Transcript Passive elements in the frequency domain  

Passive elements in the frequency
domain
• Consider a resistor, R, in the time domain
R
1kOhm
i
+ v -
i  I cos(t   )
m
i
v  R i  R I cos(t   ) 
m
v  RI
m
i
cos(t   )
i
ECE 201 Circuit Theory I
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Taking the Phasor Transform,
v  R I cos(t   )
m
i
V  RI e  RI
ji
m
I
m
m

i
 I
i
V  RI
The Phasor Voltage is equal to the
Resistance times the Phasor Current
ECE 201 Circuit Theory I
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Phase Information
V  RI  RI
m

i
The voltage has the same phase as the
current – there is no phase shift.
ECE 201 Circuit Theory I
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Consider an Inductor, L
L
1m H
i
+ v -
di
v L
dt
i  I cos(t   )
m
di
I
dt
i
m
   sin(t   )
i
di
v  L  LI sin(t   )
dt
m
ECE 201 Circuit Theory I
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v  LI sin(t   )
m
i
sin(t )  cos(t  90)
sin(t   )  cos(t    90)
i
i
V  LI e
j ( i 90 )
m
V  LI e e
ji
 j 90
m
e
 j 90
 cos 90  j sin 90   j
V  j LI e
ji 
m
V  j LI
ECE 201 Circuit Theory I
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Compare the resistor and inductor
I
L
1m H
I
R
i
i
1kOhm
+ v -
V
V
V  RI
V  j LI
+ v -
Treat jωL as if it were an R
ECE 201 Circuit Theory I
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Phasor Diagram
Im
V
j = 190
I
Re
V “leads” I by 90
or
I “lags” V by 90
ECE 201 Circuit Theory I
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Phase information
V  j LI  L 90I
V  LI
m
m

i
(  90)
i
ECE 201 Circuit Theory I
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Consider a capacitor, C
C
i
1uF
+ v -
dv
i C
dt
v  V cos(t   )
m
v
dv
 V    sin(t   ) 
dt
dv
i C
 CV sin(t   )
dt
m
v
m
ECE 201 Circuit Theory I
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i  CV sin(t   )
m
v
sin(t   )  cos(t    90)
v
v
I  CV e
j ( v 90 )
m
I  CV e e
jv
 j 90
m
e
 j 90
 cos 90  j sin 90   j
I  j CV e
jv
m
I  j CV
V 
I
j C
ECE 201 Circuit Theory I
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Compare the resistor and capacitor
i
I
R
i
I
C
1kOhm
+ v -
1uF
+ v -
V
V  RI
V
V 
I
j C
Treat 1/jωC as if it were an R
ECE 201 Circuit Theory I
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Phasor diagram
Im
I
j = 190
V
Re
I “leads” V by 90
or
V “lags” I by 90
ECE 201 Circuit Theory I
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Phase information
1
1
V 
I 
 90I
j C
C
I
V 
(  90)
C
m

i
m
i
ECE 201 Circuit Theory I
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ELI the ICE man
Phasor Diagram
Capacitor, C
Inductor, L
V
I
I
V(E) leads I
ELI
I leads V(E)
V
ECE 201 Circuit Theory I
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