l22 ac circuits reactance
TRANSCRIPT
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Lecture 21. AC Circuits, Reactance.
Outline:
Power in AC circuits, Amplitude and RMS values.
Phasors / Complex numbers.
Resistors, Capacitors, and Inductors in the AC circuits.
Reactance and Impedance.
1
Conflict final exam: December 7, 500 PM: Last day and time to request
conflict exam for Final Exam (you can request conflict exam if you have another
exam at same time OR have 3 exams in 24 hour period). You must e-mail
Professor Cizewski [email protected] with details of why you are
requesting conflict exam.November 25 (Rutgers Thursday): Required E&M Post-test during lecture
times. You can attend any lecture. Makeup post tests will be in early
December. NO Recitations the week of November 24.
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Oscillations in L-C Circuits
2
L-C circuits: the circuits with TWO elements that can store energy
(ideally, without dissipation). The energy flow back and forth
between L and C results in harmonic oscillations of and .
=
2 =
2
electric field energy
in the capacitormagnetic field energy
in the inductor
Let’s say, at t = 0 the capacitor is fully charged, =0.
2=
2+
2
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Oscillations in L-C Circuits (cont’d)
3
= 0,
≡
,
+
1
= 0
Solution: = +
amplitude angl. frequency
phase at t =0
=1
≡
1
≡
2
= 2
=
2=
2 =
41 + 2
=
= +
=
2=
2 =
41 + 2
/2
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L-C-R Circuits
6
weak damping (“small”R
) strong damping (“large”R
)
= 0,
+
+
1
= 0
Solution for weak damping
≫
4:
=
2 ∗ +
∗ =1
4
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Impedance of AC circuits
13
So far we have considered transient processes in R-C and R-L
circuits: the approach to the stationary (time-independent)
state after some perturbation (switch on / off).
Today we’ll discuss how these circuits behave being connected
to the alternating current (AC) power supply: the circuits driven
by a steady external drive, e.g. the AC voltage source.
We disregard all transient processes and instead consider the
steady-state AC currents: currents and voltages vary with time
as + , but their amplitudes are t-independent .
We’ll describe the response of an L-C-R circuit to a harmonic
drive using the notion of impedance.
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Amplitudes, rms Values, and Power in AC Circuits
14
= cos
= cos +
= cos ∙ cos + =1
2 cos 2 + + cos
≡ =1
2 cos
Currents and voltages are
NOT necessarily in phase,
is the phase shift
between V and I (the
“phase angle”).
=
2 =
2
“120V” wall outlet: = 60, = 2 ∙ 60
= 377/, =
≈ 17
= 120, ≈ 170
Root mean square (rms): the
square root of the average of
the square of the quantity:
=
Power, being expressed in the rms values: = ∙ cos
RLC
Instantaneouspower: =
Instantaneous
values:
cos -
the power factor
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Phasors / Complex Number Representation
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Problem: To find current/voltage in R-L-C circuits,
we need to solve differential equations.
Solution: The use of complex numbers / phasors
allows us to replace linear differential equationswith algebraic ones, and reduce trigonometry to
algebra.
We represent voltages and currents in the R-L-C circuits as the phase vectors
( phasors) on the 2D plane. Quantity: = cos. Corresponding
phasor: a vector of length
rotating counter-clockwise with the angular
frequency . Instantaneous value of is the projection of the phasor onto
the horizontal axis.
If all the quantities oscillate with the same , we can get
rid of the term by using the rotating (merry-go-
around) reference frame.
We’ll consider the steady state of AC circuits, when all amplitudes (the phasor lengths)
are t -independent, and the only time dependence remaining is in the single frequency
sinusoidal oscillation of voltages and currents. The angle between different phasors
represents their relative (t -independent) phase.
Instantaneous
value
phasor
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Complex Numbers, Phasors
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≡ +
= +
∗ ≡
=
=
=
≡ ∙ ∗
≡ 1
= cos + sin Euler’s
relationship
= 1 1
=
complex conjugate of
+ = + + − +
2= +
+ = + − +
2= +
= + = + =
=
− =
Imaginary
unit:
=
The absolute value (or modulus or magnitude):
Phasor : refer to either + or just . In the latter case, it is understood
to be a shorthand notation, encoding the amplitude and phase of an underlying
sinusoid.
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Complex Numbers, Phasors (cont’d)
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Addition: + + + = + + +
Multiplication:
Differentiation:
∙ = +
=
The use of complex numbers / phasors allows us to replace linear differential equations
with algebraic ones, and reduce trigonometry to algebra:
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Resistor
18
Power dissipated in a resistor: = ()
=1
2 =
1
2 =
phase 2
AC current through a resistor and AC voltage across the resistor are always in phase.
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Capacitor
19
= 0
=
=
= +
=
1
=
1
=
=
=1
2 = =
2= 0
= ()
= −
phase 2 The power IS NOT dissipated in a capacitor:
it is stored in the capacitor for half a period,
and returned to the circuit for another half.
=
= /2
For a capacitor, voltage
LAGS current by 900
.
Current (reference phasor)
Voltage= cos2 + sin 2 cos
2
+ sin
2
= sin2
0 -1
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Inductor
20
= 0
=
= +
=
+
= =
+
=1
2 = =
2= 0
= ()
= +
time t
phase
2 The power IS NOT dissipated in an inductor:
it is stored in the inductor for half a period,
and returned to the circuit for another half.
=
= /2
In an inductor, current
LAGS voltage by 900
Current (reference phasor)
Voltage
= cos2 + sin2 cos
2
+ sin
2
= sin2
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Reactance
21
= = Resistor
=
Capacitor
Inductor
=
1
=
+
=
=
=
=1
=
=
= −
= = =
= =
Wiki: “reactance is the opposition of a circuit element to a change of electric current
due to that element's inductance or capacitance”. The reactance is measured in Ohms.
Major differences between reactance and resistance: the reactance for L and C
changes with frequency, it reflects (being combined with ±) the phase shift between
V and I, and it dissipates no energy.
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Impedance
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=
Impedance is a measure of the overall opposition of a circuit to current, in other words:
how much the circuit impedes the flow of current. Both the reactance and resistance are
components of the impedance. The magnitudes of V and I are the rms voltage and
current respectively, and the various reactances behave mathematically just likeresistances, except that they are complex.
= + = + 1
For R, C , and L in series:
= + 1
= ∙
∗ = + 1
= ∙ ∗ = + 1
=
For = 120, 60, = 300Ω, = 2 , = 3:
1
= 377 ∙ 3
1
377 ∙ 2 ∙ 10−= 1131 1326 = 195Ω
=120
300 + 195Ω= 0.335
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Next time: Lecture 23. Resonance in AC circuits,
Transformers.
§§ 31.3 - 6