Transformer turns ratio
The turns ratio in an ideal transformer is defined as the quotient between the number of turns of the primary winding (N₁) and the number of turns of the secondary winding (N₂). This dimensionless parameter determines the proportion in which the alternating voltage and current are transformed between the two windings. In a real transformer, the turns ratio approximates the ideal value but is affected by core losses, leakage flux, and winding resistance. Understanding this concept is fundamental for the design, selection, and operation of transformers in power systems, electronics, and industrial applications.
A transformer operates according to Faraday’s law of electromagnetic induction. An alternating current in the primary winding produces a time-varying magnetic flux that circulates through the common ferromagnetic core. This flux links the secondary winding and induces in it an electromotive force (e.m.f.) proportional to its number of turns. In an ideal transformer, all flux generated by the primary links the secondary without leakage, and the magnetic permeability of the core is considered infinite. Under these conditions, the induced voltage in each winding is directly proportional to its number of turns.
Fundamental Equation of the Turns Ratio
Section titled “Fundamental Equation of the Turns Ratio”The relationship between primary and secondary voltages in an ideal transformer is equal to the turns ratio of the two windings, and in turn equal to the inverse ratio of the currents. The mathematical expression describing this behavior is:
V₁ / V₂ = N₁ / N₂ = I₂ / I₁
Where each variable is defined as:
| Symbol | Quantity | Unit |
|---|---|---|
| V₁ | Primary winding voltage | V |
| V₂ | Secondary winding voltage | V |
| N₁ | Number of turns of the primary winding | — (dimensionless) |
| N₂ | Number of turns of the secondary winding | — (dimensionless) |
| I₁ | Primary winding current | A |
| I₂ | Secondary winding current | A |
The quotient a = N₁ / N₂ is called the turns ratio. When a > 1, the transformer steps down the voltage; when a < 1, it steps it up.
Current Relationship
Section titled “Current Relationship”In the ideal transformer, apparent power is conserved, so the product of voltage and current in the primary equals the product in the secondary (V₁·I₁ = V₂·I₂). Combining this condition with the fundamental equation yields the current relationship:
I₁ / I₂ = N₂ / N₁ = 1 / a
The current in each winding is inversely proportional to its number of turns. A winding with more turns withstands higher voltage but carries lower current. This relationship allows proper sizing of the conductor cross-section: the low-voltage winding requires a larger gauge because it carries a higher current.
Impedance Relationship
Section titled “Impedance Relationship”When a load impedance Z₂ is connected to the secondary, the impedance reflected to the primary is transformed according to the square of the turns ratio. The expression relating both impedances is:
Z₁ = a² · Z₂
Where Z₁ is the equivalent impedance seen from the primary terminals and Z₂ is the impedance connected to the secondary. This property is used for impedance matching between stages of electronic circuits and for modeling the effect of the load on the primary.
Classification by Turns Ratio
Section titled “Classification by Turns Ratio”Transformers are classified according to the value of the turns ratio a = N₁ / N₂:
| Type | Turns Ratio (N₁ / N₂) | Effect on Voltage | Effect on Current |
|---|---|---|---|
| Step-down | a > 1 | V₂ < V₁ | I₂ > I₁ |
| Step-up | a < 1 | V₂ > V₁ | I₂ < I₁ |
| Isolation | a = 1 | V₂ = V₁ | I₂ = I₁ |
The operation of a transformer is reversible: a step-down transformer can function as a step-up if the primary and secondary windings are interchanged, provided the rated voltage and current limits of each winding are respected.
Calculation Example
Section titled “Calculation Example”A 50 kVA single-phase transformer has a primary voltage of 4000 V and a secondary voltage of 400 V. Assuming ideal behavior, determine:
- The rated currents in both windings.
- The turns ratio.
Calculation of rated currents. The primary rated current is obtained from the rated apparent power:
Primary current: I₁ = S / V₁ = 50 000 VA / 4000 V = 12,5 A
Secondary current: I₂ = S / V₂ = 50 000 VA / 400 V = 125 A
Calculation of the turns ratio. It can be determined from voltages or currents:
a = V₁ / V₂ = 4000 V / 400 V = 10
Alternatively, a = I₂ / I₁ = 125 A / 12,5 A = 10
The result a = 10 indicates a step-down transformer, in which the primary winding has 10 times more turns than the secondary.
Limitations of a Real Transformer
Section titled “Limitations of a Real Transformer”The ideal transformer model does not consider the following phenomena present in real transformers:
- Core losses: produced by magnetic hysteresis and eddy currents (Foucault currents). These losses depend on frequency, maximum flux density, and core material.
- Winding resistance: the ohmic resistance of the copper dissipates power as heat (I²R losses), reducing the output voltage under load.
- Leakage flux: part of the magnetic flux generated by the primary does not fully link the secondary. This leakage flux is modeled as leakage inductances in series with each winding.
- Magnetizing current: in a real transformer, the core requires a small excitation current even with the secondary open-circuited, to establish the magnetic flux.
These non-idealities cause the voltage ratio under load to differ slightly from the turns ratio, especially at full load and with low power factors.
Applications of the Concept
Section titled “Applications of the Concept”The turns ratio is a fundamental parameter in the following applications:
- Transmission and distribution of electrical power: step-up transformers increase voltage at generating stations to reduce resistive losses in transmission lines, and step-down transformers decrease voltage to safe levels for residential and industrial consumption.
- Impedance matching in electronics: audio and radio-frequency transformers use specific turns ratios to couple stages with different characteristic impedance, maximizing power transfer.
- Instrumentation and protection: current transformers and potential transformers use calibrated turns ratios to reduce high voltage or current signals to standardized, safe values for measuring instruments and protective relays.
- Switched-mode power supplies: pulse transformers with optimized turns ratios allow conversion of DC voltage levels in isolated DC-DC converters.
Frequently Asked Questions (FAQ)
Section titled “Frequently Asked Questions (FAQ)”What does the turns ratio represent in a transformer?
Section titled “What does the turns ratio represent in a transformer?”The turns ratio a = N₁ / N₂ represents the quotient between the number of turns of the primary and the secondary. It determines how voltage and current are modified between the two windings: V₁ / V₂ = a and I₁ / I₂ = 1 / a. A value a > 1 indicates a step-down transformer, while a < 1 corresponds to a step-up transformer.
How is the turns ratio calculated from electrical measurements?
Section titled “How is the turns ratio calculated from electrical measurements?”It can be calculated by measuring the open-circuit voltages of both windings and applying the formula a = V₁ / V₂. It can also be obtained from the rated currents, since a = I₂ / I₁. In power transformers, it is usually determined through standardized tests that correct for the effects of internal impedance.
Why is the current higher in the low-voltage winding?
Section titled “Why is the current higher in the low-voltage winding?”Apparent power is conserved in an ideal transformer (V₁·I₁ = V₂·I₂). Therefore, if the secondary winding has a lower voltage than the primary, the secondary current must be proportionally higher to keep the power product constant. This is why the low-voltage winding is manufactured with a larger conductor cross-section.
Does the turns ratio vary with load?
Section titled “Does the turns ratio vary with load?”In a real transformer, the voltage ratio under load differs slightly from the turns ratio due to the voltage drop across internal impedances (resistance and leakage reactance of the windings). This difference is greater at full load and with inductive power factors. Under no-load, the voltage ratio closely approaches the turns ratio.
What happens if a step-down transformer is connected backwards?
Section titled “What happens if a step-down transformer is connected backwards?”A step-down transformer can operate as a step-up if the low-voltage winding is supplied and the high-voltage winding is loaded. However, it is essential that the voltage applied to the new primary does not exceed its rated design value, and that the current demanded by the load does not exceed the capacity of the winding that was originally the primary. Exceeding these limits can cause core saturation or overheating.
How is the load impedance reflected to the primary?
Section titled “How is the load impedance reflected to the primary?”The impedance connected to the secondary is reflected to the primary multiplied by the square of the turns ratio: Z₁ = a² · Z₂. This property allows a transformer to act as an impedance adapter, making a load of value Z₂ appear as an equivalent load of value a²·Z₂ seen from the primary circuit.
References
Section titled “References”- engineeringtoolbox.com: https://www.engineeringtoolbox.com/transformer-d_1398.html
- allaboutcircuits.com: https://www.allaboutcircuits.com/textbook/alternating-current/chpt-9/step-up-and-step-down-transformers/
- electrical4u.com: https://www.electrical4u.com/transformer-calculator/