Lecture3 - Phasors

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Lecture 3
Phasors
Electric Circuits II
Diego Mej́ıa Giraldo
February 3, 2017
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Phasor
Example 2 — online problems
Evaluate the following complex numbers:
(a)
3− j4
2− j
=
(b)
j3 + j10 + j
j2 + j137 + 1
=
(c) (
1− j
1 + j
)3
=
2/8
Phasor
Example 2 — online problems
Evaluate the following complex numbers:
(a)
3− j4
2− j
=
(b)
j3 + j10 + j
j2 + j137 + 1
=
(c) (
1− j
1 + j
)3
=
2/8
Phasor
Example 2 — online problems
Evaluate the following complex numbers:
(a)
3− j4
2− j
=
(b)
j3 + j10 + j
j2 + j137 + 1
=
(c) (
1− j
1 + j
)3
=
2/8
Phasor
Euler’s identity again
Assume an independent voltage source whose output is
v (t) = Vm cos (ωt + φ)
Recall from Euler’s identity:
Vme
j(ωt+φ) =
Vm cos (ωt + φ) + jVm sin (ωt + φ)
Therefore
v (t) = Re
{
Vme
j(ωt+φ)
}
= Re
{
Vme
jφe jωt
}
The voltage phasor is expressed as
V = Vm φ = Vme
jφ
Q: How to represent v(t) in terms of V?
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Phasor
Euler’s identity again
Assume an independent voltage source whose output is
v (t) = Vm cos (ωt + φ)
Recall from Euler’s identity:
Vme
j(ωt+φ) = Vm cos (ωt + φ) + jVm sin (ωt + φ)
Therefore
v (t) = Re
{
Vme
j(ωt+φ)
}
= Re
{
Vme
jφe jωt
}
The voltage phasor is expressed as
V = Vm φ = Vme
jφ
Q: How to represent v(t) in terms of V?
3/8
Phasor
Euler’s identity again
Assume an independent voltage source whose output is
v (t) = Vm cos (ωt + φ)
Recall from Euler’s identity:
Vme
j(ωt+φ) = Vm cos (ωt + φ) + jVm sin (ωt + φ)
Therefore
v (t) = Re
{
Vme
j(ωt+φ)
}
= Re
{
Vme
jφe jωt
}
The voltage phasor is expressed as
V = Vm φ = Vme
jφ
Q: How to represent v(t) in terms of V?
3/8
Phasor
Euler’s identity again
Assume an independent voltage source whose output is
v (t) = Vm cos (ωt + φ)
Recall from Euler’s identity:
Vme
j(ωt+φ) = Vm cos (ωt + φ) + jVm sin (ωt + φ)
Therefore
v (t) = Re
{
Vme
j(ωt+φ)
}
= Re
{
Vme
jφe jωt
}
The voltage phasor is expressed as
V = Vm φ = Vme
jφ
Q: How to represent v(t) in terms of V?
3/8
Phasor
Can a phasor rotate?
This is the plot of Ve jωt
v(t) is the projection on the real axis. What is the projection on
the imaginary axis?
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Phasor
Can a phasor rotate?
This is the plot of Ve jωt
v(t) is the projection on the real axis. What is the projection on
the imaginary axis?
4/8
Phasor
Example 3— Practice problem 9.6 of Alexander’s book
If v1 = −10 sin (ωt − 30) V and v2 = 20 cos (ωt + 45) V, find their
sum v = v1 + v2
Sol.: v = 29.77 cos (ωt + 49.98)
Please do it yourself!
5/8
Phasor
Example 3— Practice problem 9.6 of Alexander’s book
If v1 = −10 sin (ωt − 30) V and v2 = 20 cos (ωt + 45) V, find their
sum v = v1 + v2
Sol.: v = 29.77 cos (ωt + 49.98)
Please do it yourself!
5/8
Phasor
Example 4
If i1 = −10 sin (ωt − 45) A and i2 = 10 cos (ωt + 135) A, find their
sum i = i1 + i2
Sol.: i = 10
√
2 sin t A
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Phasor
Example 4
If i1 = −10 sin (ωt − 45) A and i2 = 10 cos (ωt + 135) A, find their
sum i = i1 + i2
Sol.: i = 10
√
2 sin t A
6/8
Phasor
Time domain vs Frequency domain
These are some tricks really useful for solving integro-differential
equations. Key concept for AC circuit analysis!
I Function
v (t)⇔ V
I Derivative:
dv (t)
dt
⇔ ???
I Integral: ∫
v (t) dt ⇔ ???
Q: Do you remember the concept of leading and lagging taught
last week?
7/8
Phasor
Time domain vs Frequency domain
These are some tricks really useful for solving integro-differential
equations. Key concept for AC circuit analysis!
I Function
v (t)⇔ V
I Derivative:
dv (t)
dt
⇔ ???
I Integral: ∫
v (t) dt ⇔ ???
Q: Do you remember the concept of leading and lagging taught
last week?
7/8
Phasor
Time domain vs Frequency domain
These are some tricks really useful for solving integro-differential
equations. Key concept for AC circuit analysis!
I Function
v (t)⇔ V
I Derivative:
dv (t)
dt
⇔ ???
I Integral: ∫
v (t) dt ⇔ ???
Q: Do you remember the concept of leading and lagging taught
last week?
7/8
Phasor
Example 5 — Problem 9.24 of Alexander’s book
Find v(t) in the following integrodifferential equation using the
phasor approach:
(a)
v(t) +
∫
v dt = 10 cos (t)
(b)
v(t) + 2
dv
dt
= 4 sin (2t)
(c)
dv
dt
+ 5v(t) + 4
∫
v dt = 20 sin (4t + 10)
8/8
Phasor
Example 5 — Problem 9.24 of Alexander’s book
Find v(t) in the following integrodifferential equation using the
phasor approach:
(a)
v(t) +
∫
v dt = 10 cos (t)
(b)
v(t) + 2
dv
dt
= 4 sin (2t)
(c)
dv
dt
+ 5v(t) + 4
∫
v dt = 20 sin (4t + 10)
8/8
Phasor
Example 5 — Problem 9.24 of Alexander’s book
Find v(t) in the following integrodifferential equation using the
phasor approach:
(a)
v(t) +
∫
v dt = 10 cos (t)
(b)
v(t) + 2
dv
dt
= 4 sin (2t)
(c)
dv
dt
+ 5v(t) + 4
∫
v dt = 20 sin (4t + 10)
8/8