4.5 A.C. Transformers

For the primary-side and secondary -side induced voltages, we then have:

The no-load voltage (U₂₀) is the voltage on the secondary side when no consumer is

connected.Here, the basic transformer equation applies:

Voltage and Current Transformation

From Faraday's law it follows that:

Primary coil:

23

U1

N1

Secondary coil:

(4.18)

U2

N2

For a transformer with no load (no-load state) the voltages behave according to

The  number of windings:

U2

N2

 The undiminished ratio of the voltages is described by:

(4.21)

N2

24

4 Transformers

Ignoring losses, it then follows that:

The currents are inversely proportional to the voltages or to the number of

windIngs :

I2

I1

Transformers which serve for resistance matching are called one-to-one

transformers:

Z1

Transformer Loading

1. No-load state: The magnetization power loss (iron losses) are measured in

the  no-load state (Figure 4.9).The no-load current I0  is made up of the current

Il  and the active current component IR (heat losses).

2. Short circuit: The short circuit voltage u_k is the primary voltage with which a

transformer  with short circuited secondary winding already draws its primary

current  (Figure 4.10).  This is used to measure the short circuit losses.

The short circuit voltage is important for the determination of

-          the impedance

-           the winding power loss

-          the phase shift

-           the short circuit current

-           the arrangement of transformers in parallel

Mostly       u_k   is given as the relative short circuit voltage in percent of the

primary  voltage. It is a measure of the loading when a change of voltage occurs.

3. Short circuit current: When a short circuit arises on the secondary side of a

transformer during operation, the peak short circuit current ip will first flow.

After a certain time, this becomes the steady state short circuit current        

The magnitude of ip depends on the momentary value of the voltage and on

the magnetic state of the iron core. The worst case, zero crossing and a 

saturated iron core, results in the highest peak short circuit current at the

moment in which the short circuit arises. It consists of an A.C. component

as the steady state short circuit current and a D.C. component which arises

due to the collapse of the field and becomes zero after a time t » 5 seconds:

(4.25)

Transformers

25

26

4 Transformers

The size of the steady state short circuit current I_k  depends on the short circuit

Voltage  u_k  and the internal impedance Z.

The output voltage depends on:

.     the magnitude of the load

.     the size of the relative voltage uk

.     the phase j of the load current

The load voltage at the output is measured for the rated load. This terminal

voltage depends on the load type and the load current (Figure 4.11).

Figure 4.11       Loading of the transformer

4.6

Three-Phase Transformers

4.6.1

Design

The three-phase transformer consists of three single-phase transformers built

together, i.e. the voltages are offset by 120 from each other (Figure 4.12). As a result,

the sum of the magnetic fluxes in the middle limbs is zero at any time. For the

three-phase transformer u_k  is very small, so that the SV/PV windings are over one

another on every limb.

Figure 4.12            Design and winding

arrangement of the

three-phase transformer

4.6.2

Winding Connections

4.6 Three-Phase Transformers

27

Here, connection (Figure 4.13) is understood to mean the connection of the winding

phases  to a single winding. The upper case letters refer to the higher voltage side

(PV) and the lower case letters refer to the lower voltage side (SV).

With three-phase transformers, we must distinguish between:

1. Winding phases in delta connection (D,d)

2. Winding phases in star connection (Y,y)

3. Winding phases in zigzag connection (Z,z)

4. Winding phases in open connection (III,iii)

5. Winding phases open

28

4 Transformers

Figure 4.13       Connections

4.6.3

Connection Symbols

The connection symbols (Figure 4.14) indicate how the two windings of a

transformer are connected and by what factor of 30 the pointer of the lower voltage side

trails that of the higher voltage side with terminal designation coordinate in the

counter clock wise direction.

The following symbols are used:

Y, D, Z according to function and manufacture of the transformer

Y         for star point and high voltage

D         for isolation of the zero sequence system

Z         for low resistance zero sequence system

The preferred connection groups are Yy0, Dy5, Yd5, Yz5 and Dyn5.

For low-voltage networks, the lower voltage side is not implemented as a delta

because the neutral conductor could otherwise not be connected.

Connection symbols

Parallel Connection of Transformers

4.6 Three-Phase Transformers

29

The parallel connection of transformers serves to increase the power of a system

(Figure   4.). In order to prevent exceeding the rated current or overloading of the

individual transformers, the following conditions must be adhered to:

1. The same rated frequency and rated voltage (not ü!),

2. Approximately the same short circuit voltages (maximum ± 10% deviation),

3. The same connection symbol

4. Maximum rated power ratio 1 : 3,

5. Connection in correct phase sequence

The following relationships can be used for transformers connected in parallel:

1. for the same short circuit voltages

the load output of the transformer:

                                                                                      (4.27)

2. for unequal short circuit voltages

the load output of the transformer:

S_L1                                                                                          (4.28)

(4.27)

(

The average short circuit voltage u_krm  is given by: