November
4th
Chapter 33
RLC Circuits
Review
!
RL and RC circuits
!
!
Charge, current, and potential
grow and decay exponentially
LC circuit
!
Charge, current, and potential
change sinusoidally
!
Total electromagnetic energy is
Li 2 q 2
U = UB + UE =
+
2
2C
Review
!
Ideal LC circuit
!
!
dU
=0
dt
Total energy conserved
Solved differential equation to find
q = Q cos(ωt )
ω=
i = − I sin(ωt )
!
U
1
LC
Substituting q and i into energy equations
E
Q2
=
cos
2C
2
(ω t )
U
B
Q2
=
sin
2C
U = U B + U E = Q 2 / 2C
2
(ω t )
Review
!
RLC circuit
!
!
!
Energy is no longer conserved,
becomes thermal energy in resistor
Oscillations are damped
Solved differential equation to find
q = Qe
− Rt / 2 L
cos(ω ′t )
2
2
′
ω = ω − ( R / 2 L)
!
dU
= −i 2 R
dt
If R is very small
ω′ = ω
Energy goes as
Q 2 − Rt / L
UE =
e
cos 2 (ω ′t )
2C
Q 2 − Rt / L
Utot =
e
2C
Resistive Load
ε −v
=0
!
Apply loop rule
!
Using
!
We have
!
Amplitude across resistor is same as
across emf
!
vR = ε
ε =ε
m
R
sin ω d t
vR = ε m sin ω d t
Rewrite vR as
vR = VR sin ω d t
ε
m
= VR
Forced Oscillations
Resistive load
vR = VR sin ω d t VR = ε m
Capacitive load
vC = VC sinω d t VC = ε m
Inductive load
vL = VLsinω d t VL = ε m
Resistive load
!
Use definition of resistance to
find iR
v R = V R sin ω d t
vR VR
=
sin ω d t
iR =
R
R
!
!
Voltage and current are
functions of sin(ωdt) with φ = 0
so are in phase
No damping of vR and iR , since
the generator supplies energy
Resistive load
!
Compare current to general form
vR VR
iR =
= sin ω d t
R
R
VR = I R R
iR = I Rsin(ωd t − φ )
!
!
!
Minus sign for phase is tradition
For purely resistive load the phase
constant φ = 0
Voltage amplitude is related to
current amplitude
VR
IR =
R
VR = I R R
Capacitive load
!
Use definition of capacitance
qC = Cv C = CV C sin ω d t
!
Use definition of current and
differentiate
dq C
iC =
= ω d CV C cos ω d t
dt
!
Replace cosine term with a
phase-shifted sine term
cosωd t = sin(ωd t + 90°)
Capacitive load
!
Voltage and current relations
vC = VC sin ω d t
iC = ωd CVC sin(ωd t + 90°) = IC sin(ωd t + 90°)
VC
IC =
XC
!
1
XC =
ωd C
XC is called the capacitive reactance
and has units of ohms
Capacitive load
!
Compare vC and iC of capacitor
vC = VC sin ω d t
iC = ωd CVC sin(ωd t + 90°)
iR = I Rsin(ωd t − φ )
!
Voltage and current are out
of phase by -90°
!
Current leads voltage by T/4
Inductive load (skipped in lecture)
!
!
!
Derivation of current:
Self-induced emf across an
inductor is
di
L = vL = L
Relate
dt
ε
di
v L = VL sin ω d t = L
dt
!
di
VL
=
sin ω d t
dt
L
Want current so integrate
 VL 
VL
 cos(ωd t )
iL = ∫ sin(ωd t ) dt = −
L
 ωd L 
!
Then use
− cos( ω d t ) = sin( ω d t − 90° )
Inductive load
!
Voltage and current relations
v L = V L sin ω d t
VL
iL =
sin(ωd t − 90°) = I L sin(ωd t − 90°)
ωd L
VL
IL =
XL
!
X L = ωd L
XL is called the inductive reactance has units
of ohms
Inductive load
!
Compare vL and iL of inductor
vL = VL sin ω d t
iL = I L sin (ω d t − 90°)
! iL
!
and vL are +90° out
of phase
Current lags voltage by T/4
Summary of Forced Oscillations
Element
Resistor
Reactance/
Resistance
R
Phase of Phase Amplitude
Current angle φ Relation
In phase
0°
VR=IRR
Capacitor XC= 1/ωdC Leads vC
(ICE)
Inductor
XL=ωdL Lags vL
(ELI)
!
-90°
VC=ICXC
+90°
VL=ILXL
ELI (positively) is the ICE man
!
!
!
Voltage or emf (E) before current (I) in an inductor (L)
Phase constant φ is positive for an inductor
Current (I) before voltage or emf (E) in capacitor (C)
EM Oscillations
!
If the driving frequency, ωd , in a circuit is
increased does the amplitude voltage and
amplitude current increase, decrease or remain
the same?
!
For purely resistive circuit
!
From loop rule VR = εm
So amplitude voltage, VL stays the same
IR also stays the same VR
IR =
IR only depends on R
!
!
R
EM Oscillations
!
If the driving frequency, ωd , in a circuit is
increased does the amplitude voltage and
amplitude current increase, decrease or remain
the same?
!
For purely capacitive circuit
!
From loop rule Vc = εm
So amplitude voltage, VC stays the same
IC depends on XC which depends on ωd by
So IC increases
VC
!
!
!
IC =
XC
= ω d CVC
EM Oscillations
!
If the driving frequency, ωd , in a circuit is
increased does the amplitude voltage and
amplitude current increase, decrease or remain
the same?
!
For purely inductive circuit
!
From loop rule VL = εm
So amplitude voltage, VL stays the same
IL depends on XL which depends on ωd by
So IL decreases
VL
VL
!
!
!
IL =
XL
=
ωd L
RLC Circuits
!
LC and RLC circuits with no external emf
!
Free oscillations with natural
angular frequency, ω
1
ω=
ω ′ = ω − ( R / 2 L)
LC
! Add external oscillating emf (e.g. ac
generator) to RLC circuit
2
!
!
!
2
Oscillations said to be
driven or forced
Oscillations occur at driving
angular frequency, ωd
When ωd = ω, called resonance,
the current amplitude, I, is maximum
RLC circuits
!
!
!
!
RLC circuit – resistor, capacitor
and inductor in series
Apply alternating emf
ε =ε
m
sin ω d t
Elements are in series so same
current is driven through each
From the loop rule, at any
time t, the sum of the voltages
across the elements must
equal the applied emf
i = I sin(ωd t − φ )
ε =v
R
+ vC + vL
RLC circuits
ε = iR + iX
!
C
Taking into account the phase find current
amplitude to be
I=
!
!
+ iX L
ε
R + ( X L − XC )
Write amplitude voltage as
Where Z is the impedance
!
m
2
ε
m
2
= IZ
Like resistance and has units of Ohms
X L = ωd L
Z = R + (X L − XC )
2
2
XC =
1
ωd C
EM Oscillations
XL − XC
tan φ =
R
!
If XL > XC the circuit is
more inductive than
capacitive
!
!
!
φ is positive
Emf is before current (ELI)
If XL < XC the circuit is
more capacitive than
inductive
!
!
φ is negative
Current is before emf (ICE)
RLC Circuits
X L − XC
tan φ =
R
!
!
If XL = XC the circuit is in
resonance – emf and
current are in phase
Current amplitude I is max
when impedance, Z is min
ε
I=
m
Z
=
ε
m
R + (X L − XC )
2
X L − XC = 0
2
ε
=
m
R
EM Oscillations
!
When XL = XC the driving frequency is
1
ωd L =
ωd C
!
This is the same as the natural frequency, ω
ωd = ω =
!
ωd =
1
LC
1
LC
For RLC circuit, resonance and the max
current I occurs when ωd = ω
RLC circuits
I=
Em
1
R + ( ωd L −
ωd C
2
!
I is largest when
X L = XC
!
When ωd=ω circuit said
to be in resonance
!
When homework says
resonance frequency it
means ωd=ω
ω = ω d = 2π f d
)2