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)