Nyquist Stability Criterion Calculator

Plot the Nyquist diagram of a classic single-integrator, two-real-pole open-loop system and find its closed-loop stability using a Routh-Hurwitz array.

🌀 Nyquist Stability Criterion Calculator
Open-loop DC gain (K)10
—
0.1100
Pole time constant (T1)0.5 s
s
0.015
Pole time constant (T2)0.1 s
s
0.015
Stability verdict (Routh-Hurwitz)
—
Right-half-plane poles
—
Routh array b1
—
Closest approach to −1+j0
—
Step-by-step working

🌀 What is the Nyquist Stability Criterion?

The Nyquist stability criterion is a graphical method for determining whether a closed-loop feedback system is stable, by tracing the open-loop frequency response G(jw) as a curve in the complex plane and counting how many times that curve encircles the critical point -1+j0. Unlike a Bode plot, which shows magnitude and phase as two separate curves against frequency, a Nyquist plot combines both into a single curve, and its relationship to one specific point captures the entire stability question in one picture.

Control engineers reach for the Nyquist criterion specifically when a system's stability cannot be judged from gain and phase margins alone, systems with poles on the imaginary axis, systems with multiple crossings of the 0 dB or -180 degree lines, or conditionally stable systems where instability appears only over a specific range of gain. An aerospace engineer analyzing a flight control system with several interacting feedback loops, or a power electronics engineer studying an inverter's control loop across a wide operating range, both rely on the Nyquist criterion's ability to handle these more complex cases correctly, where a simple gain-and-phase-margin reading from a Bode plot can be ambiguous or misleading.

A common misconception is that stability can be judged by eye, just checking whether the curve looks like it passes near -1+j0. In reality, what matters is the number of encirclements, a curve that swings very close to -1+j0 without ever going around it describes a stable system with a small margin, while a curve that stays farther away on average but loops fully around -1+j0 describes an unstable one. This calculator backs its verdict with an algebraic Routh-Hurwitz test on the closed-loop characteristic equation rather than relying on a visual encirclement count, which is easy to get wrong by eye on a finite, capped plot window.

This calculator takes the DC gain K and two pole time constants T1, T2 of the same classic single-integrator, two-real-pole open-loop transfer function used by this site's Bode Plot calculator, plots the Nyquist diagram with the critical point marked, and reports the authoritative closed-loop stability verdict, the number of right-half-plane poles, and how close the curve passes to -1+j0.

📐 Formula

G(jω)  =  K / (jω(1 + jωT1)(1 + jωT2))
Let A = 1 − ω²T1T2 and B = ω(T1+T2). Then Re(G(jω)) = −KωB / (ω²(A²+B²)), Im(G(jω)) = −KωA / (ω²(A²+B²))
Characteristic equation (1 + G(s) = 0): T1T2s³ + (T1+T2)s² + s + K = 0
Routh array first column: [T1T2, T1+T2, b1, K], where b1 = [(T1+T2)×1 − T1T2×K] / (T1+T2)
Number of sign changes in that first column = number of right-half-plane poles; zero sign changes means the closed loop is stable
Example: K = 10, T1 = 0.5 s, T2 = 0.1 s → b1 ≈ 0.1667 (no sign change) → stable, closest approach to −1+j0 ≈ 0.064 (a small margin).

📖 How to Use This Calculator

Steps

1
Enter the DC gain. Type the open-loop DC gain K.
2
Enter the two pole time constants. Type the time constants T1 and T2 for the two real poles, in seconds.
3
Read the Nyquist plot and stability verdict. See the Nyquist diagram with the critical point marked, the Routh-Hurwitz stability verdict, the number of right-half-plane poles, and the closest approach to -1+j0.

💡 Example Calculations

Example 1 — Marginally Stable Default Case

K = 10, T1 = 0.5 s, T2 = 0.1 s

1
Characteristic equation: 0.0500s³ + 0.6000s² + s + 10 = 0
2
Routh first column: [0.0500, 0.6000, b1, 10], with b1 = (0.6000 − 0.0500×10) / 0.6000 = 0.1667
3
No sign changes in [0.0500, 0.6000, 0.1667, 10], so 0 right-half-plane poles
Verdict = Stable, right-half-plane poles = 0, closest approach to −1+j0 = 0.064 (a small margin, consistent with this system's low 3.94° phase margin on the Bode Plot calculator)
Try this example →

Example 2 — Unstable, High Gain

K = 50, T1 = 0.5 s, T2 = 0.1 s (same poles, higher gain)

1
Characteristic equation: 0.0500s³ + 0.6000s² + s + 50 = 0
2
b1 = (0.6000 − 0.0500×50) / 0.6000 = −3.1667, negative
3
First column [0.0500, 0.6000, −3.1667, 50] has 2 sign changes, so 2 right-half-plane poles
Verdict = Unstable (2 RHP poles), closest approach to −1+j0 = 0.476 (the curve actually loops around the critical point rather than merely passing near it)
Try this example →

Example 3 — Comfortably Stable, Low Gain

K = 2, T1 = 1 s, T2 = 0.2 s

1
Characteristic equation: 0.2000s³ + 1.2000s² + s + 2 = 0
2
b1 = (1.2000 − 0.2000×2) / 1.2000 = 0.6667, positive
3
First column [0.2000, 1.2000, 0.6667, 2] has no sign changes, so 0 right-half-plane poles
Verdict = Stable, right-half-plane poles = 0, closest approach to −1+j0 = 0.368 (a more comfortable margin than Example 1)
Try this example →

❓ Frequently Asked Questions

What is the Nyquist stability criterion?+
The Nyquist stability criterion determines closed-loop stability by counting how many times the open-loop frequency response G(jw), plotted as a curve in the complex plane, encircles the critical point -1+j0 as frequency sweeps from -infinity to +infinity. For a stable open-loop system, zero encirclements of -1+j0 means the closed loop is also stable.
How do you calculate G(jw) for a Nyquist plot?+
For G(s) = K/(s(1+sT1)(1+sT2)), substitute s = jw and rationalize: G(jw) = K/(jw(1+jwT1)(1+jwT2)). Expanding gives Re(G(jw)) and Im(G(jw)) as explicit functions of w, K, T1, and T2, which this calculator evaluates directly using complex arithmetic rather than the expanded algebraic form, then plots as a curve.
Why is the critical point -1+j0 significant?+
The critical point -1+j0 corresponds to a loop gain of exactly 1 with a phase shift of exactly 180 degrees, the precise condition where negative feedback becomes positive feedback and a signal reinforces itself indefinitely rather than settling. Encircling this point means the closed-loop system's characteristic equation has a root in the right-half plane, and is unstable.
What is the difference between the Nyquist criterion and the Routh-Hurwitz criterion?+
The Routh-Hurwitz criterion is a purely algebraic test on the closed-loop characteristic equation's coefficients, counting sign changes in a tabulated array to find the exact number of right-half-plane roots with no plotting required. The Nyquist criterion is a graphical and topological method based on encirclements of -1+j0, and additionally shows how close a stable system sits to instability, information Routh-Hurwitz alone does not directly provide.
How does this calculator determine the stability verdict?+
This calculator builds a Routh array for the closed-loop characteristic equation T1*T2*s^3 + (T1+T2)*s^2 + s + K = 0 and counts sign changes in its first column, which gives the exact number of right-half-plane roots. This algebraic result is the authoritative verdict; the Nyquist plot and its distance from -1+j0 are shown as an illustrative visual companion.
Why does the Nyquist curve go to infinity as frequency approaches zero?+
This system has a single integrator (a 1/s term) in its open-loop transfer function, and any transfer function with a pole at the origin has an open-loop magnitude that grows without bound as frequency approaches zero. The Nyquist plot's tail toward -j infinity reflects that integrator behavior directly.
What does the mirrored (dashed) branch of the plot represent?+
The full Nyquist criterion technically sweeps frequency from negative infinity through zero to positive infinity. Since G(-jw) is always the complex conjugate of G(jw) for a real-coefficient transfer function, the negative-frequency branch is exactly the mirror image of the positive-frequency branch reflected across the real axis, shown here as the dashed curve.
Can a system be stable even if its Nyquist curve passes close to -1+j0?+
Yes. Passing close to -1+j0 without encircling it means the closed loop is stable but has a small stability margin, similar to a small phase or gain margin on a Bode plot. The closest-approach distance this calculator reports is a rough visual indicator of that margin, while the Routh-Hurwitz result confirms whether the system is actually stable.
What open-loop transfer function structure does this calculator use?+
This calculator is scoped to G(s) = K/(s(1+sT1)(1+sT2)), a Type-1 system with one pole at the origin (from the integrator) and two additional real poles set by time constants T1 and T2, the same structure used by this site's Bode Plot Gain and Phase Margin Calculator for direct comparison.
How many right-half-plane poles does an unstable version of this system typically have?+
For this specific third-order characteristic equation, an unstable case (high enough K relative to T1 and T2) typically produces exactly two right-half-plane poles, reflected as two sign changes in the Routh array's first column, since a real cubic with one sign change pattern reversal generally moves a complex-conjugate pole pair across the imaginary axis together.
Tags: Nyquist Stability Criterion Calculator Nyquist Plot Calculator Routh Hurwitz Calculator Closed Loop Stability Calculator Control Systems Stability Calculator Calculator Formula Free Online

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What is the Nyquist stability criterion?

The Nyquist stability criterion determines closed-loop stability by counting how many times the open-loop frequency response G(jw), plotted as a curve in the complex plane, encircles the critical point -1+j0 as frequency sweeps from -infinity to +infinity. For a stable open-loop system, zero encirclements of -1+j0 means the closed loop is also stable.

How do you calculate G(jw) for a Nyquist plot?

For G(s) = K/(s(1+sT1)(1+sT2)), substitute s = jw and rationalize: G(jw) = K/(jw(1+jwT1)(1+jwT2)). Expanding gives Re(G(jw)) and Im(G(jw)) as explicit functions of w, K, T1, and T2, which this calculator evaluates directly using complex arithmetic rather than the expanded algebraic form, then plots as a curve.

Why is the critical point -1+j0 significant?

The critical point -1+j0 corresponds to a loop gain of exactly 1 with a phase shift of exactly 180 degrees, the precise condition where negative feedback becomes positive feedback and a signal reinforces itself indefinitely rather than settling. Encircling this point means the closed-loop system's characteristic equation has a root in the right-half plane, and is unstable.

What is the difference between the Nyquist criterion and the Routh-Hurwitz criterion?

The Routh-Hurwitz criterion is a purely algebraic test on the closed-loop characteristic equation's coefficients, counting sign changes in a tabulated array to find the exact number of right-half-plane roots with no plotting required. The Nyquist criterion is a graphical and topological method based on encirclements of -1+j0, and additionally shows how close a stable system sits to instability, information Routh-Hurwitz alone does not directly provide.

How does this calculator determine the stability verdict?

This calculator builds a Routh array for the closed-loop characteristic equation T1*T2*s^3 + (T1+T2)*s^2 + s + K = 0 and counts sign changes in its first column, which gives the exact number of right-half-plane roots. This algebraic result is the authoritative verdict; the Nyquist plot and its distance from -1+j0 are shown as an illustrative visual companion.

Why does the Nyquist curve go to infinity as frequency approaches zero?

This system has a single integrator (a 1/s term) in its open-loop transfer function, and any transfer function with a pole at the origin has an open-loop magnitude that grows without bound as frequency approaches zero. The Nyquist plot's tail toward -j infinity reflects that integrator behavior directly.

What does the mirrored (dashed) branch of the plot represent?

The full Nyquist criterion technically sweeps frequency from negative infinity through zero to positive infinity. Since G(-jw) is always the complex conjugate of G(jw) for a real-coefficient transfer function, the negative-frequency branch is exactly the mirror image of the positive-frequency branch reflected across the real axis, shown here as the dashed curve.

Can a system be stable even if its Nyquist curve passes close to -1+j0?

Yes. Passing close to -1+j0 without encircling it means the closed loop is stable but has a small stability margin, similar to a small phase or gain margin on a Bode plot. The closest-approach distance this calculator reports is a rough visual indicator of that margin, while the Routh-Hurwitz result confirms whether the system is actually stable.

What open-loop transfer function structure does this calculator use?

This calculator is scoped to G(s) = K/(s(1+sT1)(1+sT2)), a Type-1 system with one pole at the origin (from the integrator) and two additional real poles set by time constants T1 and T2, the same structure used by this site's Bode Plot Gain and Phase Margin Calculator for direct comparison.

How many right-half-plane poles does an unstable version of this system typically have?

For this specific third-order characteristic equation, an unstable case (high enough K relative to T1 and T2) typically produces exactly two right-half-plane poles, reflected as two sign changes in the Routh array's first column, since a real cubic with one sign change pattern reversal generally moves a complex-conjugate pole pair across the imaginary axis together.

📌 Quick Tips

💡The Nyquist plot shown here traces the positive-frequency branch (solid) and its mirror image for negative frequency (dashed), which is the standard simplified presentation for a teaching tool. A full rigorous Nyquist contour also includes a small detour around the pole at the origin, which this calculator does not draw.
💡Never judge stability purely by eye from how close the curve passes to -1+j0. A curve that passes very close to -1 without crossing around it is still stable, while a curve that swings wide but fully encircles -1 is not, always confirm with the Routh-Hurwitz verdict shown here.
💡This calculator uses the same open-loop structure, K/(s(1+sT1)(1+sT2)), as the Bode Plot Gain and Phase Margin Calculator. Try the same K, T1, T2 values in both tools to see how a Bode plot's gain and phase margins and a Nyquist plot's distance from -1+j0 describe the exact same stability margin from two different angles.
💡As frequency approaches zero, this system's single integrator (1/s term) sends the Nyquist curve toward infinity along the negative imaginary direction. This calculator's plot window is capped to keep the region near the critical point readable, rather than showing that unbounded tail.
💡Increasing K stretches the entire Nyquist curve outward from the origin without changing its shape, since K is a simple real multiplier. A large enough K eventually drags the curve around the critical point, which is exactly the encirclement that the Routh-Hurwitz check catches algebraically.