Critical Power W Prime Calculator
Our cycling calculator computes critical power prime instantly. Get accurate stats with historical comparisons and benchmarks.
Reviewed for accuracy by Sher, Sports Science & Nutrition Specialist
Critical Power W Prime Calculator
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Formula: P = CP + W' / t
Worked example — CP: 270W (3.60 W/kg) | W-prime: 14,400 J (14.4 kJ) | Level: Intermediate
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 Test 2: P2=290W, t2=720s, Work2 = 290 x 720 = 208,800 J CP = (63,000 - 208,800) / (180 - 720) = (-145,800) / (-540) = 270 W W' = 63,000 - 270 x 180 = 63,000 - 48,600 = 14,400 J = 14.4 kJ CP/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) 5-min: P = 270 + 14,400 / 300 = 270 + 48 = 318 W 20-min: P = 270 + 14,400 / 1200 = 270 + 12 = 282 W Time 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.
Can I improve my Critical Power and W-prime independently?
Yes, CP and W-prime respond to different types of training and can be improved somewhat independently. Critical Power improves primarily through sustained aerobic training: long endurance rides, sweet spot intervals (88 to 94 percent of CP), and threshold intervals (95 to 105 percent of CP) performed 3 to 4 times per week. Improvements in CP reflect enhanced mitochondrial density, capillarization, and oxidative enzyme activity. W-prime improves through high-intensity interval training with efforts of 120 to 200 percent of CP lasting 30 seconds to 3 minutes. Sprint training, VO2max intervals, and repeated maximal efforts increase anaerobic enzyme activity and phosphocreatine stores. Most cyclists benefit from periodizing their training to focus on CP development in base phases and W-prime development in competition phases.
What are the limitations of the two-parameter CP model?
The two-parameter CP model has several important limitations that users should understand. First, it assumes CP can be sustained indefinitely, which is not physiologically true because glycogen depletion, dehydration, and thermal regulation limit real-world endurance to several hours. Second, the model overestimates power for very short durations under 2 minutes because it does not account for neuromuscular power limits and the lag in anaerobic energy system activation. Third, pacing errors in the test efforts directly affect calculated values, making the results sensitive to test quality. Fourth, CP and W-prime are not truly constant but vary with factors like fatigue state, hydration, temperature, and altitude. Three-parameter models and exponential models address some of these limitations but require more test data. Despite these limitations, the two-parameter model remains the most practical and widely used approach for training prescription.
How often should I retest my Critical Power values?
You should retest your Critical Power and W-prime values every 6 to 8 weeks during focused training blocks, or whenever you notice that your training zones feel inappropriately easy or hard. Significant changes in fitness, body weight, or training focus all warrant retesting. After a period of reduced training or illness, retest before resuming structured training to ensure your zones reflect current fitness. Some software platforms like Golden Cheetah and WKO5 can continuously estimate CP from your power meter data without requiring formal tests, using your best recent performances across multiple durations. If formal testing, always perform tests in consistent conditions: same warm-up, similar time of day, adequate recovery beforehand, and controlled nutrition and hydration. Document your conditions to make comparisons between tests more meaningful.
References
Reviewed for accuracy by Sher, Sports Science & Nutrition Specialist · Editorial policy
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