1. Lesson 10 Calculation of Inductance LR circuits
Lesson 10: Inductance
Lesson 10
Calculation of Inductance
LR circuits
Oscillations in LC circuits
LRC circuits
EMF due to mutual inductance 1

2. I E Field in Capacitor - - q By passing an electric current through
a capacitor we store energy as an electric
E Field in Capacitor
field between the plates which has potential energy 1 U = CV 2 2
and energy density 1 u = e E 2 2 I - + q - + 2

3. By passing an electric current through a solenoid we form a magnetic Field B which also stores energy
B Field in Inductor I - + L
INDUCTOR 3

4. Charging Inductor I
Source of Back EMF 4

5. Charging an Inductor
Charging Inductor II
Current flows through coil producing a magnetic field B
As current builds up to its equilibrium value it is changing thus B is changing and hence  is changing
Changing  produces an EMF 5

6. Charging Inductor III
This EMF ind will produce a changing current Iind in the opposite direction to the charging current
Thus producing an induced magnetic field Bind that opposes B
After some time this results in equilibrium magnetic field Beq and current Ieq 6

7. Ieq is actually equal to the steady state current R --why?
and Beq is the magnetic field corresponding to this current
The effect of the inductor is to cause a gradual increase of current from zero to the steady state value
Charging Inductor IV 7

8. e Inductance = - N d F dt L dI µ changing flux in coilB dt L dI
induced emf
changing flux in coil
changing current
For soleniod BA m
nIA
thus nA n 2 Al ; l
length
hence
solenoid
which depends on geometry of coil
Inductance 8

9. SI units I e e [ ] [ ] S . I . units of Inductance L = - dI dt é ù ê úV Vs [ ] L = = = = H (
Henry ) ê dI ú A A ë û dt s
Also d F N B dt L = - dI dt é ù d F Wb ê N B ú Wb [ ] dt s L = = = = H (
Henry ) ê ú dI A A ê ú ë dt û s 9

10. SI units II 10

11. RL Circuits: Charging I- L +
Kirchoffs Law e e ( ) ( ) - I + = t R t L 11

12. e e e e e Charging II [ ] ( ) ( ) ( ) ( ) Increasing current - I + = tL e dI - I ( ) - = t R L dt e e æ ö æ ö t Rt - - Þ ( ) = I t ç - ÷ = ç - 1 e L 1 e t ÷ ç L ÷ R è ø R è ø L t =
Time constant for RL circuit : L R é ù L H [ ] t = = = ê ú s L ë û R W 12

13. Discharging 13

14. Energy Stored in the Magnetic Field
Energy in Inductor e dI - I ( ) t R - L = dt e dI Þ ( ) ( ) 2 ( ) I t - I t R - L I t = dt
Power supplied Joule heat Rate at which
by source emf dissipated energy is stored
by load by Inductor dU dI ( ) = B = LI t dt dt ò U ( t ) ò I ( t ) 1 Þ ( ) = = ( ) 2 U t dU L I d I = LI t B 2 14

15. Comparing Inductor and capacitor15

16. Energy densities ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) energy
Energy Density =
volume
Energy densities
For solenoid ( ) B t L = m n 2 = m 2 ( ) Al n
Vol
and I t = m n æ ( ) ö 2 æ ö 1 1 B t ( ) ( ) 2 U t = LI t = ç m n 2
Vol ÷ ç ÷ B è ø 2 2 è m n ø ( ) ( ) 2 U t B t 1 ( ) = ( ) u t B = = 2 B t B
Vol 2 m 2 m
compare to energy density in capacitor e ( ) ( ) 2 u t = E t E 2 16

17. Oscillations in LC circuits
V(t)
I(t)
t) L 17

18. e 1 Start with fully charged capacitor Q Q U = ; I ( ) = ; V ( ) = 2 Cm ; I ( ) = ; V ( ) = m E 2 C C 2 .
Close switch ,
charge flows from +
plate to
plate - e - =
Kirchoffs Rule : V ( t ) ( t ) Q ( t ) dI Q ( t ) d 2 Q Þ + L = Û + L = C dt C dt 2 d 2 Q 1 Þ = - Q dt 2 LC 18

19. to equation for Harmonic Oscillator of mass on springd 2 Q 1
compare = - Q dt 2 LC
to equation for Harmonic Oscillator of mass on spring
for displacement X ( t )
from equilibrium position d 2 X k k = - X = - w 2 X ; w = =
angular frequency m dt 2 m
This has solution X ( t ) = A
cos ( w t + d ) ; A =
amplitude d =
phase shift 19

20. Solution to Equations I20

21. Solution to Equations II+
+ -
+ -
Solution to Equations II
+Qm
Q(t) t
-Qm
+Im t
I(t)
-Im 21

22. Analogy to Harmonic Oscillator I22

23. Analogy to Harmonic Oscillator II+
t = 0 L
Analogy to Harmonic Oscillator II -
t = T/4 L - L
t = T/2 + L
t = 3T/4 + L
t = T - + 23 -

24. Analogy to Harmonic Oscillator III
Total Energy of Harmonic Oscillator
Analogy to Harmonic Oscillator III 1 1 = U + K = kX 2 + mv 2 2 2 æ ö 2 1 1 dX 1 = kX 2 + m ç ÷ = kA 2 ; A
is amplitude è ø 2 2 dt 2
Total Energy of LC Circuit 1 1 = U + U = Q 2 + LI 2 E B 2 C 2 æ ö 1 1 dQ 2 1 = = 2 Q 2 + L ç ÷
Qmax è ø 2 C 2 dt 2 C 24

25. PE / KE
K.E.
P.E. UB UE 25

26. In real circuits always have some resistance
RLC Circuit (dc) I
V(t) C
I(t) R
t) L 26

27. RLC Circuit (dc) II In RLC circuit get energy lost as joule heat
and oscillations decay in amplitude
and one has a damped
harmonic oscillator . 27

28. Picture 28

29. Flux
Current
and
Inductance

31. Mutual Inductance Mutual Inductance e e e e Flux through coil 2
Nearby circuits effect each other
define MUTUAL INDUCTION M
of coil 2
with 21 F
respect to coil 1 as M = N 21 2 21 I 1 F
Flux through coil 2 = 21 N =
number of turns in coil 2 2 I =
current in coil 1 1
Induced emf in coil 2
by coil 1 is e d F dI = - N 21 = - M 1 2 2 dt 21 dt
if rate at which currents change are equal e e e dI = = = - M 2 1 dt 29