› Bode Lead Lag Compensator Simulator All tools
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: wz < wp).

Pole frequency wp
rad/s

Compensator pole (lag: wz > wp).

Plant bandwidth w0
rad/s

Target plant G(s)=w0²/(s(s+w0)) bandwidth.

Presets:
Live numbers
—
Max phase lead φm
—
Frequency of φm (ωm)
—
Phase margin (after)
—
Phase margin (before)
—
Crossover ωgc
—
Compensator DC gain
Bode magnitude — original(grey) vs compensated(blue)
Bode phase — phase bump & margin
Pole-zero placement (s-plane)
Model and equations

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

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

A lead compensator (wz<wp) produces its maximum phase lead φm at the geometric mean ωm=√(ωz·ωp); placing it near the crossover frequency improves phase margin. Lag (wz>wp) boosts low-frequency gain to cut steady-state error but locally lowers phase. Magnitude is lifted by −10·log₁₀α dB at ωm.

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 (gainVal) in dB, typically 0–20 dB for industrial servo loops
  2. Set zero frequency (zeroVal) in rad/s, usually 0.5–2 rad/s below crossover for lag action
  3. Set pole frequency (poleVal) in rad/s, positioned 5–10× higher than zero for lead action
  4. Define crossover frequency (wcVal) 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 ≈ +51° (lead dominant), gain at crossover ≈ 25.6 dB (gain K = 12 dB included), pole-zero separation (ωp/ωz) = 10×. 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

Related Tools

Bode Plot Real-time Bode plot generator for transfer functions. Automatically calculates gain margin,... Control Bode Generate Bode plots from a transfer function in real time. Visualize gain margin, phase mar... Loop Shaping Simulator — Bode Plot and Stability Margins Loop Shaping Simulator — Bode Plot and Stability Margins focuses on control response, stabi... Adaptive Control MRAC Simple Simulator Adaptive Control MRAC Simple Simulator focuses on control response, stability margin, and t... Attitude Control Thruster Sizing Simulator Attitude Control Thruster Sizing Simulator compares how control response, stability margin,... Disturbance Observer (DOB) Simulator — DOB+PI vs. PI Alone Disturbance Observer (DOB) Simulator — DOB+PI vs. PI Alone focuses on control response, sta...