =I +I = C + Vdt C L dt L ∫

dI d 2V 1 d 2V 1 dI = C + V ⇔ + V = dt dt 2 L dt 2 LC dt

• Equation of a simple harmonic oscillator with pulsation: 1 ω = LC

• Or one can state that system oscillate if impedance associated to C and L are equal, i.e.: 1 Lω = Cω

P. Piot, PHYS 375 – Spring 2008 Example of a simple “tank” (LC) circuit

• What is the total impedance of the circuits?

i ω Z −1 = iLω − = i(L − Lω) = 0 ⇒ Z = ∞ Cω

• 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.,

Z = 0

P. Piot, PHYS 375 – Spring 2008 RLC series circuit

•2nd order ODE: dI Q RI + L + = V dt C Voltage across dQ capacitor with I = ; Q = CU dt d 2U (t ) dU (t ) ⇒ LC + RC + U (t ) = V (t ) dt 2 dt U(t) • Resonant frequency still

• Let’s define the parameter

d 2U ζ dU • Then the ODE rewrites + 2 +ω 2U =V dt 2 dt 0

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 d 2U ζ dU + 2 +ω 2U = 0 dt 2 dt 0

• With solutions λ U(t) = Ae +t + Beλ−t

• Where the λ are solutions of the characteristics polynomial is

• The discriminant is ∆ = R2C 2 − 4LC • And the solutions areλ 1 = (− 2ζ ± ∆ ) ± 2

P. Piot, PHYS 375 – Spring 2008 RLCλ series circuit: regimes of operation (2)

L • If ∆<0 that is if R < 2 Under damped C 1/ 2 δ  2  R  R  1 δ 2 2 ± = − ±   −  ≡ − ± −ω0 2L  2δL  LC   δ  U(t) is of the form ω

2 2 δ 2 2 U (t ) = e − t  Ae − 0 t + Be − −ω 0 t    A and B are found from initial conditions.

• If ∆=0 critical damping U(t) = Ae −δt

P. Piot, PHYS 375 – Spring 2008 RLCλ series circuit: regimes of operation (3)

L • If ∆>0 that is if R > 2 Strong damping C 1/ 2 δ  2  R 1  R  ω 2 2 = − ± i δ −    ≡ − ± i −δ ± 2L LC 2L 0   ω  δ U(t) is of the form

2 2 ω 2 2 U (t) = e − t  Ae i 0 − t + Be − i 0 −δ t    A and B are found from initial conditions.

Which can be rewritten ω −δt 2 δ 2 U(t) = De sin( 0 − t +φ)

P. Piot, PHYS 375 – Spring 2008 RLC series circuit: regimes of operation (4)

Over-damped Critical damping

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

P. Piot, PHYS 375 – Spring 2008 RLC series circuit: Impedance (1)

•2nd order ODE: dI 1 V =V +V +V = RI + L + I.dt R L C dt C ∫ d 2 I R dI 1 1 dV ⇒ + + I = dt 2 L dt LC L dt

• Resonant frequency still

• Let’s define the parameter

• Then the ODE rewrites d 2 I ζ dI 1 dV + 2 +ω 2 I = dt 2 dt 0 L dt

P. Piot, PHYS 375 – Spring 2008 RLC series circuit: Impedance (2)

• Takeω back (but could also jut compute the impedance of the system) ζω d 2 Iω ζ dI 1 dV + 2 +ω 2 I = dt 2 dt ω 0 L dt ω • Explicit I in its complexω form and deduce the current: ω 1 − 2 I + 2i I + 2 I = ζωi V 0 L I i 1 ⇒ Y ≡ = ⇒|Y |= V 2 − 2 + 2i ω 2 0 2  1  R +  L −   Cω  • Introducing x=ω/ω0 we have 1 |Y |= 2 2  1  R 1+ Q  x −   x  P. Piot, PHYS 375 – Spring 2008 RLC series circuit: resonance

Values of Q

P. Piot, PHYS 375 – Spring 2008 RLC parallel circuit : resonance

• 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)

P. Piot, PHYS 375 – Spring 2008