Notes for Oscillating Systems (Part 3) amazonaws com
Physics Part IA Cambridge University, 2012-2013
Notes for Oscillating Systems (Part 3) Simple harmonic motion in the LC circuit: For a circuit containing just an inductor and a capacitor, the behaviour of charge
q(t) with
x (t) of a particle undergoing simple harmonic
time is entirely analogous to the motion motion:
VL L
VC C
By Kirchhoff’s voltage law:
V L +V C =0
∴L
dI q 1 + =0 ∴ L q́ + q=0 dt C C
∴ q́ +
1 q=0 LC 2
∴ q́ +ω0 q=0 , ω 0=
1 √ LC
As usual, the solution to this differential equationWhich can beisexpressed in two different the equation of motion forforms: SHM, with Complex exponential form: αt
2 αt
Let q= A e ∴ ́x = A α e αt
2
2
: αt
∴ A e ( α +ω 0) =0, A e ≠ 0 ∴ α=± i ω 0
Where
A
and
B
For amplitude
Q0
and phase
ϕ , the real,
physical form of the solution, as for the mechanical oscillator, is:
∴ ℜ { q(t) } =Q 0 cos ϕ cos ω0 t+ Q0 sin ϕ sin ω 0 t
∴ q (t)= A ei ω t +B e−iω t 0
Real (physical) sinusoidal form:
0
are in
∴ ℜ { q(t) } =a0 cos (ω0 t+ ϕ) Or
ℜ { q (t )}=C cos ω 0 t + D sin ω0 t
ω0
Physics Part IA Cambridge University, 2012-2013 general complex. As before,
ω0
is the natural angular frequency of the system.
a0 =√ C 2 + D2 , ϕ=tan −1
This oscillatory behaviour arises as follows: 1. When the capacitor is fully charged (maximum
−D C
q ) no current
I
flows (zero
I ), therefore the inductor is fully discharged. Charge naturally flows out of the capacitor, generating an increasing current through the inductor and charging it 2. When the inductor is fully charged (maximum is also maximum, such that the charge
q
I ), the current through the capacitor
on its plates is zero. Thus the capacitor
is fully discharged. Due to the current, charge begins to build up on the capacitor plates with a potential difference opposing the direction of the original potential difference 3. Steps 1 and 2 then repeat in the opposite sense (the potential difference across the capacitor and the direction of the current through the circuit are reversed) 4. The process continues indefinitely with no energy dissipation
Physics Part IA Cambridge University, 2012-2013 Phasor diagram for the LC circuit: The relationship between charge
q(t) , current q́ (t) and potential
L ́q (t)
analogous to the relationship between displacement, velocity and acceleration: Charge
q( t) :
Current
q ( t )=ℜQ0 e i(ω t+ ϕ) 0
ω0 t+ ϕ Im Re
Q0 Q0 C a0 q
a0 Q0 ω0
i
a0 vL
i(t) :
is
Physics Part IA Cambridge University, 2012-2013
a0 vC
a0 Q0 ω20
a0 ∴ q ( t )=Q0 cos (ω0 t+ϕ)
ω i(¿¿ 0 t+ ϕ) Q 0 e¿ d( } ¿ dt i (t )=ℜ ¿ ω Q0 ω0 i e i(¿¿0 t +ϕ ) ∴ i ( t )=ℜ¿ ω π i (¿ ¿0 t + ϕ+ ) 2
∴ i (t )=Q 0 ω 0 e
(
∴ q ( t )=Q0 cos ω 0 t + ϕ+
π 2
)
Potential
v C (t ) and v L (t ) :
Capacitor
v C (t ) :
v C ( t )=
Inductor
Q0 cos(ω0 t+ ϕ) C v L (t ) :
ω i (¿¿ 0 t+ ϕ) ¿ Q0 ω 0 i e Energy variation in the LC circuit: The energy terms associated with the capacitor and the d inductor are analogous to the ( ¿} potential and kinetic energy terms in mechanical simple dt harmonic motion: v L ( t )=ℜ¿ Instantaneous energy stored in capacitor:
Instantaneous energy stored in inductor:
Physics Part IA Cambridge University, 2012-2013
1 1 2 q 2C
1 2 1 2 L I = L ́q 2 2
The instantaneous energy stored in the capacitor and the inductor can be used to derive an expression for the total, constant energy in the LC circuit:
EC =
1 1 2 1 2 2 q= Q cos (ω 0 t + ϕ) 2C 2C 0
1 2 1 1 2 2 2 2 2 E L= L q́ = L Q0 ω0 sin (ω 0 t + ϕ)= Q sin (ω0 t+ ϕ) 2 2 2C 0 ∴ E=EC + E L ∴ E=
1 2 1 2 2 1 2 2 2 2 Q cos (ω0 t+ ϕ)+ Q 0 sin ( ω0 t+ ϕ)= Q cos ( ω 0 t + ϕ ) +sin ( ω0 t+ ϕ ) ) 2C 0 2C 2C 0 (
∴ E=
1 2 Q 2C 0
Damped harmonic motion in the RLC circuit: Introducing a resistor
R into the LC circuit provides a means of energy dissipation which
q́ , thus is analogous to the velocity-dependent
is directly proportional to the current
viscous friction term in the damped harmonic motion of a mechanical oscillator. The behaviour of charge
q( t) with time is then analogous to the motion
x (t) of a particle
undergoing damped harmonic motion:
R
Energy in inductor and capacitor:
E= VR
∴ VC C VL L
1 1 2 1 2 q + L q́ 2C 2
Energy dissipated through resistor:
dE 2 2 =−I R=−́q R dt
dE 1 = q q́ + L q́ q́ dt C
∴−́q 2 R= ∴ q́ −
1 q q q́ + L q́ q́ ∴ ́q L q́ −R q́ + =0 C C
R q q́ + =0 L LC
(
)
Physics Part IA Cambridge University, 2012-2013
Which is the equation of motion for damped harmonic motion, with the usual solution of the
q ( t )=e−γt ( A e iωt + B e−iωt ) , ω=√ ω20−γ 2 , of which there are three cases:
form
Light damping: −γt
q ( t )=e Where
C
2
ω0 > γ
2
( C sin ωt + D cos ωt ) and
D are in
general complex, with real form: −γt q ( t )=Q0 e cos( ωt+ ϕ)
Which is oscillatory, with an amplitude which decays as −γt
e
Critical damping: 2 0
ω =γ
2
The real form of the solution becomes: −γt
q ( t )=Q0 e
2
Heavy damping: ω0 < γ
2
Defining
ω2 =iω=√ γ 2−ω20 , the solution becomes:
q ( t )=e−γt ( A e ω t +B e−ω t ) 2
Which is not oscillatory and gives a fast return to equilibrium without overshoot.
.
2
Which is not oscillatory and gives a slower return to equilibrium without overshoot.
Energy variation in the RLC circuit (light damping): Average energy stored in capacitor:
Average energy stored in inductor:
2 1 1 2 Q0 −2γt q= e cos 2( ωt+ ϕ) 2C 2C
1 1 2 L q́ 2= LQ20 e−2γt (−γ cos ( ωt + ϕ )+ ω sin ( ωt+ ϕ ) ) 2 2
cos 2( ωt+ ϕ)
1 ¿ L Q20 e−2γt ( ω20 sin 2 ( ωt+ ϕ )−ωγ sin ( 2 ( ωt+ ϕ ) ) ) 2
averages to
1/2 over a cycle, hence the average energy is: 2
Q0 −2 γt e 4C
¿
1 1 2 −2γt 1 Q0 e sin2 ( ωt+ ϕ )− ωR sin ( 2 ( ωt + ϕ ) ) 2C 2
(
)
sin 2 ( ωt+ ϕ ) averages to 1/2 over a cycle and sin ( 2 ( ωt+ ϕ ) ) averages to zero, hence the average
Physics Part IA Cambridge University, 2012-2013 energy is:
Q20 −2 γt e 4C
Both energy terms oscillate at frequency have a phase difference of
2ω with an average of (Q20 /4C)e−2γt , and
π . Thus both energies decay as e−2γt , and the total
average energy is given by:
Q20 −2 γt Q20 −2 γt Q20 −2 γt e + e = e 4C 4C 2C
Oscillations in sinusoidally-driven AC circuits: Sinusoidally-driven circuits contain a source of AC electromotive force which varies sinusoidally with time. Such circuits are particularly important, since non-sinusoidally driven circuits can still be analysed in terms of sinusoidal driving voltages, via Fourier analysis. The voltage and current in a sinusoidally-driven circuit are best represented in the complex forms:
v (t ) :
Voltage '
v ( t )=V e
Where
iωt
Current
'
,V =V 0 e
i ϕV
'
is the real voltage amplitude and
V'
ϕV
is its
is in general complex,
unless the phase of
iωt
'
i ( t )=I e , I =I 0 e
ω is the driving frequency, V 0
phase. Thus
i(t) :
v (t ) is 0 .
Where
iϕI
ω is the driving frequency,
is the real current amplitude and phase. Thus
I'
ϕI
I0 is its
is in general complex,
unless the phase of
i(t) is 0 .
Complex impedance ( Z ): The complex analogue of resistance, measured in ohms. The real part of the complex impedance is the resistance is the reactance
Z=
R , and the purely imaginary part of the complex impedance
X , also measured in ohms:
v ( t ) V ' e iωt V ' = = =R+iX i ( t ) I ' eiωt I '
Notes for Oscillating Systems (Part 3) Simple harmonic motion in the LC circuit: For a circuit containing just an inductor and a capacitor, the behaviour of charge
q(t) with
x (t) of a particle undergoing simple harmonic
time is entirely analogous to the motion motion:
VL L
VC C
By Kirchhoff’s voltage law:
V L +V C =0
∴L
dI q 1 + =0 ∴ L q́ + q=0 dt C C
∴ q́ +
1 q=0 LC 2
∴ q́ +ω0 q=0 , ω 0=
1 √ LC
As usual, the solution to this differential equationWhich can beisexpressed in two different the equation of motion forforms: SHM, with Complex exponential form: αt
2 αt
Let q= A e ∴ ́x = A α e αt
2
2
: αt
∴ A e ( α +ω 0) =0, A e ≠ 0 ∴ α=± i ω 0
Where
A
and
B
For amplitude
Q0
and phase
ϕ , the real,
physical form of the solution, as for the mechanical oscillator, is:
∴ ℜ { q(t) } =Q 0 cos ϕ cos ω0 t+ Q0 sin ϕ sin ω 0 t
∴ q (t)= A ei ω t +B e−iω t 0
Real (physical) sinusoidal form:
0
are in
∴ ℜ { q(t) } =a0 cos (ω0 t+ ϕ) Or
ℜ { q (t )}=C cos ω 0 t + D sin ω0 t
ω0
Physics Part IA Cambridge University, 2012-2013 general complex. As before,
ω0
is the natural angular frequency of the system.
a0 =√ C 2 + D2 , ϕ=tan −1
This oscillatory behaviour arises as follows: 1. When the capacitor is fully charged (maximum
−D C
q ) no current
I
flows (zero
I ), therefore the inductor is fully discharged. Charge naturally flows out of the capacitor, generating an increasing current through the inductor and charging it 2. When the inductor is fully charged (maximum is also maximum, such that the charge
q
I ), the current through the capacitor
on its plates is zero. Thus the capacitor
is fully discharged. Due to the current, charge begins to build up on the capacitor plates with a potential difference opposing the direction of the original potential difference 3. Steps 1 and 2 then repeat in the opposite sense (the potential difference across the capacitor and the direction of the current through the circuit are reversed) 4. The process continues indefinitely with no energy dissipation
Physics Part IA Cambridge University, 2012-2013 Phasor diagram for the LC circuit: The relationship between charge
q(t) , current q́ (t) and potential
L ́q (t)
analogous to the relationship between displacement, velocity and acceleration: Charge
q( t) :
Current
q ( t )=ℜQ0 e i(ω t+ ϕ) 0
ω0 t+ ϕ Im Re
Q0 Q0 C a0 q
a0 Q0 ω0
i
a0 vL
i(t) :
is
Physics Part IA Cambridge University, 2012-2013
a0 vC
a0 Q0 ω20
a0 ∴ q ( t )=Q0 cos (ω0 t+ϕ)
ω i(¿¿ 0 t+ ϕ) Q 0 e¿ d( } ¿ dt i (t )=ℜ ¿ ω Q0 ω0 i e i(¿¿0 t +ϕ ) ∴ i ( t )=ℜ¿ ω π i (¿ ¿0 t + ϕ+ ) 2
∴ i (t )=Q 0 ω 0 e
(
∴ q ( t )=Q0 cos ω 0 t + ϕ+
π 2
)
Potential
v C (t ) and v L (t ) :
Capacitor
v C (t ) :
v C ( t )=
Inductor
Q0 cos(ω0 t+ ϕ) C v L (t ) :
ω i (¿¿ 0 t+ ϕ) ¿ Q0 ω 0 i e Energy variation in the LC circuit: The energy terms associated with the capacitor and the d inductor are analogous to the ( ¿} potential and kinetic energy terms in mechanical simple dt harmonic motion: v L ( t )=ℜ¿ Instantaneous energy stored in capacitor:
Instantaneous energy stored in inductor:
Physics Part IA Cambridge University, 2012-2013
1 1 2 q 2C
1 2 1 2 L I = L ́q 2 2
The instantaneous energy stored in the capacitor and the inductor can be used to derive an expression for the total, constant energy in the LC circuit:
EC =
1 1 2 1 2 2 q= Q cos (ω 0 t + ϕ) 2C 2C 0
1 2 1 1 2 2 2 2 2 E L= L q́ = L Q0 ω0 sin (ω 0 t + ϕ)= Q sin (ω0 t+ ϕ) 2 2 2C 0 ∴ E=EC + E L ∴ E=
1 2 1 2 2 1 2 2 2 2 Q cos (ω0 t+ ϕ)+ Q 0 sin ( ω0 t+ ϕ)= Q cos ( ω 0 t + ϕ ) +sin ( ω0 t+ ϕ ) ) 2C 0 2C 2C 0 (
∴ E=
1 2 Q 2C 0
Damped harmonic motion in the RLC circuit: Introducing a resistor
R into the LC circuit provides a means of energy dissipation which
q́ , thus is analogous to the velocity-dependent
is directly proportional to the current
viscous friction term in the damped harmonic motion of a mechanical oscillator. The behaviour of charge
q( t) with time is then analogous to the motion
x (t) of a particle
undergoing damped harmonic motion:
R
Energy in inductor and capacitor:
E= VR
∴ VC C VL L
1 1 2 1 2 q + L q́ 2C 2
Energy dissipated through resistor:
dE 2 2 =−I R=−́q R dt
dE 1 = q q́ + L q́ q́ dt C
∴−́q 2 R= ∴ q́ −
1 q q q́ + L q́ q́ ∴ ́q L q́ −R q́ + =0 C C
R q q́ + =0 L LC
(
)
Physics Part IA Cambridge University, 2012-2013
Which is the equation of motion for damped harmonic motion, with the usual solution of the
q ( t )=e−γt ( A e iωt + B e−iωt ) , ω=√ ω20−γ 2 , of which there are three cases:
form
Light damping: −γt
q ( t )=e Where
C
2
ω0 > γ
2
( C sin ωt + D cos ωt ) and
D are in
general complex, with real form: −γt q ( t )=Q0 e cos( ωt+ ϕ)
Which is oscillatory, with an amplitude which decays as −γt
e
Critical damping: 2 0
ω =γ
2
The real form of the solution becomes: −γt
q ( t )=Q0 e
2
Heavy damping: ω0 < γ
2
Defining
ω2 =iω=√ γ 2−ω20 , the solution becomes:
q ( t )=e−γt ( A e ω t +B e−ω t ) 2
Which is not oscillatory and gives a fast return to equilibrium without overshoot.
.
2
Which is not oscillatory and gives a slower return to equilibrium without overshoot.
Energy variation in the RLC circuit (light damping): Average energy stored in capacitor:
Average energy stored in inductor:
2 1 1 2 Q0 −2γt q= e cos 2( ωt+ ϕ) 2C 2C
1 1 2 L q́ 2= LQ20 e−2γt (−γ cos ( ωt + ϕ )+ ω sin ( ωt+ ϕ ) ) 2 2
cos 2( ωt+ ϕ)
1 ¿ L Q20 e−2γt ( ω20 sin 2 ( ωt+ ϕ )−ωγ sin ( 2 ( ωt+ ϕ ) ) ) 2
averages to
1/2 over a cycle, hence the average energy is: 2
Q0 −2 γt e 4C
¿
1 1 2 −2γt 1 Q0 e sin2 ( ωt+ ϕ )− ωR sin ( 2 ( ωt + ϕ ) ) 2C 2
(
)
sin 2 ( ωt+ ϕ ) averages to 1/2 over a cycle and sin ( 2 ( ωt+ ϕ ) ) averages to zero, hence the average
Physics Part IA Cambridge University, 2012-2013 energy is:
Q20 −2 γt e 4C
Both energy terms oscillate at frequency have a phase difference of
2ω with an average of (Q20 /4C)e−2γt , and
π . Thus both energies decay as e−2γt , and the total
average energy is given by:
Q20 −2 γt Q20 −2 γt Q20 −2 γt e + e = e 4C 4C 2C
Oscillations in sinusoidally-driven AC circuits: Sinusoidally-driven circuits contain a source of AC electromotive force which varies sinusoidally with time. Such circuits are particularly important, since non-sinusoidally driven circuits can still be analysed in terms of sinusoidal driving voltages, via Fourier analysis. The voltage and current in a sinusoidally-driven circuit are best represented in the complex forms:
v (t ) :
Voltage '
v ( t )=V e
Where
iωt
Current
'
,V =V 0 e
i ϕV
'
is the real voltage amplitude and
V'
ϕV
is its
is in general complex,
unless the phase of
iωt
'
i ( t )=I e , I =I 0 e
ω is the driving frequency, V 0
phase. Thus
i(t) :
v (t ) is 0 .
Where
iϕI
ω is the driving frequency,
is the real current amplitude and phase. Thus
I'
ϕI
I0 is its
is in general complex,
unless the phase of
i(t) is 0 .
Complex impedance ( Z ): The complex analogue of resistance, measured in ohms. The real part of the complex impedance is the resistance is the reactance
Z=
R , and the purely imaginary part of the complex impedance
X , also measured in ohms:
v ( t ) V ' e iωt V ' = = =R+iX i ( t ) I ' eiωt I '
