ECE 493
HW #8 –
Version 1.27 January 31, 2011
Spring 2011
Univ. of Illinois
Due Tu, Jan 25
Prof. Allen
Topic of this homework:
Analytic functions of a complex variable;
Deliverable: Please, show your work.
A blue font indicates a changed from Version 1.0.
For the following the unit step function is de±ned as:
u
(
t
) =
b
1
t >
0
0
t <
0
1.
Complex functions: Domain:
s
≡
σ
+
iω
, Range:
Z
(
s
)
≡
R
(
s
) +
iX
(
s
).
The
Domain
(e.g.,
s
) and
Range
(e.g.,
Z
(
s
)) are described in the text on page 1114.
In engineering terms think of
Z
(
s
) =
X
+
iY
as an
impedance
having a real part (the
resistance)
X
, and an imaginary part (the
reactance
)
X
(
s
).
Make two axes, one for the
s
=
σ
+
iω
plane and a second for the
Z
(
s
) =
X
(
s
)+
iY
(
s
) plane.
Label the two sets of axes: On the left (
s
), the horizontal axis (abscissa) is labeled
σ
, while
the vertical axis (ordinate) is
iω
. For the
Z
(
s
) axis (on the right), the abscissa is labeled
X
and the ordinate axis is
iY
.
Plot the
Range
Z
(
s
) in terms of the speci±ed
Domain
in
s
.
(a) Domain:
s
=
σ
, Range:
Z
(
s
) = 1 +
s
.
Solution: In this speci±c example, the impedance consists of a 1 ohm [Ω] resistor
X
= 1,
in series with a
L
= 1 Henry [H] inductor of impedance
Y
=
s
. Note that
L
= 1 is not
an impedance, it is an inductance, whereas
sL
, is an impedance.
Next indicate the range
s
=
σ
on the
s
axis. This will be a line along the
σ
(
x
) axis.
Label several points on this line, including
A
=

1,
B
= 0 and
C
= 1.
On the second axis plot
Z
(
sigma
) = 1 +
σ
This will also be a line along the
x
axis, but
in this case, an axis that is labeled
R
. Note that
Z
(
σ
) =
R
=
σ
+ 1. Thus in the
Z
plane, our three points are o²set by 1. On the
x
axis of the Domain plot, the mapping
dictates
A
= 0,
B
= 1 and
C
= 2.
(b) Domain:
s
=
iω
, Range:
Z
(
s
) = 1 +
s
.
Solution: The
domain
is a vertical line de±ned
by
σ
= 0. Pick three points as

i,
0
,i
. The range here is
Z
(
iω
) = 1 +
iω
, which is
a vertical line running to the right of the
iω
axis by 1 unit. Our three points are at
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 Spring '11
 JontB.Allen
 Complex number, Polar coordinate system, CR conditions

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