Introduction
Every electrical engineer needs a solid understanding of per unit (p.u.) systems. The per unit system is a normalization method widely used in power system analysis to simplify calculations involving transformers, generators, and transmission lines. Instead of dealing with different voltage levels and base values across a power network, the per unit system allows you to work with dimensionless quantities that make comparisons and calculations far more manageable.
In this article, we will cover the fundamentals of the per unit system, walk through the conversion process step by step, and work through practical examples that mirror what you would encounter in university exams or professional practice.
Before reading this article, you should be familiar with basic circuit analysis (Ohm’s Law, Kirchhoff’s Laws) and have a working knowledge of transformer operation. If you need a refresher, check out our Transformer Basics article.
What Is the Per Unit System?
The per unit system expresses electrical quantities as a ratio of their actual value to a chosen base value. In simple terms:
This means a voltage of 220 V with a base voltage of 11 kV would be expressed as:
While this specific example uses an unusual ratio, the principle applies universally across all electrical quantities β voltage, current, power, and impedance. The beauty of this system becomes obvious when analysing multi-voltage power networks with transformers, as we’ll demonstrate shortly.
Choosing Base Values
In any power system, you need to select two independent base quantities. The standard approach is to choose:
- Base Power (Sbase) β typically 1 MVA or 100 MVA for system-level studies
- Base Voltage (Vbase) β the nominal voltage level of the section you are analysing
Once these two are chosen, the remaining base quantities are derived:
Three-Phase Considerations
When working with three-phase systems, it’s important to be consistent. The convention most commonly used in power system analysis is:
- Base power is specified as three-phase total power (S3Ο,base)
- Base voltage is specified as the line-to-line RMS voltage (VLL,base)
- Per unit values are the same whether calculated on a per-phase or three-phase basis
Always choose the same Sbase for the entire system. Choose Vbase for each voltage level based on the nominal ratings of the transformers connecting those levels. This is what eliminates transformers from the per unit equivalent circuit.
Worked Example
Let’s work through a practical example. Consider the following system:
- A generator rated at 50 MVA, 11 kV with a subtransient reactance of 0.2 p.u. on its own base
- A step-up transformer rated at 50 MVA, 11/132 kV with a leakage reactance of 0.1 p.u.
- A transmission line with an impedance of j60 Ξ©
Our goal is to draw the complete per unit impedance diagram using a common base of 100 MVA and 11 kV on the generator side.
Step 1: Convert Generator Reactance
The generator’s reactance is given on its own base (50 MVA). We need to convert it to the new base (100 MVA):
Step 2: Convert Transformer Reactance
Step 3: Calculate Transmission Line in Per Unit
First, we need the base impedance at the transmission line voltage level (132 kV):
Advantages of the Per Unit System
| Advantage | Description |
|---|---|
| Transformer elimination | Ideal transformers disappear from the per unit equivalent circuit, simplifying analysis significantly. |
| Easy comparison | Equipment of different ratings can be directly compared using their per unit impedance values. |
| Manufacturing consistency | Similar machines from different manufacturers have very similar per unit impedance values. |
| Simplified calculations | Three-phase power calculations simplify because β3 factors cancel out in per unit. |
β‘ Key Takeaway
The per unit system is not just a mathematical convenience β it is the standard language of power system engineering. Mastering base selection, base conversion, and per unit circuit construction is essential for everything from short circuit analysis to load flow studies. The examples in this article form the foundation for more advanced topics we cover in our quizzes.
Common Mistakes to Avoid
When working with per unit calculations, these are the most frequent errors we see students make:
- Forgetting to convert to a common base β Always ensure all equipment impedances are expressed on the same MVA and voltage base before connecting them in a circuit diagram.
- Mixing single-phase and three-phase bases β Be consistent. If you use three-phase base power, derive all other quantities from that.
- Using line-to-neutral voltage for Vbase β By convention, base voltage is always the line-to-line value.
- Incorrect voltage base selection across transformers β The base voltage on each side of a transformer should follow the transformer’s turns ratio.
Examination questions often deliberately give equipment data on different bases. Always be the first step β convert everything to your chosen common base before proceeding with any analysis.
Summary
In this article we covered:
- The definition and purpose of the per unit system
- How to choose and derive base values for voltage, current, power, and impedance
- The base conversion formula for changing between different MVA and voltage bases
- A complete worked example converting a generator, transformer, and transmission line to a common base
- The key advantages and common mistakes when working with per unit quantities
The per unit system is the foundation for virtually all power system analysis techniques. In our next articles, we’ll build on this foundation to cover short circuit fault analysis and load flow studies.
Ready to check your understanding? Head over to our interactive quiz platform and take the Per Unit Systems quiz. You’ll encounter similar problems with step-by-step feedback.
0 thoughts on “The per-unit System”
Very interesting, please Expand more.
Great breakdown of a critical component!
Too often, CTs are overlooked despite being the eyes and ears of the entire protection and metering scheme. I appreciate how you clearly connected their role to the three pillars, protection, measurement, and control, while also emphasizing safety.
A well-written and accessible piece for both engineers and newcomers to the power industry. Keep up the excellent content!