Page 444 - Analog and Digital Filter Design

Answers 441

4.2 The impedance of a capacitor is inversely proportional to its value.
Thus for a 600-ohm load, denormalization requires the value to be
reduced: 0.55321600 = 922pF. To scale for a frequency of 100 kHz. we
want to find a capacitor value that has the impedance of the 922pF
capacitor at 1 radls. which is 1 radlsl922pF = 1085 Q. The impedance
of a capacitor reduces with frequency, and, in order to maintain the
1085R impedance as the cutoff frequency is increased, the capacitance
value must be reduced in proportion. The denormalized capacitor is
thus 0.55321(2nFR) = 922pF1628,318 = 1467pF or 1.467nF.

4.3 They move away from the origin along a line that gasses through the
original pole position,
4.4 C1 = llo= 110.7071 = 1.4142F. C2 = GI(d + d) 0.7071/(0.5 + 8.5) =
=
0.7071F.
4.5 To denormalize C1 and C2, divide the values found in Exercise 4.4 by
impedance and frequency. C1 = 1.414212nFR = 1.4142/62,831,853 =
22.508nF. C2 = 0.707112nFR = 0.7071162,831,853 = 11.254rzF.

Chapter 5

5.1 An inductor value has to be increased in proportion to the load value,
so to denormalize for impedance we get 0.68348 x 100 = 68.3411. To
scale for cutoff frequency of 12 kHz we have to divide by 21tF radls,
where F is the cutoff frequency. In this case 2nF = 75,398 radls.
L = 0.6834 x RI27cF. This gives a denormalized value of 906pH.
5.2 The impedance of a capacitor is inversely proportional to its value. For
a 75-ohm load, denormalization requires the value to be reduced:
0.7490175 = 9.9867mF. To scale for a frequency of 10 kHz, the
capacitance value must be reduced in proportion to frequency. The
denornialized capacitor is 0.74901(2nFR) = 9.9867mF162,832 =
1 58.94t~F.
5.3 R1 = qLp, = 0.6205Q. R2 = (d,Lp) + Ui,LpJ/qLp) = (0.6205' +
0.9075')/0.6205 = 1.208510.6205 = 1.9477Q.

5.4 If C1 = C2 = lrzF and cutoff frequency = 15 kHz, R1 = 0.620512nFC =
0.6205l94.2478 x = 6.584kR. R2 = 1.941712nFC = 20.666kR.

Chapter 6

6.1 Denormalize the lowpass design to have a cutoff frequency equal ;o
the required bandwidth. Resonate each series arm with a series
connected capacitor. Resonate each shunt capacitor arm with a
parallel inductor. For both series and parallel tuned circuits, the