Modeling of Single Phase Transformers | Electrical Machines - Electrical Engineering (EE) PDF Download

Introduction

This chapter develops a practical steady-state model of a single-phase transformer by relaxing the simplifying assumptions of the ideal transformer (lossless windings and no leakage flux). The practical model accounts for

• winding resistances (ohmic losses) of primary and secondary windings;
• leakage reactances (leakage flux producing series leakage inductances) of both windings;
• magnetizing reactance representing finite core permeability (magnetizing current required to establish the core flux);
• core (iron) loss represented by a resistance across the magnetizing branch, accounting for hysteresis and eddy-current losses.

Basic circuit variables and first equations

Consider a practical transformer with primary winding turns N₁, secondary winding turns N₂, primary applied voltage v₁(t), and secondary terminal voltage v₂(t). Let the winding resistances be r₁, r₂, and the leakage inductances (series) be l₁, l₂. The winding currents are i₁(t) and i₂(t). The induced EMFs in the primary and secondary are e₁(t) and e₂(t), respectively.

The time-domain voltage equations for each winding (including series resistance and leakage inductance) are

v₁(t) = r₁ i₁(t) + l₁ (d i₁/d t) + e₁(t)

e₂(t) = r₂ i₂(t) + l₂ (d i₂/d t) + v₂(t)

As for an ideal transformer, the induced EMFs are proportional to turns:

e₁ : e₂ = N₁ : N₂

Magnetizing branch and core losses

Even when the secondary is open (i₂ = 0), a practical transformer draws a small current from the supply - the no-load current - because of finite core permeability and the core losses. To model this:

• Place a magnetizing reactance (or inductance) X_m = ωL_m (or L_m) across the primary induced EMF e₁. This branch draws the magnetizing current i_m, which produces the core flux φ_m.
• Place a core-loss resistance R₀ in parallel with X_m to represent the combined hysteresis and eddy-current losses; the corresponding branch current is i_c.

Explanation of phase relationships:

• The induced voltage e₁ is proportional to the time derivative of flux and therefore leads the flux φ_m by 90°.
• The magnetizing current i_m (through the inductance) lags e₁ by 90° and is therefore approximately in phase with flux φ_m.
• The core-loss current i_c is approximately in phase with e₁ (resistive component) because R₀ models dissipative losses.

The complete practical equivalent circuit therefore places primary resistance r₁ and leakage reactance jX_l1 in series with an ideal transformer; across the primary side of the ideal transformer is the shunt branch R₀ ∥ jX_m. The secondary has its series resistance r₂ and leakage reactance jX_l2, and the load Z_L is connected to the secondary terminals. The no-load current is i₀ = i_m + i_c, and the primary line current is the phasor sum of i₀ and the current reflected from the load through the ideal transformer.

Steady-state phasor equations

In phasor notation the winding voltage equations become

V₁ = r₁I₁ + jX_l1I₁ + E₁

E₂ = r₂I₂ + jX_l2I₂ + V₂

And the load relation is

V₂ = Z_L I₂

Magnetizing and core-loss currents (phasors) are

I_m = E₁ / (jX_m)

I_c = E₁ / R₀

The induced voltages obey

E₂ = (N₂/N₁) E₁

The current reflected into the primary by the load current is

I₂' = (N₂/N₁) I₂

Phasor diagram (qualitative)

Take the core flux phasor φ_m as the reference. Then:

• E₁ and E₂ both lead φ_m by 90°.
• The load current I₂ typically lags E₂ by the load power factor angle φ (for an inductive load), or leads for capacitive load.
• The referred load current I₂' is a scaled version of I₂ and has the same phase as I₂.
• The primary current I₁ is the phasor sum I₁ = I₂' + I_m + I_c, which gives the resultant primary phasor and, via V₁ = E₁ + I₁(r₁ + jX_l1), the applied voltage phasor V₁.

Why refer one side to the other?

The exact circuit contains an ideal transformer between the two sides. Because an ideal transformer changes voltages and currents, it is awkward to write KCL/KVL equations directly across the transformer. To enable standard circuit analysis, we eliminate the ideal transformer by referring all secondary quantities to the primary side (or vice versa). The referred circuit is algebraically equivalent and preserves voltages, currents and power relations when transformed correctly.

Referring secondary to primary (primary-referred or source-side equivalent)

Let a = N₁/N₂ be the turns ratio (primary turns per secondary turn). The secondary current I₂ produces a current in the primary of

I₂' = (N₂/N₁) I₂

From the viewpoint of the primary terminals the impedance seen on the secondary (Z_L + r₂ + jX_l2) is referred as follows:

Z_L' = a² Z_L

and similarly

r₂' = a² r₂

X_l2' = a² X_l2

Thus the exact equivalent circuit referred to the primary side places r₂', jX_l2' and the referred load Z_L' in series and connects this series branch across the same primary terminals that see E₁. The shunt branch R₀ ∥ jX_m and the primary series elements r₁, jX_l1 remain as before.

Power preservation check (algebraic demonstration):

I₂' = (N₂/N₁) I₂

r₂' = (N₁/N₂)² r₂ = a² r₂

Therefore

I₂'² r₂' = [(N₂/N₁) I₂]² · [ (N₁/N₂)² r₂ ]

The scaling factors cancel and

I₂'² r₂' = I₂² r₂

So the copper loss in the secondary is preserved in the referred circuit.

Voltage relation for the referred load:

V₂' = I₂' Z_L'

Substitute I₂' = (N₂/N₁) I₂ and Z_L' = (N₁/N₂)² Z_L

Therefore V₂' = (N₁/N₂) V₂

and the apparent power is preserved since V₂' I₂' = V₂ I₂.

Referring primary to secondary (secondary-referred or load-side equivalent)

Similarly, to refer primary quantities to the secondary side define b = N₂/N₁. Then primary impedances are transformed as follows when referred to the secondary side:

r₁' = b² r₁

X_l1' = b² X_l1

X_m' = b² X_m

R₀' = b² R₀

Currents and voltages transform inversely:

I₁' = (N₁/N₂) I₁

V₁' = (N₂/N₁) V₁

Using these transformations yields the exact equivalent circuit referred to the secondary. This circuit retains the same topological structure as the primary-referred circuit but with the transformed parameter values shown above.

Summary of common referred relationships

• Define a = N₁/N₂ (primary turns divided by secondary turns).
• To refer secondary impedances to primary: Z'_sec→prim = a² Z_sec. In particular r₂' = a² r₂, X_l2' = a² X_l2.
• To refer primary impedances to secondary: Z'_prim→sec = b² Z_prim with b = N₂/N₁. In particular r₁' = b² r₁, X_l1' = b² X_l1.
• Current transformation when referring secondary to primary: I₂' = (N₂/N₁) I₂. When referring primary to secondary: I₁' = (N₁/N₂) I₁.
• Voltage transformation when referring secondary to primary: V₂' = (N₁/N₂) V₂. When referring primary to secondary: V₁' = (N₂/N₁) V₁.
• Power and real losses are invariant under correct referring (demonstrated above for copper loss).

Exact equivalent circuits

Either of the following exact equivalent circuits may be used for steady-state analysis depending on convenience:

• The primary-referred exact equivalent circuit, where secondary series elements and load are referred to the primary (useful when solving for primary quantities given a load).
• The secondary-referred exact equivalent circuit, where primary elements (including the magnetising branch) are referred to the secondary (useful when solving for secondary terminal quantities from a known primary source).

Both circuits are topologically identical after referring (series primary resistance and leakage reactance, shunt branch representing magnetizing and core losses) and differ only in numerical values of the transformed parameters as given in the summary.

Using the equivalent circuit to solve problems

Typical procedure for steady-state analysis of a loaded single-phase transformer:

• Decide which side is convenient to refer the circuit to (usually the side where source or load is known).
• Refer the opposite side impedances using the turns-ratio square rule.
• Combine series elements and use the shunt branch R₀ ∥ jX_m as appropriate to find E₁, I₀, and the referred load current I₂'.
• Obtain primary and secondary terminal voltages and currents by inverse transformation where required.
• Compute losses and regulation using the preserved power relations.

Practical remarks

• The magnetizing reactance X_m is usually large compared with series leakage reactances; for many approximate calculations the magnetizing branch can be placed either on the primary or referred to secondary but must not be ignored for no-load conditions.
• Core losses are frequency and flux density dependent. For a given supply frequency and rated voltage these may be approximated as a constant resistance R₀ in steady-state analysis.
• Equivalent circuit parameters (r₁, r₂, X_l1, X_l2, X_m, R₀) are obtained from tests such as open-circuit (no-load) and short-circuit tests; those procedures and calculations are standard in transformer testing (see relevant practical lab and textbooks).
• The exact equivalent circuit derived here is valid for linear analysis under sinusoidal steady state. Nonlinear core magnetisation (saturation) and harmonics require more advanced models.

The model and methods given above provide the standard basis for analysing transformer voltages, currents, regulation, efficiency, and losses in steady state and are widely used in power system studies and machine design.