Ideal Transformer

Ideal Transformer

An ideal transformer is a lossless electromagnetic device that transfers alternating-current (AC) electrical energy between primary and secondary windings through mutual flux linkage. It steps RMS voltage up or down according to the turns ratio while conserving instantaneous and average real power.

The fundamental relationship between the primary (input) and secondary (output) sides is given by the turns ratio q = N_s / N_p. The secondary RMS voltage scales as V_s = q V_p, while the primary and secondary currents satisfy I_p = q I_s.

Worked Example

Consider a transformer with primary RMS voltage V_p = 12 V, primary turns N_p = 100, and secondary turns N_s = 200, connected to a purely resistive load R = 12 Ω:

1. Calculate the turns ratio: q = N_s / N_p = 200 / 100 = 2.
2. Find the secondary RMS voltage: V_s = q V_p = 2 × 12 = 24 V.
3. Determine the secondary RMS current using Ohm's law: I_s = V_s / R = 24 / 12 = 2 A.
4. Find the primary RMS current from lossless power balance: I_p = q I_s = 2 × 2 = 4 A.
5. Verify power conservation: P_in = V_p I_p = 12 × 4 = 48 W and P_out = V_s I_s = 24 × 2 = 48 W.
6. Calculate the referred input resistance seen by the source: R_in = R / q² = 12 / 4 = 3 Ω.

Experiment

You can verify these properties directly using the interactive Ideal Transformer module to explore how the turns ratio trades voltage for current and scales referred impedance:

1. Select Primary RMS voltage Vp as 24.
2. Select Primary turns Np as 200 and Secondary turns Ns as 50 to create a step-down transformer (q = 1/4 = 0.25).
3. Set Load resistance R to 6.
4. Observe that Secondary RMS voltage drops to 6 V and Secondary RMS current is 1 A, while Primary RMS current is only 0.25 A, maintaining P_in = P_out = 6 W and referred input resistance R / q² = 96 Ω.

Limitations

This idealized model assumes 100% magnetic coupling, infinite core permeability, zero winding resistance (copper loss), and zero core hysteresis or eddy-current losses. Real transformers require a nonzero magnetizing current even at no load, exhibit leakage reactance and voltage regulation droop under load, saturate at excessive flux density (V/f), and cannot transform steady DC voltages.

References

The following textbook reference covers the ideal transformer turns ratio, current scaling, power balance, and impedance referral used in this module:

• Section 15.6, Transformers — OpenStax University Physics Volume 2.