8.1.1. Operating principle of a transformer

Transformers are stationary electrical machines which transmit energy from systems with certain current and voltage values into systems with generally different current and voltage values but with identical frequency.

Two separate windings are on the same iron core.

Following connection to alternating voltage U₁ there is a standstill current I₀. The magnetomotive force Q = I₀ · N₁ generates a magnetic alternating flow (F₁) in the iron core.

The input and output winding of an alternating voltage are induced in accordance with the induction law. A self-induction voltage U₁₀ arises in the input winding. It is counter-positioned in accordance with Lenz’s law on applied voltage. During idling operation - because of mutual induction - there arises the output voltage U₂₀ which is simultaneously the terminal voltage U₂.

U_1~ ® I_0~ ® Q_0~ ® F_1~ ® U_20~

The value of the induced voltage is derived from the following equation:

	max. flow density
AFe	limb cross-section
U0	induction voltage
f	frequency
N	number of turns

The induction voltage increases along with the number of turns, the magnetic flow density in the iron core, the iron cross-section and the frequency.

Example:

Which maximum flow density occurs in an iron core of 16 cm² cross-section when a voltage of 380 V (50 Hz) is applied to the primary coil with 980 turns?

Given: A_Fe = 16 cm²; N₁ = 980; U₁ = 380 V; f = 50 Hz

Sought:

Solution:

» 1.09 V · s · m^-2

» 1.09 T

The iron core evidences a maximum flow density of 1.09 T.

8.1.2. Voltage transformation

A few field lines already close before reaching the output coil (Figure 125) so that flow F₁ can be divided into a maximum flow F_K which saturates both coils and a leakage flow F_S.

The leakage flow may be ignored in regard to the unloaded transformer (idling). Therefore the following applies:

According to the transformer equation

and

.

If we relate both equation then

Shortening gives us

During idling no current flows into the output winding, thus there is no voltage decrease. Consequently the induced voltage U₂₀ equal to the terminal voltage U₂ (Cp Figure 125):

Figure 125 - Transformer principle

1 Input winding/upper voltage winding/primary winding, 2 Output winding/under voltage winding/secondary winding

U₂₀ = U₂

In the event of minimal idling current I voltage decrease in the input winding is negligibly minimal. We therefore have

U₁₀ = U₁

which results in

The voltages behave like the numbers of turns.

The interrelationship of the numbers of turns is known as the transformation ratio We have:

The rated voltages U_1n and U_2n are indicated on the rating plate of the transformer.

Example:

What secondary terminal voltage arises in a transformer where 380 V is applied to the primary winding of 980 turns and the secondary winding has 594 turns?

Given: U₁ = 380 V; N₁ = 980; N₂ = 594

Sought: U₂

Solution:

U₂ » 230 V

8.1.3. Current transformation

Load behaviour of the transformer

If the transformer is output-loaded, current I₂ flows into coil N₂. Current I₂ generates the magnetic flow F_2K. According to Lenz’s Law this magnetic flow is counter-positioned to the cause (F_1K).

Figure 126 - Loaded transformer

In this manner the magnet flow F_1K is weakened and induction voltage U₁₀ decreases. Given uniform rated voltage, the difference increases between the two voltages U₁₀ and U₁.

Consequently, a greater input current I₁ flows whereby the magnetic flow F_1K is increased. The magnetic flow F in the iron core thus remains virtually constant:

F = F_1K - F_2K = constant

This also applies to the output voltage of the transformer.

The input current I₁ increases as the load current I₂ becomes greater.

Transformation ratio

Without heeded the losses of the transformer, the following applies according to the energy conservation law:

s₁ = s₂

U₁ · I₁ = U₂ · I₂

If we arrange the equation so that the voltage and current values appears on respective sides, then

.

The following relationships may be cited for current ratio:

Conversely the currents are proportional to the voltages or numbers of turns. A transformer converts high currents into low ones or low currents into higher ones.

Example:

A welding transformer takes up 220 (current being 10A). The output voltage is 20V. How great is the welding current?

Solution:

I₂ » 110A