HW8-sol - ECE 493 Univ. of Illinois HW #8 Version 1.27...

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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 + , 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 + 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 . 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 = , 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 ( ) = 1 + , which is a vertical line running to the right of the axis by 1 unit. Our three points are at
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HW8-sol - ECE 493 Univ. of Illinois HW #8 Version 1.27...

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