# Basic Inductance Principles in Transformers

## Transformers are critical components used in power distribution systems. Learn the principles of transformer operation and inductance through this article.

A transformer typically consists of two separate coils with different numbers of turns of conductor wound around the same closed laminated iron core (see Figure 1). The primary winding is the coil in a transformer that is energized by the source. The secondary winding is the coil that is connected to the load. The primary circuit in a transformer can be the high-voltage or the low-voltage circuit, depending on whether it is a step-up or a step-down transformer. The high-voltage leads are marked with the letter H, and the low-voltage leads are marked with the letter X. The secondary neutral is often labeled X0.

*Figure 1. A typical transformer consists of two separate coils with different numbers of turns of conductor wound around*

In a typical large industrial complex, a transmission substation may deliver electricity directly to an outside transformer vault. Conductors from the service entrance are routed through an outdoor busway to a metered switchboard from the outside transformer vault. Power is then fed through circuit breakers in the panel board and routed through busways to power distribution panels and busways with plugin sections to the points of use (see Figure 2). Depending on customer needs, the power distribution system supplies power to set points such as receptacles at standard voltage levels and fixed current ratings.

*Figure 2. Transformers are used to reduce the transmission line voltage to a level usable by the customer.*

**Note**

*Alternating current produced at generating plants is transformed to a higher voltage to allow efficient transmission of electrical power between power stations and end-users.*

## Induction

Inductance is the property of a device or circuit that causes it to store energy in the form of an electromagnetic field. Induction is the ability of a device or circuit to generate reactance to oppose a changing current (self-induction) or the ability to generate a current (mutual induction) in a nearby circuit. The current flowing in the coil produces a field that expands out of and surrounds the conductor. Energy is stored in that field. When the source voltage goes from peak to zero, the energy stored in the electromagnetic field is converted back to electrical energy in the coil conductors. The energy actually opposes the changes in the source voltage.

The three requirements for inducing a voltage through magnetism are a conductor, a magnetic field, and relative motion between the conductor and the magnetic field. In a transformer, the conductor is the wire making up the coil. The AC power flowing through a conductor generates an expanding and collapsing magnetic field. The expanding and collapsing magnetic field flows through the laminated core and provides the relative motion between the conductor in the secondary and the magnetic field.

The core is constructed of layers of a low reluctance material and offers little opposition to the magnetic lines of flux. The closed, laminated core provides a low reluctance path for the flux to flow, aligning the flux and allowing the maximum number of conductors to be cut by the magnetic field lines. This induces a voltage in the secondary. The total power is the same in the primary and the secondary circuits, except for some losses in the transformer. This means that when the voltage in the secondary is higher than in the primary, the current is proportionally lower in the secondary.

## Self-Induction

Self-induction is the ability of an inductor in a circuit to generate inductive reactance, which opposes a change in the circuit current. When an AC source voltage rises, and the magnetic flux expands around the circuit conductors, an opposing voltage, or counter-voltage, is induced in the circuit. The magnitude of the induced voltage is determined by the rate of change of the current. Lenz's law states that the polarity of the induced voltage is such that it produces a current whose magnetic field opposes the change that produced it (see Figure 3). The induced magnetic field in an inductor acts to oppose any change in current.

*Figure 3. Self-induction in a coil opposes changes to the current. Image courtesy of LouisvillePhysics*

The counter-voltage limits the circuit current, as the circuit current is determined by the impedance and the difference between the source and the counter-voltage. As the AC source voltage falls back to zero and the field collapses, the counter-voltage acts to prevent the current from falling. This shows that the first 90° of a cycle is spent charging the inductor. The electrical energy is converted into magnetic energy in the inductor. When the voltage peaks, the current is at zero, the field stops expanding, and all the energy is stored in the magnetic field. When the source voltage starts to drop from the peak, the magnetic field starts to collapse and the inductance property aids the current provided by the source. This makes the current lag 90° behind the source.

When the AC frequency and the coil inductance are known, inductive reactance is calculated as follows:

$${{X}_{L}}=2\pi fL$$

where XL = inductive reactance (in Ω)

π = 3.14

f = frequency (in Hz)

L = inductance (in H)

The total opposition or impedance is calculated as follows:

$$Z=\sqrt{{{R}^{2}}+X_{L}^{2}}$$

where

Z = impedance (in Ω)

R = resistance (in Ω)

XL = inductive reactance (in Ω)

A transformer primary is a coil, and the opposition induced in that coil can be calculated. If the winding in the coil is 500′ of AWG #22 copper wire, the wire has a resistance of approximately 8 Ω. If the coil has an inductance of 0.5 H, the inductive reactance and total impedance are calculated as follows:

$${{X}_{L}}=2\pi fL=2\pi \times 60\times 0.5=188.5\Omega $$

$$Z=\sqrt{{{R}^{2}}+X_{L}^{2}}=\sqrt{{{8}^{2}}+{{188.5}^{2}}}=188.7\Omega $$

If the wire is not wound into a coil, the resistance of the wire alone is 8 Ω. Because a coil has a higher impedance than a wire, the total impedance when the wire is wound into a coil is 188.7 Ω at 60 Hz. The current capacity in the wire alone is about 23.6 times higher than the current capacity through the coil because of the increased impedance from the inductive reactance of the coil.

The comparison of the coil and the straight conductor shows how a transformer primary can be connected to a source and act to limit the current.

The four factors that determine or control the inductance of a coil are the cross-section of the core, the number of turns, the type of the core, and the length of the coil. Many people use the acronym CoNTroL to remember these factors, where the capitalized letters represent the first letter of each of the factors.

## Mutual Induction

Mutual induction is the ability of an inductor in one circuit to induce a voltage in another circuit. When a transformer primary has alternating current flowing in the conductor, magnetic flux surrounds the conductor in proportion to the amount of the current. The expanding and contracting flux cuts the conductors in the secondary and induces a voltage in the secondary. See Figure 4.

*Figure 4. Image courtesy of Elliot Sound Products.*

When an AC source is applied to the primary of a transformer, there is a counter-voltage induced in the primary coil that is opposite and nearly equal to the applied voltage. There is a very small difference between the applied and the induced voltages that allows just enough current to magnetize the primary core. The exciting current, or magnetizing current, is the no-load current through a primary core. The exciting current causes a magnetic field that cuts across the secondary and induces a voltage in the secondary. Exciting current actually has two components. The first component is the true power no-load current (in kW) that magnetizes the core. The second component is the reactive power (in kVAR) that builds the field. For very small transformers, the exciting current may be as high as 10% of the maximum current. For very large transformers, the exciting current may be less than 1% of the maximum current.

When a load is connected across the secondary, the induced voltage causes a current to flow in the secondary. The current causes a magnetic field with polarity opposite to the field in the primary in proportion to the relative field strengths. The secondary magnetic field tends to neutralize and reduce the magnetic field in the primary because the lines of flux oppose each other. The reduced magnetic field reduces the inductance and allows more current to flow in the primary. The increased current in the primary produces a stronger magnetic field that induces an increased voltage in the secondary. The increased voltage in the secondary allows more current to flow. This continues until the load is drawing the needed current.

## Transformer Mutual Inductance

The mutual inductance, or the coefficient of coupling, of a transformer, is a measure of the efficiency by which power is transferred from the primary to the secondary coils (see Figure 5).

*Figure 5. The mutual inductance of two coils depends on their location and orientation.*

If the power transfer is perfect, the coefficient of coupling is 1. If there is no power transfer, the coefficient of coupling is 0. The coefficient of coupling depends on the design of the transformer. The most important factor is the position of each coil with respect to the other. If the coils are wound over one another and each line of flux from the primary cuts a turn in the secondary, then the coupling coefficient is very close to 1. If any flux is lost, then the coefficient of coupling is less than 1. The coefficient of coupling of typical transformers ranges from 0.95 to 0.99, depending on design and purpose.

2 CommentsIt first says the coil inductance is .5H but the formula example uses .05H….