Electricity and Magnetism-Driven LCR circuits

1 8.022 (E&M) – Lecture 17 Topics: 􀂄 Discussion of Exam 2 and make-up exam 􀂄 Back to E&M: 􀂄 RCL circuits: recap undriven RCLs, driven RCLs, inductance G. Sciolla – MIT 8.022 – Lecture 17 4 Last time 􀂄 What happens when we put inductors in circuits? 􀂄 RL circuits: exponential solutions 􀂄 LC circuits: oscillatory solution 􀂄 RCL circuits: damped oscillation 􀂄 RCL circuits are particularly interesting 􀂄 Let’s see them in some more detail… 3 G. Sciolla – MIT 8.022 – Lecture 17 5 Undriven RCL circuits: recap 􀂄 Kirchoff’s second rule: 􀂄 Does it look familiar? 􀂄 Mechanics: harmonic oscillator! 2 2 1 0 d Q dQ L R Q dt dt C + + = 1/C ~ke 􀃆 elastic term due to spring ke x 1/C Q R ~ kf 􀃆 friction (damping) term kf v = kf dx/dt R dQ/dt L ~ m: inertia term ma=m d2x/dt2 L d2Q/dt2 Interpretation Mechanics RCL 2 2 0 f e d x dx m k k x dt dt + + = G. Sciolla – MIT 8.022 – Lecture 17 6 2 ( ) NB: is a complex number, with and real = damping term, = oscillatory term Throw this into the equation and we get a quadratic equation in : 1 0 t t i t t it Q t e e e i e e RL LC β α ω α ω β α ω α ω β β β − − = = = − + + + = 􀀄 22 4 1 2R R L L LC β= − ± − ⇒ Undriven RCLs: solution 􀂄 Differential equation governing loop: 􀂄 Solve using complex number notation: 2 2 1 0 d Q R dQ Q dt L dt LC + + =4 G. Sciolla – MIT 8.022 – Lecture 17 7 RCL circuits: solution 22 22 22 22 221 L β purely real: 0 R>2 4 C β purely imaginary: 0 undamped LC 1 β truly compl 1 : 2 41 Whe ex: R>0 an n d 0 4 0 4 R 1 = and = 2L 4 RL LC R RL LC R R L L LC R R LC L L LC α β ω ⎧⎪⎪• − > ⇒ ⇒ ⎪⎪⎪⎪• ⇒ = ⇒ = ⇒ ⎪⎨⎪⎪• − < ⇒ ⎪ − ± − − − ⎪⎪⎩ = ⎪⎪ critical damping (fastest way to damp an oscillator). ( ) t t i t Qt e e e β α ω − = = 􀀄 G. Sciolla – MIT 8.022 – Lecture 17 8 RCL in weak damping limit 􀂄 Initial conditions: 􀂄 Graphical representation of solution: 2 0 0 2 0 0 0 ( )~ cos( ) ( )~ sin( ) R t L R t L Qt Qe t I t Q e t ω ω ω − − ⎧⎪⇒ ⎨⎪⎩ t I(t) 0 0 0 0 0 0 Q(0)=Q =Acos( ) and I(0)=0=A si n ; 0 A Q φ ω φ φ = ⇒ =5 G. Sciolla – MIT 8.022 – Lecture 17 9 Energy 􀂄 Energy of the circuit in the weak damping limit: 􀂄 Since Q20/2C=total energy stored initially in the system 􀃆 U decreases exponentially over time: as expected! 2 2 /2 0 0 ( ) ( ) cos 2 2 Rt L C Q Q t U t e t C C ω − = = 2 2 2 2 /2 /2 0 0 0 0 0 1 1 ( ) ( ) sin sin 2 2 2 Rt L Rt L L Q U t LI t LQ e t e t C ω ω ω − − = = = 2 2 /2 2 /0 0 0 0 ( ) ( ) ( ) (sin cos ) = 2 2 Rt L Rt L L C Q Q U t U t U t e t t e C C ω ω − − ⇒ = + = + G. Sciolla – MIT 8.022 – Lecture 17 10 Quality Factor 􀂄 Definition 1: the quality factor measures how many times the circuit oscillates before it loses a certain amount of energy 􀂄 Definition 2: the quality factor measures the ratio between energy stored (in C and L) and average power dissipated (in R) 􀂄 Q factor can be defined for any system that creates vibrations. 􀂄 Acoustics: Q of a tuning fork is much higher than the Q of a table… In the time =L/R the energy decreases by U(t)=1/e The oscillation is radian Q s L R τ ω ωτ ωτ = = ∆ ⇒ 2 0 2 0 /2 Energy For an oscill stored Q ation with frequency 2 /LI L RI R ω ω ω ω⇒ = = =2 /0 ( )=2 Rt L Q U t e C −6 G. Sciolla – MIT 8.022 – Lecture 17 11 Today’s goal: Driven RCL circuits 􀂄 ~ is an AC e.m.f. 􀂄 AC voltage supplied to the circuit: 􀂄 Convenient assumption: 􀂄 NB: V0 is purely real! 􀂄 How to solve this? Just generalize what we used for DC! 􀂄 Sum of voltage drops in loop is equal to emf (Kirchoff #2) 􀂄 The same current must pass through every circuit element 0 ( ) cos emf t V t ω = 0 ( ) Re ( ) with ( ) i t Vt Vt Vt Veω ⎡ ⎤ = = ⎣ ⎦ 􀀄 􀀄 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) emf R C L emf R C L V t V t V t V t V t V t V t V t = + + = + + 􀀄 􀀄 􀀄 􀀄 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R C L R C L I t I t I t I t I t I t I t I t = = = = = = 􀀄 􀀄 􀀄 􀀄 G. Sciolla – MIT 8.022 – Lecture 17 12 AC current 􀂄 Consider a B constant in magnitude and a loop rotating around its axis with angular velocity ω 􀂄 If S is the area of the loop: 􀂄 Faraday: 􀂄 This is how AC power is generated. In U.S.: ν=60 Hz 􀃆 ω=377 ( ) 1 . . . cos sin e m f BS t BS t c t cω ω ω ∂ = = ∂ cos cos S B da BS BS t θ ω = = ∫ 􀁇 􀁇 i ω θ B 􀁇7 G. Sciolla – MIT 8.022 – Lecture 17 13 AC emf + resistor R 􀂄 Ohm’s law holds for AC too: 􀂄 Let’s plot I(t) and V(t) on the same graph: 􀃆 In a resistor the voltage and the current are in phase (peak voltage occurs at the same time as peak current) ( ) ( ) ( ) R V t V t I tR = = ---I(t) __ V(t) ~ R V I t V(t)G. Sciolla – MIT 8.022 – Lecture 17 14 Reminder: phasor notation Any complex number can always be represented as the product of a real number (magnitude) and a complex exponential: where and given Euler’s relation: which can be easily proved using Maclaurin expansion with 1 i= -z x i y = + x y z x i y = + x y 2 2 y magnitude r= x +y and phase =arctg x (cos sin ) z r i θ θ θ ⇒ = + cos sin ie i θ θ θ = + (Phasor representation) i z reθ ⇒ =8 G. Sciolla – MIT 8.022 – Lecture 17 15 AC emf + R with phasors 􀂄 The same information can be represented with phasors in the complex plane: 􀃆 In a resistor the voltage and the current are in phase In phase means that both phasors are at the same angle ( ) ( ) V t RI t = 􀀄 􀀄 ~ R V I G. Sciolla – MIT 8.022 – Lecture 17 16 AC emf + capacitor C 􀂄 Connect AC emf across a capacitor C: 􀂄 Since V(t)=V0cosωt and I(t)= dQ/dt: 􀃆 I(t) LEADS V(t) by 90 deg /V(t) lags I(t) by 90 deg (maxima in I(t) occur before maxima in V(t)) ( ) ( ) ( ) C Q t V t V t C = = ---I(t) __ V(t) ~ C V 0 0 ( ) ( ) sin cos( ) 2 dQ t I t CV t CV t dt π ω ω ω ω = =− = +t V(t)9 G. Sciolla – MIT 8.022 – Lecture 17 17 Ohm’s law revisited and Impedance 􀂄 Relation between I(t) and V(t) becomes more obvious when using phasor notation: 􀂄 For the current: 􀂄 Combining complex currents and voltages we can write: 0 0 ( ) cos Re ( ) with ( ) i t C C V t V t V t V t Veω ω ⎡ ⎤ = = = ⎣ ⎦ 􀀄 􀀄 0 i 2 2 0 0 ( ) cos( ) Re ( ) 2 with ( ) (remember: e ) C i t i t I t CV t I t I t CV e i CV e i π π ω ω π ω ω ω ω ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ ⎡ ⎤ = + = ⎣ ⎦ = = = 􀀄 􀀄 C C ( ) ( ) where Z i (c s ompl the ex equivalent of Ohm's law)1 impe of a capacito dance Z =i : C r C V t I t Z ω = 􀀄 􀀄 G. Sciolla – MIT 8.022 – Lecture 17 18 AC emf + C: phasor representation 􀂄 Given V(t) and I(t) can easily be represented in the complex plane: NB: I(t) is ahead of V(t) by 90 degrees: I(t) leads V(t) by 90 degrees 0 0 0 ( ) and ( ) = i t i t i t C V t V e I t Z V e i CV e ω ω ω ω = = 􀀄 􀀄10 G. Sciolla – MIT 8.022 – Lecture 17 19 AC emf + inductor L 􀂄 Connect AC emf across an inductor L: 􀂄 Since V(t)=V0cosωt: 􀃆 I(t) LAGS V(t) by 90 degrees, or V(t) LEADS I(t) by 90 degrees (maxima in I(t) occur before maxima in V(t)) ( ) ( ) L dI V t V t Ldt = = ---I(t) __ V(t) ~ L V 0 0 0 cos ( ) sin cos 2 V V V dI t It t t dt L L L π ω ω ω ω ω ⎛ ⎞ = ⇒ = = − ⎜ ⎟ ⎝ ⎠ V(t) t G. Sciolla – MIT 8.022 – Lecture 17 20 Impedance of inductors 􀂄 Using phasor notation: 􀂄 The current is: 􀂄 Combining complex currents and voltages we can write: 0 0 ( ) cos Re ( ) with ( ) i t C L V t V t V t V t Veω ω ⎡ ⎤ = = = ⎣ ⎦ 􀀄 􀀄 ( ) 0 -i 1 2 0 0 2 ( ) cos( ) Re ( ) 2 with ( ) (remember: e ) i t i t V I t t I t L V V I t e e i i L i L π π ω ω π ω ω ω ω ⎛ ⎞ − ⎜ ⎟ − ⎝ ⎠ ⎡ ⎤ = − = ⎣ ⎦ = = = = − 􀀄 􀀄 L L ( ) ( ) where (complex equivalent of Ohm's law) impedance Z Z is the of an inductor: =i L V t L I t Z ω = 􀀄 􀀄11 G. Sciolla – MIT 8.022 – Lecture 17 21 AC emf + L: phasor representation 􀂄 Given V(t) and I(t) can easily be represented in the complex plane: NB: I(t) is 90 degrees behind V(t): I(t) lags V(t) by 90 degrees 0 0 0 ( ) and ( ) = i t i t i t L V V t Ve I t ZVe e i L ω ω ω ω = = 􀀄 􀀄 G. Sciolla – MIT 8.022 – Lecture 17 22 Driven RCLs using inductance 􀂄 Inductance simplifies the study of driven RCL circuits 􀂄 Let’s work with complex numbers and use Ohm’s and Kirchoff’s extensions ( ) ( )1 1 Since ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 where of the circuit total impedance is R C C emf tot L L tot V t RI t V t Z I t I t V t I t R i L I t Z i C C V t Z I t i LI t Z R i L Cω ω ω ω ω ω ⎧ = ⎪ ⎛ ⎞ ⎛ ⎞ ⎪ = = ⇒ = + − = ⎨ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎪⎪= = ⎩ ⎛ ⎞ ≡ + − ⎜ ⎟ ⎝ ⎠ 􀀄 􀀄 􀀄 􀀄 􀀄 􀀄 􀀄 􀀄 􀀄 􀀄 􀀄 􀀄 􀀄 ( ) ( ) ( ) ( ) emf R C L V t V t V t V t = + + 􀀄 􀀄 􀀄 􀀄12 G. Sciolla – MIT 8.022 – Lecture 17 23 Driven RCLs: phasor notation 􀂄 The complex current can be written as 􀂄 This can be written as: 0 ( ) ( ) 1 i t emftot V t Ve I t Z R i L C ω ω ω = = ⎛ ⎞ + − ⎜ ⎟ ⎝ ⎠ 􀀄 􀀄 * 0 0 0 0 * 2 2 0 0 2 2 -i 1 ( ) 1I = 1 Remembering that e cos sin 1 1 tg i t i t i t i t i tot tot tot tot V e V e V e I t Z R i L I e e Z Z Z C R L C V R L C i L L C R R RC ω ω ω ω φ θ ω ω ω ω ω ω θ θ ω ω ω φ ω − ⎡ ⎤ ⎛ ⎞ = = = − − = ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦ ⎛ ⎞ + − ⎜ ⎟ ⎝ ⎠ ⎧⎪⎛ ⎞ ⎪ + − ⎜ ⎟ ⎪ ⎝ ⎠ = − ⇒⎨⎪− ⎪⎪ = = − ⎩ 􀀄 G. Sciolla – MIT 8.022 – Lecture 17 24 Dependence of φ from ω 0 oo high : I lags volt1 age by 90 low : I lead N s voltage by t B: g 0 ) 9 ( i t i LR RC I t I e e ω φ ω φ ω ω ω − = → = − → 􀀄 ω φ(ω)13 G. Sciolla – MIT 8.022 – Lecture 17 25 AC motor (H26) 􀂄 2 RL circuits driven by 60 Hz AC voltage 􀂄 Coil 1: R=2.3 Ω, L=1.5mH 􀂄 Coil 2: R=2.5 Ω, L=31 mH 􀂄 What is the ∆φ between the 2 currents? 􀂄 Z1=R1+iωL1=2.3+i 377 1.5 10-3 􀂄 Z2=R2+iωL2=2.5+i 377 31 10-3 􀃆 ∆φ=64 degrees 􀂄 The difference in phase will create a rotating B field 􀃆 Eddie currents in the metal can will make it rotate! Coil 1 Coil 2 ~ ~ G. Sciolla – MIT 8.022 – Lecture 17 26 Dependence of I0 from ω 0 0 2 2 0 I = 1 1 Maximum current when 1 resonance frequency V R L C L LC C ω ω ω ω ω ⎛ ⎞ + − = = ⎜ ⎟ ⎝ ⎠ ⇒ I0 ω ω014 G. Sciolla – MIT 8.022 – Lecture 17 27 RCL resonance (Demo L8) 􀂄 RCL circuit driven with variable frequency ω 􀂄 L=50 mH 􀂄 C=0.3 µF 􀂄 Measure VR on scope and tune frequency to maximize VR 􀂄 What is the expect resonance frequency? 3 0 1 8.2 10 1.3 kHz LC ω ν = = × ⇒ = scope G. Sciolla – MIT 8.022 – Lecture 17 28 Demo L8: part 2 􀂄 Same RCL circuit driven with variable frequency ω 􀂄 Frequency is driven by a voltage Vin 􀂄 L=50 mH 􀂄 C=0.3 µF 􀂄 Display VR vs on the scope while sweeping Vin 􀂄 What do you expect to see? scope ω0=1.3 kHz15 G. Sciolla – MIT 8.022 – Lecture 17 29 Resonant RCL with light bulb (L6) 􀂄 RCL circuit driven by AC voltage 􀂄 C can be adjusted using set of switches 􀂄 L can be adjusted moving the Fe core inside a solenoid 􀂄 For each setting of C we can find an L that turn on the light bulb 􀂄 What is that L? 2 1 L C ω = G. Sciolla – MIT 8.022 – Lecture 17 30 Summary and outlook 􀂄 Today: 􀂄 Undriven RCL circuits 􀂄 Energy stored and quality factor in weak damping limit 􀂄 Driven RCL AC circuits 􀂄 Simple solution when introducing complex impedance Z 􀂄 ZR = R 􀂄 ZC = 1/(iωC) 􀂄 ZL = iωL 􀂄 Next Tuesday: 􀂄 More on driven RCLs: power, resonances, filters…