Unformatted text preview: Physics 212
Lecture 19
LC and RLC Circuits
 Oscillation frequency  Energy Energy  Damping
50 40 30 20 10 0 Confused Avg = 2.6 Confident Physics 212 Lecture 19, Slide 1 Music
Who is the Artist?
BB A) B) C) D) E) Professor Longhair Dr. John Allen Touissant Allen Touissant David Egan Henry Butler Why? I guess I’m thinking about New Orleans… Strong New Orleans piano (and vocals) If you have a chance to see him, DO
Physics 212 Lecture 19, Slide 2 Physics 212
Lecture 19
LC and RLC Circuits
 Oscillation frequency  Energy Energy  Damping
50 40 30 20 10 0 Confused Avg = 2.6 Confident Physics 212 Lecture 19, Slide 3 Some Exam Stuff
• Exam Tomorrow Night at 7:00
– – – – Covers material in Lectures 9  17 Bring your ID: Rooms determined by discussion section (see link) Conflict exam at 5:15 – sign up in your gradebook if you need to If you have conflicts with both of these, you should have heard from Prof Errede about scheduling online.physics.uiuc.edu/courses/phys212/fall09/ExamPrepHe2.html Physics 212 Lecture 19, Slide 4 LC Circuit
I
2 ddI Q VL==LL 2 dt dt L C Q Q VC = C but I = dQ dt d 2Q Q =− 2 dt LC d 2Q = −ω 2Q dt 2 ω= 1 LC Physics 212 Lecture 19, Slide 5 d 2Q = −ω 2Q dt 2 ω= 1 LC L C d 2x = −ω 2 x dt 2 k ω= m k F = kx a
m x Same thing if we notice that k↔ 1 C and m↔L Physics 212 Lecture 19, Slide 6 Time Dependence
L C Physics 212 Lecture 19, Slide 7 Preflight 2
At time t = 0 the capacitor is At the fully charged with Qmax and the and current through the circuit is 0. L C BB What is the potential difference across the inductor at t = 0 ? What A) VL = 0 B) VL = Qmax/C B) /C C) VL = Qmax/2C C) /2C VC = VL 60 50 40 The voltage across the capacitor is Qmax/C Kirchhoff's Voltage Rule implies that must also be equal to the voltage across the inductor 30 20 10 0 Pendulum… Physics 212 Lecture 19, Slide 8 Preflight 4
At time t = 0 the capacitor is At the fully charged with Qmax and the and current through the circuit is 0.
BB L C What is the potential difference across the What inductor when the current is maximum ? inductor
50 A) VL = 0 B) VL = Qmax/C B) /C C) VL = Qmax/2C C) /2C dI/dt is zero when current is maximum 40 30 20 10 0 Physics 212 Lecture 19, Slide 9 Preflight 6
At time t = 0 the capacitor is At the fully charged with Qmax and the and current through the circuit is 0.
BB L C How much energy is stored in the capacitor How when the current is a maximum ? when A) U = Qmax2/(2C) /(2C) B) U = Qmax2/(4C) B) /(4C) C) U = 0 C) Total Energy is constant ! ULmax = ½ LI2 UCmax = Qmax2/2C I = max when Q = 0
50 40 30 20 10 0 Physics 212 Lecture 19, Slide 10 10 Preflight 8
The capacitor is charged such The that the top plate has a charge +Q0 and the bottom plate Q0. +Q and At time t=0, the switch is closed t=0 the and the circuit oscillates with frequency ω = 500 radians/s. frequency What is the value of the capacitor C? A) C = 1 x 103 F B) C = 2 x 103 F B) C) C = 4 x 103 F C)
ω=
1 LC C= 1
50 40 30 20 10 0 BB L = 4 x 103 H ω = 500 rad/s rad/s L C ω 2L = 1 = 10−3 (25 × 104 )(4 × 10−3 ) Physics 212 Lecture 19, Slide 11 11 closed at t=0 Preflight 10
+Q0 Q0
BB L C Which plot best represents Which the energy in the inductor as a function of time starting just after the switch is closed? 12 U L = LI 2
2 35 30 25 Energy proportional to I ﬂ C cannot be negative Energy cannot 20 15 10 5 Current is changing ﬂ UL is not constant Current Initial current is zero current 0 Physics 212 Lecture 19, Slide 12 12 Preflight 12
When the energy stored in When the capacitor reaches its maximum again for the first time after t=0, how much time how charge is stored on the top plate of the capacitor? A) B) C) D) E) +Q0 +Q0 /2 0 Q0/2 Q0
dQ I= dt
closed at t=0 BB L C +Q0 Q0 35 30 25 20 Q is maximum when current goes to zero 15 10 5 0 Current goes to zero twice during one cycle Physics 212 Lecture 19, Slide 13 13 Add R: Damping
Just like LC circuit but energy but the oscillations get smaller because of R Concept makes sense… …but answer looks kind of complicated
Physics 212 Lecture 19, Slide 14 14 Physics Truth #1:
Even though the answer sometimes looks complicated…
Q(t ) = Q0 cos(ωt − ϕ ) …the physics under the hood is still very simple !!
d 2Q = −ω 2Q dt 2 Physics 212 Lecture 19, Slide 15 15 The elements of a circuit are very simple:
dI VL = L dt V = VL + VC + VR Q VC = C
dQ I= dt VR = IR This is all we need to know to solve for anything ! Physics 212 Lecture 19, Slide 16 16 A Different Approach Start with some initial V, I, Q, VL Now take a tiny time step dt
dI = VL dt L
(1 ms) dQ = Idt VC = Q C Repeat… VR = IR
VL = V − VR − VC Physics 212 Lecture 19, Slide 17 17 Calculation
The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. What is QMAX, the maximum charge on the capacitor? • Conceptual Analysis
– – Once switch is opened, we have an LC circuit Current will oscillate with natural frequency ω0 V R L C • Strategic Analysis
– – – Determine initial current Determine oscillation frequency ω0 Find maximum charge on capacitor Physics 212 Lecture 19, Slide 18 18 Calculation
The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.
V R IL L C What is IL, the current in the inductor, immediately AFTER the switch is opened? Take positive direction as shown. (A) IL < 0 (B) IL = 0 (B) (C) IL > 0 (C) Current through inductor immediately AFTER switch is opened Current AFTER switch IS THE SAME AS IS the current through inductor immediately BEFORE switch is opened the BEFORE switch BEFORE switch is opened: all current goes through inductor in direction shown
Physics 212 Lecture 19, Slide 19 19 Calculation
The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.
V R IL(t=0+) > 0 IL L VC=0 C The energy stored in the capacitor immediately after the switch is opened is zero. (A) TRUE (B) FALSE (B)
BEFORE switch is opened: dIL/dt ~ 0 ⇒ VL = 0 BUT: VL = VC since they are in parallel since VC = 0 AFTER switch is opened: AFTER VC cannot change abruptly cannot VC = 0 UC = ½ CVC2 = 0 !!
BB IMPORTANT: NOTE DIFFERENT CONSTRAINTS AFTER SWITCH OPENED CURRENT through INDUCTOR cannot change abruptly through INDUCTOR VOLTAGE across CAPACITOR cannot change abruptly across CAPACITOR Physics 212 Lecture 19, Slide 20 20 Calculation
The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.
V R IL(t=0+) > 0 VC(t=0+) = 0 IL L C What is the direction of the current immediately after the switch is opened? (A) clockwise (B) counterclockwise (B) Current through inductor immediately AFTER switch is opened Current AFTER switch IS THE SAME AS IS the current through inductor immediately BEFORE switch is opened the BEFORE switch BEFORE switch is opened: Current moves down through L BEFORE switch down AFTER switch is opened: Current continues to move down through L BB Physics 212 Lecture 19, Slide 21 21 Calculation
The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened.
V R IL(t=0+) > 0 VC(t=0+) = 0 L C What is the magnitude of the current right after the switch is opened? (A) I 0 = V C L (B) I 0 = V R
2 L C (C) I 0 = V R (D) I 0 = 1V 2R Current through inductor immediately AFTER switch is opened Current AFTER switch IS THE SAME AS IS the current through inductor immediately BEFORE switch is opened the BEFORE switch IL IL IL L R VL=0 C V = ILR VL = 0 BB BEFORE switch is opened: V Physics 212 Lecture 19, Slide 22 22 Calculation
The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. Hint: Energy is conserved
V R IL(t=0+) =V/R VC(t=0+) = 0 IL L C What is Qmax, the maximum charge on the capacitor during the oscillations? (A) Qmax =
Imax L C L
V LC R 1 Qmax = CV (B) 2 (C) Qmax = CV (D) Qmax = V R LC Qmax C 2 12 1 Qmax LI max = 2 2C When I is max When (and Q is 0) When Q is max When (and I is 0)
2 1 Qmax U= 2C Qmax = I max LC = V LC R BB 12 U = LI max 2 Physics 212 Lecture 19, Slide 23 23 The switch in the circuit shown has been closed for a long time. At t = 0, the switch is opened. FollowUp 1
V R Imax =V/R max =V/R IL L C Is it possible for the maximum voltage on the capacitor to be greater than V? (A) YES (B) NO (B) Qmax = V LC R Qmax = V LC R Vmax = VL RC Vmax can be greater than V IF: can IF L >R C We can rewrite this condition in terms of the resonant frequency: ω0 L > R OR 1 >R ω0C
BB We will see these forms again when we study AC circuits!! Physics 212 Lecture 19, Slide 24 24 ...
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 Fall '10
 Selig
 Energy, switch, Qmax

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