# Electrical Transformer Design !

Transformers are electrical compnents which convert alternating current or three-phase supplies to higher or lower voltages. Apart from this this voltage conversion that plays a key part in high-voltage energy transmission, transformers can also fulfil the role of converters in instrumentation applications or as repeaters for transmission of low-current signals. Transformers operate on the principle of electromagnetic induction. This course will only consider transformers as used with single-phase alternating current.

__Transformer design:__

The diagram below shows the basic design of a transformer for single-phase alternating current.

A transformer usually features a closed iron core made up of metal plates separated by insulating layers to suppress eddy currents. The plates are designed specially to minimise iron losses during re-magnetisation. Wrapped around the core are two coils, usually with windings that have a differing number of turns *N*_{1} and *N*_{2}. The coil supplied with electrical energy by an AC voltage is termed the *primary coil* (on the left in the diagram above) and the current *I*_{1} flowing through this coil is termed *primary current*. The coil that outputs the electrical energy, thus acting as a source to other loads, is termed a *secondary coil* (on the right in the diagram above) and the current flowing through it is termed *secondary current* *I _{2}*. Similarly, the voltages

*U*

_{1}and

*U*

_{2}across the coils are called the primary and secondary voltage. The terminals of the primary coil are designated

*1U*and

*1V*, those of the secondary coil

*2U*and

*2V*.

**Operating principle of a transformer:**

Since it is constantly changing in intensity and direction, the alternating current fed to the primary coil produces a fluctuating magnetic field in the iron core. Like the current, this field also changes constantly in intensity and direction. Except for a few field lines scattered in the surrounding air (stray field), most of the other magnetic field lines produced by the primary coil are constrained within the closed iron core and also pass fully through the secondary coil. The two coils are therefore coupled together closely by a shared magnetic field (of flux F). According to the law of induction, the secondary winding produces an induced voltage, the frequency of which is equal to that of the primary voltage. Connecting the secondary circuit to a load (an ohmic resistance in the diagram above) causes current to flow in the secondary circuit.

In circuit diagrams, a single-phase transformer is represented by the detailed symbol shown below. This symbol indicates the transformer's essential features, i.e. its two isolated coils.

**Operating characteristic with no load:**

With no load, i.e. when the secondary circuit is open, a transformer behaves as though the secondary coil were absent, i.e. it acts as an inductor coil. Due to the large inductance and correspondingly large inductive reactance, the current that flows when there is no-load is only very small and gives rise to an alternating magnetic field so that it is termed the *magnetisation current*. There is a phase displacement of nearly 90° between the primary supply voltage and the magnetisation current.

**No-load voltage and no-load current:**

The current in the primary winding produces a magnetic alternating field which induces a so-called *no-load voltage* in the secondary coil when no load is connected to the transformer. The peak value of the no-load voltage

depends on the peak value of the magnetic flux density,

The cross-section *A* of the iron core, the angular frequency w of the input current and the number of winding turns *N* on the secondary coil. The following equation applies in this case:

Replacing the angular frequency by the expression 2p *f* and the peak voltage value with the rms value of the no-load voltage *U*_{0} (the peak-to-peak voltage multiplied by

for a sine wave) yields the main *transformer equation: *

**Transformation ratio:**

Naturally, the main transformer equation applies not only to the secondary side of the component but also the primary side. Because the flux density, iron cross-section and frequency are identical on both sides, the following relationship is obtained if the ohmic resistance of the two coils is neglected:

Variables with an index of 1 and 2 respectively represent the transformer's primary and secondary sides. In other words:

The ratio between the primary and secondary voltages of a transformer with no load is equal to the ratio between the number of winding turns on each coil. |

Coils with many winding turns are associated with higher voltage. Coils with fewer winding turns are associated with a lower voltage. The ratio between the number of winding turns on the side of the transformer where the voltage is higher (regardless of whether this is the primary or secondary) and the number on the side where the voltage is lower is called *transformation or transformer ratio*.

If a transformer is assumed to be loss-free, then power *S*_{1} = *U*_{1}*I*_{1} fed to the primary side must be equal to the power *S*_{2} = *U*_{2}*I*_{2} consumed on the secondary side. In other words, currents behave in a fashion that is the inverse of that obeyed by the voltages. In terms of winding turns, the following relationship applies:

The following interactive animation illustrates the relationships between voltage, current and winding turns. Using the toggle control, specify different values for the primary voltage, secondary current and winding turns and observe the resulting changes in the associated variables on the other side.

**Response under load:**

When a transformer is loaded, a current flows through its secondary coil. This current attenuates the magnetic field by producing an opposing magnetic field according to *Lenz's law*. As a result, part of the field is shifted out of the iron core so that the two coils are not subjected to the full magnetic field any more. The displaced field lines are termed *stray field lines*. This leakage leads to a drop in voltage, so that the secondary voltage is lower than what the winding ratio would suggest.

The current flowing through the secondary coil also causes the input current to rise. Both currents produce voltage drops across the coils due to their resistance. Consequently, a loaded transformer behaves like a generator with an internal resistance comprising ohmic and inductive components. Shown below is a simplified equivalent circuit diagram of a transformer incorporating an ohmic resistance *R*_{L}. *I* is the load current.

The output voltage *U*_{2} is lower than the input voltage *U*_{1} by an amount equal to the transformer's internal voltage drop. The phase shift j between the two voltages depends on the type of load. The diagram below displays the relationships applicable to an ohmic load. The output voltage and load current are in phase in this case.

The transformer's internal voltage drop is composed of the partial drops *I* x *R* and *I* x *X*_{L}, represented by the grey triangle (or *Kapp triangle*) in the vector diagram.

The vector diagram below shows the relationships applicable to a pure inductive load. The load current in this case lags behind the output voltage by 90°, as does the ohmic voltage drop *I* x *R*. Because the voltage across the load is in phase with the inductive voltage drop *I* x *X*_{L}, the internal voltage drop caused by this kind of loading is quite large.

The next vector diagram shows the relationships applicable to a capacitive load. In this case, the load current leads the output voltage by 90°. The output voltage is *higher* than the input voltage, because the internal inductance and load capacitance form a series oscillating circuit. For this reason, large capacitors must not be connected unattenuated to the mains power grid.

The next vector diagram displays the relationships applicable to ohmic-inductive loads. The phase displacement between the load current and output voltage lies between 0 and 90° depending on the load resistance and inductance. This kind of loading results in the largest voltage drop, because the load's active resistance and reactance behave like a transformer's internal resistance.