11.1.4Electromagnetic Induction

The Ideal Transformer Equation

5 min read 9 parts Lesson 5 of 6 1 quiz

0%read

Section 1 of 9The turns-ratio equation

For a transformer, the ratio of the voltages equals the ratio of the numbers of turns:

V_p / V_s = N_p / N_s

What the symbols mean: V = voltage (V) · N = number of turns · I = current (A) · p = primary coil · s = secondary coil

where V_p is the primary voltage (V), V_s is the secondary voltage (V), N_p is the number of turns on the primary and N_s the number of turns on the secondary. The ratio of the turns can be given in either order as long as the primary and secondary are consistently placed; it can also be written V_s = V_p × N_s/N_p.

Section 2 of 9The power equation

For an ideal (100% efficient) transformer, the electrical power in the primary equals the power out of the secondary:

V_p I_p = V_s I_s

where I_p and I_s are the currents in amperes. This means that if the voltage is stepped up, the current is stepped down by the same factor. Since power P = IV, the equation just states power in equals power out.

Combining the two equations, I_s/I_p = N_p/N_s: the current ratio is the inverse of the turns ratio.

Section 3 of 9Efficiency

For a real transformer, efficiency is:

efficiency = (power out / power in) × 100% = (V_sI_s / V_pI_p) × 100%

It is always less than 100%, because of energy lost as heat in the coils and core. Real transformers have efficiencies around 90–99%.

Section 4 of 9Method for calculations

List the known quantities with units and subscripts p and s.

Decide if the question is about voltage and turns (use V_p/V_s = N_p/N_s) or about currents (use V_pI_p = V_sI_s).

Rearrange before substituting.

Give the answer with a unit and check: a step-up transformer must give V_s > V_p and I_s < I_p.

Section 5 of 9Examples of use in practice

A phone charger contains a step-down transformer (then a rectifier), taking 230 V to a few volts. A mains lamp on a model railway uses a transformer to produce 12 V. The grid uses step-up transformers with ratios of the order of 1:15 at the power station and step-down transformers near homes.

Section 6 of 9Further worked reasoning

Suppose a model railway transformer has 1150 turns on the primary connected to the 230 V mains and gives 12 V at the secondary.

The number of secondary turns is N_s = N_p × V_s/V_p = 1150 × 12/230 = 60 turns.

If the ideal transformer supplies 2.0 A at 12 V, the power is 24 W, and the primary current is 24/230 = 0.10 A.

It shows how a large step-down in voltage gives a large step-up in current.

In an examination always show the formula, the substitution and the answer with unit; any consistent rearrangement is acceptable.

Section 7 of 9Worked examples

A transformer has 200 turns on the primary and 50 turns on the secondary. The primary voltage is 230 V. Find the secondary voltage.
An ideal transformer steps 12 V up to 240 V. The secondary current is 0.50 A. Find the primary current.
A transformer takes 2.0 A at 230 V. The output is 11.5 V at 36 A. Find the efficiency.

Section 8 of 9Key terms

Term	Meaning
Turns ratio	N_p : N_s, equal to the voltage ratio V_p : V_s.
Ideal transformer	One with 100% efficiency.
Efficiency	Useful power out divided by power in, as a percentage.
Rectifier	Circuit that converts a.c. to d.c.

Section 9 of 9Exam technique

Mixing up primary and secondary values.

Applying V_pI_p = V_sI_s to a non-ideal transformer without allowing for losses.

Saying that the current also goes up in a step-up transformer.

Leaving out units.

Transformer Golden Equations Physics Standard

Transformer Turns Ratio & Efficiency

Ratio: V_p / V_s = N_p / N_s = I_s / I_p

Step-up reduces grid thermal loss (P = I²R).
Laminated core prevents eddy currents.