0195136047.pdf

160 TIME-DEPENDENT CIRCUIT ANALYSIS

H 2 (ω), dB

0

− 90

− 45

45

90

− 20

20

0

40

θ 2 (ω), deg

Slope = −20 dB/decade or

−6 dB/octave

Slope = − 45 ° /decade or

−13.5° /octave

0.1 ω 0 1 ω 0 10 ω 0

ω, rad/s

100 ω 0

Asymptotic

Exact Asymptotic

Exact

ω

ω

Figure 3.4.7Asymptotic Bode plot forH ̄ 2 (j ω)=

1 /( 1 +jω/ω 0 ).

Case 1: Zero or Pole at Origin

lims→ 0 H(s)=Ksn, wherencan be any positive or negative integer. The magnitude

characteristic for this factor has a slope of 20ndB/decade and passes throughKdB at

ω=1, while the component angle characteristic has a constant value ofn×90°.

Case 2: Some Zeros or Poles Occur as Complex Conjugate Pairs

Let

H(s)=

(

1 +

s

α+jβ

)(

1 +

s

α−jβ

)

= 1 + 2

α

α^2 +β^2

s+

s^2

α^2 +β^2

With a break frequency of

√

α^2 +β^2 , the magnitude is characterized by a slope of 40

dB/decade forω≥

√

α^2 +β^2 , and the angle characteristic shows a slope of 90°/decade in

the range of 0.1

√

α^2 +β^2 ≤ω≤ 10

√

α^2 +β^2. The asymptotic Bode plot for this case is

shown in Figure 3.4.8.

Note that the angle and magnitude characteristics will have the opposite sign for the case of

a complex conjugate pair of poles. It should also be pointed out that the straight-line quadratic

representations are not generally good approximations (particularly in the one-decade region on

either side of the break frequency), and a few points may be calculated to obtain the correct curves

in the questionable frequency range.

For the seriesRLCcircuit, the input admittance is given by