Per-Unit System: Changing Base Across Transformers and Networks

• fundamentalsper-unitbase-changetransformers

Purpose and Scope

This article addresses changing base quantities in per-unit systems, particularly when models span multiple voltage levels, transformers, or data sources. It builds directly on the concepts of base quantities covered in Per-Unit System: Base Quantities and Their Relationships and per-unit impedance derivation discussed in Per-Unit Impedance from Physical Parameters.

The focus is practical: understanding when a base change is required, why it works, and how to apply it correctly without corrupting physical meaning.

Why Base Changes Are Necessary

In an idealised study, all equipment data would already be expressed on a single, consistent base. In practice, this is rarely the case.

Base changes are required when:

  • Equipment data is provided on different MVA bases
  • Impedances are specified at different voltage levels
  • Transformer models connect networks with distinct base voltages
  • Models from different sources are combined

Failing to perform base changes correctly is one of the most common causes of scaling errors in both RMS and EMT studies.

Physical Interpretation of Base Changes

A base change does not alter the physical impedance of an element. It simply rescales the numerical representation of that impedance relative to a new set of base quantities.

The underlying principle is invariance: the actual voltage, current, and power relationships must remain unchanged after the base conversion.

This is why base changes can be applied algebraically, without revisiting the original physical parameters, provided the original per-unit values were defined consistently.

General Base-Change Formula

For an impedance expressed in per-unit on an old base, the equivalent per-unit value on a new base is:

Zpu,new=Zpu,old(Sbase,newSbase,old)(Vbase,oldVbase,new)2Z_{pu,new} = Z_{pu,old} \left( \frac{S_{base,new}}{S_{base,old}} \right) \left( \frac{V_{base,old}}{V_{base,new}} \right)^2

This single expression captures both power-base and voltage-base scaling.

The formula can be understood as two multiplicative factors:

  • Power scaling: Sbase,newSbase,old\frac{S_{base,new}}{S_{base,old}} — a ratio greater than 1 increases the per-unit value
  • Voltage scaling: (Vbase,oldVbase,new)2\left( \frac{V_{base,old}}{V_{base,new}} \right)^2 — note the squared term and inverted ratio

Special Case: Same Voltage Base

If the voltage base remains unchanged and only the apparent power base changes, the formula simplifies to:

Zpu,new=Zpu,old(Sbase,newSbase,old)Z_{pu,new} = Z_{pu,old} \left( \frac{S_{base,new}}{S_{base,old}} \right)

This situation commonly arises when normalising manufacturer data to a system-wide MVA base. For example, converting transformer impedance data from nameplate ratings to a 100 MVA study base.

Transformer Base Propagation

Transformers provide a natural reference for base voltage propagation. Once a base voltage is defined on one side of a transformer, the base voltage on the other side must follow the transformer ratio.

When this rule is followed, no base change is required across an ideal transformer. The per-unit impedance of the transformer remains the same on both sides.

Errors occur when:

  • Base voltages are redefined manually on each side
  • Transformer ratios are applied twice
  • Voltage bases are mixed with physical voltages

Worked Example: Base Change Across Voltage Levels

Consider a transformer with the following data:

  • Transformer impedance: 0.1 pu0.1 \, pu on a 50 MVA base
  • Rated voltages: 220 kV / 110 kV

Assume the study uses:

  • Sbase,new=100 MVAS_{base,new} = 100 \, \text{MVA}

Since base voltages follow the transformer ratio, only the power base changes:

Zpu,new=0.1×10050=0.2 puZ_{pu,new} = 0.1 \times \frac{100}{50} = 0.2 \, pu

The physical impedance remains unchanged; only the per-unit representation differs.

Common Base-Change Pitfalls

In practice, base-change errors often arise from:

  • Applying base changes when none are required
  • Failing to change base when combining models
  • Using inconsistent voltage bases across a transformer
  • Mixing per-unit values defined on different implicit bases

These issues are especially difficult to diagnose once embedded in large EMT models.

Calculator Reference

A Per-Unit Base-Change Calculator can enforce correct scaling and provide transparency when converting between bases.

👉 Per-Unit Base-Change Calculator

Position Within the Article Series

This article is part of the per-unit system series:

  1. Per-Unit System: The Concept and Why It Works — foundational theory
  2. Per-Unit System: Base Quantities and Their Relationships — deriving base values
  3. Per-Unit Impedance from Physical Parameters — converting R, L, C to per-unit
  4. This article — changing between bases
  5. Per-Unit System: Common Mistakes in EMT and RMS Studies — common mistakes and how to avoid them

Reflective Questions

  1. Are base changes explicitly documented in your models?
  2. Have transformer-related scaling errors appeared in your studies?
  3. Would an automated base-change tool reduce review effort?