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Alternating Current Ch. 31
Phasors and AC
Resistance and reactance
RLC series circuit
Power in AC circuits
Resonance in AC circuits
Transformers
C 2010 J. F. Becker
(sec. 31.1)
(sec. 31.2)
(sec. 31.3)
(sec. 31.4)
(sec. 31.5)
(sec. 31.6)
Learning Goals - we will learn: ch 31
• How phasors make it easy to describe
sinusoidally varying quantities.
• How to analyze RLC series circuits driven
by a sinusoidal emf.
• What determines the amount of power
flowing into or out of an AC circuit.
• How an RLC circuit responds to emfs of
different frequencies.
Phasor diagram -- projection of rotating vector
(phasor) onto the horizontal axis represents the
instantaneous current.
Notation:
-lower case letters
are time dependent
and
-upper case letters
are constant.
For example,
i(t) is the time
dependent current
and
I is current
amplitude;
VR is the voltage
amplitude (= IR ).
i(t) = I cos wt (source)
vR(t) = i(t) R
vR(t) = IR cos wt
where VR = IR is
the voltage amplitude.
VR = IR
Graphs (and phasors) of instantaneous voltage and
current for a resistor.
Graphs of instantaneous voltages for RLC series circuit.
(The phasor diagram is much simpler.)
i(t) = I cos wt (source)
vL(t) = L di / dt
vL(t) = L d(I cos wt )/dt
vL(t) = -IwL sin wt
E L I
VL L I
vL(t) = +IwL cos (wt + 900)
where VL = IwL (= IXL)
is the voltage amplitude
and f = +900 is the
PHASE ANGLE
(angle between voltage
across and current
through the inductor).
XL = wL
Graphs (and phasors) of instantaneous voltage and
current for an inductor.
i(t) = I cos wt (source)
i(t) = dq / dt = I cos wt
Integrating we get
q(t) = (I/w) sin wt
and from q = C vC we get
vC(t) = (I/wC) sin wt
vC(t)=(I/wC) cos (wt -
I C E
I C VC
0
90 )
where VC = I/wC (= IXC)
is the voltage amplitude
and f = -900 is the
PHASE ANGLE
(angle between voltage
across and current
through the inductor).
Graphs (and phasors) of instantaneous
voltage and current for a capacitor.
XC = 1/wC
Graphs (and phasors) of instantaneous voltage and
current showing phase relation between current (red)
and voltage (blue).
Remember: “ELI the ICE man”
Crossover network in a speaker system.
Capacitive reactance: XC =1/wC
Inductive reactance: XL = wL
Phasor diagrams for series RLC circuit
(b) XL > XC and (c) XL < XC.
Graphs of instantaneous voltages for RLC series circuit.
(The phasor diagram is much simpler.)
Graphs of instantaneous voltage, current, and power for an R,
L, C, and an RLC circuit. Average power for an arbitrary
AC circuit is 0.5 VI cos f = V rms I rms cos f.
Instantaneous
current and voltage:
Average power
depends on current
and voltage
amplitudes AND
the phase angle f:
The average power is half the product of I and the
component of V in phase with it.
The resonance
frequency is at
w = 1000 rad / sec
(where the current
is at its maximum)
Graph of current amplitude I vs source frequency w
for a series RLC circuit
with various values of circuit resistance.
A radio tuning circuit at resonance. The circles denote
rms current and voltages.
e = - dF B / dt
TRANSFORMERS
can step-up AC
voltages or stepdown AC voltages.
e2 /e1 = N /N
2
V1I1 = V2I1
1
FB = FB
Transformer: AC source is V1 and secondary provides a
voltage V2 to a device with resistance R.
(a) Primary P and secondary S windings in a transformer.
(b) Eddy currents in the iron core shown in the crosssection AA. (c) Using a laminated core reduces the
eddy currents.
Large step-down transformers at power stations are
immersed in tanks of oil for insulation and cooling.
A full-wave diode rectifier circuit. (LAB)
Review
See www.physics.sjsu.edu/becker/physics51
C 2010 J. F. Becker