Inductance, RL Circuits, LC Circuits, RLC Circuits

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Fay Haynes
6 years ago
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1 Inductance, R Circuits, C Circuits, RC Circuits

2 Inductance What happens when we close the switch? The current flows What does the current look like as a function of time? Does it look like this? I t

3 Inductance What happens when we close the switch? The current flows What does the current look like as a function of time? If there is current in the loop, then there is a magnetic field The flux would change infinitely fast if the current looked like this, so there would be an (infinite) EMF I t 3

4 Inductance In fact any loop resists having it s current changed by generating an EMF, call this (self) inductance, di E V s E =, 1 Henry (H) 1 dt di dt A 4

5 Inductance For a solenoid of N turns, we have di dφ NΦ B E = = N = solenoid dt dt I µ Φ B = solenoid BA; = B = NBA I ni = µ nna = µ n Al = µ n B V 5

6 R Circuits OK, what really happens when we close the switch Move all the inductance into a new circuit element Use Kirchhoff s Rule for voltages around a loop E E IR E = IR = solenoid di dt 6

7 E R Circuits Solving the 1 st Order Differential Equation di di IR = dt = dt E IR E I t = I E di IR = log R R ( E IR) E IR E = log e ( ) E R R t = 1 = E IR ( R ) t ( e ) I 7

8 R Circuits What happens when we disconnect the ery (but still have the loop with the resistor and inductor)? E = = R I e I e ( R ) t ( ) R t 8

9 R Circuits The important idea is that an inductor acts like an infinite resistor when you initially try to push current through it and then after the current has been flowing for a long time it acts like a wire. Once you remove the current flowing through it, it creates its own current to cancel out the change in magnetic flux through its center, but that current quickly decreases to zero (if there is resistance in the wire) 9

10 Energy in the Magnetic Field Consider the Kirchhoff loop equation again E = IR + di dt A voltage is just an energy per charge So when the current is changing the inductor is storing up or releasing energy! How much, energy? Consider the power: di P = IE = I R + I dt 1

11 Energy in the Magnetic Field Consider the Kirchhoff loop equation again E = IR + E di dt P = I = I R + I dt For the inductor du di P = = I ind dt dt Compare to capacitor I di U = du = IdI = I 1 U U C = = 1 1 CV I 11

12 C Circuits and Oscillations What happens when we have a (charged) capacitor and inductor in a circuit Start with charged capacitor, and throw the switch 1

13 C Circuits and Oscillations 1) At t =, all the energy is stored in the capacitor 13

14 C Circuits and Oscillations 1) At t =, all the energy is stored in the capacitor ) When the capacitor is discharged, all the energy is stored in the inductor 14

15 C Circuits and Oscillations 1) At t =, all the energy is stored in the capacitor ) When the capacitor is discharged, all the energy is stored in the inductor 3) The current then recharges the capacitor (oppositely) 15

16 C Circuits and Oscillations 1) At t =, all the energy is stored in the capacitor ) When the capacitor is discharged, all the energy is stored in the inductor 3) The current then recharges the capacitor (oppositely) 4) And the cycle repeats 16

17 C Circuits and Oscillations How do we find the current quantitatively? Use Kirchhoff s aw again V C + E = Q di Q d Q + = + = C dt C dt d Q 1 = Q dt C This is the same equation as a simple harmonic oscillator 17

18 C Circuits and Oscillations How do we find the current quantitatively? Use Kirchhoff s aw again V C + E = Q di Q d Q + = + = C dt C dt d Q 1 = Q dt C This is the same equation as a simple harmonic oscillator C 1 ; k m; ω = k m ω = 1 C 18

19 C Circuits and Oscillations How do we find the current quantitatively? V C + E = Q di Q d Q + = + = C dt C dt d Q 1 = Q = ω Q dt C Q( t) = Q cos ωt + ϕ ( ) dq I( t) = = ωq sin( ωt + ϕ ) dt 19

20 U C Circuits and Oscillations What about the energy? With no resistor, energy is conserved = U C + U Q = cos ωt + C Q = cos ωt + C Q = C Q 1 = = I max C 1 1 I ( ωq ) max ( cos ωt + sin ωt) sin sin ωt ωt

21 RC Circuits Now, what if we add a resistor? This is like the damped harmonic oscillator RI = du dt Q = C Q RI = C d Q = dt + C di I dt dq dt di dt 1 I Q dq + R + C dt Rt Q t = Q e ( ) d Q = dt cosω t d for small R ω d = 1 C R 1

Handout 10: Inductance. Self-Inductance and inductors

Handout 10: Inductance. Self-Inductance and inductors 1 Handout 10: Inductance Self-Inductance and inductors In Fig. 1, electric current is present in an isolate circuit, setting up magnetic field that causes a magnetic flux through the circuit itself. This