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

The given Bode plot indicates slope changes at different frequencies. The transfer function of the system is given by:

\( G(s) = \frac{k(s+z)^a}{s^b(s+p)^c} \)

To find the sum \(a + b + c\), follow these steps:

1. The initial slope is \(-20 \, \text{dB/decade}\), indicating either a single zero (\(s+z\)) or a single pole (\(s^b\)). This suggests \(a - b = -1\).
2. The middle section is flat, indicating the zero and pole effects cancel each other out around this frequency. This is consistent with a combination of a zero \(z\) being canceled by poles or vice versa.
3. The final slope is \(-40 \, \text{dB/decade}\), indicating an increase of two slopes (e.g., two more poles). This provides the equation \(a - (b + c) = -2\).

We have two equations:

• \(a - b = -1\)
• \(a - b - c = -2\)

Solving these:

• From \(a - b = -1\), rearrange to get \(a = b - 1\).
• Substitute in \(a - b - c = -2\) to get \((b - 1) - b - c = -2\).
• This simplifies to \(-1 - c = -2\).
• Solve for \(c\): \(c = 1\).
• Substitute \(c = 1\) back into the equation for \(a\): \(a = b - 1\).

Thus, with \(b = 2\), \(a = 1\), so:

\(a + b + c = 1 + 2 + 1 = 4\).

This value of 4 fits the expected range \(4,4\).