• Lecture 21: Phasors
ECEN 1400 Introduction to Analog and Digital Electronics
Lecture 21
Phasors
https://xkcd.com/179/
Robert R. McLeod, University of Colorado
185
• Lecture 21: Phasors
ECEN 1400 Introduction to Analog and Digital Electronics
Euler’s formula
A proof of the “most remarkable formula in mathematics” (Feynman):
Geometrical interpretation. A complex number is a vector in the 2D plane:
i is unit vector of imaginary axis
Add vectors in Cartesian form:
( AR + iA I ) + ( BR + iB I ) = ( AR + BR ) + i ( AI + BI )
Multiply vectors in polar form:
(A
M
eiφA ) ( BM eiφB ) = ( AM BM ) ei(φA +φB )
Examples:
i
π
i
π
i ⋅i = e 2 ⋅ e 2 = eiπ = cos π + i sin π = −1
(1+ i )2 = 1− 2i − 1 = 2i
2
π
i ⎞
⎛
= ⎜ 2e 4 ⎟ =
⎝
⎠
Robert R. McLeod, University of Colorado
( )
2
2 e
i 2⋅
π
4
i
π
= 2e 2 = 2i
http://en.wikipedia.org/wiki/Euler’s_formula
186
• Lecture 21: Phasors
ECEN 1400 Introduction to Analog and Digital Electronics
Phasor representation of
sinusoidal functions
•  Imagine a complex circuit of R, C and L driven by a sinusoidal
voltage.
•  C and L take derivatives and integrals of the voltage.
•  These functions produce new sines and cosines.
•  For example:
dVC ( t )
dt
d
= C V cos (ω t + φ )
dt
= −CV ω sin (ω t + φ )
IC (t ) = C
•  You need to keep track of the precise functional form of every
quantity in the circuit. Awkward.
•  Note that you can rewrite the last function as a cosine:
I C ( t ) = −CV ω sin (ω t + φ )
π⎞
⎛
= CV ω cos ⎜ ω t + φ + ⎟
⎝
2⎠
•  Now write this cosine as the real part of a complex exponential:
π⎞
⎛
⎛ π⎞
i⎜ ω t+φ + ⎟ ⎤
i⎜ φ + ⎟
⎡
⎡
⎤
π⎞
⎛
⎝
⎝ 2 ⎠ iω t
2⎠
CV ω cos ⎜ ω t + φ + ⎟ = Re ⎢CV ω e
e ⎥
⎥ = Re ⎢CV ω e
⎝
2⎠
⎣
⎦
⎣
⎦
•  The phasor representation of this function is just the complex number
CV ω e
⎛ π⎞
i⎜ φ + ⎟
⎝ 2⎠
π
consisting of an amplitude CVω and a phase φ + 2
Robert R. McLeod, University of Colorado
187
• Lecture 21: Phasors
ECEN 1400 Introduction to Analog and Digital Electronics
Why are phasors cool?
1) Taking a derivative.
Let’s do the capacitor problem again but in phasor notation
dVC ( t )
dt
d
= C Re ⎡⎣Veiφ eiω t ⎤⎦
dt
d
⎡
⎤
= Re ⎢CVeiφ eiω t ⎥
dt
⎣
⎦
IC (t ) = C
= Re ⎡⎣CVeiφ iω eiω t ⎤⎦
π
iφ +
⎡
⎤
= Re ⎢CV ω e 2 eiω t ⎥
⎣
⎦
So the phasor representation of IC is
IC = CV ω e
iφ +
π
2
= iω CVeiφ = iω CVC
The time derivative operation in phasor notation is thus just multiplication
d
A ( t ) ⇒ iω A
dt
Robert R. McLeod, University of Colorado
188
• Lecture 21: Phasors
ECEN 1400 Introduction to Analog and Digital Electronics
Why are phasors cool?
2) Doing an integral.
What if we’d been given the current through a capacitor and asked to find
the voltage?
1
VC ( t ) =
I ( t ) dt
∫
C
C
1
iφ iω t
⎡
Re
Ie
e ⎤⎦ dt
C∫ ⎣
⎡1
⎤
= Re ⎢ Ieiφ ∫ eiω t dt ⎥
⎣C
⎦
=
1
⎡1
⎤
= Re ⎢ Ieiφ eiω t ⎥
iω
⎣C
⎦
⎡ 1 iφ − π2 iω t ⎤
= Re ⎢
Ie e ⎥
ω
C
⎣
⎦
So the phasor representation of VC is
1 iφ − π2
1
1
VC =
Ie
=
Ieiφ =
IC
ωC
iω C
iω C
The time integral operation in phasor notation is thus just division
∫ A (t ) dt ⇒
Robert R. McLeod, University of Colorado
1
A
iω
189
• Lecture 21: Phasors
ECEN 1400 Introduction to Analog and Digital Electronics
Why are phasors cool?
3) Generalizing Ohm’s law to L and C
Write the IV relationship for inductors and capacitors in phasor notation
1
1
I
t
dt
⇒
V
=
IC
(
)
C
C
C∫
iω C
dI ( t )
VL ( t ) = L L
⇒ VL = iω L I L
dt
VC ( t ) =
which look a lot like Ohm’s law
VR = R I R
Now we can treat any of these as a resistor:
Z = R+
I=
Robert R. McLeod, University of Colorado
1
iω C
V
V
=
Z R+ 1
iω C
190