Linear Physical Systems

Numerator Polynomial, N(s):
Numerator Polynomial, D(s):
Multiplicative Constant, C:

We start with the transfer function (note: the process here follows that on the second page of the file BodeRules.pdf):

H (s) = C \frac{s^{2} + 5 s + 6}{s^{5} + 4 s^{4} + 7 s^{3} + 6 s^{2} + 2 s}

$H(s)=C\frac{s^2 + 5s + 6}{s^5 + 4s^4 + 7s^3 + 6s^2 + 2s}$

We rewrite it by factoring into real poles & zeros, complex poles & zeros and poles & zeros at the origin.

H (s) = C (\frac{1}{s}) \frac{(s + ω_{z 1}) (s + ω_{z 2})}{(s + ω_{p 1})^{2} (s^{2} + 2 ζ_{p 1} ω_{0 p 1} s + ω_{0 p 1}^{2})}

$H(s) = C\left( \frac{1}{s} \right)\frac{(s + \omega_{z1})(s + \omega_{z2})}{(s + \omega_{p1})^2(s^2 + 2\zeta_{p1}\omega_{0p1}s + \omega_{0p1}^2)}$

With:

Constant: C=10
A real pole at s=-1.00, of muliplicity 2.
This is the $(s + \omega_{p1})^2$ term in the denominator, with $\omega_{p1}$ =1.
A real zero at s=-2.00.
This is the $(s + \omega_{z1})$ term in the numerator, with $\omega_{z1}$ =2.
A real zero at s=-3.00.
This is the $(s + \omega_{z2})$ term in the numerator, with $\omega_{z2}$ =3.
Complex poles, at s = -1.00 ± 1.00j.
This is the $(s^2 + 2\zeta_{p1}\omega_{0p1}s + \omega_{0p1}^2)$ term in the denominator, with $\omega_{0p1}$ =1.41, $\zeta_{p1}$ =0.707.
A pole at the origin.

Next we write all the poles and zeros is our standard form.

H (s) = C \frac{ω_{z 1} ω_{z 2}}{ω_{p 1}^{2} ω_{0 p 1}^{2}} (\frac{1}{s}) \frac{(1 + s / ω_{z 1}) (1 + s / ω_{z 2})}{(1 + s / ω_{p 1})^{2} (1 + 2 ζ_{p 1} (s / ω_{0 p 1}) + (s / ω_{0 p 1}^{2}))}

$H(s) = C\frac{\omega_{z1}\omega_{z2}}{\omega_{p1}^2\omega_{0p1}^2}\left( \frac{1}{s} \right)\frac{(1 + s/\omega_{z1})(1 + s/\omega_{z2})}{(1 + s/\omega_{p1})^2(1 + 2\zeta_{p1}(s/\omega_{0p1}) + (s/\omega_{0p1}^2))}$

Rewrite the constant:

K = C \frac{ω_{z 1} ω_{z 2}}{ω_{p 1}^{2} ω_{0 p 1}^{2}} = 30.0 = 29.6 d B

$K = C\frac{\omega_{z1}\omega_{z2}}{\omega_{p1}^2\omega_{0p1}^2} = 30.0 = 29.6dB$

H (s) = K (\frac{1}{s}) \frac{(1 + s / ω_{z 1}) (1 + s / ω_{z 2})}{(1 + s / ω_{p 1})^{2} (1 + 2 ζ_{p 1} (s / ω_{0 p 1}) + (s / ω_{0 p 1}^{2}))}

$H(s) = K\left( \frac{1}{s} \right)\frac{(1 + s/\omega_{z1})(1 + s/\omega_{z2})}{(1 + s/\omega_{p1})^2(1 + 2\zeta_{p1}(s/\omega_{0p1}) + (s/\omega_{0p1}^2))}$

Now the transfer function is in the form we need to apply our rules to draw the Bode plot.

Putting It All Together

Magnitude

Phase

Calculate Sinusoid