Physics 169 Luis anchordoqui Kitt Peak National Observatory Tuesday, April 6, 21 1 Chapter 9
Inductance
9.19.1 Inductance Inductance An inductor stores energy in magnetic field An inductor stores energy injust the asmagnetic a capacitor field storesjust as energy a capacitor in electricstores field energy in the electric field. AWe changing have shown B-field earlier will that leada to changing an induced B-field emf will in lead a circuit to an induced emf in a circuit. Question QuestionIf a circuit: If generates a circuit generates a changing a changing magnetic magnetic field field, does it lead to an induced emf in the samedoes circuit? it lead to YES! an inducedSelf-Inductance emf in same circuit? YES! Self-Inductance The inductance L of any current element is Inductance L of any current element is di di = V = L The negative sign = V = L EL NegativeL signdt comes comesfrom fromLenz Lenz Law Law. EL L dt VS VsVs Unit of L : HenryUnit (H) of L: Henry(H) 1H = 1 1H=1 1H=1 A · A · A All circuit elements (including resistors) have some inductance. • • All circuit elements (including resistors) have some inductance •Commonly Commonly used used inductors: inductors: solenoids, solenoids toroidsand toroids • Vs •circuit circuit symbol: symbol 1H=1 • A Tuesday, April 6, 21 2 Example : Solenoid
di di = V V = L < 0 = V V = L > 0 EL B A dt EL B A dt VB
Inductance
9.1 Inductance
An inductor stores energy in the magnetic field just as a capacitor stores energy in the electric field. We have shown earlier that a changing B-field will lead to an induced emf in a circuit. Question : If a circuit generates a changing magnetic field, does it lead to an induced emf in the same circuit? YES! Self-Inductance The inductance L of any current element is di = V = L The negative sign EL L dt comes from Lenz Law.
Unit of L: Henry(H) 1H=1 Vs · A All circuit elements (including resistors) have some inductance. • Commonly used inductors: solenoids, toroids • circuit symbol: • ExampleExample: SolenoidSolenoid
di di = V V = L < 0 = V V = L > 0 EL B A dt EL B A dt VB
Recall Faraday’s Law d d = N B = (N ) EL dt dt B
where B is magnetic flux ☛ N B is flux linkage ) Alternative definition of Inductance d di N (N )= L L = B dt B dt ) i ) Inductance is also flux linkage per unit current
Tuesday, April 6, 21 3 9.1. INDUCTANCE 108
Recall Faraday’s Law, d d = N B = (N ) EL dt dt B
where B is magnetic flux, N B is flux linkage. Alternative definition of Inductance:
d di N (N )= L L = B dt B dt ⇥ i
Inductance is also flux linkage per unit current.
Calculating Inductance:
(1) Solenoid: To first order approximation, Calculating Inductance: B = µ ni ① Solenoid 0 where n = N/L = Number of To first order approximation B = µcoils0ni per unit length. Consider a subsection of length l of the solenoid: n = N/` ☛ number of coils per unit length Flux linkage = N B = nl BA where A is Consider a subsection of length l · of solenoidcross-sectional area N Flux linkage = N BL = B = µ n2lA i 0 L 2 = nl BA= µ0n A where= Inductance A per is unit cross-sectional length area · l Notice : (i) LN n2 B 2 L = ⇤ = µ0n lA (ii) Thei inductance, like the capacitance, depends only on geometric ) factors, not on i. L = µ n2A = Inductance per unit length l 0 Note ❑ L n2 / ❑Inductance (like capacitance) depends only on geometric factors (not on i)
Tuesday, April 6, 21 4 Compute the self-inductance of a solenoid with N turns, length l , and radius R with a current I flowing through each turn, as shown in Figure 11.2.2.
Figure 11.2.2 Solenoid
Solution: Ignoring edge effects and applying Ampere’s law, the magnetic field inside a solenoid is given by Eq. (9.4.3) ! NI µ0 ˆ ˆ B = k = µ0nI k , (11.2.4) l
where n = N / l is the number of turns per unit length. The magnetic flux through each turn is ! = BA = µ nI "(# R2 ) = µ nI# R2 . (11.2.5) B 0 0
Thus, the self-inductance is N! L = B = µ n2" R2l . (11.2.6) I 0
② We see that L depends only on the geometrical factors ( n , R and l ) and is independent Toroid of the current I .
Inside toroid ExampleRecall 11.3 Self ☛-Inductance B-field of linesa Toroid are concentric circles µ iN B = 0 2⇡r Outside toroid
B =0 9.1. INDUCTANCE 109 (a) (b) Flux linkage through toroid (2) Toroid: ~ ~ FigureB 11.2.3 dA~ atoroid with N turns N B = N B d~a k KEY Recall:· B-field linesda are= concentrichdr cir- Z cles. + 2 Insideb the toroid: 11-7 µ0iN hdr n µ iN = B = 0 2⇡ r 2⇥r Za (NOT a constant) µ iN 2h b = 0 wherelnr is the distance from center. 2⇡ Outside thea toroid: N ⇣ µ⌘NB =02h b L = B = 0 ln ) Inductance i 2⇡ a 2 Again L N Flux linkage through the toroid⇣ ⌘ / ⇤ Tuesday, April 6, 21 ⇤ Notice B d⇤a 5 N B = N B d⇤a ⇤ KEY ˆ · ⇤ Write da = hdr ⌅ µ iN 2 b hdr = 0 2⇥ ˆa r µ iN 2h b = 0 ln 2⇥ a ⇥ N µ N 2h b Inductance L = B = 0 ln i 2⇥ a ⇥ Again, L N 2 ⇥ Inductance with magnetic materials : We showed earlier that for capacitors:
E⇤ E/⇤ (after insertion of e ⇤ C eC dielectric e > 1) For inductors, we first know that
B⇤ B⇤ (after insertion of m magnetic material) N Inductance L = B i However = B⇤ dA⇤ B ˆ · m B 1 2 U B = LI . (11.13.27) 2
Thus, when the current is at its maximum, all the energy originally stored in the capacitor is now in the inductor
1 2 1 2 C! = LI0 . (11.13.28) 2 2
This implies a maximum current C I0 = ! . (11.13.29) L
(d) At any time, the total energy in the circuit would be equal to the initial energy that the capacitance stored, that is
1 2 U = U E + U B = C! . (11.13.30) 2
11.14 Conceptual Questions
1. How would you shape a wire of fixed length to obtain the greatest and the smallest inductance?
2. If the wire of a tightly wound solenoid is unwound and made into another tightly wound solenoid with a diameter 3 times that of the original one, by what factor does the inductance change?
3. What analogies can you draw between an ideal solenoid and a parallel-plate capacitor?
4. In the RL circuit show in Figure 11.14.1, can the self-induced emf ever be greater 9. 2 LR Circuits than the emf supplied by the battery?
(A) Charging an inductor
When switch is adjusted to position a By loop rule (clockwise)
V + V =0 0 R L a E Figure 11.14.1
di 0 iR L =0 E dt 11-53
di R 0 ) + i = E First Order Differential dt L L Equation
Similar to equation for charging a capacitor!
Tuesday, April 6, 21 6 changing variables
x =( /R) i dx = di E0 L dx x + =0 R dt
x dx R t 0 = dt x L Zxo 0 Z0
ln(x/x )= Rt/L 0 Rt/L x = x0e i =0@t =0 x = /R ) 0 E0 0 0 Rt/L E i = E e R R Tuesday, April 6, 21 7 9.2. LR CIRCUITS 110
L L (after insertion of ⇤ m magnetic material) To increase inductance, fill the interior of inductor with ferromagnetic materials.( 103 104) ⇥ 9.2 LR Circuits
(A) ”Charging” an inductor
When the switch is adjusted to position a, By loop rule (clockwise) : V + V =0 E0 R L ⌅⌅di iR L =0 E0 dt di R 0 First Order Di er- + i = E dt L L ential Equation 0 t/⌧ SimilarSolution to ☛ thei equation(t)=E for(1 charginge aL capacitor!) (Chap5) R
0 t/ Solution: i(t)=E 1 e L ⌧L = L/R ☛ RInductive time constant ⇥ where ⇥L = Inductive time constant = L/R t/⌧ V = iR = (1 e L ) R 0 t/ L | | VR = iR E= 0(1 e ) | | di di E 1 1 0 0 t/t/⌧ LL t/ L t/⌧L VL = VLL = L= L= LE E ee == 0 e 0e ) | | | | dt dt · R· R· ⌧·L⇥L· · E E
Tuesday, April 6, 21 8 9.2. LR CIRCUITS 111
(B) an inductor (B) ”Discharging”Discharging an inductor WhenWhen theswitch switch is adjusted is adjusted at atposition position b after b after inductor the inductor has been has charged been ”charged”(i.e. current i = 0/R is flowing in the circuit.). i.e. current i = 0 /R is Eflowing in circuit E ByByloop loop rule rule:
V VL VRV =0=0 L R ⇤⇤di di L iR =0 L dt iR =0 (Treatdt inductor as source of emf) Treat inductor as source of emf di R Discharging a capacitor di + Ri =0 dt+ L i =0Discharging(Chap5) an inductor ) dt L t/ L i(t)=i0 e t/⌧L i(t)=i0 e
where i0 = i(t = 0) = Current when the circuit just switch to position b.
where i 0 = i ( t = 0) = Current when circuit just switch to position b
Tuesday, April 6, 21 9
Summary : During charging of inductor,
1. At t = 0, inductor acts like open circuit when current flowing is zero. 2. At t , inductor acts like short circuit when current flowing is stablized⇥⌅ at maximum.
3. Inductors are used everyday in switches for safety concerns. 9.2. LR CIRCUITS 111
(B) ”Discharging” an inductor When the switch is adjusted at position b after the inductor has been ”charged”(i.e. current i = 0/R is flowing in the circuit.). 9.2. LR CIRCUITSE 111 By loop rule: (B) ”Discharging” an inductor VL VR =0 When the switch is adjusted at position b after the inductor has been ”charged”(i.e. current⇤⇤di i = 0/R is flowing in the circuit.). L EiR =0 By loop rule: dt (Treat inductor as source of emf) V V =0 L R ⇤⇤di di R Discharging a capacitor L iR =0 + i =0 (Chap5) dt dt L
(Treat inductor as source of emf) t/ L i(t)=i0 e di R Discharging a capacitor + i =0 where i0 = i(t = 0)dt =L Current when(Chap5) the circuit just switch to position b.
t/ L i(t)=i0 e
where i0 = i(t = 0) = Current when the circuit just switch to position b.
Summary SummaryDuring: During charging charging of of inductor,inductor 1. At t = 0, inductor acts like open circuit when current flowing is zero. 1. At t =0 inductor acts like open circuit when current flowing is zero Summary : During2. At chargingt , of inductor inductor, acts like short circuit when current flowing is stablized⇥⌅ at maximum. 2. At t 1. At t =inductor 0, inductor acts like likeopen short circuit when circuitcurrent when flowing iscurrent zero. flowing !1 2. At t , inductor acts like short circuit when currentis flowing stabilized is at maximum stablized⇥⌅ at maximum.
3. Inductors are used everyday in switches for safety concerns.
3. Inductors are used everyday in switches for safety concerns. 3. Inductors are used everyday in switches for safety concerns
Tuesday, April 6, 21 10 Summary
2 dI (b) 11.6.1 Rising Current(a) I! = I R + LI . (11.6.6) dt
FigureThe 11.6.1 left-hand Modified side represents Kirchhoff’s the rule rate for at inductorswhich the (a) battery with increasingdelivers energy current, to theand circuit. (b) withOn decreasing the other current. hand, the See first Section term on11.4.2 the for right cautions-hand side about is the poweruse of this dissipated modified in the rule.resistor in the form of heat, and the second term is the rate at which energy is stored in the inductor. While the energy dissipated through the resistor is irrecoverable, the The magnetic modified energy rule for stored inductors in the inductor may be obtainedcan be released as follows: later. The polarity of the self- induced emf is such as to oppose the change in current, in accord with Lenz’s law. If the rate of change of current is positive, as shown in Figure 11.6.1(a), the self-induced emf ! sets11.6.2 up anDecaying induced Curren currentt I moving in the opposite direction of the current I to L ind oppose such an increase. The inductor could be replaced by an emf Next we consider the RL circuit shown in Figure 11.6.5. Suppose the switch S has been | ! | = L | dI / dt | = +L(dI / dt) with the polarity shown in Figure 11.6.1(a). On the 1 other L closed for a long time so that the current is at its equilibrium value ! / R . What happens hand, if RL circuit, as shown with in risingFigure 11.6.1(b current), the inducedand equivalent current I set circuit up by the Figure 11. dI6.2/ dt(a)< RL0 Circuit with rising current. (b) Equivalent circuitin dusing the modified to the current when at t = 0 switches S is opened and S closed? 1 2 Kirchhoff’sself-induced loop emf rule. ! flows in the same direction as I to oppose such a decrease. L
ConsiderWe see thatthe whetherRL circuit the rateshown of change in Figure of current 11.6. 2in. Atincrea t =sing0 the( dI /switchdt > 0 )is or closed. decreasing We find that( dIthe/ dtcurrent< 0 ), in does both not cases, rise the immediately change in potential to its maximum when moving value from ! / Ra .to This b along is due the to the presencedirection of of the the self current-induced I is Vemf! Vin =the! Linductor.(dI / dt) .Using Thus, thewe havemodified Kirchhoff’s rule for b a increasing current, dI / dt > 0 , the RL circuit is described by the following differential equation:Kirchhoff's Loop Rule Modified for Inductors (Misleading, see Section 11.4.2): dI ! " IR" |! | = ! " IR " L = 0 . (11.6.1) If an inductor is traversed in the directionL of the current,dt the “potential change” is !L(dI / dt). On the other hand, if the inductor is traversed in the direction opposite of the
current,RL the circuit “potential with change” decaying is L(dI / dt current). and equivalent circuit Note that thereFigure is 11. an6.5 important(a) RL circuit + distinction with decaying between current an, and inductor (b) equivalent and a circuit. resistor. The Tuesday, April 6, 21 11 potentialAppl differenceying the modified across a Kirchhoff resistor ’dependss loop rule on to I , thewhile right the loop potential for decreasing difference current, across an Useinductor of this depends modified on Kirchhoff’s dI / dt . Therule selfwill- giveinduced the correct emf does equations not oppose for circuit the problemscurrent itself, that dcontainI / dt < inductors.0 , yields However, keep in mind that it is misleading at best, and at some butlevel the changewrong in of terms current of thedI physics./ dt . Eq. Again, (11.6 .we1) canemphasizedI be rewritten that Kirchhoff's as loop rule was ! |! L | "IR = "L " IR = 0 , (11.6.7) originally based on the fact that the line integraldt of E around a closed loop was zero. dI dt Withwhich time can-changing be rewritten magnetic as fields, this is= no! longer. so, and thus the sum of the(11 .6.2) “potential drops” around the circuit, if we take that to mean the negative of the closed ! I ! " /dIR Ldt/ R loop integral of E, is no longer zero in fact= it! is L(. dI / dt) . (11.6.8) − I L /+R Integrating over both sides and imposing the condition I(t = 0) = 0 , the solution to the differentialThe solution equation to theis above differential equation is
! "t/# 11-26 I(t) = (1"! e t/ ). (11.6.3) I(Rt) = e" # , (11.6.9) R
This solution reduces to what we expect for large times, that is , but it also where ! = L / R is the same time constant as in the case of rising I (current!) = ". /AR plot of the showscurr a continuousent as a function rise of of thetime current is shown from in Figure I(t = 11.0) 6.6= 0. initially to this final value, with a
characteristic time ! L defined by
L 11-29 ! L = . (11.6.4) R
This time constant is known as the inductive time constant. This is the effect of having a non-zero inductance in a circuit, that is, of taking into account the “induced” electric 11-27
9.3 Energy Stored in Inductors Inductors stored magnetic energy through magnetic field stored in circuit Recall equation for charging inductors di iR L =0 E0 dt Multiply both sides by i di i = i2R + Li E0 dt Power input by emf Joule’s heating (Energy supplied (Power dissipated Power stored in inductor one charge = |{z} q ) by resistor)|{z} E0 | {z } ) Power stored in inductor Integrating both sides and use initial condition
At t =0,i(t = 0) = UB(t = 0) = 0 1 ) Energy stored in inductor ☛ U = Li2 B 2
Tuesday, April 6, 21 12 Energy Density Stored in Inductors Consider an infinitely long solenoid of cross-sectional area A For a portion l of solenoid 2 L = µ0n lA ) Energy stored in inductor: 1 1 U = Li2 = µ n2i2 lA B 2 2 0 Volume of solenoid ) Energy density (= Energy stored per unit volume)|{z} inside inductor U 1 u = B = µ n2i2 B lA 2 0 Recall magnetic field inside solenoid B = µ0ni B2 ) uB = 2µ0 This is a general result of energy stored in a magnetic field Tuesday, April 6, 21 13 9.4 Mutual Inductance Inductance and Magnetic Energy Very often the magnetic flux through the area enclosed by a circuit varies with time because of time-varying currents in nearby circuits 11.1 Mutual Inductance
Suppose two coils are placed near each other, as shown in Figure 11.1.1 mutual inductance depends on interaction of two circuits
Consider two closely wound coils of wire shown in cross-sectional view
Current I 1 in coil 1 which has N 1 turns creates a magnetic field
Some magnetic field lines pass through coil 2 which has N 2 turns
The magnetic flux caused by the currentFigure in coil11.1.1 1 Changingand passing current through in coil 1coil produces 2 is changing 12 magnetic flux in coil 2.
We define the mutual inductance ofThe coil first 2 withcoil has respect N turns to andcoil carries 1 a current I which gives rise to a magnetic field ! 1 1 B . The second coil has N turns. Because the two coils are close to each other, some of 1 N2 12 2 the magnetic field lines through coil 1 will also pass through coil 2. Let ! denote the M12 12 ma⌘gnetic fluxI1 through one turn of coil 2 due to I . Now, by varying I with time, there Tuesday, April 6, 21 1 14 1 will be an induced emf associated with the changing magnetic flux in the second coil:
d# d ! ! ! = "N 12 = " B $ dA . (11.1.1) 12 2 dt dt %% 1 2 coil 2
The time rate of change of magnetic flux ! in coil 2 is proportional to the time rate of 12 change of the current in coil 1:
d!12 dI1 N2 = M12 , (11.1.2) dt dt
where the proportionality constant M is called the mutual inductance. It can also be 12 written as N ! M = 2 12 . (11.1.3) 12 I 1
11-3
The SI unit for inductance is the henry [H]:
2 1 henry = 1 H = 1 T!m /A . (11.1.4)
We shall see that the mutual inductance M depends only on the geometrical properties 12 of the two coils such as the number of turns and the radii of the two coils. If current I 1 varies with time In a similar manner, suppose instead there is a current I in the second coil and it is we see from Faraday’s law that emf induced by coil 1 in coil 2 is 2 varying with time (Figure 11.1.2). Then the induced emf in coil 1 becomes
d# d ! ! ! = "N 21 = " B $ dA , (11.1.5) d 12 d M12I1 21 1 dt dI1dt %% 2 1 2 = N2 = N2 = M12 coil 1 E dt dt N2 dt and a current✓ is induced in◆ coil 1. If current I 2 varies with time ☛ emf induced by coil 2 in coil 1 is
dI = M 2 E1 21 dt
In mutual induction emf induced in one coil is always proportional toFigure rate at11.1.2 which Changing current current in inother coil 2 coilproduces is changing changing magnetic flux in coil 1.
This changing flux in coil 1 is proportional to the changing current in coil 2, It is easily seen that M12 = M21 = M Tuesday, April 6, 21 15 d!21 dI2 N1 = M21 , (11.1.6) dt dt
where the proportionality constant M is another mutual inductance and can be written 21 as N ! M = 1 21 . (11.1.7) 21 I 2
The mutual inductance reciprocity theorem states that the constants are equal
M = M ! M . (11.1.8) 12 21
11-4
11.13.6 LC Circuit
Consider the circuit shown in Figure 11.13.6. Suppose the switch that has been connected to point a for a long time is suddenly9. 5 LC Circuitthrow (Electromagneticn to b at t = Oscillator)0 .
Figure 11.13.6 LC circuit After the capacitor is charged we move the switch to position b Find the following quantities: Tuesday, April 6, 21 16 (a) the frequency of oscillation of the LC circuit.
(b) the maximum charge that appears on the capacitor.
(c) the maximum current in the inductor.
(d) the total energy the circuit possesses at any time t .
Solution:
(a) The angular frequency of oscillation of the LC circuit is given by ! = 2" f = 1/ LC . Therefore, the frequency is
1 f = . (11.13.24) 2! LC
(b) The maximum charge stored in the capacitor before the switch is thrown to b is
Q = C! . (11.13.25)
(c) The energy stored in the capacitor before the switch is thrown is
1 2 U E = C! . (11.13.26) 2
On the other hand, the magnetic energy stored in the inductor is
11-52
9.4. LC CIRCUIT (ELECTROMAGNETIC OSCILLATOR) 113
9.4 LC Circuit (Electromagnetic Oscillator)
Initial charge on capacitor = Q Initial charge on capacitor = Q InitialInitial current current=0 = 0 No battery. No battery
AssumeAssume current current i toi to be be in in thedirection direction that that increases decreasescharge charge on the positive capacitor plate. on positive capacitor plate dQ dQi = (9.1) i =⇤ dt (10.1) By Lenz Law, we also) know the ”poles”dt of the inductor. By Lenz Law we also know poles of inductor Loop rule: VC + VL =0 Q di L = 0 (9.2) Loop rule ☛ VC + VL =0 C dt Combining equations (9.1) and (9.2), we get Q di (10.2) L d2Q=01 + Q =0 C dtdt2 LC Combining equations (10.1) and (10.2) we get This is similar to the equation of motion of a simple harmonicd2Q oscillator1 : 2 + Q =0 dt LC d2x k + x =0 Tuesday, April 6, 21 dt2 m 17 Another approach (conservation of energy) Total energy stored in circuit:
U = UE + UB
Q⇥⇥2 1 U = + Li2 2C 2 Since the resistance in the circuit is zero, no energy is dissipated in the circuit. Energy contained in the circuit is conserved. dU =0 dt Q dQ⇥ di dQ ⇥ + L i =0 (⇥ i = ) ⇤ C · ⇥dt dt dt 9.4. LC CIRCUIT (ELECTROMAGNETIC OSCILLATOR) 113
9.4 LC Circuit (Electromagnetic Oscillator)
Initial charge on capacitor = Q Initial current = 0 No battery.
Assume current i to be in the direction that increases charge on the positive capacitor plate. dQ i = (9.1) ⇤ dt By Lenz Law, we also know the ”poles” of the inductor.
Loop rule: VC + VL =0 Q di L = 0 (9.2) C dt Combining equations (9.1) and (9.2), we get
d2Q 1 + Q =0 dt2 LC This is similar to equation of motion of This is similara simple to the harmonic equation oscillator of motion of a simple harmonic oscillatord2x k: + x =0 dt2 m Another approach (conservationd2x kof energy) Total energy stored in circuit2 + x =0 dt Um = UE + UB Another approach (conservation of energy) Total energy stored in circuit: Q2 1 U = + Li2 2C 2 Since resistance in circuitU = is zeroUE no+ energyUB is dissipated in circuit ) Energy contained in circuit ⇥⇥is2 conserved dU Q 1 2 ) U=0= + Li dt 2C 2 Since the resistance in theQ circuitdQ is zero,dino energy is dissipateddQ in the circuit. + Li =0 * i = Energy contained) in theC circuit· dt is conserveddt . dt ⇣ ⌘ Tuesday, April 6, 21 dU 18 =0 dt Q dQ⇥ di dQ ⇥ + L i =0 (⇥ i = ) ⇤ C · ⇥dt dt dt di Q L + =0 ) dt C d2Q 1 + Q =0 ) dt2 LC Solution to this differential equation is in form
Q(t)=Q0 cos(!t + ) dQ = !Q sin(!t + ) ) dt 0 d2Q = !2Q cos(!t + ) ) dt2 0 = !2Q d2Q + !2Q =0 ) dt2 1 !2 = Angular frequency of LC oscillator ) LC Tuesday, April 6, 21 19 9.4. LC CIRCUIT (ELECTROMAGNETIC OSCILLATOR) 114
di Q L + =0 ⇥ dt C d2Q 1 + Q =0 ⇥ dt2 LC
The solution to this di erential equation is in the form
Q(t)=Q0 cos(⇥t + ) dQ = ⇥Q sin(⇥t + ) dt 0 d2Q = ⇥2Q cos(⇥t + ) dt2 0 = ⇥2Q d2Q + ⇥2Q =0 dt2 Q0, are constants derived from initial conditions 1 ⇥2 = Angular frequencydQ (Two initial conditions, e.g. Q (t = 0)LCand iof(t the= LC 0) oscillator = are required) dt t=0 Also, Q , are constants derived from the initial2 conditions.2 (Two initial condi- 0 Q Q0 2 Energytions, e.g.storedQ(t =in 0),capacitor and i(t =0)=C =dQ are required.)= cos (!t + ) dt t=0 2C 2C Q2 Q12 1 Energy storedEnergy in stored inductance in C = L == 0 Licos22(⇥=t + ) L!2Q2 sin2(!t + ) 2C 2C2 2 0 1 2 1 22 2 2 Energy2 stored1 in L = Li = QL⇥ Q0 sin (⇥t + ) ( Since* L ! = ) 2 = 2 0 sin2(!t + ) C 1 Q2 L⇥2 = = 20 Csin2(⇥t + ) ⇥ C 2C 2 Q 2 0 Q0 ) Total energy Total stored energy stored= = 2C 2C = Initial energy stored in capacitor = Initial energy stored in capacitor
Tuesday, April 6, 21 20 Energy oscillations in LC system and mass-spring system
LC Circuit Mass-spring System Energy
Tuesday, April 6, 21 21 Figure 11.7.5 Energy oscillations in the LC Circuit and the mass-spring system In Figure 11.7.5 we illustrate the energy oscillations in the LC Circuit and the mass- spring system (harmonic oscillator).
11-35
9. 6 RLC Circuit (Damped Oscillator) In real life circuit ☛ there’s always resistance energy stored in LC oscillator is NOT conserved 11.8 The RLC Series Circuit and We now consider a series RLC circuit that contains a resistor, an inductor and a dU capacitor, as shown in Figure2 11.8.1. = Power dissipated in resistor = i R dt
Negative sign shows that energy U is decreasing
i FigureJoule’s 11.8.1 A series heating RLC circuit
di Q dQ 2 ) Li + = i R dt C · z}|{dt d2Q R dQ 1 + + Q =0 ) dt2 L · dt LC
Tuesday, April 6, 21 22
11-34
This is similar to equation of motion of a damped harmonic oscillator (e.g. if a mass-spring system faces a frictional force F ~ = b ~v ) Solution to equation is of form R t Q(t)=Q0 e 2L cos(!0t + )
exponential oscillating term decay term | {z } | {z } 1 R 2 !0 = LC 2L s ✓ ◆ 2 2 R !0 = ! 0 2L s ✓ ◆ R = damping factor 2L
There are three possible scenarios depending on the relative values of and ! 0 Tuesday, April 6, 21 23 There are three possible scenarios, depending on the relative values of and ! . ! 0
Case 1: Underdamping
When ! > " , or equivalently, ' is real and positive, the system is said to be 0 ! underdamped. This is the case when the resistance is small. Charge oscillates (the cosine function) with exponentially decaying amplitude Q e!" t . However, the frequency of this 0 damped oscillation is less than the undamped oscillation, ! ' < ! . The qualitative 0 behavior of the chargeCase on I:the Underdamping capacitor as a! function0 > of time is shown in Figure 11.10.1.
Figure 11.10.1 Underdamped oscillations
As an example, suppose the initial condition is Q(t = 0) = Q . The phase constant is then Underdamped oscillator always oscillates 0 ! = 0 , and at a lower frequency than natural frequency of oscillator Tuesday, April 6, 21 24 Q(t) = Q e!" t cos(# 't) . (11.10.5) 0 The corresponding current is
dQ I(t) Q 'e!# t $sin( 't) ( / ')cos( 't)& . (11.10.6) = ! = 0" % " + # " " ' dt
For small R , the above expression may be approximated as
Q I(t) ! 0 e"# t sin($ 't +%) (11.10.7) LC where % # ( ! = tan"1 . (11.10.8) ' '* & $ )
The derivation is left to the readers as an exercise.
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Case 2: Overdamping
In the overdamped case, ! < " , implying that ' is imaginary. There is no oscillation in 0 ! this case. By writing ! ' = i" , where ! = " 2 #$ 2 , one may show that the most general 0 solution can be written as Q(t) = Q e!(" +# )t + Q e!(" !# )t , (11.10.9) 1 2 where the constantsCase Q II:and Overdamping Q can be ! determined0 < from the initial conditions. 1 2
FigureCase III: 1 Critical1.10.2 damping Overdamping !0 = and critical damping
Case 3: Critical damping Tuesday, April 6, 21 25 When the system is critically damped, ! = " , ' 0 . Again there is no oscillation. The 0 ! = general solution is Q(t) = (Q + Q t)e!" t , (11.10.10) 1 2 where Q and Q are constants which can be determined from the initial conditions. In 1 2 this case one may show that the energy of the system decays most rapidly with time. The qualitative behavior of Q(t) in overdamping and critical damping is depicted in Figure 11.10.2.
11.10.1 Quality Factor
When the resistance is small, the system is underdamped, and the charge oscillates with decaying amplitude Q e!" t . The “quality” of this underdamped oscillation is measured by 0 the so-called “quality factor,” Q (not to be confused with charge.) The larger the value qual of Q , the less the damping and the higher the quality. Mathematically, Q is defined qual qual as
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Tuesday, April 6, 21 26 Tuesday, April 6, 21 27