Critical Power W Prime Calculator
Our cycling calculator computes critical power prime instantly. Get accurate stats with historical comparisons and benchmarks.
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Power-Duration Predictions
Time to Exhaustion Above CP
Formula
The two-parameter Critical Power model defines the power-duration relationship. CP is the asymptotic power that can be sustained indefinitely. W-prime is the finite anaerobic work capacity above CP in joules. Power for any duration equals CP plus W-prime divided by time in seconds.
Last reviewed: December 2025
Worked Examples
Example 1: CP and W-prime from 3-min and 12-min Tests
Example 2: Predicting 5-minute and 20-minute Power
Background & Theory
The Critical Power & W Prime applies the following established principles and formulas. Sports statistics and performance metrics represent one of the most data-rich domains of applied mathematics available to the general public. Baseball, in particular, has developed an exceptionally dense vocabulary of calculated metrics. Earned run average (ERA) quantifies a pitcher's effectiveness as (earned runs × 9) / innings pitched, normalising performance to a nine-inning standard regardless of how many complete games were pitched. WHIP, or walks and hits per inning pitched, is computed as (walks + hits) / innings pitched and provides a complementary measure of how frequently a pitcher allows baserunners. Batting average, one of the oldest statistics in the sport, is simply hits / at-bats, though more modern metrics such as on-base percentage and slugging percentage have largely supplanted it as primary performance indicators. The NFL passer rating formula is considerably more complex, combining completion percentage, yards per attempt, touchdown rate, and interception rate into a composite score scaled to a 0–158.3 range. Golf handicap calculation, now governed by the World Handicap System introduced in 2020, uses a Handicap Differential formula applied to the best 8 of a player's most recent 20 score differentials, with adjustments for course rating and slope. The Elo rating system, originally developed by physicist Arpad Elo for chess ranking in the 1960s, has become a widely adopted framework for competitive ranking in sports ranging from football to table tennis. It updates each player's rating after every match based on the margin of expected versus actual result. In endurance sports, pace calculation converts total time to a per-mile or per-kilometre rate, informing training intensity and race strategy. In cycling, power-to-weight ratio (watts per kilogram) is the primary determinant of climbing performance and is central to both professional race analysis and amateur fitness tracking. Fantasy sports scoring systems synthesise multiple individual statistics into aggregate point totals, requiring participants to understand the relative value of different performance categories across sports.
History
The history behind the Critical Power & W Prime traces back through the following developments. Organised athletic competition has roots extending to ancient Greece, where the Olympic Games were held at Olympia beginning around 776 BCE. These early games were embedded in religious observance and civic identity, featuring events such as sprinting, wrestling, and the pentathlon. The codification of modern sport rules accelerated dramatically in 19th century Britain, where industrialisation created both the leisure time and the institutional infrastructure for organised competition. The Football Association formalised the rules of association football in 1863, and similar governing bodies for cricket, rugby, tennis, and athletics followed in subsequent decades. Pierre de Coubertin, a French educator inspired by the English model of sport as character-building, campaigned to revive the Olympic Games as a modern international institution. The first modern Summer Olympics were held in Athens in 1896, establishing the template for international multi-sport competition that has continued to the present. FIFA, the international governing body for association football, was founded in Paris in 1904 with seven member nations. The serious statistical analysis of baseball, later termed sabermetrics, was pioneered by writers and analysts including Bill James beginning in the late 1970s. James self-published his Baseball Abstract annuals starting in 1977, introducing rigorous empirical methods to a domain previously dominated by traditional counting statistics and subjective scouting. His work influenced a generation of analysts and front-office executives. The publication of Michael Lewis's Moneyball in 2003, documenting the Oakland Athletics' 2002 season and their use of on-base percentage and other undervalued metrics, brought sports analytics to mainstream attention. The subsequent analytics revolution reshaped hiring practices and game strategy across professional sports leagues. Fantasy sports, which require participants to engage directly with statistical outputs, grew from a hobby practised by a few thousand enthusiasts in the 1980s into a multi-billion dollar industry by the 2010s, with tens of millions of participants across football, baseball, basketball, and other sports.
Frequently Asked Questions
Formula
P = CP + W' / t
The two-parameter Critical Power model defines the power-duration relationship. CP is the asymptotic power that can be sustained indefinitely. W-prime is the finite anaerobic work capacity above CP in joules. Power for any duration equals CP plus W-prime divided by time in seconds.
Worked Examples
Example 1: CP and W-prime from 3-min and 12-min Tests
Problem: A cyclist produces 350W average over 3 minutes and 290W average over 12 minutes. Weight is 75 kg. Calculate CP and W-prime.
Solution: Test 1: P1=350W, t1=180s, Work1 = 350 x 180 = 63,000 J\nTest 2: P2=290W, t2=720s, Work2 = 290 x 720 = 208,800 J\nCP = (63,000 - 208,800) / (180 - 720) = (-145,800) / (-540) = 270 W\nW' = 63,000 - 270 x 180 = 63,000 - 48,600 = 14,400 J = 14.4 kJ\nCP/kg = 270 / 75 = 3.60 W/kg
Result: CP: 270W (3.60 W/kg) | W-prime: 14,400 J (14.4 kJ) | Level: Intermediate
Example 2: Predicting 5-minute and 20-minute Power
Problem: Using CP = 270W and W-prime = 14,400 J, predict maximum power for 5 min and 20 min efforts.
Solution: Power = CP + W' / time (in seconds)\n5-min: P = 270 + 14,400 / 300 = 270 + 48 = 318 W\n20-min: P = 270 + 14,400 / 1200 = 270 + 12 = 282 W\nTime to exhaustion at 370W = 14,400 / (370 - 270) = 144 seconds = 2:24
Result: 5-min Power: 318W | 20-min Power: 282W | TTE at 370W: 2:24
Frequently Asked Questions
What is Critical Power and how does it differ from FTP?
Critical Power (CP) is a physiological parameter representing the highest power output that can theoretically be sustained indefinitely without progressive fatigue. It marks the boundary between the heavy and severe exercise intensity domains. Unlike FTP (Functional Threshold Power), which is typically estimated as the power sustainable for 60 minutes and often approximated as 95 percent of 20-minute power, CP is derived mathematically from the power-duration relationship using at least two maximal efforts of different durations. In practice, CP tends to be 3 to 8 percent higher than FTP because CP represents a true physiological threshold rather than a practical one-hour limit. Both metrics are useful for training, but CP provides a more rigorous foundation for the two-parameter model that also yields the anaerobic work capacity.
What is W-prime and what does it represent physiologically?
W-prime (written as W') represents the finite amount of work that can be performed above Critical Power before exhaustion occurs. It is measured in joules or kilojoules and typically ranges from 15 to 30 kJ for trained cyclists. Physiologically, W-prime represents the total anaerobic energy reserve available for efforts above CP, including phosphocreatine stores, anaerobic glycolysis capacity, and oxygen stores in muscle myoglobin. When you ride above CP, you deplete W-prime at a rate equal to the difference between your current power and CP. Once W-prime reaches zero, you cannot sustain power above CP and must reduce intensity to allow W-prime to reconstitute. The reconstitution of W-prime occurs during recovery periods below CP, though it recovers more slowly than it depletes, following an exponential time constant.
How do I perform the two-test protocol to determine CP and W-prime?
The two-test protocol requires two all-out efforts of different durations, typically performed on separate days or with at least 30 minutes of recovery between them. The short test should last 2 to 5 minutes and the long test should last 10 to 20 minutes. Both efforts must be truly maximal, meaning you pace them to produce the highest possible average power for the entire duration. Before each test, perform a thorough warm-up of at least 15 minutes including some high-intensity efforts. Record your average power for each test duration using a calibrated power meter. The calculator then solves the two equations simultaneously to find CP and W-prime. For more accurate results, performing three or more tests at different durations reduces the impact of pacing errors on the calculated values.
What are typical CP and W-prime values for different cycling levels?
Critical Power values vary enormously based on training status, genetics, and body size. For recreational cyclists, CP typically ranges from 150 to 220 watts (2.2 to 3.2 W/kg). Intermediate club cyclists achieve 220 to 300 watts (3.2 to 3.8 W/kg). Advanced racers produce 280 to 350 watts (3.8 to 4.5 W/kg). Elite and professional cyclists reach 350 to 450 watts (4.5 to 6.0 W/kg). W-prime values are less dependent on endurance fitness and more related to anaerobic capacity. Typical W-prime ranges from 10 to 15 kJ for endurance-focused riders, 15 to 25 kJ for all-around cyclists, and 25 to 40 kJ for sprint-oriented and track cyclists. A high CP with a relatively low W-prime indicates an endurance profile, while a lower CP with high W-prime suggests a sprinter or anaerobic athlete.
How does the power-duration curve predicted by the CP model work?
The CP model predicts maximum sustainable power for any duration using the hyperbolic formula: Power = CP + W-prime / Time. This creates a curve that starts very high for short durations (where W-prime is spread over few seconds) and asymptotically approaches CP for longer durations. At 1 minute, the predicted power might be CP plus several hundred watts. At 5 minutes, the addition from W-prime is much smaller. Beyond 30 minutes, the predicted power is essentially equal to CP. The model is most accurate for durations between 2 and 30 minutes. For very short durations under 2 minutes, it overestimates sustainable power because it does not account for the kinetics of oxygen uptake and phosphocreatine depletion. For very long durations over 30 minutes, factors like glycogen depletion, dehydration, and thermal stress cause actual performance to fall below CP.
How does W-prime balance work during interval training?
During interval training, W-prime acts like a rechargeable battery that depletes during efforts above CP and reconstitutes during recovery periods below CP. The depletion rate is linear: riding at CP plus 100 watts depletes W-prime at 100 joules per second. The reconstitution rate follows an exponential pattern with a time constant (tau) typically between 300 and 600 seconds. This means W-prime recovers quickly at first and then more slowly. After depleting 50 percent of W-prime, approximately one tau of recovery time (5 to 10 minutes) at well below CP restores most of it. However, after complete or near-complete depletion, full reconstitution can take 15 to 30 minutes. This explains why repeated hard intervals become progressively harder even with rest periods: each interval starts with less available W-prime than the one before.
References
Reviewed by Sher, Sports Science & Nutrition Specialist · Editorial policy