Engineering 6, Winter 2010
Midterm Solutions
Regrade requests must be submitted in writing to Dr. Lagerstrom no later than the lecture section
on Wednesday, February 24. Regrades will only be considered in cases where it looks like the
grader missed something. That is, requests along the lines of "I think I deserve more points" will
not get very far, because the the grading scale on each problem was applied consistently for all
students. Note also
: Your exam may have been photocopied before it was returned, so please don't
risk your engineering career here at UCD by changing an answer and submitting it for a regrade.
Exam Version A (tan and yellow copies)
1. (20 points total)
(a) (6 points) Give the numerical value of y after each of the following Matlab expressions is executed (if
square roots are involved, it’s okay to leave it in square root form):
(i)
y = exp(j*pi)
Answer: 1
(ii)
y = exp(–j*pi/2)
Answer: j (or –i)
(iii)
y = exp(j*pi/4)
Answer:
j
2
2
2
2
+
, or
j
2
1
2
1
+
(b) (6 points) Write the three forms of a sinusoidal signal x(t): the cosine form, the single phasor form, and
the dual phasor form. Define the variables you use, i.e., say what each represents. You can use the same
variables in each of the three forms. (Write the expressions in regular math notation, not in Matlab code.)
cosine form:
)
cos(
)
(
ϕ
ω
+
=
t
A
t
x
single phasor form:
( )
t
j
j
e
Ae
t
x
Re
)
(
=
dual phasor form:
t
j
j
t
j
j
e
e
A
e
e
A
t
x


+
=
2
2
)
(
where A = amplitude,
ω
= angular frequency, t = time, and
φ
= phase angle.
(c) (8 points) Explain how the the single phasor form and the dual phasor form of x(t) give the same result.
To do so, draw a sketch of each in the complex plane (two sketches, one for each) and explain via the
sketch (and in words) how the value of x(t) varies as t varies.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentIn the single phasor form, the
( )
t
j
j
e
Ae
ω
ϕ
part is a complex number that moves counterclockwise in a
circle of radius A in the complex plane as t varies (all the other quantities are constants). The real part of
this "rotating phasor" at any given time t is the value of x(t) at that time t.
In the dual phasor form, there are two phasorlike terms. The first one is a complex number that moves
counterclockwise in a circle of radius A/2 in the complex plane as t varies. The second term is a complex
number that moves clockwise in a circle of radius A/2. At any given time t, the imaginary part of the first
complex number is equal and opposite to the imaginary part of the second complex number. So when the
two terms are added together, the imaginary parts cancel and one is left with a real value, which is the
value of x(t) at that time t (and is the same value you would get using the single phasor version).
For further explanation and sketches, see the "Phasors 1" and "Phasors 2" video clips. (In the first case, the
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Largerstrom
 Complex number

Click to edit the document details