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 4π • 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 dI V =V +V = RI + L = (R + iωL)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 V 1 1 1 I =I +I = + Vdt = ( + )V ⇒ Z −1 = R−1 + ()iωL −1 R L R L ∫ R iωL 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: 1 1 Z = i Lω − =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 Z −1 = iLω − = i(Lω − 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 dU • Then the ODE rewrites + 2ζ +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 dU + 2ζ +ω 2U = 0 dt 2 dt 0 • With solutions U(t) = Aeλ+t + Beλ−t • Where the λ are solutions of the characteristics polynomial is • The discriminant is ∆ = R2C 2 − 4LC • And the solutions are 1 λ± = (− 2ζ ± ∆ ) 2 P. Piot, PHYS 375 – Spring 2008 RLC series circuit: regimes of operation (2) L • If ∆<0 that is if R < 2 Under damped C 1/ 2 2 R R 1 2 2 λ± = − ± − ≡ −δ ± δ −ω0 2L 2L LC U(t) is of the form 2 2 2 2 U (t ) = e −δt Ae δ −ω 0 t + Be − δ −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 RLC series circuit: regimes of operation (3) L • If ∆>0 that is if R > 2 Strong damping C 1/ 2 2 R 1 R 2 2 λ± = − ± i − ≡ −δ ± i ω0 −δ 2L LC 2L U(t) is of the form 2 2 2 2 U (t) = e −δt Ae i ω 0 −δ t + Be − i ω 0 −δ t A and B are found from initial conditions. Which can be rewritten −δt 2 2 U(t) = De sin( ω0 −δ 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 dI 1 dV + 2ζ +ω 2 I = dt 2 dt 0 L dt P. Piot, PHYS 375 – Spring 2008 RLC series circuit: Impedance (2) • Take back (but could also jut compute the impedance of the system) d 2 I dI 1 dV + 2ζ +ω 2 I = dt 2 dt 0 L dt • Explicit I in its complex form and deduce the current: 1 −ω 2 I + 2iζωI +ω 2 I = i ωV 0 L I iω 1 ⇒ Y ≡ = ⇒|Y |= V ω 2 −ω 2 + 2iζω 2 0 2 1 R + Lω − 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.