The two-parameter critical-power model
In Monod and Scherrer's 1965 model, total work in a maximal effort is a line against duration: work = CP·t + W′. Power against duration is then a hyperbola with Critical Power as its asymptote.
From two efforts, CP = (P1·t1 − P2·t2) ÷ (t1 − t2) and W′ = (P1 − CP)·t1, in watts and kilojoules.
What W′, the anaerobic battery, means
W′ (W-prime) is the finite work you can do above CP before exhaustion, typically 10–30 kJ in trained cyclists. Time above CP drains it, time below recharges it. That is why 120% of CP lasts minutes, not an hour.
Testing it correctly
Use two maximal efforts of clearly different durations, about 3 and 10–12 minutes, fully recovered. Pace evenly and go to exhaustion. A held-back short effort over-estimates CP and collapses W′.
Critical Power vs FTP
CP and FTP usually agree within a few percent. CP adds W′, so it also predicts time above threshold.
Worked example
300 W for 3 minutes (180 s) and 250 W for 10 minutes (600 s):
| Total work, effort 1 | 300 × 180 = 54,000 J |
| Total work, effort 2 | 250 × 600 = 150,000 J |
| Critical Power | (54,000 − 150,000) ÷ (180 − 600) ≈ 229 W |
| W′ anaerobic battery | (300 − 229) × 180 ≈ 12.9 kJ |
CP of about 229 W is the ceiling. W′ of 12.9 kJ fuels efforts above it.
Frequently asked questions
What is the difference between critical power and FTP?
Critical power and FTP mark the same boundary and usually agree within a few percent. FTP is one number from one test. Critical power comes from two efforts and adds W′, so it also models time above threshold.
What is W′ (W-prime)?
W′ is the fixed work, in joules, you can do above critical power before exhaustion, typically 10–30 kJ in trained cyclists. It drains above CP and recharges below it, so repeated surges eventually empty it.
How do I test my critical power?
Test critical power with two maximal, evenly paced efforts of different durations, for example 3 and 10–12 minutes, fully recovered. The calculator solves CP = (P1·t1 − P2·t2) ÷ (t1 − t2) and W′ = (P1 − CP)·t1.
Why do the two efforts need different durations?
The model fits a line through two points on the work–duration plot. Efforts too close in length nearly overlap, so CP (the slope) and W′ (the intercept) become unstable. One short and one long effort give a stable fit.
Can runners use critical power?
Yes. Running power meters such as Stryd report a running critical power with the same math. Running and cycling power are not interchangeable, since running power includes vertical motion and form.
Sources
- Monod & Scherrer (1965). “The work capacity of a synergic muscular group.” Ergonomics 8(3):329–338, the critical-power / W′ two-parameter model.
- Jones, Vanhatalo et al. (2010). “Critical power: implications for determination of V̇O₂max and exercise tolerance.” Med Sci Sports Exerc 42(10):1876–1890.
- Allen, Coggan & McGregor, Training and Racing with a Power Meter. 3rd ed. (2019). Definition of Functional Threshold Power and the seven-zone power model.