Understanding of the principles of magnetism led to a major breakthrough in transformer design. Hans Christian Oersted discovered that current flowing in the conductor produces magnetic field. This discovery helped Andre’-Marie Ampere to come up with an idea that material magnetism results from localized currents [1] . Ampere observed that multiple numbers of small current loops oriented appropriately induced magnetic fields associated with magnetic materials and permanent magnets [2] .

Transformer cores are made from materials with low reluctance to the flow of magnetic flux. Transformer cores are supposed to give easiness to the flow of magnetic flux. Magnetic material can be described as soft materials when it can easily be magnetized and demagnetized. Hard magnetic materials are those which cannot be easily magnetized and demagnetized [1] .

Magnetic material is one of the most important parts of the transformer. The transformer cores are constructed from magnetic materials. There are different types of magnetic materials. Transformer Design Engineers are therefore supposed to know the properties of different types of magnetic materials.

Iron is the most common material used in core construction. In some cases, small amount of silicon is added to iron in order to improve magnetic properties which reduce core losses [2] . Other materials used in core construction are: nickel, cobalt and iron alloys (permalloys) and iron oxides (ferrites). These materials are classified in a group of ferromagnetic materials [2] .

In summary, magnetic cores have the following properties, which need to be considered when making a decision for selecting core material for a specific transformer design: resistivity, relative permeability, saturation flux density, Hysteresis and Eddy current losses per unit volume, Curie temperature and upper operating frequency [1] .

Ampere discovery proved that a varying current, flowing through a conductor created a varying magnetic field, around that conductor [2] . Ampere’s law is then summarized using the formula which presents the parameters of the closed loop [2] .

dI is the vector length in the direction of the path, J is the current density, S is the surface where current I is flowing, and is enclosed current in the path C [2] .

If a coil has N turns and the current flowing through it is I, then ampere’s Law can be applied to the coil using the formula:

This law states that a time varying magnetic flux flowing through a closed loop generates voltage in that loop [2] .

where λ is flux linkage, N is the number of turns of the coil, is magnetic flux, is reluctance of magnetic circuit and L is the inductance of the coil and is the current. The formula can further be substituted by flux density, magnetic field intensity and permeability as follows:

According to Faraday’s Law, flux density is dependent to the applied voltage to the primary winding. The volt-seconds, applied to the primary winding for half cycle [1] ;

The change in the flux value in the magnetic core will be;

Therefore, to achieve the required change in flux density, the number of turns of primary winding must be;

The current in the inductor at a given specific time can be represented by the formula:

This is another important law in magnetic circuits. Lenz noted that the voltage created by a time-varying magnetic flux had a direction that induced periodic current in the closed loop which also induced magnetic flux that opposed the change in the initial flux. Increasing the initial magnetic flux the induced current creates flux which opposes the increase of the initial flux and vice-versa [2] .

Magnetic circuit requires a source of energy to create a flow of magnetic flux in the magnetic circuit. Magneto-motive force, is the source of that energy which makes magnetic flux to flow in the magnetic circuit. Magneto-motive force is created by current flowing in the coil. Magneto-motive force is directly proportional to the current flowing in the coil and the number of turns of the coil.

The unit used to measure mmf is “A.turns”. Increasing either current or number of turns will increase the mmf which in return will increase the flow of magnetic flux in magnetic circuit.

As stated earlier, magnetic flux in the magnetic circuit is analogous to the current flowing in electrical circuit. Magnetic flux on the surface area of “s” is given by [2]

where “B” is flux per unit area.

The unit used to measure magnetic flux is “weber (wb)”.

Magneto-motive force of the coil can be defined as per unit length of the coil. This type of measure is described as magnetic field intensity, H. The unit used to measure magnetic field intensity is “Amps per meter, (A/m)”. Therefore, H can be presented by a formula:

where the length of the coil in meters, N is the number of turns of the coil and is the current flowing in the coil.

Magnetic flux density is analogous to current density in electrical circuit. It is the amount of magnetic flux per unit cross-sectional area. Magnetic flux density can be calculated as follows:

where, is the total flux in magnetic circuit and A is the cross-sectional area of magnetic path.

Magnetic flux can also be calculated in relation to magnetic field intensity;

where, is permeability of free space and is the relative permeability.

As the current is increased in the coil, the flux density in the core increases exponentially. The rate of the increase in flux density in the core is very small when a current is increased up to a certain specified value. The core goes into saturation. The flux density is said to reach its saturation point,. Therefore, the flux at saturation point is,

It is the sum of flux enclosed by individual turn of wound around the magnetic core [2] . Flux linkage, can be presented by the formulas below;

Note that the unit for flux linkage is Volt-second.

Magnetic circuit has some opposition to the flow of magnetic flux the same way it is with resistance in the electrical circuit. The opposition to magnetic flux flow in the magnetic circuit is call reluctance,. The easiness of magnetic flux to flow in a magnetic circuit is called permeance, P which is analogous to conductivity in electrical circuit. Reluctance has, therefore an inverse relationship to the magnetic material permeability and cross-sectional area of the magnetic path. Reluctance is also directly proportional to the length of magnetic circuit.

Thus, reluctance of magnetic circuit can be calculated by [2] :

where is magnetic core length, is cross-sectional area of the core, P permeance of magnetic circuit. The unit for reluctance is “turns per Henry, (turns/H)”.

Based on the formula for reluctance, we can drive the formula for permeance as follows;

The unit for permeance is “Henry per turns, (H/turns)”.

We can also drive formulas from for magnetic flux from Equations (3.6.1) and (3.6.2), thus:

The magnetic core inductance, L can then be calculated using reluctance of the core.

Therefore, inductive reactance is;

Some magnetic core may have an air gap by design or because of some joints made in laminated core. The effective cross-sectional area of the air gap is not the same as the core cross-sectional area because the magnetic flux does not take a straight path in the air gap.

Making and assumption that the cross-sectional area of an air gap is the same as that of the core; we can visualize the reluctances of the core and the air gap to be in series with each other. Therefore, the total magnetic circuit reluctance is;

Since we made an assumption that both air gap and core have the same cross-sectional area, and then we can calculate the total reluctance of magnetic circuit using the equation;

Note that the term in the parenthesis in Equation (3.6.6) is defined as gap factor, [2] . Gap factor can also be defined as the ratio of the total circuit reluctance to the total reluctance of the core, thus;

Based on the Equation (3.7.7), there is a linear relationship between the length of the air gap and the gap factor. Figure 3 shows the relationship between length of the gap and the gap factor.

From the Equation (3.7.7) we can then come up with a relationship of the core relative permeability and air gap factor called effective relative permeability,;

Since permeability of free space is very low, introducing an air gap reduces effective permeability of a magnetic circuit. The advantage of an air gap in a magnetic circuit is that it produces stable effective permeability and reluctance which in turn produces predictable and stable inductance [2] . Figure 4 demonstrates the relationship between the air gap and effective permeability.

The length of the gap can therefore be calculated using the equation:

From Equation (3.6.9) we can drive the equation for inductance for the core with air gap.

As the gap length increases, the inductance of the coil reduces as shown in Figure 5.

It must be noted that the inductance of the gap is directly proportional to the effective permeability,. As shown in Figure 4, effective permeability displays a graphical exponential decrease as the air-gap of the core increases. The same graphical exponential decrease in inductance is displayed in Figure 5 because of the aforementioned relationship between Inductance and effective permeability.

From Equation (3.7.9) we can then drive an equation to calculate the number of turns of the windings:

Magnetic flux lines have a tendency to repel each other when they are passing through a nonmagnetic material. When magnetic circuit has an air gap, the magnetic lines tend to bulge outwards, thus increases the cross-sectional area of the effective air gap which in returns reduces flux density [2] . The bulged magnetic flux lines in an air gap are described as fringing flux. Since the sum of fringing flux and the air gap flux must equal to the flux in the magnetic core, fringing reluctance and air gap reluctance must be viewed as being in parallel. Therefore, magnetic flux in the core is:

where air is gap magnetic flux and is fringing magnetic flux.

Since fringing and gap reluctances are in parallel and in series with core reluctances, total magnetic circuit reluctance can be presented by an equation:

But

Substituting Equation (3.8.2) with Equations (3.8.3), (3.8.4) and (3.8.5) and then factorize, we get the final equation;

Therefore, the inductance of the magnetic circuit taking into consideration fringing effect will be [2] ;

The ratio of the inductance of the air gap with fringing flux lines to that of without fringing flux lines is described as fringing factor,. Figure 6 shows the effect of fringing on inductance.

Based on the Equation (3.8.7), and Equation (3.8.8) the total magnetic circuit inductance increases with an increase in air gap. Because of the increase in the inductance due to the air gap, likewise, the number of turns required will be increased too. Thus;

To avoid saturation, the maximum number of primary windings of the transformer with air gap must be;

There are also some factors which may affect the size selection of the winding conductor, like frequency due to skin effect. At very high frequency, the current flowing in the conductor has a tendency of concentrating on the surface of the conductor and not in the inner part of the conductor. The skin depth, of a conductor with resistivity, is calculated by;

Therefore, as the frequency increases, the current density on the surface of the conductor increases and the skin depth decreases [2] , thus;

The maximum current flowing in the conductor whose current density is;

The product of and is the effective cross-sectional area of the conductor,.

The resistance of the conductor with length is;

Both the inductive reactance and the DC resistance are the same and are proportional to the square root of frequency,.

The time-average power loss in winding strip of length is [2] ;

The inductance then can be calculated, since we have the value of inductive reactance and frequency, thus;

The resistivity of a winding conductor is also affected by temperature. Conducting materials have a tendency to increase in resistivity due to increase in temperature. It is therefore to consider the effect of temperature on a winding more especially on transformers which will be operating at high temperatures [2] . Thus;

where is temperature coefficient of a conductor and is the resistivity of a conductor at temperature,.

The size of conductor to be used in winding the transformer must be capable to curry continuous full load current at the specified operating frequency and temperature. American Wire Gauge, AWG is commonly used in industries to define the wire sizes. Using the American Wire Gauge, the diameter of the wire can be calculated using the formula [2] :

By knowing the diameter and the length of wire we would like to use, we can therefore calculate the total DC resistance of the wire we are going to use.

Transformer winding wire has a layer of insulation. It is therefore necessary to also calculate the thickness of the insulation since it will also occupy some space in the core window of the transformer.

It is there, the bare conductor cross-sectional area is divided by the cross- sectional area of the insulated conductor. This ratio is defined as wire insulation factor,.

For a round conductor, the wire insulation factor can be calculated by;

where is the diameter of uninsulated conductor and is the thickness of the insulation.

When winding the transformer, not all space is filled with insulation and a wire. There is some are pockets within the winding. The ratio of the area filled by insulated wire to the area filled by both insulated wire and air pockets is defined as air factor.

The windings of the transformer are not directly wound on the core. Most of the transformer’s windings are made on the bobbin. It is necessary to determine the area occupied by the bobbin so that it can be added to the area occupied by the windings and the air. The total volume of windings, air and bobbin will help to size the area of the window required for the windings. The ratio of the area of the bobbin window to the area of the core window is called bobbin factor, [2] .

It is also important to know that not entire area of the window is used by the windings. There is always an area which is not used. The ration of the area of the window used to the total area of the core window is called window utilization factor, [2] .

The product of the window area and a winding conductor are is called, area product,. Area product is also known as ferrite-copper area product [2] .

A well designed core must be able to handle the maximum load power with minimal losses. It must be able to handle the maximum current without going into saturation. More importantly, the core must be able to accommodate the windings flexibly.

Given the transformer power rating, , peak flux density, and the operating frequency, we can calculate the number of the primary turns, required.

where for a sine wave.

The number of secondary turns can be calculated in relation to the voltage ratio and primary turns, thus;

We can now calculate the cross-sectional area of the wire required for both primary and secondary windings;

where is the current density of the type of wire selected.

The total cross-sectional area to be occupied by both primary and secondary windings can be calculated by the formula give below.

If the current density of the primary wire is the same as secondary wire, the area product of the transformer is;

For an ideal transformer with sinusoidal voltages and currents,.

Notice that as the frequency is increased the area product of a transformer reduces.

The length of each turn of the transformer winding can be calculated from the perimeter of the core. It must also be known that each layer of the winding will have a different winding length. For a round core, the estimated length of each turn of the winding for the first layer, is

The perimeter of each following layer is increased proportionally to the diameter of the winding wire. Therefore, the length of the second layer, of the turn is;

where is the diameter of the core and is the cross-sectional area of the primary winding wire.

The total length of the primary winding wire is;

The total length of the secondary winding wire is;

To minimize the margin of error, it is good to calculate the total length of each winding layer and then add them up to find the total length of the primary winding and the secondary winding.

The cross-sectional area required by primary winding:

The cross-sectional area required by secondary winding:

Since the total cross-sectional area assigned to all the windings is, the fraction of the area occupied by the primary windings is;

The fraction of the area occupied by the secondary windings is

Therefore, rearranging Equations (6.2.7) and (6.2.8);

Knowing the length, cross-sectional area and the type of the wire to be used in the winding, the total DC resistance of the winding can be calculated [2] .

We can therefore calculate the dc winding power losses in both primary and secondary windings since we know the maximum operating current of the transformer and DC resistance of the windings;

The secondary winding DC power loss is;

Therefore, the total DC winding power loss for the transformer is;

For a two winding transformer;

By substituting in Equation (6.2.15) by Equation (6.2.16) and factorize, we get;

The derivative of Equation (6.2.17) will give the value of which will minimize the winding losses.

The minimum DC winding loss occurs when:

By substituting in Equation (6.2.17) with the value of Equation (6.2.19);

Substituting in Equation (6.2.20) with Equation (6.1.1)

As the peak flux density increases, the optimum winding losses reduce exponentially.

The minimum transformer winding losses are achieved when;

Figure 7 shows graphically the point where optimum transformer losses are achieved in relation to flux density.

The designer must also make sure that transformer must not be saturated when operating at normal voltage. Therefore, there must be a limit for maximum flux density. Thus the maximum flux density can be calculated by;

where is the effective cross-sectional area of the core.

Considering the DC components;

Since the windings of the transformer are insulated and that they are side by side separated by the insulation, they act like capacitors. The transformer winding

capacitance may have an effect on changes in voltage phase angle and may also contribute to surge phenomena during system switching [3] .

The capacitance between windings can be calculated by the formula;

where is permittivity through space, is mean diameter between two windings, H is the height of the winding, is the thickness of the insulation between the conductors and is the permittivity of the insulation.

The capacitance between cylindrical conductors and ground plane is [3]

where R is the radius of the cylindrical conductor, H is the length of the cylindrical conductor and S is the distance of center of cylindrical conductor from the plane.

The capacitance between the primary and secondary windings is given by;

where is the average diameter of the winding, w is the width of the bare conductor, is the total thickness of the insulation.

Electrical energy stored in a capacitor is;

For the RF transformers, resonance frequency, is achieved when inductive reactance is equal to the capacitive reactance. Thus

The series elements like winding resistance and leakage reactance cause phase shift. The out-of-phase component of voltage drop across these series elements is usually caused by in-phase load current flowing through the leakage inductance or out-of-phase load current flowing through the winding resistance or a combination [3]

Therefore, the transformer phase shift is;

For the core to be magnetized there is a continuous alignment and reversal of magnetic dipoles in the core material. This process requires energy hence contributes to the core loss called hysteresis loss. The hysteresis loss is frequency dependent loss. Energy lost per cycle is;

where is cross-sectional area of the core and is the length of the core.

Empirical equation (Steinmetz equation)

where is hysteresis loss in watts, (for a sine voltage) is a Hysteresis constant, is frequency in Hertz and is maximum flux density.

Based on Faraday’s Law, changing magnetic flux induces voltage in a conducting material which in return causes current flow in that material.

Since the core is within the vicinity of magnetic flux, there will be an induced voltage in the core which will cause current circulation in the core. This circulating current will cause power loss in the core in form of heat due to the resistance of the core. The formula for calculating Eddy current is;

where the cross-sectional area of the core is, is the resistivity of the core, and is the frequency.

There are other transformer losses called stray losses. Some of the stray losses may be caused by some of the magnetic flux inked to the transformer tank. Stray losses in a transformer are difficult to estimate, hence this paper will only cover hysteresis and eddy current losses for the core.

Estimated total core losses will the sum of hysteresis and eddy current losses, thus.

When ac voltage is applied to the windings, the magnetic fields built up in the windings. Some of the input power to the transformer is used up for this magnetic field built-up.

The magnetic field on the primary are supposed to link the secondary winding so the secondary voltage is induced in the coil. Not all the primary magnetic flux is linked to the secondary. Therefore, this also contributes to energy losses in the transformer.

That is,

Transformer turns ratio test is one of the most important tests. Since there is a direct relationship between turns ratio and transformer voltage ratio, any huge era in turns ratio will be more damaging to the circuit supplied by such a transformer because the secondary voltage of the transformer will be different. Therefore, every transformer will go under TTR test after being wound. The circuit diagram in Figure 15 demonstrates how TTR test can be done.

Transformers xfmr1 and xfmr2 are both connected in parallel and are supplied by voltages V_source. Transformer xfmr1 is the transformer under taste. Transformer xfmr2 has a known variable turns ratio.

Set the turns ratio of the transformer xfmr2 to the calculated turns ratio of the transformer xfmr1. Connect the transformers as shown in the circuit diagram in Figure 15. If the output voltages of transformers xfmr1 and xfmr2 are the same then the transformer xfmr1 is wound correctly. If the voltages are different then the transformer xfmr1 is not correctly wound.

This test is used to measure transformer core losses. The measured values can be used to calculate the active component and reactive components of the transformer core, and respectively. Figure 16 shows how this test is set up.

In the circuit diagram, xfmr1 is the transformer under test, A is ammeter, W is wattmeter with both current and voltage probes connected to the circuit to measure active power, V is the voltmeter.

The primary rated voltage, V_in is applied to the transformer as shown in the circuit. The open circuit current in the primary, is measured by ammeter A, open circuit voltage, is measured by the voltmeter V and active power is measured by wattmeter W. It must be noted that is the active core power loss [1] .

Therefore, the impedance angle is

Open circuit impedance is

Therefore, we can find the values of and of the transformer core from the short circuit results. Thus

This test is important because it measures the winding power losses in the transformer. The measured values of the current and voltages can be used to calculate the active and reactive components of the windings. Figure 17 shows the circuit set up of the test.

The input primary voltage, V_in is gradually increased from zero volts until the ammeter; “A” reading reaches the maximum value of the current rating of the transformer. The voltage reading, and wattmeter reading, are recorded when the maximum current rating, is reached. The input voltage for the maximum current of the transformer is also recorded by voltmeter V.

The impedance angle is

Therefore, the primary winding total impedance is

Therefore, we can find the values of and of the primary winding of the transformer from the short circuit test results. Thus

We can do the same short circuit test on the secondary of the transformer to find the winding impedance,.

The chances to have the insulation of the windings scratched are high during the winding of the transformer. It is therefore necessary to measure the resistance of the winding and compare it with the calculated one. If the measured resistance is very low compared to the calculated one, then some of the turns are shorting due to the damaged winding insulation. This test is not as reliable since it uses DC voltage hence does not take into count skin deep effect. Although this test is not as accurate compared to TTR test, it is very easy to perform.

There are two main methods to measure the winding resistance. It can be measure directly using ohmmeter. The winding resistance can also be measured using voltmeter and ammeter as shown in the circuit in Figure 18.