Control Loop Bandwidth Feasibility Calculator — Sampling, Delay, Phase Margin & Mechanical Limit Validator
Calculate maximum achievable bandwidth, phase margin, and bandwidth ratio using sampling frequency, system delay, mechanical frequency, damping, sensor resolution, and noise level.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Control configuration
System parameters
Target performance
Sensor & noise
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
maximum achievable bandwidth formula:
fmax = min(fsampling, fdelay, fmech,limit, fnoise, fquant)
Limit equations:
- fsampling = ksampling · fsample
- ksampling = 0.05 for position loop
- ksampling = 0.15 for velocity loop
- tdelay,s = tdelay / 1000
- fdelay = 1 / (2π · tdelay,s) when tdelay,s > 0
- fdelay = fsampling when tdelay,s = 0
- fmech,base = fmech · √(1 − 2ζ²) when ζ < 0.7 and (1 − 2ζ²) > 0
- fmech,base = fmech otherwise
- fmech,limit = fmech,base · kmode
- kmode = 0.5 for position loop
- kmode = 1.0 for velocity loop
- knoise = 1 / (1 + noise / 20) when noise > 0
- knoise = 1 when noise = 0
- fnoise = fsampling · knoise
- kquant = resolution / 5000 when resolution < 5000
- kquant = 1 when resolution ≥ 5000
- fquant = fsampling · kquant
phase margin formula:
- φdelay = −360 · ftarget · tdelay,s
- φsampling = −45 · (ftarget / fsample)
- ωn = 2π · fmech
- ω = 2π · ftarget
- rω = ω / ωn
- φplant = −atan2(2ζ · rω, 1 − rω²) · 180 / π
- φm = 180 + φdelay + φsampling + φplant
- φm is clamped to the range 0° to 180°
bandwidth ratio formula:
- r = ftarget / fmax
where:
- fmax — maximum achievable bandwidth [Hz]
- φm — estimated phase margin [deg]
- r — bandwidth ratio ftarget / fmax [—]
- fsample — sampling frequency [Hz]
- tdelay — total system delay [ms]
- tdelay,s — total system delay converted to seconds [s]
- fmech — mechanical natural frequency [Hz]
- ζ — damping ratio [—]
- ftarget — target bandwidth [Hz]
- resolution — sensor counts per revolution [counts/rev]
- noise — measurement noise level [—]
- ksampling — sampling bandwidth factor [—]
- kmode — control mode factor [—]
- knoise — noise penalty factor [—]
- kquant — quantization penalty factor [—]
When to use this calculator
When to use this calculator:
- Check whether a target position-loop bandwidth is below the calculated sampling, delay, mechanical, noise, and quantization limits.
- Check whether a target velocity-loop bandwidth remains feasible when the sampling factor is 0.15 of the sampling frequency.
- Identify whether sampling frequency, total system delay, mechanical natural frequency, measurement noise, or sensor resolution is the dominant bandwidth constraint.
- Estimate phase margin from delay phase, sampling phase, and plant phase at the selected target bandwidth.
- Compare target bandwidth against maximum achievable bandwidth using the bandwidth ratio.
How to interpret the result
Maximum achievable bandwidth is defined as the lowest valid bandwidth limit from sampling, delay, mechanical response, measurement noise, and sensor quantization.
Phase margin is defined as the remaining phase margin after delay phase, sampling phase, and plant phase are evaluated at the target bandwidth.
Bandwidth ratio is defined as target bandwidth divided by maximum achievable bandwidth. A higher bandwidth ratio means the requested target bandwidth is closer to the calculated feasibility limit.
- Safe — bandwidth ratio is ≤ 0.70 and phase margin is ≥ 45°.
- Warning — bandwidth ratio is > 0.70 or phase margin is < 45°, while bandwidth ratio is ≤ 0.85 and phase margin is ≥ 30°.
- Limit — bandwidth ratio is > 0.85 or phase margin is < 30°, while bandwidth ratio is ≤ 1.00 and phase margin is ≥ 20°.
- Unsafe — bandwidth ratio is > 1.00 or phase margin is < 20°.
- Invalid — input conditions fail validation, including target bandwidth ≥ half of sampling frequency, sampling frequency < 10 × target bandwidth, target bandwidth > mechanical natural frequency, damping ratio > 5, damping ratio < 0.05, sensor resolution < 10, or total system delay > 1 s.
Increasing sampling frequency increases the sampling-based bandwidth limit. Increasing total system delay decreases the delay-based bandwidth limit and reduces phase margin. Increasing measurement noise decreases the noise-limited bandwidth. Increasing sensor resolution increases the quantization-limited bandwidth when resolution is below 5000 counts per rev.
Calculation example
Example:
A user wants to check whether a position control loop can run at a target bandwidth of 20 Hz with a 1000 Hz sampling frequency, 2 ms total system delay, 80 Hz mechanical natural frequency, damping ratio of 0.8, 10000 counts per rev sensor resolution, and measurement noise level of 5.
- Control mode = Position loop
- fsample = 1000 Hz
- tdelay = 2 ms
- fmech = 80 Hz
- ζ = 0.8
- ftarget = 20 Hz
- Resolution = 10000 counts/rev
- Noise level = 5
The calculated maximum achievable bandwidth is 40 Hz, the estimated phase margin is about 124.6°, and the bandwidth ratio is 0.50.
Assumptions and limitations
- The control mode is either position loop or velocity loop.
- The position-loop sampling limit is 0.05 × sampling frequency.
- The velocity-loop sampling limit is 0.15 × sampling frequency.
- Total system delay is entered in milliseconds and converted to seconds before delay and phase calculations.
- The delay bandwidth limit is calculated as 1 / (2π · delay in seconds) when delay is greater than zero.
- The mechanical limit is reduced by √(1 − 2ζ²) only when damping ratio is below 0.7 and the square-root term is positive.
- The mechanical limit is multiplied by 0.5 for position loop and by 1.0 for velocity loop.
- Measurement noise above 200 is clamped to 200 before calculating the noise penalty.
- The quantization penalty applies only when sensor resolution is below 5000 counts per rev.
- Only positive, finite bandwidth candidates not exceeding half of the sampling frequency are eligible for the maximum achievable bandwidth calculation.