The per-unit system is a normalization method used in power system analysis to simplify calculations by expressing system quantities as dimensionless ratios of their actual values to chosen base values. This technique eliminates the complexity of dealing with different voltage levels, makes transformer calculations transparent, and reveals the true electrical characteristics of power system components independent of their rated values. Engineers in utilities, industrial power distribution, and equipment manufacturing rely on per-unit analysis for fault studies, stability analysis, and equipment specification.
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System Diagram
Per-Unit System Interactive Calculator
Per-Unit System Equations
Fundamental Per-Unit Conversion
Quantitypu = Quantityactual / Quantitybase
Where:
Quantitypu = dimensionless per-unit value
Quantityactual = actual physical value (V, A, Ω, VA, etc.)
Quantitybase = chosen reference value in same units
Three-Phase Base Quantities
Sbase,3φ = chosen MVA base (three-phase)
Vbase,L-L = chosen kV base (line-to-line)
Ibase = Sbase,3φ / (√3 × Vbase,L-L)
Zbase = (Vbase,L-L)2 / Sbase,3φ
Where:
Sbase,3φ = base apparent power in MVA
Vbase,L-L = base line-to-line voltage in kV
Ibase = base current in amperes
Zbase = base impedance in ohms
Impedance Conversion to Per-Unit
Zpu = ZΩ / Zbase = ZΩ × Sbase / (Vbase)2
Where:
Zpu = impedance in per-unit
ZΩ = impedance in ohms
Zbase = base impedance in ohms
Sbase = base power in VA (same units as Vbase2)
Vbase = base voltage in volts
Base Change for Impedance
Zpu,new = Zpu,old × (Sbase,new / Sbase,old) × (Vbase,old / Vbase,new)2
Where:
Zpu,new = impedance on new base
Zpu,old = impedance on old base
Sbase,new, Sbase,old = new and old power bases
Vbase,new, Vbase,old = new and old voltage bases
Transformer Impedance Conversion
Zpu,system = (Z% / 100) × (Sbase,system / Srated,transformer)
Where:
Zpu,system = transformer impedance on system base
Z% = transformer percent impedance (nameplate)
Sbase,system = system MVA base
Srated,transformer = transformer MVA rating
Power Base Change
Spu,new = Spu,old × (Sbase,old / Sbase,new)
Power quantities scale linearly with base change
Theory & Engineering Applications
The per-unit system represents one of the most elegant normalization methods in power systems engineering, transforming complex multi-voltage-level networks into dimensionless systems where equipment characteristics can be compared directly regardless of their rated values. By expressing voltages, currents, impedances, and powers as fractions of chosen base quantities, engineers eliminate the need to track voltage transformation ratios through transformers and can identify abnormal operating conditions at a glance—values significantly exceeding 1.0 pu indicate overload or fault conditions, while values near 1.0 represent normal operation.
Mathematical Foundation and Base Quantity Selection
The per-unit system begins with the selection of two independent base quantities in a power system—typically base power (Sbase) and base voltage (Vbase). Once these are chosen, all other base quantities are mathematically determined. For three-phase systems, the base power is usually selected as a round number MVA value (commonly 100 MVA for large systems, or the largest equipment rating in smaller systems), while base voltage is chosen as the nominal line-to-line voltage at each voltage level in the network. The base current then follows from Ibase = Sbase / (√3 × Vbase), and crucially, the base impedance is Zbase = Vbase2 / Sbase when using consistent units (kV and MVA yield base impedance in ohms).
A critical but often overlooked aspect of per-unit calculations is that the base must be chosen separately for each voltage level in a multi-level system. The base MVA remains constant throughout the system, but the base kV changes at each transformer. This creates a natural impedance scaling that makes transformer impedances appear identical on either side of the transformation—a 0.08 pu reactance transformer has the same per-unit value whether viewed from the high-voltage or low-voltage side, even though the ohmic values differ by the square of the turns ratio. This property dramatically simplifies fault calculations and stability studies.
Impedance Transformations and the Quadratic Scaling Law
When changing impedance values from one per-unit base to another, the conversion follows a quadratic relationship with voltage and a linear relationship with power: Zpu,new = Zpu,old × (Snew/Sold) × (Vold/Vnew)2. This non-intuitive scaling often confuses engineers transitioning from ohmic calculations. The quadratic voltage term arises because impedance inherently relates voltage to current, and both of these quantities scale with the base choice. When voltage base doubles, the current base halves (for constant power base), causing the impedance base to quadruple. Understanding this relationship is essential when combining manufacturer data given on equipment rating bases with system studies performed on a common system base.
Transformer percent impedance values, commonly provided on equipment nameplates, represent a special case of per-unit values where the base is the transformer's own MVA and voltage ratings. Converting these to a system per-unit base requires only multiplication by the ratio Ssystem/Stransformer, since the voltage bases are already aligned. A 50 MVA transformer with 8% impedance operating in a system with 100 MVA base has an impedance of 0.08 × (100/50) = 0.16 pu on the system base—it appears "stronger" in the larger base reference frame.
Practical Advantages in Fault Analysis
Per-unit analysis reveals its greatest value in short-circuit calculations, where equipment damage potential must be assessed across voltage levels. When a three-phase bolted fault occurs, the fault current in per-unit equals the reciprocal of the total impedance to the fault point: Ifault,pu = 1 / Ztotal,pu. This simple relationship holds regardless of voltage level. Converting back to actual amperes requires multiplication by the base current at that location, but the per-unit value immediately indicates severity—a fault current of 20 pu suggests the impedance to the fault is only 0.05 pu, indicating a close-in fault on a strong system requiring high interrupting capacity.
Generator and motor impedances typically range from 0.10 to 0.25 pu (subtransient reactance), transformers from 0.05 to 0.15 pu, and transmission lines from 0.001 to 0.05 pu per mile on 100 MVA base. These characteristic ranges allow engineers to quickly estimate fault levels and identify data errors—a transformer impedance of 2.5 pu would immediately flag a base mismatch or incorrect conversion.
Worked Example: Multi-Voltage Industrial Distribution System
Consider an industrial facility supplied by a 138 kV utility connection with a fault MVA of 2,500 MVA at the point of common coupling. The facility has a 138/13.8 kV, 75 MVA transformer with 9.5% impedance connecting to a 13.8 kV distribution bus. A 13.8/0.48 kV, 2.5 MVA transformer with 5.8% impedance supplies a critical motor control center. We need to determine the available fault current at the 480V bus using 100 MVA system base.
Step 1: Calculate utility source impedance. The utility short-circuit capacity of 2,500 MVA represents the reciprocal of source impedance on its own base. Converting to 100 MVA system base: Zutility,pu = Sbase,system / Ssc,utility = 100 / 2500 = 0.04 pu. This impedance is referenced to the 138 kV base voltage.
Step 2: Convert main transformer impedance to system base. The 75 MVA transformer has Z = 0.095 pu on its own rating. On the 100 MVA system base: ZT1,pu = 0.095 × (100/75) = 0.1267 pu. The voltage bases (138 kV and 13.8 kV) are already correctly selected as the transformer nominal voltages, so no voltage correction is needed.
Step 3: Convert distribution transformer impedance to system base. The 2.5 MVA transformer has Z = 0.058 pu on its own rating. On the 100 MVA system base: ZT2,pu = 0.058 × (100/2.5) = 2.32 pu. This high impedance value is typical for small transformers when viewed on large system bases.
Step 4: Calculate total impedance to 480V bus. The impedances add in series: Ztotal,pu = 0.04 + 0.1267 + 2.32 = 2.4867 pu. Notice how the small distribution transformer dominates the total impedance, illustrating a key power system principle—fault currents are primarily limited by the smallest transformer in the path.
Step 5: Calculate fault current in per-unit and amperes. The three-phase fault current: Ifault,pu = 1 / 2.4867 = 0.4021 pu. To convert to amperes at 480V, calculate the base current: Ibase,480V = (100 × 106) / (√3 × 480) = 120,281 A. Therefore, Ifault,actual = 0.4021 × 120,281 = 48,365 amperes. This 48.4 kA fault level requires circuit breakers rated for at least 50 kA interrupting capacity, allowing for safety margin and future system growth.
Step 6: Validate the result. As a sanity check, the 2.5 MVA transformer base current is (2.5 × 106) / (√3 × 480) = 3,007 A. The fault current represents 48,365 / 3,007 = 16.1 times the transformer rated current, which corresponds to 1 / 0.058 = 17.2 times on the transformer's own base (the slight difference is due to upstream impedance). This confirms our calculation consistency.
Per-Unit System in Stability and Dynamic Analysis
Beyond fault studies, per-unit representation enables power system stability analysis by normalizing the swing equations that govern generator rotor dynamics. The mechanical power input, electrical power output, and inertia constant are all expressed in per-unit on the machine's own MVA base, creating differential equations with well-behaved numerical coefficients. Generator inertia constants typically range from 2 to 9 MW-seconds per MVA, and expressing these on a per-unit basis allows direct comparison of rotational energy storage across machines of vastly different sizes. A 500 MW generator with H = 4.0 seconds has the same relative inertial response as a 50 MW generator with H = 4.0 seconds when subjected to proportional disturbances.
Motor starting analysis also benefits from per-unit normalization. Induction motors draw 5 to 7 times rated current at starting, expressed as 5 to 7 pu. The voltage dip during starting can be estimated as ΔVpu = Istarting,pu × Zsystem,pu, immediately revealing whether adjacent loads will experience unacceptable voltage sag. This calculation, trivial in per-unit, becomes cumbersome when tracking actual voltages and impedances across transformers.
Advanced Considerations and Common Pitfalls
One subtlety that catches even experienced engineers is the treatment of three-phase versus single-phase quantities. Per-unit calculations in three-phase systems inherently use three-phase MVA bases and line-to-line kV bases, but the resulting impedances automatically apply in the positive-sequence network for balanced analysis. Sequence impedances (positive, negative, zero) all use the same base impedance value, though their actual magnitudes differ dramatically—zero-sequence impedances are typically 2 to 10 times larger than positive-sequence values for transformers and rotating machines.
Another critical consideration is off-nominal tap positions on transformers. A transformer with a 1.05 pu tap setting effectively changes the voltage transformation ratio, creating a 1:1.05 turns ratio on the per-unit system rather than the ideal 1:1. This requires insertion of an ideal transformer element in the per-unit model, complicating load flow calculations but accurately representing the physical system behavior.
When working with manufacturer data, always verify whether impedances are given in ohms, percent, or per-unit, and confirm the base quantities. Mixing bases is the most common source of catastrophic errors in fault studies—using a machine's per-unit impedance directly on a system base without conversion can underestimate fault currents by an order of magnitude, leading to grossly undersized protective equipment. Modern power system analysis software automates these conversions, but understanding the underlying mathematics remains essential for validating results and troubleshooting anomalies.
Practical Applications
Scenario: Protection Engineer Coordinating Overcurrent Relays
Marcus, a protection engineer at a regional utility, is coordinating time-overcurrent relays across a 230/115/34.5 kV substation. The relay settings must be specified in multiples of pickup current, but the fault study report provides fault currents at each bus in amperes at different voltage levels—18,400 A at 230 kV, 41,200 A at 115 kV, and 12,800 A at 34.5 kV. To establish consistent coordination curves, Marcus converts all fault currents to per-unit on a common 100 MVA base. At 230 kV, base current is 251 A, making the fault current 73.3 pu. At 115 kV, base current is 502 A, yielding 82.1 pu. At 34.5 kV, base current is 1,674 A, resulting in 7.64 pu. This per-unit representation immediately reveals that the 115 kV bus has the highest fault contribution relative to system impedance, requiring the most robust protective settings. The normalized values allow Marcus to set relay pickup at consistent per-unit multiples (typically 1.5 to 2.0 pu) across voltage levels, ensuring proper discrimination without laboriously tracking transformation ratios.
Scenario: Design Engineer Sizing a Generator Step-Up Transformer
Jennifer is designing the electrical system for a 65 MW combined-cycle power plant. The generator produces 13.8 kV and must connect to a 138 kV transmission system through a step-up transformer. Generator manufacturers quote reactances in per-unit on machine rating (X"d = 0.18 pu on 72 MVA base), while the utility requires fault study data on a 100 MVA system base. Jennifer uses the per-unit calculator to convert the generator reactance: 0.18 × (100/72) = 0.25 pu on system base. She's evaluating two transformer options: a 75 MVA unit with 8.5% impedance and an 80 MVA unit with 9.2% impedance. Converting both to system base yields 0.085 × (100/75) = 0.113 pu and 0.092 × (100/80) = 0.115 pu respectively. The total impedance from generator to transmission system will be approximately 0.25 + 0.113 = 0.363 pu with the first option, limiting fault contribution to 2.75 pu (about 13,900 A at 138 kV). This per-unit analysis allows Jennifer to quickly verify that either transformer choice will limit fault current below the utility's maximum interconnection requirement of 3.0 pu without tedious conversion between ohmic values at different voltage levels.
Scenario: Facility Manager Investigating Motor Starting Voltage Dip
Roberto manages a manufacturing plant experiencing dimming lights and nuisance trips when a large 800 HP compressor motor starts. The 460V motor draws 4,200 amperes during starting, while the 2,000 kVA service transformer is fed from a utility supply with 0.06 pu source impedance and has 5.2% transformer impedance, both referenced to the transformer's own 2 MVA base. Roberto uses the per-unit calculator to analyze the voltage dip. The motor starting current is 4,200 A, compared to a base current at 460V of (2,000,000)/(√3 × 460) = 2,509 A, giving 4,200/2,509 = 1.674 pu starting current. The total system impedance is 0.06 + 0.052 = 0.112 pu. The voltage drop during starting is approximately 1.674 × 0.112 = 0.188 pu, or 18.8%, dropping the bus voltage to 0.812 pu (374V). This explains the dimming lights and confirms that adding a reduced-voltage starter (lowering starting current to perhaps 0.8 pu) would reduce voltage dip to 0.8 × 0.112 = 0.090 pu (9%), keeping voltage above 0.91 pu and eliminating the nuisance trips. The per-unit calculation revealed the problem severity in minutes rather than hours of ohmic impedance calculations across the transformer.
Frequently Asked Questions
▼ Why do we use per-unit instead of actual ohms, volts, and amperes?
▼ How do I choose appropriate base values for my system?
▼ Why does impedance change with the square of voltage base but power changes linearly?
▼ What's the difference between percent impedance and per-unit impedance?
▼ Can I use per-unit for unbalanced systems and sequence networks?
▼ What are typical per-unit impedance values for different equipment types?
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About the Author
Robbie Dickson — Chief Engineer & Founder, FIRGELLI Automations
Robbie Dickson brings over two decades of engineering expertise to FIRGELLI Automations. With a distinguished career at Rolls-Royce, BMW, and Ford, he has deep expertise in mechanical systems, actuator technology, and precision engineering.
