Gain and Phase Crossover Frequency Calculator

Find the gain crossover and phase crossover frequencies of a classic single-integrator, two-real-pole open-loop system, and see how each shifts with the DC gain K.

📶 Gain and Phase Crossover Frequency 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
Gain crossover (ωgc)
—
Phase crossover (ωpc)
—
Phase at ωgc
—
Magnitude at ωpc
—
Step-by-step working

📶 What is Gain and Phase Crossover Frequency?

The gain crossover frequency and phase crossover frequency are the two reference frequencies used to define the classic stability margins of a feedback control loop. The gain crossover frequency, omega_gc, is where the open-loop magnitude |G(jw)| equals exactly 1 (0 dB). The phase crossover frequency, omega_pc, is where the open-loop phase angle equals exactly -180 degrees. Phase margin is measured at the gain crossover frequency, and gain margin is measured at the phase crossover frequency, so finding these two frequencies is the necessary first step before either margin can be computed.

Control engineers locate these crossover frequencies constantly when tuning a feedback loop. A servo engineer checks the gain crossover frequency because it roughly sets the closed-loop bandwidth and response speed of the system. A process control engineer checks the phase crossover frequency to know how much extra gain the loop can tolerate before instability sets in. Anyone comparing two candidate controller gains for the same plant reads off both crossover frequencies to see how each choice shifts the tradeoff between speed and stability margin.

A common point of confusion is assuming both crossover frequencies move together as the loop gain K changes. They do not: for the standard structure G(s) = K/(s(1+sT1)(1+sT2)), the phase crossover frequency depends only on the two time constants T1 and T2, never on K, because a positive real gain contributes zero phase shift at any frequency. Only the gain crossover frequency shifts when K changes, since K directly scales the magnitude curve up or down. Increasing K therefore pushes the gain crossover frequency toward the fixed phase crossover frequency, which is exactly why raising loop gain erodes both margins simultaneously.

This calculator takes the DC gain K and the two pole time constants T1 and T2 for this standard open-loop family, locates both crossover frequencies with a numeric bisection search (since neither generally has a simple closed-form solution once K is involved), reports the phase at the gain crossover and the magnitude at the phase crossover, derives the phase margin and gain margin from them, and plots how both crossover frequencies shift as K is varied.

📐 Formula

G(s)  =  K / (s(1 + sT1)(1 + sT2))
|G(jω)| = K / (ω × √(1+(ωT1)²) × √(1+(ωT2)²))
∠G(jω) = −90° − arctan(ωT1) − arctan(ωT2)
Gain crossover: ωgc solves |G(jω)| = 1, found by bisection since K makes this transcendental
Phase crossover: ωpc solves ∠G(jω) = −180°; for this family ωpc = 1/√(T1T2), independent of K
Example: K=10, T1=0.5, T2=0.1 → ωgc ≈ 4.0776 rad/s, ωpc = 1/√(0.05) ≈ 4.4721 rad/s

📖 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 crossover frequencies and margins. See the gain crossover frequency, phase crossover frequency, the phase at gain crossover, the magnitude at phase crossover, and the crossover-vs-gain chart.

💡 Example Calculations

Example 1 — Standard Type-1 Loop (Matches the Bode Plot Calculator Default)

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

1
Bisection search for |G(jω)| = 1 converges to ωgc = 4.0776 rad/s
2
Bisection search for ∠G(jω) = −180° converges to ωpc = 4.4721 rad/s, matching the closed form 1/√(0.5×0.1) = 1/√0.05 = 4.4721
3
∠G(jωgc) = −176.06°, so phase margin = 180 − 176.06 = 3.94°; |G(jωpc)| = 0.8333, so gain margin = −20log₁₀(0.8333) = 1.58 dB
ωgc = 4.0776 rad/s, ωpc = 4.4721 rad/s, PM = 3.94°, GM = 1.58 dB
Try this example →

Example 2 — Slower Loop, Same Margins

K = 5, T1 = 1 s, T2 = 0.2 s

1
Bisection search for |G(jω)| = 1 converges to ωgc = 2.0388 rad/s, exactly half of Example 1's value since T1, T2, and 1/K all doubled together
2
Bisection search for ∠G(jω) = −180° converges to ωpc = 2.2361 rad/s = 1/√(1×0.2)
3
∠G(jωgc) = −176.06°, so PM = 3.94°; |G(jωpc)| = 0.8333, so GM = 1.58 dB, identical to Example 1 because this system is a pure time-scaled copy of it
ωgc = 2.0388 rad/s, ωpc = 2.2361 rad/s, PM = 3.94°, GM = 1.58 dB
Try this example →

Example 3 — Fast, High-Gain Loop with Larger Margins

K = 50, T1 = 0.05 s, T2 = 0.01 s

1
Bisection search for |G(jω)| = 1 converges to ωgc = 27.9917 rad/s
2
Bisection search for ∠G(jω) = −180° converges to ωpc = 44.7214 rad/s = 1/√(0.05×0.01)
3
∠G(jωgc) = −160.09°, so PM = 19.91°; |G(jωpc)| = 0.4167, so GM = 7.6 dB, noticeably healthier than Examples 1 and 2 because ωgc sits further from the fixed ωpc
ωgc = 27.9917 rad/s, ωpc = 44.7214 rad/s, PM = 19.91°, GM = 7.6 dB
Try this example →

❓ Frequently Asked Questions

What is the gain crossover frequency?+
The gain crossover frequency, written omega_gc, is the frequency at which the open-loop magnitude |G(jw)| equals exactly 1 (0 dB). It is the frequency used to measure phase margin, since phase margin is defined as 180 degrees plus the open-loop phase evaluated at this specific frequency.
What is the phase crossover frequency?+
The phase crossover frequency, written omega_pc, is the frequency at which the open-loop phase angle equals exactly -180 degrees. It is the frequency used to measure gain margin, since gain margin is defined from the open-loop magnitude evaluated at this specific frequency.
How is the gain crossover frequency calculated when there is no algebraic formula?+
For most open-loop transfer functions there is no closed-form solution for the frequency where |G(jw)| = 1, so this calculator uses a numeric bisection search: it repeatedly narrows a bracket in log-frequency space, checking whether the magnitude is above or below 1 at the midpoint, until the bracket converges to the crossover frequency to high precision.
Why does the phase crossover frequency not depend on the gain K?+
A real, positive gain constant K contributes exactly zero phase shift at every frequency, since multiplying a complex number by a positive real number only scales its magnitude, never its angle. Because the phase curve of G(jw) = K/(jw(1+jwT1)(1+jwT2)) comes entirely from the integrator and the two real poles, none of which involve K, the phase crossover frequency depends only on T1 and T2.
What happens to the crossover frequencies if K is increased?+
Increasing K shifts the entire magnitude curve upward (in dB, by 20*log10 of the ratio) without moving the phase curve at all. This pushes the gain crossover frequency to a higher value, closer to the fixed phase crossover frequency, which is why raising K generally shrinks both the phase margin and the gain margin.
What if the gain crossover frequency does not exist for my inputs?+
If K is very small, the open-loop magnitude can stay below 1 (0 dB) at every frequency in the search range, meaning there is no gain crossover frequency to find within that range. This calculator detects that condition and reports it explicitly instead of returning a misleading number.
How do gain crossover and phase crossover relate to gain margin and phase margin?+
Phase margin is 180 degrees plus the open-loop phase measured at the gain crossover frequency. Gain margin is minus 20*log10 of the open-loop magnitude measured at the phase crossover frequency. Both stability margins are only meaningful once their corresponding crossover frequency has been located.
Is there a closed-form formula for the phase crossover frequency in this transfer function family?+
Yes, for the specific structure K/(s(1+sT1)(1+sT2)), the phase reaches exactly -180 degrees when omega equals 1/sqrt(T1*T2). This calculator uses numeric bisection rather than this closed form so the same code generalizes to other open-loop structures, but the closed form is a useful independent check for this particular family.
Why does this calculator use the same K/(s(1+sT1)(1+sT2)) structure as the Bode Plot calculator?+
Both calculators analyze the identical open-loop transfer function family, a single integrator plus two real poles, which is the standard textbook Type-1 system used to teach gain and phase margin. Using the same structure lets the same default example (K=10, T1=0.5 s, T2=0.1 s) produce matching crossover frequencies and margins across both tools.
How does the gain crossover frequency change if the DC gain K is doubled?+
Doubling K raises the magnitude curve by 20*log10(2) is approximately 6.02 dB at every frequency, which pushes the gain crossover frequency higher (the magnitude now stays above 1 for longer before falling to 0 dB). The exact new value depends on T1 and T2 as well, so this calculator solves it numerically rather than by a simple ratio, since the magnitude curve is not a straight line on a log-log plot for a two-pole system.
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What is the gain crossover frequency?

The gain crossover frequency, written omega_gc, is the frequency at which the open-loop magnitude |G(jw)| equals exactly 1 (0 dB). It is the frequency used to measure phase margin, since phase margin is defined as 180 degrees plus the open-loop phase evaluated at this specific frequency.

What is the phase crossover frequency?

The phase crossover frequency, written omega_pc, is the frequency at which the open-loop phase angle equals exactly -180 degrees. It is the frequency used to measure gain margin, since gain margin is defined from the open-loop magnitude evaluated at this specific frequency.

How is the gain crossover frequency calculated when there is no algebraic formula?

For most open-loop transfer functions there is no closed-form solution for the frequency where |G(jw)| = 1, so this calculator uses a numeric bisection search: it repeatedly narrows a bracket in log-frequency space, checking whether the magnitude is above or below 1 at the midpoint, until the bracket converges to the crossover frequency to high precision.

Why does the phase crossover frequency not depend on the gain K?

A real, positive gain constant K contributes exactly zero phase shift at every frequency, since multiplying a complex number by a positive real number only scales its magnitude, never its angle. Because the phase curve of G(jw) = K/(jw(1+jwT1)(1+jwT2)) comes entirely from the integrator and the two real poles, none of which involve K, the phase crossover frequency depends only on T1 and T2.

What happens to the crossover frequencies if K is increased?

Increasing K shifts the entire magnitude curve upward (in dB, by 20*log10 of the ratio) without moving the phase curve at all. This pushes the gain crossover frequency to a higher value, closer to the fixed phase crossover frequency, which is why raising K generally shrinks both the phase margin and the gain margin.

What if the gain crossover frequency does not exist for my inputs?

If K is very small, the open-loop magnitude can stay below 1 (0 dB) at every frequency in the search range, meaning there is no gain crossover frequency to find within that range. This calculator detects that condition and reports it explicitly instead of returning a misleading number.

How do gain crossover and phase crossover relate to gain margin and phase margin?

Phase margin is 180 degrees plus the open-loop phase measured at the gain crossover frequency. Gain margin is minus 20*log10 of the open-loop magnitude measured at the phase crossover frequency. Both stability margins are only meaningful once their corresponding crossover frequency has been located.

Is there a closed-form formula for the phase crossover frequency in this transfer function family?

Yes, for the specific structure K/(s(1+sT1)(1+sT2)), the phase reaches exactly -180 degrees when omega equals 1/sqrt(T1*T2). This calculator uses numeric bisection rather than this closed form so the same code generalizes to other open-loop structures, but the closed form is a useful independent check for this particular family.

Why does this calculator use the same K/(s(1+sT1)(1+sT2)) structure as the Bode Plot calculator?

Both calculators analyze the identical open-loop transfer function family, a single integrator plus two real poles, which is the standard textbook Type-1 system used to teach gain and phase margin. Using the same structure lets the same default example (K=10, T1=0.5 s, T2=0.1 s) produce matching crossover frequencies and margins across both tools.

How does the gain crossover frequency change if the DC gain K is doubled?

Doubling K raises the magnitude curve by 20*log10(2) is approximately 6.02 dB at every frequency, which pushes the gain crossover frequency higher (the magnitude now stays above 1 for longer before falling to 0 dB). The exact new value depends on T1 and T2 as well, so this calculator solves it numerically rather than by a simple ratio, since the magnitude curve is not a straight line on a log-log plot for a two-pole system.

📌 Quick Tips

💡The gain crossover frequency depends on all three parameters, K, T1, and T2, because K directly shifts the magnitude curve up or down. The phase crossover frequency depends only on T1 and T2, since a real positive gain K contributes zero phase shift at any frequency.
💡Because the phase crossover frequency never moves as K changes, raising K only ever pushes the gain crossover frequency toward the fixed phase crossover frequency, shrinking the phase margin as it goes, exactly why gain margin and phase margin move together in a Type-1, two-real-pole loop.
💡For this transfer function family, the phase crossover frequency has the closed form omega_pc = 1/sqrt(T1*T2), since the phase reaches exactly -180 degrees when tan(atan(w*T1)) times tan(atan(w*T2)) equals 1, which happens at w squared = 1/(T1*T2).
💡If the gain crossover frequency never occurs within a practical search range, the magnitude never falls to 0 dB, which happens when K is large enough that |G(jw)| stays above 1 out to the search bound; this calculator reports that case explicitly rather than showing a stale number.
💡A wider gap between the gain crossover and phase crossover frequencies generally corresponds to healthier margins; if the two crossover frequencies converge toward each other as K increases, both margins are shrinking toward zero simultaneously.