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PCB Thermal Via Array Resistance & Temperature Rise Formula
Calculate Hollow Plated-Copper Barrel Ring Area (mm²), Single-Via Vertical Thermal Resistance (°C/W), Parallel N-Via Array Resistance (°C/W), and Exposed-Pad Board Delta_T (°C).
Read the guide →Financial & AccountingPDURATION & RRI Formulas: Periods to Reach Target Value & Equivalent Return
Calculate the exact number of compounding periods required for an investment to grow from PV to FV (PDURATION) and the equivalent compound growth rate over N periods (RRI) using exact logarithmic and power identities.
Read the guide →Private Equity, Venture Capital & Institutional LP ReportingPE & Venture Fund DPI, RVPI, Gross TVPI & Net TVPI Formula
Calculate ILPA Institutional LP Fund Multiples: Realized Cash-on-Cash DPI (Distributions / Paid-In Capital), Unrealized RVPI (Residual NAV / PIC), Gross Total Value to Paid-In (TVPI), and Accrued-Carry-Adjusted LP Net TVPI.
Read the guide →Private Equity Partnership Agreements, LPA Waterfalls & GP ClawbackPE American Deal-by-Deal Waterfall & GP Clawback Formula
Calculate Private Equity American (Deal-by-Deal) Early Carried Interest Paid, True Whole-Fund Earned Carry After Later Realized Write-Offs, Gross GP Clawback Liability, and Out-of-Pocket GP Cash Beyond Escrow.
Read the guide →Private Equity & Fund AccountingPE European Waterfall, GP Catch-Up & Carried Interest Formula
Calculate Limited Partner and General Partner distributions across all four European Waterfall tiers: Return of Capital, Preferred Hurdle, 100% GP Catch-Up Tranche H * c / (1 - c), and 80/20 Carried Interest.
Read the guide →Private Equity Fund Finance, Portfolio NAV Facilities & Credit RiskPE NAV Loan LTV Covenant, Drawdown Buffer & DPI Formula
Calculate Late-Life PE Portfolio NAV Facility Initial LTV (Debt / NAV), Hard Maintenance Covenant Minimum NAV (Debt / Max_LTV), Allowable Portfolio Valuation Drawdown Buffer, and Synthetic Recap DPI Uplift.
Read the guide →Private Equity Fund Finance, ILPA Reporting & Performance AttributionPE Subscription Line (Subline) IRR Boost vs TVPI Formula
Calculate Unlevered Deal IRR (%), Delayed LP Capital Call With Subline Interest ($M), Subline-Levered LP IRR (%), and Net LP TVPI Multiple Dilution (x).
Read the guide →Private Equity Fund Finance, ILPA Reporting & Subscription FacilitiesPE Subscription Line IRR Uplift vs TVPI Dilution Formula
Compare Unlevered Day-1 Capital-Call IRR against Subscription-Credit-Facility ("Sub-Line") Deferred LP Capital-Call IRR, Financed LP Drawdown Including Interest, and Sub-Line Net TVPI Multiple Dilution.
Read the guide →Dairy Standardization, Beverage/Feed Formulation & Mass-Balance BlendingPearson Square Dairy Butterfat & Two-Stream Blend Formula
Calculate Pearson Diagonal Parts of High-Concentration Stream, Diagonal Parts of Low-Concentration Stream, Exact High-Stream Mass (kg), and Exact Low-Stream Mass (kg).
Read the guide →Healthcare & ClinicalPECARN Pediatric Head Injury / Trauma CT Algorithm (Children >= 2 Years)
Apply Kuppermann's 42,412-child PECARN Pediatric Head Trauma Rule (children >= 2 years with GCS 14–15) to avoid unnecessary ionizing head CT radiation while catching clinically important TBI (ciTBI).
Read the guide →Healthcare & Clinical CalculatorsPediatric Appendicitis Score (PAS — Samuel 0–10 Scale & Ultrasound Triage)
Calculate Madan Samuel’s (2002) 10-point Pediatric Appendicitis Score weighting cough/percussion/hopping RLQ tenderness (+2) and RLQ palpation tenderness (+2) to guide ultrasound-first imaging vs. surgical consult.
Read the guide →Healthcare & ClinicalPediatric Trauma Score (PTS) Tepas Field Triage Formula (-6 to +12)
Calculate Joseph Tepas’s -6 to +12 Pediatric Trauma Score across Weight, Airway, Systolic BP, CNS Level, Cutaneous Wound, and Skeletal Fracture (scored +2, +1, or -1)—flagging PTS <= 8 for designated Pediatric Trauma Center transport.
Read the guide →Learn with AIPeer review a spreadsheet answer: value, formula and explanation
Use a two-person practice exercise to distinguish a correct number from a reusable formula. Discuss a cell, propose a correction and verify it before acceptance.
Read the guide →FinancialPEG Ratio & Benjamin Graham Number SQRT(22.5 * EPS * BVPS) stock valuation
Practice PEG and the Graham-number heuristic using consistent per-share inputs. Neither calculation establishes fair value or investment suitability.
Read the guide →Healthcare & ClinicalPELD Score Calculator: Pediatric End-Stage Liver Disease Transplant Formula
Calculate the UNOS Pediatric End-Stage Liver Disease (PELD) score for children under 12 from bilirubin, INR, albumin, age < 1 year, and growth failure (< -2 SD) using floor-clamped natural logarithms.
Read the guide →Healthcare & Clinical CalculatorsPELOD-2 (Pediatric Logistic Organ Dysfunction-2) Score & Mortality
Sum 5 pediatric organ-system dysfunction subscores (Neurologic, Cardiovascular, Renal, Respiratory, Hematologic; 0–33) and compute Leteurtre et al.’s (2013) PICU mortality probability = 100 / (1 + EXP(6.61 - 0.47 * PELOD-2)).
Read the guide →Engineering & SciencePeltier Thermoelectric Cooler (TEC) Net Cooling Power (Qc) & Optimal Current (Imax)
Model solid-state Peltier laser/CCD cooling modules via the 3-term cold-plate energy balance Q_c = S_m * T_c * I - 0.5 * I^2 * R_m - K_m * (T_h - T_c) (Watts) and maximum-cooling current I_max = (S_m * T_c) / R_m.
Read the guide →Electrochemical Engineering, Green Hydrogen & PEM / Alkaline ElectrolyzersPEM Electrolyzer Faradaic H2 Rate & LHV Efficiency Formula
Calculate Multi-Cell PEM / Alkaline Electrolyzer Stack DC Input Power (MW), Faradaic Hydrogen Mass Production Rate (kg H2/hr), Specific Stack Energy Consumption (kWh/kg H2), and Lower Heating Value (33.33 kWh/kg) Efficiency.
Read the guide →Electrochemical Powertrains, Hydrogen Fuel Cells & Polarization ModelingPEM Fuel Cell Polarization, LHV Efficiency & Heat Formula
Calculate Hydrogen PEM Fuel Cell Operating Voltage V_cell (V) After Activation and Area-Specific Ohmic Losses, Power Density (W/cm²), Anode-Utilization-Adjusted LHV Electrical Efficiency, and Coolant Radiator Waste-Heat Density (W/cm²).
Read the guide →EngineeringPeng-Robinson EOS: Acentric Kappa & Alpha(T) Cohesion Function
Calculate Peng–Robinson kappa, reduced temperature and alpha with unrounded helper cells, validated inputs and an explicit choice of acentric-factor correlation.
Read the guide →Engineering & Fluid MechanicsPenman Combination Equation Formula: Reservoir & Open-Water Evaporation (mm/day)
Combine net solar radiation energy balance E_rad and aerodynamic wind-vapor deficit drying power E_aero without needing surface water temperature via Howard L. Penman’s 1948 equation E_0 = (Delta * E_rad + gamma * E_aero) / (Delta + gamma).
Read the guide →Clinical Medicine & PhysiologyPenn State 2003b & Modified 2010 ICU Ventilator Calorie Formula: Mifflin + T_max + V_E
Estimate Resting Energy Expenditure in mechanically ventilated ICU patients when indirect calorimetry is unavailable using David Frankenfield’s ASPEN-recommended Penn State 2003b and Modified 2010 (Age >= 60 & BMI >= 30) equations.
Read the guide →Retirement Planning & Corporate Pension Risk TransferPension Lump-Sum Buyout vs Annuity Breakeven Formula
Compare a Corporate Pension Lump-Sum Buyout Offer against a Guaranteed Annuity by contrasting Naive Undiscounted Payback (Lump / Pmt) with True Discounted Breakeven Longevity -LN(1 - Lump*r/Pmt)/LN(1 + r) and Present-Value Advantage.
Read the guide →Healthcare & ClinicalPERC Rule-Out Criteria & Revised Geneva Score for Pulmonary Embolism
Evaluate Jeffrey Kline's 8-item Pulmonary Embolism Rule-Out Criteria (PERC = 0 rules out PE without D-dimer in low-pretest patients) alongside the objective Revised Geneva PE Score.
Read the guide →FinancialPercent of goal and budget variance formulas (Revenue vs Expense sign rules)
Calculate quota attainment %, remaining amount to goal, and favorable vs unfavorable FP&A variances for both revenue and cost centers.
Read the guide →Reporting & AggregationsPercent Sold, Remaining Allocation & Portfolio Weighted Completion Formula
Calculate SKU or event-tier percent sold (Sold / Allocation) with zero-allocation protection, oversold waitlist detection, and true volume-weighted portfolio percent sold.
Read the guide →PercentagesPercentage change in Excel: increases, decreases and a zero starting value
Compare monthly workshop bookings using the original month as the base. Distinguish a percentage change from an absolute difference and show an explicit result when the starting value is zero.
Read the guide →Descriptive statisticsPERCENTILE.INC function: calculate P50 and P90 service-level thresholds
Report 50th and 90th percentile latency or delivery times using a 0-to-1 k fraction, understand linear interpolation between points, and catch #NUM! errors.
Read the guide →Architectural Acoustics & Auditorium Timber Baffles, Ingard-Fok Perforated Resonant PanelPerforated Acoustic Panel Porosity & Peak Hz Formula
Calculate Square-Grid Perforation Open-Area Porosity (%), Ingard-Fok Hole-Interaction Effective Neck Length teff (mm), Resonant Peak Absorption Frequency f0 (Hz), and Wavelength (m).
Read the guide →Operations & LogisticsPeriodic Order Quantity (POQ) Time-Supply Interval & Discrete MRP Lot Sizing
Convert fractional Wilson EOQ into an integer MRP order cycle T_POQ = MAX(1, ROUND(EOQ / D_period, 0)) and evaluate discrete Periodic Order Quantity lot size and cycle cost.
Read the guide →Math & TrigonometryPermutations Without and With Repetition (PERMUT & PERMUTATIONA Formulas)
Count ordered sequences without replacement using FACT(n) / FACT(n - k) and with replacement using POWER(n, k), plus compute no-repeat collision probabilities.
Read the guide →StatisticalPERT three-point estimates and a fixed path’s duration uncertainty
Convert Optimistic, Most Likely, and Pessimistic estimates into PERT Weighted Mean (O + 4M + P)/6, Task SD (P - O)/6, and Quadrature Path Risk.
Read the guide →Panel Econometrics & Time SeriesPesaran CD Test for Cross-Sectional Dependence in Panels
Compute Pesaran’s (2004) cross-sectional dependence statistic CD = SQRT(2*T / (N*(N-1))) * SUM(rho_ij) and compare it against Breusch-Pagan LM across panel residual columns.
Read the guide →Operations & AnalyticsPeterson-Silver Variability Coefficient (V): EOQ vs. Dynamic Lot Sizing
Calculate Rein Peterson and Edward Silver’s squared-CV demand variability index V = (N * SUM(D^2)) / (SUM(D)^2) - 1 to decide when steady Economic Order Quantity (V < 0.25) beats Silver-Meal or Wagner-Whitin dynamic lot sizing (V >= 0.25).
Read the guide →Statistical & Clinical AnalysisPeto One-Step Pooled Odds Ratio Formula for Rare-Event Clinical Trials
Pool rare-event and zero-cell randomized trials without arbitrary 0.5 continuity corrections using Richard Peto’s O - E score and hypergeometric variance V: OR_Peto = EXP(Σ(O - E) / ΣV).
Read the guide →Engineering & SciencePetukhov Smooth-Pipe Darcy Friction Factor & Turbulent Nusselt Formula
Cut turbulent heat-exchanger tube error from ±25% (Dittus-Boelter) down to ±5% using Boris Petukhov’s (1970) momentum-heat analogy Nu_D and explicit Darcy friction factor f = (0.790 * LN(Re) - 1.64)^(-2).
Read the guide →Electrical & Energy Systems EngineeringPeukert Battery Discharge Runtime and Effective Capacity Formula
Calculate high-current battery discharge runtime t = H * (C_H / (I * H))^k and effective delivered amp-hours C_eff = I * t using the dimensionally balanced Peukert law.
Read the guide →EngineeringpH conversion and the logarithm of mean estimated hydrogen concentration
Compare arithmetic mean pH with the pH-scale value of a mean concentration under a dilute-solution approximation.
Read the guide →Civil, Structural & Geotechnical EngineeringPhilip Two-Term Soil Infiltration Formula: Sorptivity S, Gravity Factor A & Rate i(t)
Calculate cumulative rainfall/irrigation infiltration I(t) = S * SQRT(t) + A * t and instantaneous infiltration capacity i(t) = 0.5 * S / SQRT(t) + A using John R. Philip’s (1957) analytical solution of Richards’ unsaturated flow equation.
Read the guide →SciencePhotoelectric Effect Kinetic Energy & Compton X-Ray Wavelength Shift
Calculate a single-photon photoelectric energy threshold and maximum kinetic energy, and separately explore the ideal free-electron Compton wavelength shift.
Read the guide →Healthcare & ClinicalPHQ-9 Depression Severity Score (0–27) & Item-9 Safety Alert Formula
Calculate Kurt Kroenke & Robert Spitzer's 9-item Patient Health Questionnaire (PHQ-9, 0 to 27 points), classify depression severity into Minimal (0–4), Mild (5–9), Moderate (10–14), Moderately Severe (15–19), and Severe (20–27), and flag any positive Item 9 safety response.
Read the guide →MathPI function: calculate circle area, pipe circumference, and tank volume
Use full 15-digit precision PI() instead of hardcoded 3.14 for engineering areas, circumferences, and cylindrical volumes.
Read the guide →Lookup & ReferencePick a Value From a Fixed Set of Options (INDEX + MOD / RANDBETWEEN)
Select an allowed tier, shift size, or discrete test parameter (such as {10, 25, 50, 100}) from a fixed lookup list using INDEX(Options, selector)—either deterministically via MOD(ID - 1, COUNTA(Options)) + 1 or randomly via RANDBETWEEN(1, COUNTA(Options)).
Read the guide →Math & StatisticalPie & Doughnut Chart Percentage Share & Small-Slice "Other" Rollup Formula
Compute exact category percentage shares ROUND(B2 / SUM($B$2:$B$6) * 100, 2) and consolidate unreadable < 5% tail categories into a single "Other" slice using SUMIFS(B2:B6, B2:B6, "<" & (0.05 * SUM(B2:B6))).
Read the guide →Leveraged Finance, Private Credit & Mezzanine DebtPIK-Toggle Mezzanine Note Balloon Accretion & Cash-Save Formula
Calculate Payment-in-Kind (PIK) Toggle Mezzanine Annual Cash Liquidity Preserved, Compounded Balloon Redemption Principal P0*(1 + r_cash + r_stepup)^T, Cumulative PIK Accretion, and Excess Compounding Cost vs Cash Pay.
Read the guide →Healthcare & Clinical CalculatorsPIM-3 (Pediatric Index of Mortality 3) PICU Admission Risk Formula
Calculate first-hour PICU standardized mortality probability from Straney et al.’s (2013) PIM-3 logistic model incorporating pupillary reaction, mechanical ventilation, base excess, and diagnosis risk tier.
Read the guide →Optoelectronics, Fiber-Optic Receivers & Photon CountingPIN Photodiode Responsivity & Shot-Noise SNR Formula
Convert Photodiode Quantum Efficiency eta and Wavelength lambda (um) into Responsivity R = (eta * lambda_um)/1.24 (A/W), Signal Photocurrent (uA), Combined Poisson Shot + TIA Thermal RMS Noise, and Electrical SNR (dB).
Read the guide →Probabilistic Forecasting & Supply Chain AnalyticsPinball Quantile Loss & Winkler Interval Score Formula
Evaluate probabilistic quantile forecasts via asymmetric Pinball Loss MAX(tau*(y - q), (tau - 1)*(y - q)) and audit (1 - alpha) prediction intervals with Robert Winkler’s (1972) Interval Score.
Read the guide →Process Heat Integration, Pinch Technology & Energy EfficiencyPinch Analysis Minimum Hot & Cold Utility (Delta Tmin) Formula
Calculate Hot-Stream Cooling Duty (kW), Maximum Process-to-Process Heat Recovery (kW), Minimum Cooling-Water Utility Q_C,min (kW), and Minimum Steam Hot Utility Q_H,min (kW).
Read the guide →FinancialPiotroski F-Score formula: 9-point fundamental value investing checklist
Practice a nine-signal accounting screen using explicitly prepared financial metrics. A high score is not a recommendation or a guaranteed return.
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