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Interactive simulator

Bode Lead Lag Compensator Simulator

Link magnitude, phase, and pole-zero views to see how zero-pole spacing changes stability margin.

Parameters
Compensator Gain K
dB

Overall compensator gain.

Zero Frequency wz
rad/s

Compensator zero (lead compensation has wz < wp).

Pole Frequency wp
rad/s

Compensator pole (lag compensation has wz > wp).

Plant Bandwidth w0
rad/s

Bandwidth of the target plant G(s)=w0²/(s(s+w0)).

Presets:
Results
—
Maximum phase lead φm
—
Frequency of φm ωm
—
Phase margin (compensated)
—
Phase margin (uncompensated)
—
Gain-crossover frequency ωgc
—
Compensator DC gain
Bode magnitude plot
Bode phase plot
Pole-zero placement
Model and equations

$$C(s)=K\,\frac{1+s/\omega_z}{1+s/\omega_p},\qquad \alpha=\frac{\omega_z}{\omega_p}$$

$$\phi_m=\arcsin\!\frac{1-\alpha}{1+\alpha}\quad\text{at}\quad \omega_m=\sqrt{\omega_z\,\omega_p}$$

Lead compensation (wz<wp) produces the maximum phase lead φm at the geometric mean ωm=√(ωz·ωp); placing it near the crossover frequency improves the phase margin. Lag compensation (wz>wp) increases low-frequency gain and reduces steady-state error, but locally decreases the phase. At ωm, the magnitude is boosted by −10·log₁₀α dB.

How to read it

The magnitude plot shows gain change around crossover.

The phase plot shows the lift between zero and pole.

The pole-zero view shows why close pole-zero placement gives weak compensation.

Learn Bode Lead Lag Compensator by dialogue

🙋
When reading Bode Lead Lag Compensator, where should I look first? Moving Compensator gain K changes both the plots and the result cards.
🎓
Start with Phase contribution, but do not treat the number as the whole answer. Use Bode magnitude plot to confirm the assumed state, then read Bode phase plot for the distribution or trend. The magnitude plot shows gain change around crossover.
🙋
I can see why Compensator gain K changes Phase contribution. How should I judge the influence of Zero frequency wz?
🎓
Move Zero frequency wz in small steps and watch Gain at crossover. That reveals which term is controlling the result. A lead compensator places the zero below the pole to add phase around crossover. Lag compensation raises low-frequency gain but affects bandwidth and response speed. A single operating point is not enough; sweep the realistic scatter range.
🙋
What is Pole-zero placement for? It feels like the ordinary curve already tells the story.
🎓
Pole-zero placement is for finding boundaries where the condition becomes risky or margin collapses quickly. The phase plot shows the lift between zero and pole. In Improving phase margin around PID loops, the important question is often what happens after a small change, not only the nominal value.
🙋
So if Phase contribution is within the target, can I accept the condition?
🎓
Treat this as a first-pass review. It helps with Initial compensator design from Bode plots and Checking bandwidth versus stability margin tradeoff, but final decisions still need standards, measured data, detailed analysis, and vendor limits. The pole-zero view shows why close pole-zero placement gives weak compensation.

Practical use

Improving phase margin around PID loops.

Initial compensator design from Bode plots.

Checking bandwidth versus stability margin tradeoff.

FAQ

Start with Phase contribution and Gain at crossover. Then use Bode magnitude plot to confirm the assumed state and Bode phase plot to read distribution or bias. The magnitude plot shows gain change around crossover
Move Compensator gain K alone, then move Zero frequency wz by a comparable amount and compare the change in Phase contribution. Pole-zero placement shows combinations where margin or performance changes quickly.
Use it for Improving phase margin around PID loops. Instead of trusting a single point, widen the input range and check whether Phase contribution keeps enough margin before moving to detailed analysis.
A lead compensator places the zero below the pole to add phase around crossover. Lag compensation raises low-frequency gain but affects bandwidth and response speed. Final decisions still require standards, measured data, detailed analysis, and vendor limits.

How to Use

  1. Enter compensator gain in dB, typically 0–20 dB for industrial servo loops
  2. Set zero frequency in rad/s, usually 0.5–2 rad/s below crossover for lag action
  3. Set pole frequency in rad/s, positioned 5–10× higher than zero for lead action
  4. Define crossover frequency in rad/s where magnitude should equal 0 dB
  5. Run simulator to display Bode magnitude/phase curves and real-time margin estimates

Worked Example

DC motor speed control with target crossover at wc=8 rad/s. Configure gain=12 dB, zero=1.5 rad/s, pole=15 rad/s. Simulator computes phase contribution ≈ +35° (lead dominant), gain at crossover ≈ 0.2 dB, pole-zero separation = 13.5 rad/s. Phase margin improves from 25° (uncompensated) to 48° (compensated), meeting automotive tier-1 stability requirement of ≥45°.

Practical Notes

  1. Pole-zero separation >10 rad/s ensures clean lead action; spacing <5 rad/s risks lag-only behavior and reduced bandwidth
  2. For hydraulic proportional valve loops (wc≈20 rad/s), use gain offset of +6–8 dB to compensate for inherent attenuation
  3. Phase peak occurs near geometric mean of zero and pole; verify peak occurs within ±1 decade of target crossover
  4. Negative phase contributions signal pole placement too close to zero—increase pole frequency by factor of 2–3