Hz to rad/s Converter for Control Loops and Bode Plots

Turning Measured Hertz into the rad/s a Bode Plot Expects

A loop analyser, a scope FFT and a plant datasheet all report in hertz, because counting complete cycles in a second is what an instrument physically does. Control theory refuses to speak that language. Transfer functions are written in the Laplace variable s, and along the imaginary axis s = jω, where ω is angular frequency in radians per second. Every pole location, every compensator corner and every phase-margin reading in a textbook or in a bode command is anchored to that axis. The number you measured and the number your model wants therefore differ by one fixed factor, and getting it wrong shifts an entire design by almost eight tenths of a decade.

Conversion factor: ω = 2πf. One cycle carries the phase through one complete revolution, and one revolution is exactly 2π radians, so hertz are multiplied by 2π ≈ 6.283185307. A velocity loop closed at 100 Hz sits at 100 × 6.283185307 = 628.3185 rad/s, and a 1 kHz current loop sits at 6 283.1853 rad/s.

One breakpoint, two notations

Why the Laplace Axis Insists on Radians

The exponential e^jωt underneath every transfer function takes its argument as an angle, so ω has to arrive in rad/s or the phase term is wrong by a factor of 2π. That is why a first-order lag is written 1/(1 + s/ω_p) and never with a frequency in hertz in the denominator.

Why the Analyser Insists on Hertz

A frequency-response analyser injects a sine, counts its cycles against a clock and reports cycles per second. Sample rates, anti-alias filters and drive switching frequencies on the same bench are specified the same way, so the raw evidence always lands in the other unit.

Gain Crossover Named Twice

The point where open-loop gain passes 0 dB is the same physical frequency whichever unit labels it, and phase margin is read exactly there. Confusing a measured 100 Hz crossover with a modelled 100 rad/s crossover is a 6.28× design error, not a rounding detail.

Placing a Loop Breakpoint Step by Step

The usual sequence starts at a measurement and ends in a compensator coefficient, with the 2π sitting squarely in the middle of it.

Enter the frequency the analyser reported

Type the measured gain-crossover or plant-corner frequency into the hertz field. The rad/s side follows keystroke by keystroke, so a swept measurement can be transcribed one breakpoint at a time. Decimal points and commas are both accepted, and stray spaces inside the number are ignored.

Copy ω straight into the model

The copy button on the rad/s field yields the bare number, with no unit symbol and no thousands spacing, which is exactly what a transfer-function denominator, a state-space entry or a discrete filter coefficient needs. Pressing Ctrl+C with the cursor inside the field does the same thing.

Reverse it when the simulation answers back

Simulation output arrives in rad/s, but the next bench sweep needs a hertz setpoint. Press the swap button (↔) to flip the direction, or simply type into the rad/s field instead — both fields are editable, so the conversion runs whichever way you push it.

Reach the neighbouring units when a loop crosses domains

The searchable dropdown on each side carries the whole frequency list, so a mechanical resonance quoted in RPM or a modulation rate quoted in kHz can be brought onto the same angular axis without leaving the page.

Loop Bandwidths, Poles and Compensator Corners

Cascaded servo architectures separate their loops by roughly a decade each, and switch-mode converters place crossover well below the switching-stage dynamics. Written out in both units, the spread of a realistic design becomes easy to sanity-check at a glance.

Loop or breakpoint	Frequency (Hz)	Angular frequency ω (rad/s)
Outer temperature loop	0.05 Hz	0.3142 rad/s
Position loop, servo axis	10 Hz	62.8319 rad/s
Lead compensator zero	20 Hz	125.6637 rad/s
Velocity loop crossover	100 Hz	628.3185 rad/s
Anti-alias filter pole	500 Hz	3 141.5927 rad/s
Current (torque) loop bandwidth	1 000 Hz	6 283.1853 rad/s
Converter voltage-loop crossover	5 000 Hz	31 415.9265 rad/s
Switching-stage pole region	20 000 Hz	125 663.7061 rad/s

Transcribing a Whole Sweep

Breakpoints from one measurement session go through one after another, each result copied as a plain number for the model file, so the log sheet and the plant description end up describing the same corner.

Seven Decades of Cascade

A thermal outer loop and a switching-stage pole live enormously far apart. Answers carry up to eight decimals and roll over to scientific notation past 10¹⁰, so both ends of the cascade stay legible.

Time Constants and −3 dB Points Reconciled

A pole quoted as a time constant becomes ω = 1/τ, while a filter quoted by its −3 dB point arrives in hertz. Both are pushed onto one axis here before they meet in a single Bode overlay.

Control Loop Tuning Questions

Why does control theory work in rad/s while my instruments read hertz?

The two sides are describing different aspects of the same signal. An instrument counts complete cycles against a clock, which is a pure count per second. A transfer function evaluates e^jωt, whose argument is an angle, and angles inside the Laplace and Fourier machinery are radians. Substituting hertz into s = jω leaves the phase term short by a factor of 2π, turning what should be a 45° contribution into roughly 7°. Keep the bench log in hertz and convert once, at the moment the number enters the model.

Is a decade on a rad/s Bode axis the same width as on a hertz axis?

Yes. A decade is a factor of ten in whichever unit labels the axis, and multiplying every point by 2π slides the whole logarithmic axis sideways by log₁₀(2π) ≈ 0.798 decades without stretching it. The span from 10 to 100 Hz occupies precisely the same screen width as 62.8319 to 628.3185 rad/s. That is why asymptotic slopes such as −20 dB per decade, and the −6 dB per octave version of the same statement, are quoted without naming a unit at all: only the tick labels move.

What rise time follows from a given closed-loop bandwidth?

Where one real pole dominates the response, the 10–90% rise time is 2.2/ω, which is the familiar t_r ≈ 0.35/f once the 2π has been folded in. A loop closed at 100 Hz has ω = 628.3185 rad/s, giving 2.2 ÷ 628.3185 = 3.50 ms, in agreement with 0.35 ÷ 100. Treat it as a sanity estimate rather than a specification: a lightly damped second-order closed loop overshoots and rises faster than the rule predicts, while extra lag inside the loop drags it out.

Where does a first-order pole's corner land in each unit?

At the corner the gain is 3 dB down and the phase has rotated 45°, and that point is ω = 1/τ in radians per second. A 1 kΩ resistor with a 1 µF capacitor gives τ = 1 ms, so the pole sits at 1 000 rad/s, which this page reports as 159.1549 Hz. Watch the mirror-image mistake: dividing 1/τ by 6.2832 and then labelling the answer rad/s places the pole 39.5 times too low in the s-plane, because the error is applied twice.

How much faster than the loop bandwidth must a digital controller sample?

Nyquist is only the floor. Practical designs sample ten to twenty times the closed-loop bandwidth, because the zero-order hold behaves like a dead time of half a sample period, and dead time costs phase in direct proportion to ω. That lag is ω·T_s/2 radians. A 100 Hz loop sits at 628.3185 rad/s, so sampling at 2 kHz costs 628.3185 × 0.00025 = 0.157 rad, or 9° of phase margin; halving the rate to 1 kHz doubles that to 18°. Computation delay and filter group delay pile on top, so budget the phase in radians and see what survives at crossover.