10/26/09
1
Phasors
for AC Circuit Analysis
Theory and Lab data
obtained with
Sinusoidal signals
© 2009 A. Ganago
1
Phasors in Ac±on
DC and AC Voltages and Currents
•
DC currents and voltages are constant values
•
AC currents and voltages are sinusoids, which
diFer by their amplitude A and phase shiK φ;
we need to keep track of both A and φ
© 2009 A. Ganago
Phasors in Ac±on
2
Lab example: Green=IN, Blue=OUT
© 2009 A. Ganago
Phasors in Ac±on
3
Lab example: Opposite Sign of φ
© 2009 A. Ganago
Phasors in Ac±on
4
Real and Complex Numbers for Circuits
•
We calculate DC currents and voltages by
using real numbers
•
We use complex numbers
for AC voltages and
currents, because
complex numbers have
both magnitude and phase
•
A
phasor
is a shorthand nota±on for the
magnitude and phase of
sinusoidal
voltage or
current at the chosen frequency ω
© 2009 A. Ganago
Phasors in Ac±on
5
Phasor Nota±on for Sine Waves
•
We keep in mind the frequency ω but we do
not include it in the phasor nota±on
•
Phasor describes the whole sinusoidal signal,
at all moments of ±me
•
²or example,
© 2009 A. Ganago
Phasors in Ac±on
6
X
=
A
∠
φ
is shorthand notation for
:
x
(
t
)
=
A
⋅
cos(
ω
t
+
)
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2 types of problems with phasors
•
(1) The frequency ω can be always the same,
like in the power grid
•
Then we calculate a phasor for the parCcular
voltage or current as a constant
•
(2) The frequency ω can vary, as in an audio
signal
•
Then we have to calculate the dependence of
the current or voltage on the frequency ω
© 2009 A. Ganago
Phasors in AcCon
7
Phasors and Impedances
•
Phasors are used for voltages and currents
•
Impedances are used as resistances
•
With phasors
V
for voltage and
I
for current,
and impedance
Z
, we can use
Ohm’s law for
resistors,
inductors and capacitors
•
V
=
Z I
© 2009 A. Ganago
Phasors in AcCon
8
Formulas for Impedances (1)
•
Impedance
Z
R
of a resistor equals resistance R
and does not depend on frequency
•
For a capacitor:
Z
C
= 1/(jω±), where j =√(‐1)
•
If ω is in rad/sec, and ± in farads,
Z
C
is in ohms
•
For example, ± = 1.5 nF, f = 3 MHz; note that
ω=2πf = 1.885e7 rad/sec; thus
Z
C
= ‐j 35.37 Ω
•
Z
C
decreases with the frequency ω
•
Z
C
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 Spring '07
 Ganago
 Alternating Current, Volt, Electrical impedance, A. Ganago, A. Ganago Phasors

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