download searchable PDF of Circuit Design book - IEEE Global ...

Problems
6S
magnitude. The program provides the residues corresponding to the order in
which the roots were furnished and in descending root multiplicity. The
residue output order can be understood best by running the example appearing
in Figure 3.9 and comparing the results. A Laplace transformation to the
time domain was illustrated as one of many important applications of the
partial fraction expansion program.
Problems
3.1. Differentiate
using the calculus formula
Differentiate
f(z)=(z-I)(z-2)2
d(uvw)=vwdu+uwdv+ uvdw.
fez) =z'-5z 2 +8z-4
and evaluate f'(2). Note why the derivative of polynomials with multiple
roots is zero at the root.
3.2. Given the polynomial
fez) =z' = (x + jY)' = (x' - 3xy2) + j(3x 2 y - y') = u + jv.
(a)
(b)
(c)
Find derivative df/ dz by differentiation.
Find derivative df/dz using the Cauchy-Riemann identity.
Show, using fez), that
au = av and av = _ au
ax ay ax ay'
3.3. Given the complex polynomial
fez) = 5+3z+2z 2 +4z'-2z 4 =u+jv
for z=x+jy, use the Mitrovic method to find numerically the values of
u, v, and the following derivatives when z = I + j3:
au av au d av
ax' ax' ay' an ay .
3.4. A root finder has located root Z. = -!- jff/2 of the polynomial equation
f(z) = 2z'+9z 4 + 13z'+z2-13z+4=0.