Electrical Machines  Basic vocational knowledge (Institut für Berufliche Entwicklung, 144 p.) 
8. Transformer 
8.1. Transformer principle 

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 U_{1} there is a standstill current I_{0}. The magnetomotive force Q = I_{0} · N_{1} generates a magnetic alternating flow (F_{1}) in the iron core.
The input and output winding of an alternating voltage are induced in accordance with the induction law. A selfinduction voltage U_{10} arises in the input winding. It is counterpositioned in accordance with Lenz’s law on applied voltage. During idling operation  because of mutual induction  there arises the output voltage U_{20} which is simultaneously the terminal voltage U_{2}.
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 
A_{Fe} 
limb crosssection 
U_{0} 
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 crosssection and the frequency.
Example:
Which maximum flow density occurs in an iron core of 16 cm^{2 }crosssection when a voltage of 380 V (50 Hz) is applied to the primary coil with 980 turns?
Given: A_{Fe} = 16 cm^{2}; N_{1} = 980; U_{1} = 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.
A few field lines already close before reaching the output coil (Figure 125) so that flow F_{1} 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 U_{20} equal to the terminal voltage U_{2} (Cp Figure 125):
Figure 125  Transformer principle
1 Input winding/upper voltage winding/primary winding, 2 Output winding/under voltage winding/secondary winding
U_{20} = U_{2}
In the event of minimal idling current I voltage decrease in the input winding is negligibly minimal. We therefore have
U_{10} = U_{1}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: U_{1} = 380 V; N_{1} = 980; N_{2} = 594
Sought: U_{2}
Solution:
_{}
_{}
_{}U_{2} » 230 V
Load behaviour of the transformer
If the transformer is outputloaded, current I_{2} flows into coil N_{2}. Current I_{2} generates the magnetic flow F_{2K}. According to Lenz’s Law this magnetic flow is counterpositioned to the cause (F_{1K}).
Figure 126  Loaded transformer
In this manner the magnet flow F_{1K} is weakened and induction voltage U_{10} decreases. Given uniform rated voltage, the difference increases between the two voltages U_{10} and U_{1}.
Consequently, a greater input current I_{1} 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 I_{1} increases as the load current I_{2} becomes greater.
Transformation ratio
Without heeded the losses of the transformer, the following applies according to the energy conservation law:
s_{1} = s_{2}U_{1} · I_{1} = U_{2} · I_{2}
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:
_{}
_{}I_{2} » 110A