Bode plots show how a linear system responds to sinusoidal inputs across frequency. This cheat sheet helps students sketch magnitude and phase plots quickly from a transfer function. It is useful in controls, circuits, and signal processing because frequency response reveals stability, bandwidth, resonance, and robustness.
Clear construction rules make it easier to estimate system behavior without relying only on software.
Key Facts
- For a transfer function G(s), evaluate frequency response by substituting s = jω, then plot 20 log10 |G(jω)| in dB and angle G(jω) in degrees versus log10 ω.
- A constant gain K contributes 20 log10 |K| dB to magnitude and contributes 0 degrees phase if K > 0 or plus or minus 180 degrees if K < 0.
- A pole at the origin, 1/s, contributes a magnitude slope of -20 dB/decade and a constant phase of -90 degrees.
- A zero at the origin, s, contributes a magnitude slope of +20 dB/decade and a constant phase of +90 degrees.
- A first-order pole 1/(1 + s/ωc) changes slope by -20 dB/decade after ωc and shifts phase from 0 degrees to -90 degrees across about 0.1ωc to 10ωc.
- A first-order zero 1 + s/ωc changes slope by +20 dB/decade after ωc and shifts phase from 0 degrees to +90 degrees across about 0.1ωc to 10ωc.
- Gain margin is GM = 1/|L(jωpc)|, or GMdB = -20 log10 |L(jωpc)|, where ωpc is the phase crossover frequency with angle L(jωpc) = -180 degrees.
- Phase margin is PM = 180 degrees + angle L(jωgc), where ωgc is the gain crossover frequency with |L(jωgc)| = 1 or 0 dB.
Vocabulary
- Bode Plot
- A pair of frequency-response graphs showing magnitude in decibels and phase in degrees versus logarithmic frequency.
- Corner Frequency
- The frequency ωc where a pole or zero begins to significantly change the magnitude slope and phase.
- Gain Crossover Frequency
- The frequency ωgc where the open-loop magnitude equals 1, which is 0 dB.
- Phase Crossover Frequency
- The frequency ωpc where the open-loop phase equals -180 degrees.
- Phase Margin
- The extra phase lag needed at the gain crossover frequency to reach -180 degrees.
- Gain Margin
- The factor by which open-loop gain can increase before the system reaches the stability boundary at the phase crossover frequency.
Common Mistakes to Avoid
- Using linear frequency spacing on a Bode plot is wrong because Bode construction assumes a logarithmic frequency axis with equal spacing per decade.
- Forgetting to convert magnitude to decibels is wrong because Bode magnitude uses 20 log10 |G(jω)|, not the raw amplitude ratio.
- Adding slopes before the corner frequency for a first-order pole or zero is wrong because the asymptotic slope change starts at the break frequency ωc.
- Computing phase margin at the phase crossover frequency is wrong because phase margin must be measured at the gain crossover frequency where magnitude is 0 dB.
- Reporting gain margin from the gain crossover frequency is wrong because gain margin must be measured at the phase crossover frequency where phase is -180 degrees.
Practice Questions
- 1 For G(s) = 10/(1 + s/100), find the low-frequency magnitude in dB and the high-frequency magnitude slope after 100 rad/s.
- 2 For L(s) = 50/(s(1 + s/10)), list the slope of the magnitude plot before 10 rad/s and after 10 rad/s.
- 3 An open-loop system has |L(jωpc)| = 0.25 at the phase crossover frequency. Find the gain margin as a ratio and in dB.
- 4 If two systems have the same gain crossover frequency but one has a larger phase margin, explain which system is generally more robust to added delay and why.