Factors in the Functionality of Transformer in Inductors

The functionality of a transformer

A transformer consists of at least two windings, with the winding turns N_P on the primary side and N_S on the secondary side. For the sake of simplicity, we will look at an ideal transformer with a turns ratio of 1 : 1.

In a first step, we will look at a transformer with an open secondary winding N_S (Figure 1). A U_P voltage pulse is created at winding N_P. Due to the inductance of the
winding, this pulse generates a linearly progressive current I_P. The following applies:

Figure:  EQ-1_40

Figure 1.:  Principle of a transformer with zero load. This ideal transformer is wound as a bifilar so as to ignore parasitic effects.

Winding N_S also wraps around this magnetic flux. Changing the magnetic flux creates voltage.

Figure:  EQ-1_41

If you solve both equations after changing the voltage and then equate them, you will get the following for the voltage transformation:

Figure:  EQ-1_42

Current does not flow in winding N_S because the winding is open. If we now connect winding N_S to a load resistor R_L (Figure 2.), the voltage induced in N_S generates a current flow through the load resistor:

Figure:  EQ-1_43

Figure 2.: The same transformer but with a load

The primary current now consists of the transformer secondary current and the linearly progressive magnetisation current that is already available without load.

Figure : EQ-1_44

As no power can be generated, the transforming power is the same as the primary power put into the system. If the magnetisation current is disregarded, the following applies:

Figure : EQ-1_45

Currents are thus transformed in the reversed direction as voltage. The following also applies:

Figure : EQ-1_46

Resistances are thus transformed with the transformation ratio squared. This also applies to inductances, capacitances, and impedances. So the magnetizing current is not transferred to the secondary side. It is required to generate the magnetic field. The aim of the transformer design must, therefore, be to keep the magnetizing current as small as possible.

There are two possibilities here:

Insertion of a highly permeable core to increase the primary inductance. This causes the magnetizing current to rise less steeply and is therefore smaller.

Figure 3:  Magnetizing current of a transformer with and without a highly permeable core

Shorter voltage pulses with higher frequency are generated, as the rise in current stops at the end of the voltage pulse and start again at the original point for the next pulse.

Figure 4:  Magnetizing current for a transformer at different driving frequencies

Parasitic effects

In reality, there are other factors that affect the behavior of transformers. The most important ones are:

• Leakage inductance
• Coupling capacitance (capacitance between windings)
• Winding capacitance (capacitance within a winding)

Leakage inductance
Looking at two windings, we see that the entire flux is not coupled to the other winding. A part of the streamlines of the magnetic field closes outside of the other winding. This part of the inductance is called “leakage inductance.” To understand how to minimize leakage inductance, you must know the parameters that influence it.

If you look at a long concentric coil, its inductance results from L=:

Figure: EQ-1_47

l_W Length of the coil
N Winding turns
A Cross section of the coil

If a second winding is wound on top of it (Figure 6 named as Long solenoid with second winding), the leakage inductance results from (1.48)

Figure :  EQ-1_48

Figure 6: Long solenoid with second winding

Where A here is the surface between the two windings. It can be calculated using:

Figure: EQ-1_49

MLT Mean length of the turn
H_ins Distance between the windings (isolation)
H₁, H₂ Winding height of windings 1 and 2

The leakage inductance is thus independent of the core material and air gap. To minimize leakage inductance you must either increase the length of the coil (broad windings) or reduce the distance between the windings (e.g. bifilar wind).

Figure 1.38 shows various more or less ideal winding constructions. With existing geometry, the most commonly used means is a sandwich construction (Figure 7d), in which the secondary winding is wound between the primary winding that is divided into two halves. This doubles the length of the winding.

Figure 7: Different winding structures

Coupling capacitance

You can picture the coupling capacitance between the two windings as a plate capacitor between the two windings. From this, it follows that you can reduce this capacitance either by increasing the distance or reducing the surface. Both directly increase leakage inductance.

Winding capacitance

Each winding builds up winding capacitance because they are isolated from each other and rest on different potential. This capacitance increases with the number of layers that are required within a winding. It can be reduced by means of various winding technologies, e.g. Z-wind (wire is returned after each layer).

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