Plot the open-loop frequency response on a single diagram with phase on the horizontal axis and gain on the vertical axis. Adjust the gain, time constants and dead time to see the gain margin, phase margin and resonant peak update in real time, and judge how stable the closed loop is at a glance.
Parameters
Open-loop gain K
Overall proportional gain. Higher is faster but cuts the margins
Time constant τ₁
s
Time constant of the dominant first-order lag
Time constant τ₂
s
Time constant of the second first-order lag (higher frequency)
Dead time T_d
s
Transport / computation delay. Adds pure phase lag and erodes the margins
Results
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Gain margin GM (dB)
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Phase margin PM (°)
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Gain crossover ωgc (rad/s)
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Phase crossover ωpc (rad/s)
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Resonant peak Mr (dB)
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Stability
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Nichols chart — open-loop locus
Horizontal axis is open-loop phase (degrees), vertical axis is open-loop gain (dB). The cross marks the critical point (−180°, 0 dB). A marker travels along the locus as frequency ω increases.
The open-loop transfer function under analysis (a type-1 system). K: open-loop gain, τ₁/τ₂: time constants, T_d: dead time. It contains one integrator 1/(jω).
The phase margin PM uses the phase at the gain-crossover frequency ωgc (|L| = 1); the gain margin GM uses the gain at the phase-crossover frequency ωpc (∠L = −180°).
The closed-loop transfer function T and the resonant peak Mr. The critical point (−180°, 0 dB) is the boundary of closed-loop stability; how far the locus stays from it is the margin.
What is the Nichols Chart Simulator?
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I have never heard of a "Nichols chart". How is it different from a Bode plot or a Nyquist plot?
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All three look at the "frequency response of the open-loop transfer function L(jω)". A Bode plot draws gain and phase on two separate graphs. A Nyquist plot puts real and imaginary parts on the complex plane. A Nichols chart sits in between: phase (degrees) on the horizontal axis, gain (dB) on the vertical axis, all on one diagram. Think of it as the two Bode graphs merged into one. As you sweep the frequency ω, that point moves along a single curve across the diagram.
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Is there any benefit to merging them into one diagram?
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The biggest benefit is that you can "read off" the gain margin and phase margin by eye. See the point marked with a cross on the diagram? That is the critical point (−180°, 0 dB). It is the boundary of closed-loop stability — the closer the locus comes to it, the more dangerous. The phase margin shows up as the horizontal gap from the locus to the critical point at the 0 dB line, and the gain margin as the vertical gap along the −180° line. Raise the "open-loop gain K" on the left: you will see the whole locus lift upward and approach the critical point.
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You're right — when I raised K to about 14 the verdict turned "unstable". What happened?
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Raising the gain shifts the whole locus upward. The phase does not change but the height (dB) increases. So the point where it crosses 0 dB — the gain-crossover frequency ωgc — has its phase creep ever closer to −180°, and eventually overshoots it. The instant the phase margin PM = 180° + ∠L(ωgc) goes negative, that closed loop diverges. On real hardware that is a motor that hums and never stops vibrating. So if you naively raise the gain "to make it faster", you will always hit instability somewhere. That trade-off is the nature of feedback control.
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Why does increasing the dead time T_d cut the margins so sharply? I'm not even raising the gain.
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Good question. The dead time e^(−jωT_d) does not change the gain (magnitude) at all, but it lags the phase by −ωT_d in one direction only. And the higher the frequency ω, the larger the lag. On the Nichols chart, the locus gets pulled further and further to the left at high frequencies and starts grazing the critical point. So adding just a little T_d shaves off a lot of phase margin. In processes with transport lag, control over a network, or coarse-sampled digital control, dead time is the single biggest enemy of stability. Look at the closed-loop response chart below — you will see the resonant peak Mr rise as you increase T_d.
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So in the end, how should I read the resonant peak Mr?
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Mr is the height of the hump in the closed-loop frequency response T(jω) = L/(1+L). It tells you how strongly the input is amplified at some frequency; the larger the Mr, the more oscillatory the closed loop. Roughly speaking, an Mr of 0-3 dB is good, while above 5 dB the overshoot is visibly large. Mr is almost the flip side of the phase margin: a small PM means a large Mr. So in design the standard target is something like "phase margin around 45°, Mr below 3 dB". A handy feature of the Nichols chart is that you can overlay these Mr contours (M-circles) directly on it.
Frequently Asked Questions
A Nichols chart plots the open-loop transfer function L(jω) with phase (degrees) on the horizontal axis and gain (dB) on the vertical axis, all on one diagram. Sweeping the frequency ω traces a single curve that merges the gain and phase plots of a Bode diagram into one locus. Its key advantage is that the gain margin and phase margin can be read directly as the distance of the locus from the critical point (−180°, 0 dB): the phase margin is the horizontal gap to the critical point at the 0 dB line, and the gain margin is the vertical gap along the −180° line.
First sweep the frequency ω logarithmically and compute the magnitude and phase of the open-loop L(jω) at each point. The gain-crossover frequency ωgc is where |L| = 1 (0 dB); with the phase there written ∠L(ωgc), the phase margin is PM = 180° + ∠L(ωgc). The phase-crossover frequency ωpc is where ∠L = −180°, and from the gain there the gain margin is GM = −20·log10|L(ωpc)| dB. This tool interpolates around each crossing to display these values in real time.
A common rule of thumb is a phase margin of 30-60° and a gain margin of at least 6 dB (about a factor of two). A small phase margin makes the closed-loop response oscillatory, with more overshoot and a longer settling time; too large a margin makes the response sluggish. A phase margin of 45-60° usually gives a good balance of speed and stability. This tool warns of a 'marginal' design when PM is below 30° or GM is below 6 dB, and reports 'unstable' when PM ≤ 0 or GM ≤ 0.
The resonant peak Mr is the maximum magnitude of the closed-loop frequency response T(jω) = L/(1+L), expressed in dB. A larger Mr means the input is strongly amplified at some frequency, that is, the closed-loop response is more oscillatory with greater overshoot. Empirically, an Mr of roughly 0-3 dB is good, while above 5 dB the oscillation becomes noticeable and corresponds to a small phase margin. Mr is closely tied to the phase margin: the smaller the PM, the larger the Mr.
Real-World Applications
Servo and motion control: Machine-tool feed axes, robot joints and hard-disk head positioning are servo loops where engineers must keep the gain and phase margins while pushing the gain as high as possible for speed. A Nichols chart shows at a glance how the locus approaches the critical point as the gain is raised, so it can be used directly to set the upper limit on gain. Overlaying the M-contours lets you read off the gain that keeps the resonant peak below a target value.
Process control (chemical / plant): Temperature, flow and pressure loops carry significant dead time from pipe transport lag and slow measurement response. Because dead time lags only the phase and erodes the phase margin, on a Nichols chart the high-frequency part of the locus stretches to the left. When tuning PID gains, this diagram helps confirm that enough phase margin remains once the dead time is included.
Power supplies and power electronics: The voltage and current loops of switching power supplies and inverters are checked for stability margin using the frequency response in the final design review. Because computation and sampling delays in digital control act as dead time, a Nichols chart or Bode plot is used to verify criteria such as a phase margin above 45° and a gain margin above 6 dB.
Control education and loop shaping: A Nichols chart is an excellent teaching aid for loop shaping. You can see visually how the locus moves when a compensator is added, and how lifting the phase with a lead compensator restores the phase margin. Moving the gain, time constants and dead time in this tool to feel the relationship between the locus and the margins is a good entry point for building control-engineering intuition.
Common Misconceptions and Pitfalls
A common misconception is that "if both the gain margin and phase margin are sufficient, the system is always stable and performs well". GM and PM are "local" measures based on a single crossing each. For complex transfer functions, the gain curve may cross 0 dB several times, or the phase may move back and forth through −180°, producing multiple crossings. In that case you must evaluate by the distance to the nearest critical point (the vector margin, or the closest approach on the Nichols chart); otherwise a design that looks good in GM and PM may actually pass dangerously close to the critical point. This tool handles a simple type-1 system with a single pair of crossings, but on real hardware always check the shape of the whole locus.
Next is the assumption that "dead time is small, so it can be ignored". The dead time e^(−jωT_d) keeps the gain at unity and lags the phase by −ωT_d. At low frequency the lag is tiny, but the higher the gain-crossover frequency ωgc, the larger ωgc·T_d becomes, and the more phase margin is shaved off. "Raising the gain to speed things up" lifts ωgc, so the same dead time costs more phase loss — a vicious circle. Do not forget to include effects such as an equivalent dead time of about half the sampling period and the lags of sensors and actuators.
Finally, there is the misconception that "the Nichols chart is an open-loop diagram, so it has nothing to do with closed-loop performance". The Nichols chart is indeed the locus of the open-loop L(jω), but its real strength is that you can overlay contours of the closed-loop gain |T| (M-contours) and closed-loop phase (N-contours) on top of it. Reading where the open-loop locus passes among the M-contours directly tells you the closed-loop resonant peak Mr and the bandwidth. In other words, a Nichols chart bridges "open-loop manipulation" and "closed-loop performance" on a single diagram — it is not purely an open-loop story. That is exactly why this tool also displays the closed-loop response |T(jω)| and Mr alongside the open-loop locus.
How to Use
Enter the controller gain K using kNum and kRange sliders to adjust loop gain magnitude.
Set time constants t1Num, t1Range (lead/lag numerator pole) and t2Num, t2Range (denominator pole) to shape the frequency response curve.
Input time delay tdNum and tdRange (seconds) to model transport lag effects on phase margin.
Observe the Nichols plot trace updating in real-time as the open-loop transfer function L(jω) maps across phase (horizontal) versus magnitude (vertical) axes.
Read stability metrics: gain margin GM in dB, phase margin PM in degrees, crossover frequencies ωgc and ωpc in rad/s, and resonant peak Mr indicating closed-loop overshoot tendency.
Worked Example
For a servo motor control system: set K=2.5, t1=0.05s (lead compensator zero), t2=0.2s (plant pole), td=0.008s transport delay. The Nichols curve traces through phase angles from -180° toward 0°. At ωgc≈8.3 rad/s, gain reaches 0 dB; at ωpc≈5.7 rad/s, phase crosses -180°. Expected results: GM≈6.2 dB, PM≈32°, Mr≈2.8 dB. This 32° phase margin ensures acceptable damping (ζ≈0.35) and rise time around 0.44 seconds for the closed-loop step response.
Practical Notes
Phase margin below 30° indicates marginal stability; aerospace control systems typically require PM≥45° to handle model uncertainty and sensor noise.
Transport delay td directly rotates the curve leftward on the Nichols chart; increase t1 zero frequency to compensate by advancing phase at mid-range frequencies.
Gain margin peaking above 12 dB suggests over-conservative tuning; industrial PID loops often target GM between 6–10 dB for responsiveness.
Resonant peak Mr above 3 dB predicts closed-loop oscillations; use t2 pole placement to attenuate high-frequency gain if Mr exceeds specification.