1 / 26

Summary: RC and LC circuits Mechanical Harmonic Oscillator – prototype

Electricity and Magnetism Lecture 13 - Physics 121 ( for ECE and non-ECE ) Electromagnetic Oscillations in LC & LCR Circuits, Y&F Chapter 30, Sec. 5 - 6 Alternating Current Circuits, Y&F Chapter 31, Sec. 1 - 2. Summary: RC and LC circuits Mechanical Harmonic Oscillator – prototype

bannon
Download Presentation

Summary: RC and LC circuits Mechanical Harmonic Oscillator – prototype

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Electricity and MagnetismLecture 13 - Physics 121 (for ECE and non-ECE)Electromagnetic Oscillations in LC & LCR Circuits, Y&F Chapter 30, Sec. 5 - 6Alternating Current Circuits, Y&F Chapter 31, Sec. 1 - 2 • Summary: RC and LC circuits • Mechanical Harmonic Oscillator – prototype • LC circuits oscillate forever • LCR circuits oscillate with damping • AC circuits oscillate at forced frequency • AC circuits look simpler using “phasors” • Currents and voltages have fixed relative phasing that differs for simple resistive, capacitive, inductive components in circuits.

  2. R R E E C L Voltage across C related to Q - integral of current. Voltage will LAG current Voltage across L related to derivative of current. Voltage will LEAD current Growth Phase Decay Phase Now: LCR in same circuit + periodic EMF –> New effects R • Resonant oscillations in LC circuit – ignore damping L • Damped oscillation in LCR circuit - not driven • Circuit driven by external AC frequency wD=2pf • which could be the ‘resonant’ frequency. C • New descriptive quantities: • Generalized resistances: reactances, impedance R L C Time dependent effects - RC and RL circuits with constant EMF

  3. Definition: oscillating system • State ( t ) = state( t + T ) = ……. = state( t + NT ). • T is the “period” -Time to complete one complete cycle • f = 1/T is the frequency. w = 2pf is the “angular frequency” • “State” can mean: position and velocity, electric & magnetic fields,… Mechanical Example: Simple harmonic oscillator (undamped) Mechanical energy is conserved` no friction Systems that oscillate obey equations like this... With solutions like this... What oscillates? position & velocity, acceleration, spring force, Energy oscillates between 100% kinetic and 100% potential: Kmax = Umax Model: Resonant mechanical oscillations (SHM)

  4. a + b Kirchhoff loop equation: L E C Substitute: An oscillator equation results solution: derivative: What oscillates? Charge, current, B & E fields, UB, UE Peaks of current and charge are out of phase by 900 Electrical Oscillations in an LC circuit, zero resistance Charge capacitor fully to Q0=CEthen switch to “b” Note: could have chosen sin rather than cos, with phase constant f adjusted

  5. Total electrical potential energy is constant: The peak values UE0 and UB0 are equal: Electrical Oscillations in an LC circuit, zero resistance

  6. The definition and imply that: oscillator equation • Either (no current ever flows) OR: • Oscillator solution: • To evaluate w0: plug the first and second derivatives of the solution into the differential equation. • The resonant oscillation frequency w0 is: Details: Derive Oscillator Equation from Energy Conservation • The total energy: • It is constant so:

  7. Oscillations Forever? 13 – 1: What do you think will happen to the oscillations in an ideal LC circuit (versus a real circuit) over a long time? • They will stop after one complete cycle. • They will continue forever. • They will continue for awhile, and then suddenly stop. • They will continue for awhile, but eventually die away. • There is not enough information to tell what will happen.

  8. Potential Energy alternates between all electrostatic and all magnetic – two reversals per period t • C is fully charged twice • each cycle (opposite polarity) • So is L • Peaks are 90o out of phase

  9. C L a) Find the oscillation period and frequency f = ω / 2π b) Find the maximum (peak) current (differentiate Q(t) ) c) When does current i(t) have its first maximum? When |sin(w0t)| = 1 First One at: Maxima of Q(t): All energy is in E field Maxima of i(t): All energy is in B field Maxima of current at T/4, 3T/4, … (2n+1)T/4 Others at: A 4 µF capacitor is charged to E = 5.0 V, and then connected to a 0.3 Henry inductance in an LC circuit Example: Use preceding solutions with f = 0

  10. Find the expression for the current in the circuit: c) How long until the capacitor charge is exactly reversed? That occurs every ½ period, given by: Example • Find the voltage across the capacitor in the circuit as a function of time. L = 30 mH, C = 100 mF.The capacitor has charge Q0 = 0.001 C. at time t = 0. The resonant frequency is: The voltage across the capacitor has the same time dependence as the charge: At time t = 0, Q = Q0, so chose phase angle f = 0 to match initial condition.

  11. Which Current is Greatest? 13 – 2: The expressions below could represent the charge on a capacitor in an LC circuit as a function of time. Which one has the greatest peak current magnitude? • Q(t) = 2 sin(5t) • Q(t) = 2 cos(4t) • Q(t) = 2 cos(4t+p/2) • Q(t) = 2 sin(2t) • Q(t) = 4 cos(2t)

  12. I. II. III. LC circuit oscillation frequencies 13 – 3: The three LC circuits below have identical inductors and capacitors. Order the circuits according to their oscillation frequency in ascending order. • I, II, III. • II, I, III. • III, I, II. • III, II, I. • II, III, I.

  13. a R + b E C The resistor dissipates stored energy. The power is: L Modified oscillator equation has damping (decay) term: Find transient solution: oscillations with exponential decay (under- damped case) Shifted resonant frequency w’ can be real or imaginary “damped frequency” Underdamped: Overdamped Critically damped Q(t) LCR Series Circuit adds series resistance: Shifted Oscillations but with Damping Charge capacitor fully to Q0=CEthen switch to “b”

  14. Resonant frequency with damping 13 – 4: How does the resonant frequency w0 for an ideal LC circuit (no resistance) compare with w’ for an under-damped LCR circuit whose resistance is not negligible? • The resonant frequency for the damped circuit is higher than for the ideal one (w’ > w0). • The resonant frequency for the damped circuit is lower than for the ideal one (w’ < w0). • The resistance in the circuit does not affect the resonant frequency—they are the same (w’ = w0). • The damped circuit has an imaginary value of w’.

  15. Apply external voltage Real, instantaneous current flows • Sinusoidal EMF appears in rotating coils in a magnetic field (Faraday’s Law) • Slip rings and brushes take EMF from the coil and produce sinusoidal AC voltage and current. • External circuits are driven by sinusoidal EMF • Current in external circuit may lead or lag EMF according to character of the load connected. Alternating External Voltage (AC) Source • Commercial electric power (home or office) is AC, with frequency = 60 Hz in U.S, 50 Hz in most other countries. • High transmission voltage  Lower i2R loss than DC, and is reduced for distribution to customers by using transformers. • “Steady State” sinusoidal signal has frequency wD: • “Phase angle” F is between current and EMF in the driven circuit. • You might (wrongly) expect F = 0 • F is positive when Emax leads Imax and negative otherwise (convention)

  16. load the driving frequency STEADY STATE RESPONSE (after transients die away): . Current in load is sinusoidal, has same frequency as source ... but . Current may be retarded or advanced relative to E by “phase angle” F for the whole circuit(due to inertia L and stiffness 1/C). resonant frequency peak applied voltage DEFINE: IMPEDANCE IS THE RATIO OF PEAK (or RMS) EMF TO PEAK (or RMS) CURRENT. peak current • Impedance for a circuit is a function of resistances, reactances, and how they are interconnected Apply Sinusoidal AC Source to circuit KIRCHHOFF RULES APPLY: . In a series branch, current has the same amplitude and phase everywhere. . Across parallel branches, voltage drops have the same amplitudes and phases . At nodes (junctions), instantaneous net currents in & out are conserved

  17. Peak amplitudes  lengths of the phasors. • Measurable, real, instantaneous values of i( t ) and • E( t ) are projections of the phasors onto the x-axis. F “phase angle”: the angle from current phasor to EMF phasor in the driven circuit. wDt and are independent Real X • Current is the same (phase included) everywhere • in a single essential branch of any circuit. • Current vector is reference for series LCR circuit. • Applied EMF Emax(t) leads or lags the current by a • phase angle F in the range [-p/2, +p/2] Relative phases for individual series circuit elements (to be shown): • Voltage across R is in phase with the current. • Voltage across C lags the current by 900. • Voltage across L leads the current by 900. VR VC VL XC>XL XL>XC represent currents and potentials as vectors rotating at frequency wD in complex plane Phasors Imaginary Y

  18. Peak voltage and current amplitudes are just the coefficients out front • represented by lengths of rotating phasors • Simplistic time averaging of periodic quantities gives zero (and useless) results). • example: integrate over a whole number of periods – one t is enough (wt=2p) Integrand is odd over a full cycle The meaningful types of quantities in AC circuits can be instantaneous, peak, or average (RMS) • Instantaneous voltages and currents refer to a specific time: • oscillatory, depend on time through argument “wt” • possibly advanced or retarded relative to each other by phase angle • represented by x-component of rotating “phasors” – see slides below So how should “average” be defined?

  19. “RMS averages” are used the way instantaneous quantities were in DC circuits: • “RMS” means “root, mean, squared”. NOTE: • “Rectified average values” are useful for DC but seldom used in AC circuits: • Integrate |cos(wt)| over one full cycle or cos(wt) over a positive half cycle Integrand on a whole cycle is positive after squaring Prescription: RMS Value = Peak value / Sqrt(2) Definitions of Average AC Quantities

  20. Current/Voltage Phases in Resistors, Capacitors, and Inductors VR& Im in phase Resistance VC lags Im by p/2 Capacitive Reactance VL leads Im by p/2 Inductive Reactance Same Phase currrent • Sinusoidal current i(t) = Imcos(wDt). Peak is Im • vR(t) ~ i(t) vL(t) ~ derivative of i(t) Vc(t) ~ integral of i(t) • Peak voltages across R, L, or C lead/lag current by 0, p/2, -p/2 radians • Ratios of peak voltages to peak currents are called “reactances” in C’s and L’s • Phases of voltages in a series branch are relative tothe current phasor.

  21. i(t) vR( t ) Applied EMF: From Kirchhoff loop rule: Time dependent current: The phases of vR(t) and i(t) coincide The ratio of the AMPLITUDES (peaks) VRto Im is the resistance: Peak current and peak voltage are in phase across a resistance, rotating at the driving frequency wD AC current i(t) and voltage vR(t) in a resistor are in phase Extra Discussion From voltage drop across R:

  22. i(t) Applied EMF: L vL(t) From Kirchhoff loop rule: Time dependent current: Note: Definition: inductive reactance Voltage phasor across an inductor leads current by +p/2 (F positive) • Inductive Reactance • limiting cases • w 0: Zero reactance. • Inductor acts like a wire. • w  infinity: Infinite reactance. • Inductor acts like a broken wire. The RATIO of the peaks (AMPLITUDES) VL to Im is the inductive reactance XL: AC current i(t) lags the voltage vL(t) by 900 in an inductor Sine from derivative From voltage drop across L (Faraday Law): so...phase angle F = + p/2 for inductor Extra Discussion

  23. Applied EMF: From Kirchhoff loop rule: i vC ( t ) C Time dependent current: Note: Definition: capacitive reactance • Capacitive Reactance • limiting cases • w 0: Infinite reactance. DC blocked. C acts like broken wire. • w  infinity: Reactance is zero. Capacitor acts like a simple wire The RATIO of the AMPLITUDES VC to Im is the capacitive reactance XC: AC current i(t) leads the voltage vC(t) by 900 in a capacitor From voltage drop across C (proportional to Q(t)): Sine from integral so...phase angle F = - p/2 for capacitor Voltage phasor across capacitor lags the current by p/2 (F negative) Extra Discussion

  24. Applied EMF: vR Current: E R L vL C vC F Im Em wDt+F wDt Instantaneous voltages add normally (Loop Rule) VL Em Im • Phasors for VR, VL, & VC all rotate at wD : • VR has same phase as Im • VClags Im by p/2 • VLleads Im by p/2 F wDt VR Phasor magnitudes add like vectors VC along Im perpendicular to Im Phasors applied to a Series LCR circuit Same current including phase everywhere in the single branch • Refer voltage phasors to current phasor • Everything oscillates at the driving frequency wD • Same phase for the current in all the component, but...... • Current leads or lags E (t) by a constant phase angle F

  25. Impedance Definition: applied to the Series LCR Circuit Voltage peak addition rule in series circuit: Em Im F VL-VC Z Reactances: XL-XC VR wDt R Same peak current flows in each component: Divide each voltage in Em by (same) peak current: Applies to a single series branch with L, C, R Magnitude of Z: F measures the power absorbed by the circuit: Phase angle F: See phasor diagram • RESONANCE: XL=XC Im parallel to Em  F = 0  Z = R  maximum current F is zero  circuit acts purely resistively. • Otherwise: F >0 (current lags EMF) or F<0 (current leads EMF) • R ~ 0  Im normal to Em  F ~ +/- p/2  tiny losses, no power absorbed

  26. vR Circuit Element Symbol Resistance or Reactance Phase of Current Phase Angle Amplitude Relation E R Resistor R R In phase with VR 0º (0 rad) VR = ImR L vL C Capacitor C XC=1/wdC Leads VC by 90º -90º (-p/2) VC = ImXC vC Inductor L XL=wdL Lags VL by 90º +90º (p/2) VL = ImXL Em Im F VL-VC Z VR XL-XC wDt R sketch shows XL > XC Summary: AC Series LCR Circuit

More Related