# Characterized ideal LC

```Review
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Characterized ideal LC circuit
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Charge, current and voltage vary sinusoidally
Oscillations become damped
Charge, current and voltage still vary
sinusoidally but decay exponentially
Added ac generator to circuits with just a
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Resistor
Capacitor
Inductor
Review
Element
Resistor
Reactance/ Phase of Phase Amplitude
Resistance Current angle φ Relation
R
In phase
0&deg;
VR=IRR
Inductor
XL=ωdL
Lags vL
-90&deg;
VC=ICXC
+90&deg;
VL=ILXL
EM Oscillations
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RLC circuit – resistor, capacitor
and inductor in series
Apply alternating emf
E = E m sin ωd t
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Elements are in series so same
current is driven through each
i = I sin ωd t − φ
From the loop rule, at any
time t, the sum of the voltages
across the elements must
E = vR + vC + vL
equal the applied emf
(
)
EM Oscillations
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Want to find amplitude I
and the phase constant φ
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Using phasors, represent
the current at time t
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Length is amplitude I
Projection on vertical axis is
current i at time t
Angle of rotation is the phase
at time t
ωd t − φ
EM Oscillations
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Draw phasors for voltages
of R, C and L at same time t
Orient VR, VL, &amp; VC phasors
relative to current phasor
Resistor – VR and I are in
phase
Inductor – VL is ahead of I
by 90&deg;
Capacitor – I is ahead of VC
by 90&deg;
vR, vC, &amp; vL are projections
EM Oscillations
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Draw phasor for applied emf
E = E m sin ωd t
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Length is amplitude Em
Projection is E at time t
Angle is phase of emf ω d t
From loop rule the projection
E = the algebraic sum of
projections vR, vL &amp; vC
E = VR + VC + VL
EM Oscillations
E = VR + VC + VL
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Phasors rotate together so
equality always holds
Phasor Em = vector sum of
voltage phasors
r
r r
r
Em = VR + VC + VL
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Combine VL &amp; VC to form
single phasor
V L − VC
EM Oscillations
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Using Pythagorean theorem
E = V + (VL − VC )
2
m
z
2
R
2
From amplitude relations
replace voltages with
VR = IR VL = IX L VC = IX C
E = ( IR) + (IX L − IX C )
2
m
z
2
Rearrange to find
amplitude I
I=
2
Em
R + (X L − XC )
2
2
EM Oscillations
I=
z
Em
R 2 + ( X L − X C )2
Define impedance, Z to be
Z = R 2 + ( X L − X C )2
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Using reactances
rewrite current as
I=
X L = ωd L
Em
R + (ωd L − 1 / ωd C )
2
2
1
XC =
ωd C
Em
I=
Z
EM Oscillations
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Using trig find the phase
constant φ
VL − VC
tan φ =
VR
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Using amplitude relations
IX L − IX C
tan φ =
IR
X L − XC
tan φ =
R
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Examine 3 cases:
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XL &gt; XC
XL &lt; XC
XL = XC
EM Oscillations
X L − XC
tan φ =
R
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If XL &gt; XC the circuit is
more inductive than
capacitive
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φ is positive
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Emf is before current
If XL &lt; XC the circuit is
more capacitive than
inductive
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φ is negative
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Current is before emf
EM Oscillations
X L − XC
tan φ =
R
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If XL = XC the circuit is in
resonance – emf and
current are in phase
Current amplitude I is max
when impedance, Z is min
X L − XC = 0
Em
I=
=
Z
Em
Z=R
Em
=
2
2
R
R + (X L − XC )
EM Oscillations
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When XL = XC the driving frequency is
1
ωd L =
ωd C
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This is the same as the natural frequency, ω
ωd = ω =
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ωd =
1
LC
1
LC
For RLC circuit, resonance and the max
current I occurs when ωd = ω
EM Oscillations
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For small driving
frequency, ωd &lt; ω
I=
z
XL is small but XC is large
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Circuit capacitive
For large driving
frequency, ωd &gt; ω
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XC is small but XL is large
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Circuit inductive
For ωd = ω, circuit is in
resonance
Em
R + (ωd L − 1 / ωd C )
2
2
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