Lesson 3: RLC circuits & resonance

Transcription

1 P. Piot, PHYS 375 Spring 008 esson 3: RC circuits & resonance nductor, nductance Comparison of nductance and Capacitance nductance in an AC signals R circuits C circuits: the electric pendulum RC series & parallel circuits Resonance

2 Start with Maxwell s equation ntegrate over a surface S (bounded by contour C) and use Stoke s theorem: The voltage is thus S V r r r E. da emf S C Φ t nductor r r E. dl r r r B E t S r B r. da t d V dt V Z i ω Φ t Magnetic flux in Weber P. Piot, PHYS 375 Spring 008 Wihelm Weber (804-89)

3 P. Piot, PHYS 375 Spring 008 nductor 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: ntegrate over a surface S the magnetic flux is going to be of the form The voltage is thus r µ r r db 0 dl ˆ 4π r Φ B nductance measured in Henri (symbol H) Φ V t d Joseph Henri ( )

4 P. Piot, PHYS 375 Spring 008 nductor Case of loop made with an infinitely thin wire µ B l. 4π δ f the inductor is composed of n loop per meter then total B-field is So inductance is Φ BA B µ 4π µ An 4π Area of the loop n µ An 4π ncrease magnetic permeability (e.g. use metallic core instead of air) ncrease number of wire per unit length increase

5 P. Piot, PHYS 375 Spring 008 nductor in an AC Circuit d V dt V Z V T V iω ntroduce reactance for an inductor: X ω V V P

6 P. Piot, PHYS 375 Spring 008 nductor, Capacitor, Resistor Resistance friction against motion of electrons Reactance inertia that opposes motion of electrons X ω Z X C ωc X mpedance is a generally complex number: R Z R + Note also one introduces the Admittance: Y G + ib Z conductance ix susceptance

7 P. Piot, PHYS 375 Spring 008 nductor versus Capacitor CAPACTOR NDUCTOR

8 P. Piot, PHYS 375 Spring 008 R series Circuits d V T V R+ V R + ( R + iω ) Z R + iω R + ix V T V V T For the above circuit we can compute a numerical value for the impedance: V R Z ( i) Ω Z , Θ 37.0

9 P. Piot, PHYS 375 Spring 008 R parallel Circuits V R+ + V ( + ) V Z R + R R iω ( iω ) V For the above circuit we can compute a numerical value for the impedance: Z (.8+.40i) Ω Z 3.0, Θ 5.98

10 P. Piot, PHYS 375 Spring 008 nductor: Technical aspects nductors 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)

11 RC series/parallel Circuits P. Piot, PHYS 375 Spring 008

12 P. Piot, PHYS 375 Spring 008 RC series/parallel Circuits: an example Compute impedance of the circuit below Step : consider C in series with Z Step : consider Z in parallel with R Z Step 3: consider Z in series with C et s do this: Z i ω i Z i Cω + i Z R Z3 Z i Cω Current in the circuit is V i ma, Z 3 And then one can get the voltage across any components

13 C circuit: An electrical pendulum P. Piot, PHYS 375 Spring 008

14 P. Piot, PHYS 375 Spring 008 C circuit: An electrical pendulum Mechanical pendulum: oscillation between potential and kinetic energy Electrical pendulum: oscillation between magnetic (/ ) and electrostatic (/CV ) energy n practice, the C circuit showed has some resistance, i.e. some energy is dissipated and therefore the oscillation amplitude is damped. The oscillation frequency keeps unchanged. C circuit are sometime called tank circuit and oscillate (resonate)

15 P. Piot, PHYS 375 Spring 008 Example of a simple tank (C) circuit ODE governing this circuit? dv C+ C + V d C d V + V d V + C V d Equation of a simple harmonic oscillator with pulsation: ω C Or one can state that system oscillate if impedance associated to C and are equal, i.e.: ω Cω

16 P. Piot, PHYS 375 Spring 008 Example of a simple tank (C) circuit What is the total impedance of the circuits? i Z iω i( ω ω) 0 Cω Z So the tank circuit behaves as an open circuit at resonance! n a very similar way one can show that a series C circuit behaves as a short circuit when driven on resonance i.e., Z 0

17 P. Piot, PHYS 375 Spring 008 RC series circuit nd order ODE: d Q R + + V C dq with ; Q CU d U ( t ) du ( t ) C + RC + Resonant frequency still Voltage across capacitor U ( t ) V ( t ) U(t) et s define the parameter Then the ODE rewrites d U + du ζ + ω0u V

18 P. Piot, PHYS 375 Spring 008 RC series circuit: regimes of operation () et s consider V(t) to be a dirac-like impulsion (not physical ) at t0. Then for t>0, V(t)0 and the previous equation simplifies to d U du + ζ + ω0u 0 With solutions U( t) Ae λ+ t + Be λ t Where the λ are solutions of the characteristics polynomial is The discriminant is And the solutions are R C 4C λ± ζ ( ± )

19 P. Piot, PHYS 375 Spring 008 RC series circuit: regimes of operation () f <0 that is if Under damped λ ± R ± U(t) is of the form R < C R C / δ ± δ ω 0 U ( t ) e δt δ ω t Ae + 0 Be 0 δ ω t A and B are found from initial conditions. f 0 critical damping U( t) Ae δt

20 P. Piot, PHYS 375 Spring 008 RC series circuit: regimes of operation (3) R > C / R R λ± ± i δ ± i C f >0 that is if Strong damping U(t) is of the form U ( t ) e δt i ω t Ae + 0 δ Be i ω δ 0 0 ω δ t A and B are found from initial conditions. Which can be rewritten U( t) De sin ( ) ω δ t + φ δt 0

21 P. Piot, PHYS 375 Spring 008 RC series circuit: regimes of operation (4) 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 /Q where the quality factor is defined as: Q R C

22 P. Piot, PHYS 375 Spring 008 RC series circuit: mpedance () nd order ODE: d V VR + V + VC R + + C d R d dv + + C. Resonant frequency still et s define the parameter Then the ODE rewrites d d + ζ + ω0 dv

23 P. Piot, PHYS 375 Spring 008 Take back (but could also jut compute the impedance of the system) Explicit in its complex form and deduce the current: ntroducing xω/ω 0 we have RC series circuit: mpedance () dv d d ω ζ ω ω ζω ω ω ω ω ω ζω ω C R Y i i V Y V i i + x x Q R Y

24 P. Piot, PHYS 375 Spring 008 RC series circuit: resonance Values of Q

25 P. Piot, PHYS 375 Spring 008 RC parallel circuit : resonance The same formalism as before can be applied to parallel RC circuits. The difference with serial circuit is: at resonance the impedance has a maximum (and not the admittance as in a serial circuit)