Question

In a Bode magnitude plot, which one of the following slopes would be exhibited at high frequencies by a 4^th order all-pole system –

1. -80 dB/dec
2. -40 dB/dec
3. +40 dB/dec     
4. +80 dB/dec

Answer 1

Correct Answer - Option 1 : -80 dB/dec

Concept:

Bode plot transfer function is represented in standard time constant form as

\(T\left( s \right) = \frac{{k\left( {\frac{s}{{{ω _{{c_1}}}}} + 1} \right) \ldots }}{{\left( {\frac{s}{{{ω _{{c_2}}}}} + 1} \right)\left( {\frac{s}{{{ω _{{c_3}}}}} + 1} \right) \ldots }}\)

ωc1, ωc2, … are corner frequencies.

In a Bode magnitude plot,

• For a pole at the origin, the initial slope is -20 dB/decade
• For a zero at the origin, the initial slope is 20 dB/decade
• The slope of magnitude plot changes at each corner frequency
• The corner frequency associated with poles causes a slope of -20 dB/decade
• The corner frequency associated with poles causes a slope of -20 dB/decade
• The final slope of Bode magnitude plot = (Z – P) × 20 dB/decade

Where Z is the number of zeros and P is the number of poles.

Calculation:

Each pole adds a -20 dB/dec slope and each zero adds a +20 db/dec slope

Hence the overall slope at high frequency is given by:

Slope at high freq. = (-20 × No. of poles + 20 × No. of zeros) dB/dec

for P = 4 and Z = 0

= -80 dB/dec