Question

The asymptotic magnitude Bode plot of a minimum phase system is shown in the figure. The transfer function of the system is $(s) = \frac{k(s+z)^a}{s^b(s+p)^c}$, where k, z, p, a, b and c are positive constants. The value of $(a+b+c)$ is ___________ (rounded off to the nearest integer). 

 

Hint:

In a Bode magnitude plot: - A pole at the origin ($1/s^b$) causes an initial slope of $-20b$ dB/decade. - A simple pole ($1/(s+p)$) causes the slope to decrease by 20 dB/decade at $\omega=p$. - A simple zero ($(s+z)$) causes the slope to increase by 20 dB/decade at $\omega=z$. The total slope at any frequency is the sum of the contributions from all poles and zeros at lower frequencies.

Answer 1

Correct Answer: 4

We can determine the values of a, b, and c by analyzing the changes in the slope of the asymptotic Bode plot.
1. Initial Slope (for $\omega \to 0$): The plot starts with a slope of -20 dB/decade. An initial slope of $-20 \times b$ dB/decade is caused by a term $s^b$ in the denominator (b poles at the origin). $-20b = -20 \implies b=1$.
2. Change in slope at $\omega_1$: At the corner frequency $\omega_1$, the slope changes from -20 dB/decade to 0 dB/decade. Change in slope = (New Slope) - (Old Slope) = $0 - (-20) = +20$ dB/decade. A positive change in slope is caused by a zero. A zero term $(s+z)^a$ contributes $+20 \times a$ dB/decade to the slope after its corner frequency. $+20a = +20 \implies a=1$.
3. Change in slope at $\omega_2$: At the corner frequency $\omega_2$, the slope changes from 0 dB/decade to -40 dB/decade. Change in slope = (New Slope) - (Old Slope) = $-40 - 0 = -40$ dB/decade. A negative change in slope is caused by a pole. A pole term $(s+p)^c$ contributes $-20 \times c$ dB/decade to the slope after its corner frequency. $-20c = -40 \implies c=2$.
Now we have the values: $a=1, b=1, c=2$.
The value of the expression $(a+b+c)$ is: $a+b+c = 1 + 1 + 2 = 4$.
The value is 4, which is an integer.