Physics 227: Lecture 22
AC Circuits I
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Lecture 21 review:
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Mutual inductance of two conductors: ξ2 induced = -M di1/dt or
ξ1 induced = -M di2/dt with the same M.
LC circuit: energy goes back and forth between capacitor E
field and inductor B field, ``SHM’’ with ω2 = 1/LC.
LRC circuit: LC with energy lost to damping. E.g.: Q = A e-(R/2L)t
cos(ω’t+φ) with ω’2 = (1/LC) - (R2/4L2):
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Monday, November 28, 2011
ω’2 > 0: damped oscillation
R2 = 4L/C → ω’ = 0: critical damping - Q → A’ e-(R/2L)t
R2 > 4L/C → ω’2 < 0: overdamped
Physics 227: Lecture 22
AC Circuits I
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Previous lecture
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Voltage = constant, or changed instantaneously in a step from 0
to V or from V to 0
This lecture: AC voltage: V = Vmax cos(ωt)
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Why? Power around the world is almost always generated
and supplied to homes / businesses / offices as AC power
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Monday, November 28, 2011
Westinghouse beats Edison
Much of the world uses AC electrical grids operating at
f = ω/2π = 50 Hz; the US uses 60 Hz.
Home voltages vary from about 120 - 240 V - we will define
this better later.
AC
symbol for
AC source
Monday, November 28, 2011
What happens if you put AC
voltage over a resistor?
V(t) = Vmax cos(ωt)
A. I = 0.
B. I = Vmax/R.
C. I is undefined.
D. I = (Vmax/R) cos(ωt).
E. I = Vmax R cos(ωt).
Monday, November 28, 2011
Power with AC over a resistor
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V(t) = Vmax cos(ωt)
P = IV = I(t) V(t) = (Vmax/R)cos(ωt) Vmax cos(ωt) = (Vmax2/R)cos2(ωt)
P = IV = I(t) V(t) = Imaxcos(ωt) ImaxRcos(ωt) = Imax2Rcos2(ωt)
Note I dislike the
book’s usage of I vs i.
Monday, November 28, 2011
Average Power with AC over a resistor
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What is the average cos2(x)?
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``Obviously’’ 1/2 - recall
cos2(x) + sin2(x) = 1.
Thus, average of
P(t) = (Vmax 2/R)cos2(ωt) is
Vmax 2/2R.
For purposes of power, we can
define a Vave = Vmax/√2, but
more typically we would write
Vrms = Vmax/√2, since for
V(t) = Vmaxcos(ωt), Vave = 0.
rms = root mean square - the
square root of the average
square.
Pave = IrmsVrms = Vrms2/R = Irms2R
Monday, November 28, 2011
V(t) = v cos(ωt)
One way to convert AC to DC - Diodes
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For voltage below some amount,
no current flows through diode
- as if R = ∞.
Idealized diode
I
For voltage above ≈ 0.6 V, any
current can flow through the
diode - as if R = 0.
A little more realistically, one
has an exponential increase in
the current with voltage.
Monday, November 28, 2011
V
Circuit symbol
One way to convert AC to DC - Diodes
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Current
flows along
purple path
when V
positive to
left, along
pink path
when V
positive to
right
Monday, November 28, 2011
Diode to ``rectify’’ current
- current through G
Inductors & Capacitors with AC voltage?
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V = L dI/dt
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For fixed voltage, as ω → ∞, i → 0 - acts more like open
circuit. But as ω → 0, i → ∞ - acts more like short circuit.
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If I(t) = Imaxcos(ωt), V = -ωLImaxsin(ωt) = -ωLImaxcos(ωt-90)
The voltage ``leads’’ the current by 90o - we have to subtract
90o from it.
Q = CV → I = C dV/dt.
For V(t) = Vmaxcos(ωt), I(t) = -ωCVmaxcos(ωt-90)
The current ``leads’’ the voltage by 90o.
For fixed voltage, as ω increase, i increases - acts more like
short circuit, but as ω decreases, i decreases - acts more like
open circuit
How do we unify the concepts of
resistance, capacitance, and inductance?
Monday, November 28, 2011
Phasor Diagrams - Not Star Trek!
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Graph the current or
voltage as a vector.
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Monday, November 28, 2011
The length is the maximum
current or voltage.
The angle is θ = ωt.
The actual current or
voltage at some point in
time is the ``x’’
component of the vector.
For a resistor, the I and V
vectors overlay one another
- they have the same
phase / ωt argument.
Phasor Diagrams for Capacitor and Inductor
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Inductor I(t) = Imaxcos(ωt), V = -ωLImaxcos(ωt-90)
Capacitor: V(t) = Vmaxcos(ωt), I(t) = -ωCVmaxcos(ωt-90)
I=Imaxcos(ωt)
V=-ωLImaxcos(ωt-90)
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θ=ωt
V or I
V=Vmaxcos(ωt)
I=-ωCVmaxcos(ωt-90)
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θ=ωt
V or I
Monday, November 28, 2011
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As time passes, the
vectors rotate CCW.
For the inductor (top),
the voltage leads the
current.
For the capactor
(bottom), the current
leads the voltage.
For the resistor (not
shown), current and
voltage are in phase.
Phasor Diagram for Inductor
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Inductor I(t) = Imaxcos(ωt), V = -ωLImaxcos(ωt-90)
Monday, November 28, 2011
Phasor Diagram for
Capacitor
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Capacitor: V(t) = Vmaxcos(ωt),
I(t) = -ωCVmaxcos(ωt-90)
Monday, November 28, 2011
Reactance
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Resistor: V(t) = Vmaxcos(ωt), I(t) = (Vmax/R)cos(ωt).
Inductor: I(t) = Imaxcos(ωt), V = -ωLImaxcos(ωt-90).
Capacitor: V(t) = Vmaxcos(ωt), I(t) = -ωCVmaxcos(ωt-90).
We now further unify the concepts of resistance, capacitance, and
inductance by defining a reactance as X = Vmax/Imax. Reactance
tells us about the relative amplitudes of voltage and current, but
not the phase.
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Resistor: XR = R.
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V and I in phase.
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V leads I by 90o.
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V lags I by 90o.
Inductor: XL = ωL.
Capacitor: XC = 1/ωC.
Monday, November 28, 2011
Voltage DIvider
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Consider an AC voltage applied over two circuit elements
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X1
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VAC
X2
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Monday, November 28, 2011
With two resistors:
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V1 = R1/(R1+R2) VAC.
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V1 = R1/(R1+ωL) VAC.
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V1 = R1/(R1+1/ωC) VAC = ωCR1/(1+ωCR1) VAC.
V2 = R2/(R1+R2) VAC.
What about, e.g. X2 = ωL?
V2 = ωL/(R1+ωL) VAC.
What about, e.g. X2 = 1/ωC?
V2 = (1/ωC)/(R1+1/ωC) VAC = 1/(1+ωCR1) VAC.
Voltage DIvider
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Monday, November 28, 2011
R+L: (woofer)
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VR = R1/(R1+ωL) VAC.
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VR = ωCR1/(1+ωCR1) VAC.
VL = ωL/(R1+ωL) VAC.
R+C: (tweeter)
VC = 1/(1+ωCR1) VAC.
Review RL iClicker
For the circuit shown, when is the
current i at a maximum compared to
Vab?
A. It depends on whether i is iinductor or isupply.
B. 1/4 cycle after Vab is maximum.
C. 1/4 cycle before Vab is maximum.
D. The same time as Vab is maximum.
E. 1/2 off from when Vab is maximum.
Monday, November 28, 2011
LC iClicker
VAC
For the circuit shown, which condition gives
the maximum voltage across the capacitor?
A. XC = 1/ωC = XL = ωL. (ω2=1/LC.)
B. ω = 0.
C. ω = ∞.
D. C/L is very very large.
E. The voltage across the capacitor cannot be maximized.
Monday, November 28, 2011
Thursday: Impedance, RLC circuits
Some summer job opportunites internships with Federal government:
www.orau.gov/dhsinternships
www.orau.org/ornl
http://science.energy.gov/wdts/suli/
Monday, November 28, 2011