Nyquist Stability Interactive Calculator

If you want a closed-loop system that doesn’t break into oscillation, you have to know your margins—there’s no substitute for doing the math the right way. This interactive Nyquist calculator gives you direct gain and phase margin, encirclements, and the critical gain—all from open-loop frequency data. That’s what matters if you’re designing for fields like aerospace, process control, or robotics. What works on paper might take off into instability at some corner case if you’re not careful. Below you’ll find the working Nyquist criterion, a sample calculation, core stability margin theory and practical issues, and a FAQ with specific use cases.

What is Nyquist Stability?

Nyquist stability uses the way your open-loop frequency response wraps around a point (-1, 0) in the complex plane to tell you if your closed-loop system will stay stable. If the plot takes the wrong path, your system goes unstable—you get runaway oscillation or worse.

Simple Explanation

Imagine steering a car and trying to keep it straight; if every correction is a bit too much, you eventually swerve out of control. The Nyquist criterion is a way to size up just how much feedback (gain and phase) you can use before your system actually starts to feed on its own errors and gets worse. It checks that your system fixes problems instead of causing bigger ones.

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System Diagram

Nyquist Stability Calculator

How to Use This Calculator

1. Pick your calculation mode—gain & phase margin, encirclements, critical gain, etc.—from the dropdown.
2. Plug in the frequency and magnitude values the mode needs (fields update automatically).
3. Use Try Example if you want to see how it works before using your own data.
4. Hit Calculate for results.

Equations & Formulas

Use the formula below to calculate the number of closed-loop right-half-plane poles and determine system stability.

Use the formula below to calculate phase margin from the open-loop phase at the gain crossover frequency.

Use the formula below to calculate gain margin in decibels from the open-loop magnitude at the phase crossover frequency.

Use the formula below to calculate the critical gain — the maximum proportional gain before the system becomes unstable.

Use the formula below to calculate gain margin in decibels from a linear gain ratio.

Simple Example

Using the Gain & Phase Margin mode:

• Gain crossover frequency ω_gc = 10 rad/s
• Phase crossover frequency ω_pc = 5 rad/s
• Phase at ω_gc = -135°
• Gain at ω_pc = -10 dB

Result: PM = -135° + 180° = 45° — GM = -(-10) = 10 dB — System is STABLE with excellent margins.

Theory & Engineering Applications

Nyquist’s method—first put forward in 1932—is a practical way to check closed-loop stability for any linear time-invariant system. Unlike algebraic-only approaches like Routh-Hurwitz, this method makes use of frequency response mapped in the complex plane to judge stability directly. This visual method isn’t just for theory; it shows you the effect of each design tweak at a glance, including how changes in the open-loop system echo into the closed loop.

Mathematical Foundation and Cauchy's Argument Principle

Nyquist stability traces back to Cauchy’s argument principle. With a closed-loop equation 1 + G(s)H(s) = 0, stability comes down to where the roots fall in the s-plane. Cauchy’s principle relates how a mapped contour wraps around the origin (in the F-plane) to the zeros and poles it encircles. The Nyquist contour itself covers the full imaginary axis plus a big arc at infinity in the right half-plane—so in effect, it includes every point that matters for possible instability. When you look at this through G(s)H(s), those encirclements of the origin map over to encirclements of the -1 point in the G(s)H(s) plane. In practice: Z = N + P tells you if your system is headed for trouble or not.

Stability Margins: Phase and Gain

The Nyquist criterion tells you if you’re stable, but as an engineer you need to know “how stable?” That’s where phase and gain margin come in. They’re the real gauges for how much wiggle room you have before things go off the rails.

Phase margin is how much extra phase lag at the gain crossover frequency (where |G(jω)H(jω)| = 1) you could take on before you cross over into instability. Calculate it: PM = ∠G(jω_gc)H(jω_gc) + 180°. If you’re running a phase margin of 45-60°, you’re usually in good shape—expect some overshoot but nothing wild. Below about 30°, your system will feel loose and oscillatory. There’s a rough thumb rule: for second-order systems, phase margin in degrees is about 100 times the damping ratio if you’re between 0.3 and 0.7 for ζ. It’s not exact, but helpful on napkin sketches.

Gain margin says how much you can crank loop gain up at the critical frequency before you hit instability. It’s measured at the phase crossover—where loop phase hits -180°. You get GM = -20log₁₀|G(jω_pc)H(jω_pc)| in dB. If you’re positive (say 6 dB or higher), you have some room for parts drift and real-world surprises. More demanding environments (for example, aerospace) usually work at 10 dB plus, because you want to sleep at night when the plant gain changes due to weather, wear, or switching system configuration.

Practical Limitations and Non-Minimum Phase Systems

Standard Nyquist analysis assumes all open-loop zeros are in the left half-plane (minimum-phase), but that doesn’t always match reality. When you’ve got right-half-plane zeros (non-minimum phase), phase can drop with frequency even though the system isn’t unstable. This shows up, for example, in some aircraft, chemical plants, and digital or delayed feedback systems. You can’t get the same bandwidth without losing margin; phase drops off quickly and you have to slow things down to stay stable.

Time delays are a real limitation, not just a theory footnote. Pure time delay (e^(-sτ)) adds phase lag that grows with frequency without reducing gain. The Nyquist plot draws a tight inward spiral toward zero—so as frequency increases, phase margin evaporates fast. Control bandwidth becomes limited by the delay; as a rule of thumb, don’t let ω_gc exceed about π/(2τ), or you’ll be chasing phantom instabilities. This is the reason remote or networked control—like telerobotics over the internet—feels sluggish: delays eat your phase margin fast, and sometimes there’s just no way around it.

Worked Example: DC Motor Position Control System

Let’s look at a CNC-grade DC motor positioner, holding to ±5 micrometer. Suppose the tested plant has this open-loop transfer (with integral gain K):

G(s)H(s) = K / [s(s + 8.3)(s + 47.2)]

We’re after the maximum gain K for stability, pick K for PM = 50°, and check margins. Start with critical gain using Nyquist.

Step 1: Find phase crossover frequency ω_pc

At phase crossover, open-loop phase is -180°. For three poles:

∠G(jω_pc) = -90° - arctan(ω_pc/8.3) - arctan(ω_pc/47.2) = -180°

So, arctan(ω_pc/8.3) + arctan(ω_pc/47.2) = 90°. Numeric solution gives ω_pc = 19.87 rad/s.

Step 2: Calculate magnitude at phase crossover

At ω = 19.87 rad/s, K = 1:

|G(j19.87)| = 1 / [19.87 × √(19.87² + 8.3²) × √(19.87² + 47.2²)] = 1 / 21,920 = 4.564 × 10⁻⁵

Step 3: Find critical gain

Set K_cr × |G(j19.87)| = 1. So, K_cr = 1 / (4.564 × 10⁻⁵) = 21,920

At K = 21,920, you’re on the edge—oscillation at ω_pc = 19.87 rad/s (3.16 Hz).

Step 4: Pick gain for PM = 50°

Find ω_gc where phase = -180° + 50° = -130°:

-90° - arctan(ω_gc/8.3) - arctan(ω_gc/47.2) = -130°, so arctan sums to 40°.

Solve numerically: ω_gc = 9.24 rad/s.

Plug in at K = 1: |G(j9.24)| = 1.876 × 10⁻⁴.

Set K × 1.876 × 10⁻⁴ = 1, so K = 5,330.

Step 5: Check gain margin at K = 5,330

At phase crossover ω_pc = 19.87 rad/s: |G(j19.87)H(j19.87)| = 5,330 × 4.564 × 10⁻⁵ = 0.243

GM = -20log₁₀(0.243) = 12.3 dB (gain margin as a linear ratio: 4.11).

This means you can nearly quadruple your gain before risking instability—a healthy margin for high-precision, variable-load work.

Step 6: Engineering notes

With K = 5,330, you get PM = 50° and GM = 12.3 dB—better than typical standards. The gain crossover (ω_gc = 9.24 rad/s ≈ 1.47 Hz) caps your bandwidth; you can expect roughly 0.68-second settling (settling ~ 4/ω_gc). Overshoot stays around 16%. The extra gain margin keeps you out of trouble if friction, inertia, or drive gain changes, so you stay reliably in spec year after year.

Industrial Applications Across Engineering Disciplines

Flight control in aerospace depends on proper Nyquist margin—not just as a box-check, but live in operation. For example, the Boeing 787 checks these margins nonstop, and in test work, Nyquist ensures stability against things like flutter. Relaxed-stability military jets might demand margins up to 60° or more for all possible combinations—flight regime, structural shifts, etc.

Long-delay process control (think distillation) faces a tough Nyquist limit—delays push the critical point dangerously close, so bandwidth gets capped well below what you’d want for quick corrections. That’s why process industries accept slow, overdamped loops that would be unacceptable in, say, robotic arms.

Robotics needs tuning to the particular job. If you’re doing force or contact work, you might run with very high phase margin (up to 80°) for safe, deadbeat behavior. High-speed pick-and-place, though, might limp by with just 35° margin to get more bandwidth—if you accept a little overshoot. Nyquist makes these tradeoffs clear up front, rather than discovering them late in commissioning.

Motor drives and inverters are a good place to see why stability margin matters. Sometimes grid impedance changes unpredictably. If you design to 8–10 dB gain margin and 45° phase margin at minimum for all expected loads, you minimize the chance of new instability as grid stiffness varies. For more control tools, you can use the full calculator collection to get numbers for mechanical and electrical details as well.

Practical Applications

Frequently Asked Questions

📹 Video Walkthrough — How to Use This Calculator

📹 Video Walkthrough — How to Use This Calculator