# The Asymptotic Bode Diagram for Non-Minimum Phase Poles and Zeros

## A Real Pole with Negative ω0

Elsewhere we have discussed how to make Bode plots for a real pole. You should be familiar with that analysis. The discussion there assumed that the value of ω0 was positive; here we discuss the case if ω0 is negative. We start with

$$H(s)=\frac{1}{1+\frac{s}{\omega_0}}$$

#### Magnitude

If you carefully examine the analysis (here) of the "Magnitude" plot you'll see that the only time ω0 is used, it is squared (e.g., $\left( \frac{\omega}{\omega_0} \right)^2$). Therefore, the magnitude plot does not depend on the sign of ω0, only its absolute value, so we don't need to change anything to accomodate a negative value of ω0.

#### Phase

The phase however does change. The phase of a single real pole is given by is given by

$$\angle H\left( {j\omega } \right) = \angle \left( {{1 \over {1 + j{\omega \over {{\omega _0}}}}}} \right) = - \angle \left( {1 + j{\omega \over {{\omega _0}}}} \right) = - \arctan \left( {{\omega \over {{\omega _0}}}} \right)$$

Let us again consider three cases for the value of the frequency, but we assume ω0 is negative:

Case 1) ω<<ω0.  This is the low frequency case with ω/ω0→0, and doesn't depend on the sign of ω0.  At these frequencies We can write an approximation for the phase of the transfer function

$$\angle H\left( {j\omega } \right) \approx -\arctan \left( 0 \right) = 0^\circ = 0\;rad$$

Case 2) ω>>ω0.  Here we will consider the cases of positive and negative ω0 side by side.

ω0 > 0    (the minimum phase case, discussed previously)

This is the high frequency case with ω/ω0 → +∞.  We can write an approximation for the phase of the transfer function

$$\angle H\left( {j\omega } \right) \approx - \arctan \left( \infty \right) = - 90^\circ$$

The high frequency approximation is at shown in green on the diagram below.  It is a horizontal line at -90°.

ω0 < 0    (the non-minimum phase case)

This is the high frequency case with ω/ω0 → -∞.  We can write an approximation for the phase of the transfer function

$$\angle H\left( {j\omega } \right) \approx - \arctan \left(- \infty \right) = + 90^\circ$$

The high frequency approximation is at shown in green on the diagram below.  It is a horizontal line at +90°.

Case 3) ω=ω0. Again consider the cases of positive and negative ω0 side by side.

ω0 > 0    (the minimum phase case, discussed previously)

$$\angle H\left( {j\omega } \right) = - \arctan \left( 1 \right) = - 45^\circ$$

ω0 < 0    (the non-minimum phase case)

$$\angle H\left( {j\omega } \right) = - \arctan \left( -1 \right) =+ 45^\circ$$

From the above discussion you can see that the only effect of the pole having a negative value of ω0 is that the phase is inverted (it increases from 0 → +90° as ω increases from 0 → ∞. The images below show the Bode plots for

$$H_1(s)=\frac{1}{1+\frac{s}{10}}, \quad\quad H_2(s)=\frac{1}{1-\frac{s}{10}}$$

H1(s) has a positive ω00=+10, so H1(s)=1/(1+s/10)) and H2(s) has a negative ω00=-10, so H2(s)=1/(1-s/10)). The pole of H1(s) is at s=-10 (a negative real part, the left half of the s-plane; a minimum phase zero) and the pole of H2(s) is at s=+10 (a positive real part, the right half of the s-plane; a non-minimum phase zero)
H1(s) is plotted as a solid blue line, and H2(s) as a dotted pink line. Note that the magnitudes are identical, but the phases are opposites.

The same conclusion holds for first order poles and second order poles and zeros (see below).

## A Real Zero with Negative ω0

The images below show the Bode plots for two functions, one with a positive ω00=+10) and one with a negative ω00=-10). The zero of H1(s) is at s=-10 (a negative real part, the left half of the s-plane; a minimum phase pole) and the pole of H2(s) is at s=+10 (a positive real part, the right half of the s-plane; a non-minimum phase zero).

$$H_1(s)=1+\frac{s}{10}, \quad\quad H_2(s)=1-\frac{s}{10}$$

H1(s) is plotted as a solid blue line, and H2(s) as a dotted pink line. Note that the magnitudes are identical, but the phases are opposites. Recall that 360° is equivalent to 0° so you can think of the plot for the angle of H2(s) as starting at 0° and dropping by 90° (though the plot shows it as starting at 360°).

## A Second Order Pole with Negative ζ

The images below show the Bode plots for (note the sign of the middle term in the numerator is different)

$$H_1(s)=\frac{1}{1+0.1\cdot\frac{s}{10}+\left(\frac{s}{10}\right)^2}, \quad\quad H_2(s)=\frac{1}{1-0.1\cdot\frac{s}{10}+\left(\frac{s}{10}\right)^2}$$

The poles of H1(s) are at s=-0.5±j9.987 (a negative real part, the left half of the s-plane; a minimum phase pole) and the pole of H2(s) is at s=+0.5±j9.987 (a positive real part, the right half of the s-plane; a non-minimum phase pole). H1(s) is plotted as a solid blue line, and H2(s) as a dotted pink line. Note that, again, the magnitudes are identical, but the phases are opposites.

## A Second Order Zero with Negative ζ

The images below show the Bode plots for (note the sign of the middle term in the numerator is different)

$$H_1(s)=1+0.1\cdot\frac{s}{10}+\left(\frac{s}{10}\right)^2, \quad\quad H_2(s)=1-0.1\cdot\frac{s}{10}+\left(\frac{s}{10}\right)^2$$

The zeros of H1(s) are at s=-0.5±j9.987 (a negative real part, the left half of the s-plane; a minimum phase zero) and the pole of H2(s) is at s=+0.5±j 9.987(a positive real part, the right half of the s-plane; a non-minimum phase zero). H1(s) is plotted as a solid blue line, and H2(s) as a dotted pink line. Note that the magnitudes are identical, but the phases are opposites.

##### Key Concept: For poles and zeros with positive real parts, phase is inverted

If a 1st order pole has a positive real part (i.e., a nonminimum phase system, pole is in right half of s-plane) at say s=+5, so

$$H(s)=\frac{1}{1 - \frac{s}{5}}$$

the magnitude of the Bode plot is unchanged from the case of a corresponding pole with negative value, at s=-5 (pole is in left half of s-plane)

$$H(s)=\frac{1}{1 + \frac{s}{5}}$$

but the phase of the plot is inverted.

The same rule holds Bode plots for 2nd order (complex conjugate) poles, and for 1st and 2nd order zeros.

References

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