Short note: Bode plot

A Bode plot is a pair of graphs of a system's frequency response $H(j\omega)$ against $\log_{10}\omega$:

• Magnitude plot: $20\log_{10}|H(j\omega)|$ in dB,
• Phase plot: $\angle H(j\omega)$ in degrees.

Because a product of factors becomes a sum of dB values, the plot is drawn by adding simple straight-line (asymptotic) approximations of each factor.

Rules for sketching:

• A constant $K$ adds $20\log K$ dB.
• Each pole at the origin adds $-20$ dB/decade and $-90^\circ$; each zero at the origin $+20$ dB/dec and $+90^\circ$.
• A simple pole $1/(1 + j\omega/a)$: 0 dB up to the corner frequency $a$, then $-20$ dB/dec; phase goes from $0^\circ$ to $-90^\circ$ ($-45^\circ$ at $a$). At the corner frequency, the true curve is 3 dB below the asymptote.

Example: $H(j\omega) = \dfrac{1}{1 + j\omega RC}$ (RC LPF): flat 0 dB up to $\omega = 1/RC$, then $-20$ dB/dec.

Uses: read bandwidth and filter type, and find gain and phase margins for stability of control systems.