E E 2415 Lecture 8 - Introduction to Phasors. The Sinusoidal Function
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Transcript of E E 2415 Lecture 8 - Introduction to Phasors. The Sinusoidal Function
E E 2415Lecture 8 - Introduction to Phasors
The Sinusoidal Function
1.05 0 1.05 2.09 3.14 4.19 5.24 6.281.2
0.8
0.4
0
0.4
0.8
1.2
f t( )
t
f
Euler Functions
cos2
j je e
sin2
j je ej
cos sinje j
cos sinje j
1j
: an angle in radians
Sinusoid to Phasor
Exponential form:
Separate exponents:
Phasor transform:
Also:
fV~Im
Re
Simple Harmonic Motion - 0• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real axis
plotted vs time• Phasor shown at 0 degrees
Simple Harmonic Motion - 30• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 30 degrees
Simple Harmonic Motion - 60• Phasor rotates CCW about origin
of complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 60 degrees
Simple Harmonic Motion - 90• Phasor rotates CCW about origin
of complex plane• Blue dot shows projection on real
axis plotted vs time• Phasor shown at 90 degrees
Simple Harmonic Motion - 120• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 120 degrees
Simple Harmonic Motion - 150• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 150 degrees
Simple Harmonic Motion - 180• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 180 degrees
Simple Harmonic Motion - 210• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 210 degrees
Simple Harmonic Motion - 240• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 240 degrees
Simple Harmonic Motion - 270• Phasor rotates CCW about origin of
complex plane• Blue dot shows projection on real
axis plotted vs time• Phasor shown at 270 degrees
Simple Harmonic Motion - 300• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 300 degrees
Simple Harmonic Motion - 330• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 330 degrees
Simple Harmonic Motion - 360• Phasor rotates CCW about origin of
complex plane• Blue line shows projection on real
axis plotted vs time• Phasor shown at 360 degrees
Impedance of an Inductor
( ) Re j t jm
div t L LI j e edt
L+ v -
i
jL
+ V -~I~
Phasor Diagram of Inductor V-I Relationship
Voltage leads current by 90°
I~
V~
90o
Im
Re
Impedance of a Capacitor
( ) Re j t jm
dvi t C CV j e edt
f
C
+ v -
i
jCI~
~+ V -
Phasor Diagram of Capacitor V-I Relationship
Current leads Voltage by 90°
I~
V~f
90o
Im
Re
Phasor R-L Example (1/3)
Vs
R jL
~ + VR -~ + VL -~
I~
Phasor R-L Example (2/3)
KVL now in complex plane.
Phasor R-L Example (3/3)
Vs
R jL
~ + VR -~ + VL -~
I~
~VL ~VR
~Vs
I~
f Re
Im
Numerical Phasor R-L Example (1/2)
100 0 4 36.8720 15
Ij
20 4 36.87 80 36.87RV
15 90 4 36.87 60 53.13LV
Numerical Phasor R-L Example (1/2)