Bode Form

Consider a transfer function $H(s)$$H(s)$ with real poles and zeros:

This can be written in Bode form:

where $K_0$$K_0$ is the DC gain,

Bode Plots

Constant

Let $H(s)=K_0$$H(s) = K_0$. Then,

Bode Plot for ${10}^2$$\large 10^2$:

Bode Plot for $\frac{1}{10}$$\large \frac{1}{10}$:

Zeros & Poles At Origin

Let $H(s)=s^q$$H(s) = s^q$. Then,

Bode Plot for $s^2$$\large s^2$:

Bode Plot for $s^{-1}$$\large s^{-1}$:

Real Zeros & Poles

Let $H(s)=(\frac{s}{z}+1{)}^{\pm 1}$$H(s) = (\frac{s}{z}+1)^{\pm1}$. Then,

For convenience, we draw the magnitude as a linear transition of slope $\pm 20$$\pm20$ dB/decade starting at $\omega =z$$\omega=z$.

For convenience, we draw the phase as a linear transition starting at $\omega =0.1z$$\omega=0.1z$ and ending at $\omega =10z$$\omega=10z$.

Bode Plot for $(\frac{s}{z}+1)$$\large (\frac{s}{z}+1)$:

Bode Plot for $(\frac{s}{z}+1{)}^{-1}$$\large (\frac{s}{z} + 1)^{-1}$:

Complex Zeros & Poles

Let $H(s)=((\frac{s}{{\omega }_n}{)}^2+2\zeta (\frac{s}{{\omega }_n})+1{)}^{\pm 1}$$H(s) = ((\frac{s}{\omega_n})^2+2\zeta(\frac{s}{\omega_n})+1)^{\pm1}$. Then,

For convenience, we draw the magnitude as a linear transition of slope $\pm 40$$\pm40$ dB/decade starting at $\omega ={\omega }_n$$\omega=\omega_n$.

Bode Plot for $((\frac{s}{{\omega }_n}{)}^2+2\zeta (\frac{s}{{\omega }_n})+1)$$\large ((\frac{s}{\omega_n})^2+2\zeta(\frac{s}{\omega_n})+1)$:

Bode Plot for $((\frac{s}{{\omega }_n}{)}^2+2\zeta (\frac{s}{{\omega }_n})+1{)}^{-1}$$\large ((\frac{s}{\omega_n})^2+2\zeta(\frac{s}{\omega_n})+1)^{-1}$: