Peak Time Calculator
Compute peak time for a second-order underdamped system with tp = π / (ωn·√(1−ζ²)) — three-way solving for tp, ωn or ζ, with a live step-response schematic and companion time-domain metrics.
The peak time is exactly half the damped period
For an underdamped second-order system, the response oscillates about the final value at the damped natural frequency ωd = ωn√(1−ζ²). The peak time — the moment of maximum overshoot — occurs exactly one half-period after the step is applied, so tp = π/ωd. This is why the formula has a π and not a 2π: it marks the first hump of the decaying sinusoid, not a full cycle. The same reasoning explains why the second peak occurs at t ≈ 3π/ωd, the third at 5π/ωd, and so on, each one smaller than the last by the factor exp(−πζ/√(1−ζ²)).
Higher damping pushes the peak later and lower
As ζ increases toward 1, the factor √(1−ζ²) shrinks, so the denominator ωd shrinks and tp grows. At the same time, the overshoot Mp = exp(−πζ/√(1−ζ²)) shrinks to zero. The two effects are two sides of the same coin: damping removes energy from the oscillation, so the system peaks later and at a lower value. At ζ = 1 the peak disappears entirely — the response never overshoots, and the formula tp = π/ωd becomes undefined because ωd = 0.
Peak time is not the settling time
tp is when the response reaches its maximum, not when it finishes settling. A well-damped system may peak at 0.1 s but not settle to within ±2 % of the final value until 0.4 s. The settling time ts ≈ 4/(ζωn) is typically 4–6 times longer than tp for ζ ≈ 0.7. Both are important for controller design: tp tells you how quickly the system overshoots, ts tells you when you can trust the output to hold its steady state. Together with rise time and overshoot, they form the four classic time-domain specifications.
Peak time scales inversely with ωn
Because ζ and ωn are independent parameters, and tp = π/(ωn√(1−ζ²)), the peak time is proportional to 1/ωn at fixed ζ. Doubling the natural frequency halves the peak time; halving it doubles the peak time. This is why control engineers treat ωn as the "speed" parameter and ζ as the "shape" parameter — ωn sets all the time scales (rise, peak, settling) proportionally, while ζ shapes how the response behaves at that speed.
How to Use This Calculator
The peak time relation links three quantities: tp, ωn and ζ. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick tp, ωn or ζ from the toggle. The calculator rearranges tp = π/(ωn√(1−ζ²)) automatically for whichever variable you choose.
2. Use realistic presets
Presets cover peak times from 0.5 ms to 30 s, natural frequencies from 0.5 to 1000 rad/s, and damping ratios from 0.1 to 0.99 — spanning fast amplifier loops through large thermal and process control.
3. Remember the underdamped constraint
The formula only applies for 0 < ζ < 1. At ζ = 1 the system is critically damped and has no overshoot — the peak disappears. Above ζ = 1 (overdamped), the response still overshoots is false; it approaches the final value monotonically. The calculator will refuse invalid inputs.
4. Read the result and verdict
The result card shows the solved value plus the equivalent oscillation frequency 1/tp. A colour-coded verdict classifies the response speed from ultrafast through very fast, fast, moderate, slow, very slow and extremely slow.
5. Check the schematic and charts
The schematic shows the live step response curve with a red marker at the first peak, an amber marker at the second peak, dashed reference lines at y = 0.5 and y = 1.0, and the peak-time interval highlighted on the horizontal axis. The main chart plots tp against ωn for three damping ratios.
6. Switch to Advanced for depth
Advanced mode adds a tp vs. ζ chart, a sensitivity analysis, and companion metrics including rise time, settling times, overshoot percentage, damped frequency, damped period, and the second- and third-peak times with their magnitudes.
What This Tool Offers
Built around the classic second-order peak time relation tp = π/(ωn√(1−ζ²)), the fundamental link between natural frequency, damping, and the first overshoot of a step response.
Three-way solving
Solve for peak time, natural frequency or damping ratio. The calculator rearranges the formula for whichever variable you choose.
Live step response schematic
A real second-order step response curve for the current ζ and ωn, with a red marker at the first peak, an amber marker at the second peak, and the peak-time interval highlighted on the horizontal axis.
Speed-class verdict
Each result is classified from ultrafast (RF and audio) through very fast, fast, moderate, slow, very slow and extremely slow, with a coloured badge.
Companion metrics
Rise time, 2% and 5% settling times, maximum overshoot, damped natural frequency, damped period, second and third peak times with their magnitudes — all live.
Dual comparison charts
Peak time vs. natural frequency for three damping ratios, plus an advanced peak time vs. damping ratio chart for three natural frequencies.
Reference tables
A peak-time lookup grid, typical ζ and ωn values by application, and a full variable reference with the classic time-domain formulas.
Reference Tables & Conversions
A quick two-field peak-time solver, a lookup grid, and typical values by application.
Quick peak-time solver — enter ωn and ζ
Peak Time Lookup — ωn vs. ζ
Computed from tp = π / (ωn·√(1−ζ²)). Values in seconds.
| ωn (rad/s) | ζ = 0.1 | ζ = 0.3 | ζ = 0.5 | ζ = 0.707 | ζ = 0.9 | ζ = 0.99 |
|---|
Typical ωn, ζ and tp by Application
Representative values for common second-order systems. Actual values depend on plant dynamics and controller tuning.
| Application | Typical ωn (rad/s) | Typical ζ | Typical tp (s) |
|---|---|---|---|
| Audio power amplifier | 1,000 – 10,000 | 0.5 – 0.9 | 0.0003 – 0.003 |
| Fast digital servo / robot joint | 200 – 2,000 | 0.6 – 1.0 | 0.002 – 0.02 |
| CNC axis position loop | 50 – 500 | 0.5 – 0.9 | 0.007 – 0.07 |
| Motor speed control | 10 – 200 | 0.4 – 1.0 | 0.02 – 0.4 |
| Switching voltage regulator | 100 – 1,000 | 0.4 – 0.9 | 0.003 – 0.03 |
| Quadcopter attitude loop | 50 – 300 | 0.5 – 0.9 | 0.01 – 0.07 |
| Aircraft pitch autopilot | 1 – 10 | 0.5 – 0.8 | 0.3 – 3 |
| Hydraulic position servo | 30 – 300 | 0.4 – 0.8 | 0.01 – 0.1 |
| Flow control loop | 0.5 – 5 | 0.5 – 1.0 | 0.7 – 7 |
| Thermal process (small oven) | 0.05 – 0.5 | 0.6 – 1.2 | 7 – 70 |
| Large thermal process (kiln) | 0.001 – 0.05 | 0.7 – 1.5 | 70 – 3,500 |
| Ship steering autopilot | 0.05 – 1 | 0.7 – 1.2 | 3 – 70 |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| tp | Peak time | 10⁻⁵ – 10³ | s |
| ωn | Natural (undamped) frequency | 0.001 – 10⁶ | rad/s |
| ζ | Damping ratio (0 < ζ < 1 for tp) | 0 – 1 | — |
| ωd | Damped natural frequency | 0 – ωn | rad/s |
| Mp | Maximum overshoot | 0 – 100 | % |
| tp2 | Second peak time (3π/ωd) | — | s |
| tp3 | Third peak time (5π/ωd) | — | s |
| Td | Damped period (2π/ωd) | — | s |
What Does tp = π / (ωn√(1−ζ²)) Measure?
The peak time of a second-order underdamped system is the moment of maximum overshoot — the instant when the response y(t) climbs above its final value, reaches its highest point, and starts to fall back. It is one of the four classic time-domain specifications used in control engineering (alongside rise time, settling time and overshoot percentage), and it is the most direct way to characterise how quickly a system reacts to a step input. For a well-damped second-order system, the peak time is roughly two to three times the rise time, so a design specification that says "peak within 100 ms" implicitly sets a rise time of about 30–50 ms.
The formula tp = π/ωd has a clean physical interpretation: it is exactly half the damped period of the oscillation. If you could watch a single impulse response ring down, the first peak would arrive at t = π/ωd, the second at t = 3π/ωd, the third at t = 5π/ωd, and so on. Each peak is smaller than the last by the factor exp(−πζ/√(1−ζ²)), which is exactly the overshoot percentage expressed as a fraction.
Peak Time Formula
The defining relationship is:
Or equivalently, using the damped natural frequency ωd = ωn√(1−ζ²):
Rearranged to solve for the other two unknowns:
ζ = √( 1 − ( π / ( tp · ωn ) )² )
Related peak quantities for the same system:
3rd peak: tp3 = 5π / ωd = 5 · tp
Peak value: y(tp) = 1 + Mp where Mp = exp( −πζ / √(1−ζ²) )
Damped period: Td = 2π / ωd = 2 · tp
Second-Order Variables
| Symbol | Meaning | Notes |
|---|---|---|
| tp | Peak time | Time from step input to the first overshoot maximum |
| ωn | Natural frequency | Sets the overall speed of the response; the dominant tuning parameter |
| ζ | Damping ratio | Must be 0 < ζ < 1 for the peak time formula to apply |
| ωd | Damped natural frequency | ωn√(1−ζ²); the actual oscillation frequency |
| Mp | Maximum overshoot | Fraction by which y(tp) exceeds the final value |
| tp2 | Second peak time | 3·tp; useful for verifying the damping ratio from measurement |
| Td | Damped period | 2·tp; the full period of the ringing oscillation |
How Each Variable Affects the Result
The relation is inverse in ωn and inverse-square-root in (1−ζ²). Doubling ωn halves the peak time — the system reaches its maximum twice as fast. Increasing ζ from 0 to 0.707 has a subtler effect: the term √(1−ζ²) falls from 1 to 0.707, so tp grows by a factor of about 1.41. Pushing ζ to 0.9 extends tp by a further 1.7×, and at ζ = 0.99 the peak arrives nearly 7× later than it would with zero damping. In the limit ζ → 1, √(1−ζ²) → 0 and tp → ∞ — the "peak" never arrives because the response never overshoots.
In practice, the natural frequency is set by the plant and the controller bandwidth, while the damping ratio is set by the controller's phase margin. Designers usually target ζ ≈ 0.7 for a good compromise between speed (short tp), stability (moderate overshoot), and robustness. Going to ζ = 0.5 makes the system faster but with 16 % overshoot; going to ζ = 0.9 makes it slower but with only 0.15 % overshoot. The trade-off is fundamental — no controller can give both minimal peak time and zero overshoot at the same time.
The peak time is also useful for experimental identification. If you measure the step response of a real system and observe two successive peaks separated by Δt, then Δt = tp2 − tp = 2π/ωd. From that you can back out ωd, then combine it with the observed overshoot ratio Mp = exp(−πζ/√(1−ζ²)) to solve for ζ and finally ωn. This is the standard "peak-to-peak" method for second-order system identification.
Peak Time Calculation Examples
ωd = 10 × √(1 − 0.5) = 10 × 0.7071 = 7.071 rad/s
tp = π / 7.071 = 0.4443 s
√(1 − 0.25) = 0.866
ωn = π / (0.1 × 0.866) = 36.28 rad/s
π / (tp · ωn) = π / 5 = 0.6283
ζ = √(1 − 0.6283²) = √(1 − 0.3948) = √0.6052 = 0.778
ωd = 100 × √(1 − 0.09) = 100 × 0.9539 = 95.39 rad/s
tp = π / 95.39 = 0.03293 s = 32.93 ms
Mp = exp(−π × 0.3 / 0.9539) = 37.2 %
Frequently Asked Questions
Because the peak of the damped sinusoid occurs exactly one half-period after the step is applied. The response of a second-order underdamped system is y(t) = 1 − e−ζωnt·[cos(ωdt) + (ζ/√(1−ζ²))·sin(ωdt)], and the maximum of this expression occurs when the derivative vanishes, which happens at t = π/ωd. Since ωd = ωn√(1−ζ²), the formula can be written either as tp = π/ωd or as tp = π/(ωn√(1−ζ²)). Both forms are used in textbooks; the second is convenient when ωn and ζ are the design parameters.
The formula breaks down. At ζ = 1, the system is critically damped: √(1−ζ²) = 0, ωd = 0, and the peak time is infinite — meaning there is no peak at all. The response rises monotonically to the final value without ever overshooting. For ζ > 1 (overdamped), the response is a sum of two decaying exponentials with real poles; again it never overshoots, and the concept of a "peak time" as a maximum above the final value does not exist. Some textbooks redefine peak time for overdamped systems as the time to reach the maximum rate of change or the time to reach 50 % of the final value, but this is non-standard. If you need a metric for critically damped or overdamped systems, use rise time or settling time instead.
They measure different things: rise time is how long the response takes to go from 10 % to 90 % of the final value, while peak time is when the response reaches its maximum above the final value. For a well-damped second-order system with ζ ≈ 0.7, the ratio is approximately tp/tr ≈ 2.5. So if a specification calls for a rise time of 100 ms, the peak time will be around 250 ms. At lower damping (ζ ≈ 0.3), the ratio drops to about 1.6 — the response overshoots quickly. At higher damping (ζ ≈ 0.9), the ratio rises to 4 or more — the response climbs slowly and peaks late. Both metrics are needed for a complete picture of the step response.
After the first overshoot, the response decays back through the final value and overshoots again — but by less. The second peak occurs at tp2 = 3π/ωd = 3·tp, and the third at tp3 = 5π/ωd = 5·tp. The ratio between successive peak heights is constant and equal to exp(−2πζ/√(1−ζ²)) = Mp², so the overshoot of the second peak is roughly Mp² times the first. This is the basis of the "logarithmic decrement" method for measuring ζ from a physical step response: measure the ratio of two successive peaks, take the natural log, and solve for ζ. The second peak time itself is also used to estimate ωd directly — the time between peaks is exactly one damped period Td = 2π/ωd.
For a linear time-invariant (LTI) system, no. The peak time of a step response is independent of the step amplitude — whether the input is 1 V, 5 V or 100 V, the response peaks at the same time. This is a fundamental property of linear systems: scaling the input scales the output but does not change its shape in time. The peak time formula only involves ωn and ζ, both of which are properties of the system, not the input. In real systems with saturation or rate limits, however, a large enough step can push the system into nonlinearity, and the effective peak time will shift.
Yes, and this is one of the most useful applications of the peak time formula. Two methods are common. First, the overshoot method: measure Mp from the first peak, then solve Mp = exp(−πζ/√(1−ζ²)) for ζ. Second, the logarithmic decrement method: measure the heights of two successive peaks y1 and y2, compute δ = ln(y1/y2), and then ζ = δ / √(4π² + δ²). Once ζ is known, you can back out ωd from the peak time (ωd = π/tp) and finally ωn = ωd/√(1−ζ²). This gives you the full second-order model of your system from a single step test — a remarkably powerful experimental technique.
In classical control design, the peak time is one of the four specifications that translate directly into desired pole locations. If a specification calls for tp ≤ 0.2 s and Mp ≤ 10 %, you can solve for the required ζ and ωn: from Mp = 10 % you get ζ ≈ 0.59, and from tp = π/ωd = 0.2 s you get ωd ≈ 15.7 rad/s, so ωn ≈ 19.3 rad/s. Those are the target closed-loop pole locations. Then you design a controller (PID, lead-lag, state feedback) that places the actual closed-loop poles at or near those locations. This pole-placement approach works well for plants that are approximately second-order and is the foundation of classical servo design.
Because a lightly damped second-order system can overshoot multiple times, and the "peak time" strictly refers to the first overshoot. Some authors use "time to first peak" to be explicit; others use "peak time" and assume the first peak is meant. In most control textbooks the two are synonymous, and the formula tp = π/ωd always gives the time of the first peak. The subsequent peaks occur at odd multiples of tp: 3tp, 5tp, 7tp, etc. In practice the first peak is the only one that matters for most specifications, because subsequent peaks are typically well within the settling band and do not violate overshoot or timing requirements.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| t_p | Time to reach peak response | s |
| omega_n | Natural (undamped) frequency of the system | rad/s |
| zeta | Damping ratio of the system (0 < zeta < 1) |
What it means
Peak time is the time it takes for the response to reach its first maximum. It is inversely proportional to the damped natural frequency. It is a performance metric for transient response. Understanding it helps in setting expectations for response times.
Worked example
Peak Time (2nd‑Order) – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| ζ (damping ratio) | 0.5 |
| ω_n (rad/s) | 2 |
| Parameter | Value |
|---|---|
| ζ | 0.1 |
| ω_n | 1 |
Common mistakes
- Peak time: t_p = π / (ω_n·√(1−ζ²)) – the time of the first peak (overshoot).
- Units: ω_n in rad/s, t_p in seconds.
- Damped natural frequency: ω_d = ω_n·√(1−ζ²) – this is the frequency of oscillation.
- Applies to: Underdamped second‑order systems (0 < ζ < 1).
- Relation: t_p occurs at the first maximum of the step response.
Applications
Peak time for a second‑order system, t_p = π/(ω_n·√(1−ζ²)), is the time at which the response reaches its first peak. This is used to characterise the speed of response and to tune controllers. Engineers use it to ensure that the system meets response time constraints. A smaller peak time indicates a faster but potentially more oscillatory response. By designing for a specific peak time, they can meet dynamic performance requirements. This formula is widely used in control system design and analysis.
- Specification of transient response speed
- Controller tuning for desired peak time
- Optimisation of response characteristics in servos
- Analysis of mechanical and electrical systems
- Educational application of time‑domain metrics
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