Electrical Machines

Fundamental Electricity and Magnetism [1]


Conventional and electron current flow

In a dry cell having one positive terminal (+) and one negative terminal (-), the potential difference between them cause an electron current to flow in the circuit; from negative to positive terminal. This is due to electron excession at the negative terminal compared to the positive terminal. Therefore, when we connect a wire between the two terminals, electrons will move along the wire from the negative terminal and reenter the cell by the positive terminal.      

Meanwhile, back in the 17th century before the electron theory was fully understood, the scientists decided that electric current flows from the positive terminal to the negative terminal. This 'conventional current flow' is still used today and is accepted direction of current flow in electrical machines topics. 

Distinction between sources and loads

We can simply distinguish between the source and the load by observing circuit elements; a source delivers electrical power, where the load absorbs it.


Sign notation

We use symbols (+) and (-) to indicates the direction of electric current, mechanical force, or rotational speed. If the sign changes, it means that the direction is reversed.

Double-subscript notation for voltages

Consider a source having a positive terminal A and a negative terminal B ;

The potential difference and the relative polarities of the terminals A and B can be designated by the double-subscript notation; 

➡ The volatge between A and B is 100 V, and A is positive with respect to B.

➡The voltage between A and B is 100 V, and B is negative with respect to A.

Sign notation for voltages

Although we can represent the value and the polarity of volatges by the double-script notation, we can often prefer to use the sign notation. It consises of designating the voltage by a symbol and identifying one of the terminals by a positive (+) sign.

If E₂₁ = - 100V, terminal 2 is negative with respect to terminal 1.



Sign notation to designate a voltage.

Graph of an alternating voltage

The figure above shows a graph of an alternating voltage having a peak of 100 V.

In the following contents, we will be dealing with sources whose voltages change polarity periodically. Such alternating voltages may be represented by means of graph (Fig. 1.) The vertical axis indicates the voltage at each instant, while the horizontal axis indicates the corresponding time. Voltages are positive when they are above the horizontal axis and negative when they are below.

Starting from zero, E₂₁ gradually increases, attaining +100 V after 0.5 seconds. It then gradually falls to zero at the end of one second. During this one-second interval, terminal 2 is positive with respect to terminal 1. 

Positive and negative currents

The signs for current flow are allocated with respect to a reference direction given on the circuit diagram.

The figure shown is a circuit element showing positive direction of current flow. The positive direction is shown arbitrarily by means of an arrow.

Sinusoidal voltage

The ac voltage generated by alternators can be expressed by the equation

e = Eₘcos(2πft + θ)


e = instantaneous voltage [V]

Eₘ = peak valus of the sinusoidal voltage [V]

f = frequency [Hz]

t = time [s]

θ = a fixed angle [rad]


The expression 2πft and θ are angles, expressed in radians. However, it is often more convenient to express the angle in degrees, as follows;

e = Eₘcos(360ft + θ)


e = Eₘcos(Φ + θ)

where Φ (= 360 ft) is expressed in degrees along with θ.

Converting cosine functions into sine functions

We can convert a cosine fuction of voltage or current into a sine function and vise versa.

Converting cosine to sine by adding 90° to θ.

Converting sine to cosine by subtracting 90° from θ.

Effective value of an AC voltage

Although the peak value Eₘ are specified, it is much more common to use the effective value.

For a voltage that varies sinusoidally, the relationship between the effective value and Eₘ is given by the expression

The effective value of an ac voltage is sometimes called the RMS (root mean square) value of the voltage. It is measured of the heating effect of as compared to that of an equivalent dc voltage.

The same remarks apply to the effective value of an ac current. Thus a current that varies sinusoidally and whose peak value is Im possesses an efective value Ieff given by

Phasor representation

  • length = i/e
  • angle = electrical phase angle between i and e
  • in phase ➡ two vectors are parallel, pointing in the same direction, angle = 0.
  • out of phase ➡ two vectors point in diffrent way. 


The voltages and currents in a power circuit are frequently not pure sine waves. 

Energy in an inductor

A coil stores energy in its magnetic field when it carries a current I. The energy is given by


W = energy stored in the coil [J]

L = inductance of the coil [H]

I = current [A]

If the current varies, the stored energy rises and falls in step with the current. Thus, whenever the current increases, the coil absorbs energy and whenever the current falls, energy is released.

Energy in a capacitor

A capacitor stores energy in its electric field whenever a voltage E appears across its terminals. The energy is given by


W = energy stored in the capacitor [J]

C = capacitance of the capacitor [F]

E = voltage [V]

[Table] Impedance of some common AC circuits

Circuit diagram Impedance

Equivalent Circuit

Below are the equivalent circuit for Electric and Magnetic;



E = Volatage 

I = Current 

R = Resistance      

where R = ρl/A

mmf = Magnetic Force

Φ = Magnetic Flux 

R = Reluctance          

where R = l/μA


Magnetic field intensity H and flux density B

Whenever a magnetic flux Φ exists in a body or component, it's due to the presence of a magnetic filed intensity H, given by


B = flux density [T or Wb/m²]

Φ = flux in the component [Wb]

A = cross section of the component [m²]


H = magnetic field intensity [A/m]

U = magnetomotive force acting on the component [A] (or ampere turn)

l = length of the component [m]


There is a definite relationship between the flux density (b) and the magnetic filed intensity (H) of any material. This relationship is usually expressed graphically by the B-H curve of the material.

B-H curve of vacuum

In vacuum, the magnetic flux density B is expressed by the equation

B = μ₀ × H


B = blux density [T]

H = magnetic field intensity [A/m]

μ₀ = magnetic constant or permeability of vacuum [= 4π ×10⁻⁷]

The figure above shows the B-H curve of vacuum and non magnetic materials.

The B-H curve of vacuum is a straight line. A vacuum never saturates, no matter how great the flux density maybe. 

B-H curve of a magnetic material

The flux density in a magnetic material also depends upon the magnetic field intensity to which it is subjected. Its value is given by

B = μ₀μᵣH

The value of μᵣ is not constant but varies with the flux density in the material, therefore the relationship between B and H is not linear.


Determining the relative permeability

The relative permeability μᵣ of a material is the ratio of flux density in the material to the flux density that would be produced in vacuum, under the same magnetic field intensity H.

Faraday's law of electromagnetic induction

It revealed a fundamental relationship between the voltage and flux in a circuit. Faraday;s law states:

  • If the flux linking loop (ot turn) varies as a function of time, a voltage is induced between its terminals.
  • The value of the induced voltage is proportional to the rate of change of flux.

Consequently, if the flux varies inside a coil of N turns, the voltage induced is given by


E = induced voltage [V]

N = number of turns in the coil

ΔΦ = change of flux inside the coil [Wb]

Δt = time interval during which the flux changes [s]

Voltage induced in a conductor

In many motors and generators, the coils move with respect to a flux that is fixed in space. The relative motion produces a change in the lux linking the coils, and consequently, a voltage is induced according to Faraday's Law. However, it is easier to calculate the induced voltage with reference to the conductors, rather than with reference to the coil itself. In effect, whenever a conductor cuts a magenetic field, a voltage is induced across its terminals. The value of the induced voltage is given by


E = induced voltage [V]

B = flux density [T]

l = active length of the conductor in the magnetic field [m]

v = relative speed of the conductor [m/s]

Lorentz force on a conductor

When a current-carrying conductor is placed in a magnetic field, it is subjected to an electromagnetic force, or Lorentz force. It constitutes of the force depends upon the orientation of the conductor with respect to the direction of the field. The force is greatest when the conductor is perpendicular to the field and zero when it is parallel to it.

The maximum force acting on a straight conductor is given by


F = force acting on the conductor [N]

B = flux density of the field [T]

l = active length of the conductor [m]

I = current in the conductor [A]

where  F = Lorentz force on a conductor

The upward direction for can be obtained by changing the polar of the voltage source.

Direction of the force acting on a straight conductor

When there exists current in a conductor, the conductor will be surrounded by a magnetic field.

     a. Magnetic field due to magnet and conductor

     b. Resulting magnetic field pushes the conductor downward.

It can be observed that the lines of force created respectively by the conductor and the permanent magnet act in the same direction above the conductor and in opposite directions below it. Therefore, the number of lines above the conductor must be greater than the number below. 

Residual flux density and coercive force

A current source, connected to the coil, produces a current whose value and direction can be changed at will. When I is gradually increased, B and H increased.

The maximum I will generates maximum H. When I is decreasing, there will be flux residue in the magnetic core. The value depends on μᵣ, if μᵣ is high, there will be more Bᵣ left after we discharge the circuit.When the graph of current is negative, there occurs negative maximum current which effect this graph of magnetization curve.

Here, if we let the current run in the circuit for a period of time then we open the circuit, there will be Bᵣ left in the material, the value depending on μᵣ.

Hysteresis Loop

The area shaded under the Bᵣ curve is the amount of coercive force, that reduces I to zero. This coercive force acts like one kind of energy.

The direction of the loop will be from a to e respectively.


Hysteresis losses caused by rotation

Hysteresis losses occur when iron cores in an AC generator are subject to effects from a magnetic field. The magnetic domains of the cores are held in alignment with the field in varying numbers, dependent upon field strength.   

The magnetic domains rotate, with respect to the domains not held in  alignment, one complete turn during each rotation of the rotor. This rotation  of magnetic domains in the iron causes friction and heat.   

The heat produced by this friction is called magnetic hysteresis loss. To reduce hysteresis losses, most AC armatures are constructed of heat-treated silicon steel, which has an inherently low hysteresis loss. After the heat-treated silicon steel is formed to the desired shape, the lamination are heated to a dull red and then allowed to cool. This process, known as annealing, reduces hysteresis losses to a very low value.

Eddy currents

Consider an ac flux Φ that links a rectangular-shaped conductor. According to Faraday's law, an ac voltage E is induced across its terminals.

If the conductor is short-circuited, a substantial alternating current I₁ will flow, causing the conductor is placed inside the first, a smaller voltage is induced because it links a smaller flux. Consequently, the short-circuit current I₂ is less than I₁ and so, too, is the power dissipated in this loop.

Here, we have North and South pole of magnet to generate magnetic flux. Then if the open circuit are shorted, there will be a current called eddy current from the magnetic induction.

The thin mica layer will block the current, with very small amount of current in a particular plate, it will minimize the eddy current.

Eddy currents in a stationary iron core

We can reduce the losses by splitting the core in two along its length, insulating the two sections from each other. The voltage induced in each section is one half of what it was before, with the result that the eddy currents, and the corresponding losses, are considerably reduced.

If we continue to subdivide the core, we find that the losses decrease progressively. Furthermore, a small amount of silicon is alloyed with the steel to increase its resistivity, thereby reducing the losses still more.


Eddy-current losses in a revolving core

The stationary field in direct-current motors and generators produces a constant dc flux. This constant flux induces eddy current in the revolving armature. Consider a solid cylindrical iron core that revolves between the poles of a magnet. As it turns, the core cuts the flux lines and, according to Faraday's law, a voltage is induced along its length having the polarities shown. These eddy currents produce large power losses which are immediately converted into heat, in which the power loss is proportional to the square of the speed and the square of the flux density.

To reduce the eddy-current losses, we laminate the armature using thin circular laminations that are insulated from each other.

Current in an inductor

In an inductive circuit, the voltage and current are related by the equation


e = instantaneous voltage induced in the circuit [V]

L = inductance of the circuit [H]

Δi/Δt = rate of change of current [A/s]

This equation enables us to calculate the instantaneous voltage e, when we know the rate of change of current, However, it often happens that e is known and we want to calculate the resulting current I.

Question [1.1]

  • The peak voltage is 138 V.
  • The peak voltage is 219 V.
  • The peak voltage is 339 V.
  • The peak voltage is 457 V.
  • The peak current is 2.75 A.
  • The peak current is 5.17 A.
  • The peak current is 7.93 A.
  • The peak current is 14.1 A.
A 60 Hz source having an effective voltage of 240 V delivers an effective current of 10 A to a circuit. The current lags the voltage by 30 degree.

Question [1.2]

  • Phasor E is said to lead phasor I.
  • Phasor E is said to lag phasor I.
If a phasor E has to be rotated clockwise to make it point in the same direction as phasor I, then...

Question [1.3]

  • F = 100 N
  • F = 200 N
  • F = 300 N
  • F = 400 N
A conductor 3 m long carrying a current of 200 A is placed in a magnetic field whose density is 0.5 T. Calculate the force on the conductor if it is perpendicular to the lines of force.

Magnetic Circuit [2]

Magnetic Circuits

Magnetic circuits

In most electrical machines, the magnetic field (or flux) is produced by passing through coils wound on ferromagnetic materials.

i-H Relation

When a conductor carries current a magnetic field is produced around it. The direction of flux lines or magnetic field intensity H can be determined by 'Thumb rule', which states that if the conductor is held with the right hand, the thumb indicates the direction of current in the conductor and the fingertips indicate the direction of magnetic field intensity. Hence, the relationship between current and field intensity can be obtained by using Ampere's circuit law.

If θ is the angle between vectors H and dl, then

Consider a circular contour for the magnetic field intensity H at a distance r from the conductor. At each point on the contour, H and dl are in the same direction, that is, θ = 0. Because of symmetry, H will be the same at all points on this contour. Therefore,

B-H Relation

The magnetic field intensity H produces a magnetic flux density B everywhere it exists. These quantities are functionally related by

B = μH weber/m² or tesla

B = μᵣμ₀H  Wb/m² or T      

where   μ is a characteristic of the medium and is called the permeability of the medium

               μ₀ is the permeability of free space and is 4π10⁻⁷ henry/meter

               μᵣ is the relative permeability of the medium

For ferromagnetic materials, the value of μᵣ varies in the range of 2000 to 6000. A large value of μᵣ implies that a small current can produce a large flux density in the machine.

Magnetic Equivalent Circuit

Consider a ring-shaped magnetic core, called a toroid, and a coil that wound around the entire circumference.

When current i flows through the coil of N turns, magnetic flux is mostly confined in the core material. And the leakage flux is so small that for all practical purposes it can be neglected. Therefore

The quantity Ni is called the magnetomotive force (mmf) F, and its unit is ampere-turn.

From B = μH

If we assume that all the fluxes are confined in the toroid (no leakage), the flux crossing the cross section of the toroid is

where B is the average flux density in the core and A is the area of cross section of the toroid. If H is the magnetic intensity for this path, then


is called the reluctance of the magnetic path and P is called the permeance of the magnetic path.


Magnetization curve

If the magnetic intensity in the core is increased by increasing current, the flux density in the core changes in the way shown in the figure. The flux density B increases almost linearly in the region of low values of the magnetic intensity H. However, at higher values of H, the change of B is nonlinear. The magnetic material shows the effect of saturation. The B-H curve, shown in the figure, is called the magnetization curve. The reluctance of the magnetic path is dependent on the flux density.


The flux densities are

In the air gap the magnetic flux lines bulge outward known as fringing of the flux. The effect of the fringing is to increase the cross-sectional area of the air gap and could be neglected for small air gaps


A coil wound on a magnetic core is frequently used in electric circuits. This coil may be represented by an ideal circuit element, called inductance, which is defined as the flux linkage of the coil per ampere of its current.




Consider the coil-core assembly in the figure.

When the current i is increase, H will increase and also B will increase until H reaches H₁. After that, if H is decreased, B would also decrease. But when H is made zero, the core has retained flux density B, called as residual flux density. Then, if we reverse H by reversing i, B would also decrease. And the value of H when the residual flux density is removed is coercivity or coercive force of the magnetic core. If H is further increased in the reverse direction, B will decrease to point e. After that, H is now decreasing in the reverse direction and increase to the value of H₁ again, but this time, the loop does not close and H is now varied for another cycle. After a few of magnetization the loop almost closes and it is called the hysteresis loop. Throughout the whole cycle of magnetization, the flux density lags behind the magnetic intensity. This lagging phenomenon in the magnetic core is called hysteresis.

Smaller hysteresis loops are obtained by decreasing the amplitude of variation of the magnetic intensity. A family of hysteresis loops is shown in the figure. The locus of the tip of the hysteresis loop is called the magnetization curve. However, in some magnetic cores, the hysteresis loop is very narrow and the hysteresis effect is neglected for such cores, the B—H characteristic is represented by the magnetization curve.

Deltamax Cores

Special ferromagnetic alloys are sometimes developed for special applications.The hysteresis loops for these alloys have shapes that are significantly different. Cores made of alloys having this type of almost square B-H loop are known as deltamax cores. A coil wound on a deltamax core can be used as a switch.

Hysteresis Loss

The hysteresis loops are obtained by slowly varying the current i of the coil over a cycle. However, the energy flowing in is greater than the energy returned. This energy loss heats the core and is called hysteresis loss.

Assume that the coil has no resistance and the flux in the core is Φ. The voltage e across the coil, according to Faraday’s law, is

The energy transfer during an interval of time t₁ to t₂ is



The energy transfer over one cycle of variation is

where Wₕ = ∮ H dB is the energy density of the core.The power loss in the core due to the hysteresis effect is

where f  is the frequency of variation of the current i.W

It is difficult to evaluate the area of the hysteresis loop with simple mathematical expression. Hence, for magnetic materials used in electric machines, an approximate relation is given by

and the hysteresis loss is
where Kₕ is a constant whose value depends on the ferromagnetic material and the volume of the core.

Eddy Current Loss

Another power loss occurs in a magnetic core when the flux density changes rapidly in the core is an eddy current, which will flow around the path of B. Because core material has resistance, a power loss i²R will be caused by the eddy current and will appear as heat in the core.

The eddy current loss can be reduced in two ways.

  1. A high-resistivity core material may be used. Addition of a few percent of silicon to iron will increase the resistivity significantly.
  2. A laminated core may be used. The thin laminations are insulated from each other. The lamination is made in the plane of the flux.

The eddy current loss in a magnetic core subjected to a time-varying flux is

The lamination thickness varies from 0.5 to 5 mm in electrical machines and from 0.01 to 0.5 mm in devices used in electronic circuits operating at higher frequencies.

Core Loss

The hysteresis loss and the eddy current loss are lumped together as the core loss of the coil-core as 

If the current through the coil changes rapidly, the B-H loop becomes broader because of the pronounced effect of eddy currents induced in the core. This enlarged loop is called a hystero-eddy current loop or dynamic loop.

The core loss can also be computed from the area of the dynamic B-H loop

Sinusoidal Excitation

Sinusoidal Excitation

In ac electric machines, the voltages and fluxes vary sinusoidally with time. Consider the coil-core with its flux Φ(t) varies sinusoidally with time. Thus,


From Faraday's law, the voltage induced in the N-turn coil is

The root-mean-square (rms) value of the induced voltage is

Exciting Current

If the coil is connected to a sinusoidal voltage source, a current flows in the coil to establish a sinusoidal flux in the core. This current is called the exciting current. If the B-H characteristic of the ferromagnetic core is nonlinear, the exciting current will be nonsinusoidal.

No Hysteresis

consider a B-H characteristic with no hysteresis loop. The B-H curve can be rescaled (Φ = BA, i = Hl/N) to obtain the Φ-i curve for the core. From the sinusoidal flux wave and the Φ-i curve, the exciting current waveform is obtained. The excitation current is therefore a purely lagging current and the exciting winding can be represented by a pure inductance. The phasor diagram for fundamental current and applied voltage is shown in the figure.

With Hysteresis

The waveform of the exciting current is obtained from the sinusoidal flux waveform and the multivated Φ-i characteristic of the core. The exciting current can be split into two components, one in phase with voltage accounting for the core loss and the other in phase with and symmetrical with respect to e, accounting for the magnetization of the core. This magnetizing component is the same as the exciting current if the hysteresis loop is neglected. The hysteresis loop is neglected. The exciting coil can therefore be represented by a resistance, to represent core loss, and a magnetizing inductance, to represent the magnetization of the core.

Permanent Magnet

Permanent Magnet

A permanent magnet is capable of maintaining a magnetic field without any excitation mmf provided to it. Permanent magnets are normally alloys of iron, nickel, and cobalt. They are characterized by a large B-H loop, high retentivity, and high coercive force. Permanent magnets are often referred to as hard iron and other magnetic materials as soft iron.

Magnetization of Permanent Magnets

Consider the magnetic circuit shown in the figure. Assume that the magnet material is initially unmagnetized. A large mmf is applied, and on its removal the flux density will remain at the residual value Bᵣ on the magnetization curve, point a in the figure. If a reversed magnetic field intensity of magnitude H₁ is now applied to the hard iron, the operating point moves to point b. If H₁ is removed and reapplied, the B-H locus follows a minor loop as shown in the figure. The minor loop is narrow and for all practical purposes can be represented by the straight line bc, known as the recoil line. This line is almost parallel to the tangent xay to the demagnetizing curve at point a. The slope of the recoil line is called the recoil permeability μᵣₑ. For alnico magnets it is in the range of 3-5μ₀, whereas for ferrite magnets it may be as low as 1.2μ₀.


As long as the reversed magnetic field intensity does not exceed H₁, the magnet may be considered reasonably permanent. If a negative magnetic field intensity greater than H₁ is applied, such as H₂ , the flux density of the permanent magnet will decrease to the value B₂. If H₂ is removed, the operation will move along a new recoil line de.

Approximation Design of Permanent Magnets

Let the permanent magnet be magnetized to the residual flux density denoted by point a. If the small soft iron keeper is removed, the air gap will become the active region for most applications.

In order to determine the resultant flux density in the magnet and in the air gap, the following assumptions should be made.

1. There is no leakage or fringing flux. 
2. No mmf is required for the soft iron.

From Ampere’s circuit law,For continuity of flux,



This equation represents a straight line through the origin, called the shear line. The intersection of the shear line with the demagnetization curve at point b determines the operating values of B and H of the hard iron material with the keeper removed. If the keeper is now reinserted, the operating point moves up the recoil line bc. This analysis indicates that the operating point of a permanent magnet with an air gap is determined by the demagnetizing portion of the B-H loop and the dimensions of the magnet and air gap.

The volume of the permanent magnet material is

The final operating point is located such that the BₘHₘ product is a maximum. This quantity BₘHₘ is known as the energy product of the hard iron.

Permanent Magnet Materials

A family of alloys called alnico (aluminum-nickel-cobalt) has been used for permanent magnets since the 1930s. Alnico has a high residual flux density.

Ferrite permanent magnet materials have been used since the 1950s. These have lower residual flux density but very high coercive force.

Since 1960 a new class of permanent magnets known as rare-earth permanent magnets has been developed. They combined the relatively high residual flux density of alnico-type materials with greater coercivity than the ferrites. These materials are compounds of iron, nickel, and cobalt with one or more of the rare-earth elements, such as samarium—cobalt and neodymium—iron—boron.

Question [2.1]

More question for practice:

1. In the magnetic circuit shown in the figure below, the second coil carries a current of 2 A. If flux in the core is to be made zero, the current I in the first coil should be

(A) + 4 A

(B) – 2 A

(C) – 4 A

(D) + 2


2. A magnetic circuit has a continuous core of a ferromagnetic material. Coil is supplied from a battery and draws a certain amount of exciting current producing a certain amount of flux in the core. If now an air gap is introduced in the core, the exciting current will:

(A) increase.

(B) remain same.

(C) decrease

(D) become 0.


3. For the magnetic circuit shown in figure below, the reluctance of the central limb (PS) is 10 × 105 AT/Wb and the reluctance of the outer limbs (PTS and PQS) are same and equal to 15 × 105 AT/Wb. To produce 0.5 mWb in PQS, the mmf to be produced by the coil is:

(A) 750 AT

(B) 1750 AT

(C) 250 AT

(D) 1500 AT


4. For the magnetic circuit shown in figure below, the reluctance of the central limb (PS) is 10 × 105 AT/Wb and the reluctance of the outer limbs (PTS and PQS) are same and equal to 15 × 105 AT/ Wb. To produce 0.5 mWb in PQS, the mmf to be produced by the coil is:

(A) 2625 AT

(B) 1125 AT

(C) 750 AT

(D) 1875 AT


5. A magnetic circuit draws a certain amount of alternating sinusoidal exciting current producing a certain amount of alternating flux in the core. If an air gap is introduced in the core path, the exciting current will:

(A) increase

(B) remain same.

(C) decrease.

(D) vanish.

  • Air gap reluctance = 1.017 × 10⁶ A/Wb
  • Air gap reluctance = 3.762 × 10⁶ A/Wb

A magnetic circuit with a single air gap is shown below

The core dimensions are:

Cross-sectional area , 1.8 × 10-3 m²

Mean core length, 0.6 m

Gap length, 2.3 x 10-3 m

= 83 turns

Assume that the core is of infinite permeability (μ > ∞) and neglect the effects of fringing fields at the air gap and leakage flux. Calculate the reluctance of the air gap.

Transformer [3]

Ideal Transformer


A transformer is a static machine. Although it is not an energy conversion device, it is indispensable in many energy conversion systems. It is a simple device, having two or more electric circuits coupled by a common magnetic circuit. Analysis of transformers involves many principles that are basic to the understanding of electric machines. Transformers are so widely used as electrical apparatus that they are treated along with other electric machines.

A transformer essentially consists of two or more windings coupled by a mutual magnetic field. Ferromagnetic cores are used to provide tight magnetic coupling and high flux densities. Such transformers are known as iron core transformers. They are invariably used in high-power applications. Air core transformers have poor magnetic coupling and are sometimes used in low-power electronic circuits. In this chapter we primarily discuss iron core transformers.

Two types of core constructions are normally used, as shown in the figure. In the core type (Left), the windings are wound around two legs of a magnetic core of rectangular shape. In the shell type (Right), the windings are wound around the center leg of a three-legged magnetic core. To reduce core losses, the magnetic core is formed of a stack of 0.014 inch thick silicon-steel laminations.

Transformers are widely used in low-power electronic or control circuits to isolate one circuit from another circuit or to match the impedance of a source with its load for maximum power transfer. Transformers are also used to measure voltages and currents; these are known as instrument transformers.

Ideal Transformer

Consider a transformer with two windings, primary winding of N₁ turns and a secondary winding of N₂ turns. An ideal transformer that has the following properties:

  1. The winding resistances are negligible.
  2. No leakage fluxes are present. Core losses are assumed to be negligible.
  3. The exciting current required to establish flux in the core is negligible; that is, the net mmf required to establish a flux in the core is zero.

When the primary winding is connected to a time-varying voltage v₁, a time-varying flux Φ is established in the core. A voltage e₁ will be induced in the winding and will equal the applied voltage if resistance of the winding is neglected.

The core flux also links the secondary winding and induces a voltage e₂, which is the same as the terminal voltage v₂:


where a is the turns ratio.

This indicates that the voltages in the windings of an ideal trans- former are directly proportional to the turns of the windings.

Let us now close the switch to connect the load to the secondary winding. A current i₂ will flow in the secondary winding, and the secondary winding will provide an mmf Ni₂ for the core. This will immediately make a primary winding current i, flow so that a counter mmf Ni₁ can oppose Ni₂. Otherwise Ni₂ would make  the core flux change and the balance between v₁ and e₁ would be disturbed.

Because all power loss are neglected in an ideal transformer, the instantaneous power input to the transformer equals the instantaneous power output from the transformer. If the supply voltage v₁ is  sinusoidal, then:

Impedance Transformer

Consider the circuit in the figure, the transformer circuit could be reduced into only one load circuit. By, first, consider the input impedance with the turns ratio.


Impedance can also be transferred from secondary to primary in case that its value has to be divided by the square of the turns ratio:

This impedance transfer is very useful because it eliminates a coupled circuit in an electrical circuit and thereby simplifies the circuit.


Windings on transformers or other electrical machines are marked to indicate terminals of like polarity. Consider the two windings. Terminals 1 and 3 are identical, because currents entering these terminals produce fluxes in the same direction in the core that forms the common magnetic path. For the same reason, terminals 2 and 4 are identical. If these two windings are linked by a common time-varying flux, voltages will be induced in these windings such that, if at a particular instant the potential of terminal 1 is positive with respect to terminal 2, then at the same instant the potential of terminal 3 will be positive with respect to terminal 4. 

Identical terminals such as 1 and 3 or 2 and 4 are sometimes marked by dots or ±. These are called the polarity markings of the windings. They indicate how the windings are wound on the core.

If the windings can be visually seen in a machine, the polarities can be determined. However, usually only the terminals of the windings are brought outside the machine. Nevertheless, it is possible to determine the polarities of the windings experimentally as shown in the figure below.

The voltages across 1-2, 3-4, and 1-3 are measured by a voltmeter. Let these voltage readings be called V₁₂ , V₃₄ , and V₁₃ , respectively. If a voltmeter reading V₁₃ is the sum of voltmeter readings V₁₂ and V₃₄, terminals 1 and 4 are identical (or same polarity) terminals. If the voltmeter reading V₁₃ is the difference between voltmeter readings V₁₂ and V₃₄, then 1 and 3 are terminals of the same polarity.

Polarities of windings must be known if transformers are connected in parallel to share a common load. The left figure shows the parallel connection of two single-phase (1Φ) transformers. This is the correct connection because secondary voltages e₂₁ and e₂₂ oppose each other internally. While the connection in the right figure is wrong, because e₂₁ and e₂₂ aid each other internally and a large circulating current will flow in the windings and may damage the transformers. Also, for three-phase connection of transformers, the winding polarities must also be known.

Practical Transformer

Practical Transformer

Unlike an ideal transformer, a practical transformer has many imperfections. For example, in a practical transformer the windings have resistances, not all windings link the same flux, permeability of the core material is not infinite, and core losses occur when the core material is subjected to time-varying flux. And all these must be considered.

Two methods of analysis can be used to account for the departures from the ideal transformer:

  1. An equivalent circuit model based on physical reasoning.
  2. A mathematical model based on the classical theory of magnetically coupled circuits.

Both methods will provide the same performance characteristics for the practical transformer. However, the equivalent circuit approach provides a better appreciation and understanding of the physical phenomena involved, and this technique will be presented here.

Consider a practical winding with a resistance, and this resistance can be shown as a lumped quantity in series with the winding. When currents flow through windings in the transformer, they establish a resultant mutual flux Φₘ that is confined essentially to the magnetic core. A small amount of flux, known as leakage flux, varies linearly with current, could be accounted by an inductance, called leakage inductance.

In a practical magnetic core having finite permeability, a magnetizing current Iₘ, is required to establish a flux in the core. This effect can be represented by a magnetizing inductance Lₘ . Also, the core loss in the magnetic material can be represented by a resistance. A practical transformer is therefore equivalent to an ideal transformer plus external impedances that represent imperfections of an actual transformer.

Referred Equivalent circuits

The ideal transformer can be moved to the right or left by referring all quantities to the primary or secondary side. For convenience, the ideal transformer is usually not shown and the equivalent circuit is drawn with all quantities referred to one side. The referred quantities are indicated with primes and the actual quantities can be determined if the turn ratio is known.

Approximate Equivalent Circuits

The voltage drops I₁R₁ and I₁X ₗ₁ are normally small and |E₁| = |V₁|. If this is true then the shunt branch can be moved to the supply terminal. This approximate equivalent circuit simplifies computation of currents, because both the exciting branch impedance and the load branch impedance are directly connected across the supply voltage. Besides, the winding resistances and leakage reactances can be lumped together. The equivalent circuit below is frequently used to determine the performance characteristics of a practical transformer.

In a transformer, the exciting current, is a small percentage of the rated current of the transformer. A further approximation of the equivalent circuit can be made by removing the excitation branch.

Determination of Equivalent Circuit Parameters

The parameters R₁, X ₗ₁,R ₗ, Xₘ₁, R₂, Xₗ₂ and a (= N₁/N₂) must be known so that the equivalent circuit model can be used. These parameters can be calculated from the dimensions and properties of the materials used. These parameters can be directly and more easily determined by performing two tests that involve little power consumption will provide information for determining the parameters of the equivalent circuit of a transformer.

No-Load Test (Open circuit)

This test is performed by applying a voltage to either the high-voltage side or low-voltage side, whichever is convenient. A wiring diagram for open-circuit test of a transformer is shown in the figure below. Note that the secondary winding is kept open.

And the equivalent circuit under open-circuit conditions is

The primary current is the exciting current and the losses measured by the wattmeter are essentially the core losses and the parameters R and X can be determined from the voltmeter, ammeter, and wattmeter readings.

Short-Circuit Test

This test is performed by short-circuiting one winding and applying rated current to the other winding.

If the secondary terminals are shorted,the high impedance of the shunt branch can be neglected. the parameters R and X can be determined from the voltmeter, ammeter, and wattmeter. Note that voltage applied under the short-circuit condition is small, the core losses are neglected and the wattmeter reading can be taken entirely to represent the copper losses in the windings.

Voltage Regulation

Voltage Regulation

Most loads connected to the secondary of a transformer are designed to operate at essentially constant voltage. However, voltage drops in the internal impedance of the transformer changes the load terminal voltage. If a load is not applied to the transformer the load terminal voltage is

If the load switch is now closed and the load is connected to the transformer secondary, the load terminal voltage is

The load terminal voltage may go up or down depending on the nature of the load. This voltage change is due to the voltage drop (IZ) in the internal impedance of the transformer. To reduce the magnitude of the voltage change, the transformer should be designed for a low value of the internal impedance Z.

Voltage regulation is used to identify this characteristic of voltage change in a transformer with loading. The voltage regulation is defined as the change in magnitude of the secondary voltage as the load current changes from the no-load to the loaded condition.

Consider the equivalent circuit referred to the primary. The equation above can also be written as

And in percentages

Therefore the maximum voltage regulation occurs if the power factor angle of the load is the same as the transformer equivalent impedance angle and the load power factor is lagging.



Losses in transformer are small, because the transformer is a static device, there are no rotational losses such as windage and friction losses in a rotating machine. The efficiency is defined as follow

The copper loss can be determined if the winding currents and their resistances are known:


Normally, load voltage remains fixed. Therefore, efficiency depends on load current (I₂) and load power factor (cos θ₂).

Maximum Efficiency

For constant values of the terminal voltage V₂ and load power factor angle θ₂, the maximum efficiency occurs when

Hence, the condition for maximum efficiency is

that is, core loss = copper loss. For full-load condition,



For constant values of the terminal voltage V₂ and load current I₂ , the maximum efficiency occurs when

Lastly, the condition for maximum efficiency is

Therefore, maximum efficiency in a transformer occurs when the load power factor is unity and load current is such that copper loss equals core loss.

All-day (or Energy) Efficiency

The transformer in a power plant usually operates near its full capacity and is taken out of circuit when it is not required. Such transformers are called power transformers, and they are usually designed for maximum efficiency occurring near the rated output. A transformer connected to the utility that supplies power to your house and the locality is called a distribution transformer. Such transformers are connected to the power system for 24 hours a day and operate well below the rated power output for most of the time. It is therefore desirable to design a distribution transformer for maximum efficiency occurring at the average output power.

A figure of merit that will be more appropriate to represent the efficiency performance of a distribution transformer is the “all-day” or “energy” efficiency of the transformer. This is defined as follows:

If the load cycle of the transformer is known, the all-day efficiency can be determined.



This is a special connection of the transformer from which a variable ac voltage can be obtained at the secondary. A common winding is mounted on a core and the secondary is taken from a tap on the winding. The primary and secondary of an autotransformer are physically connected. However, the basic principle of operation is the same as that of the two winding transformer.

Since all the turns link the same flux in the transformer core,

If the secondary tapping is replaced by a slider, the output voltage can be varied over the range 0 < V₂ < V₁.

The ampere-turns provided by the upper half are

The ampere-turns provided by the lower half are

For ampere-turn balance, from the first two equations,

Therefore, viewed from the terminals of the autotransformer, the voltages and currents are related by the same turns ratio as in a two-winding transformer.

The advantages of an autotransformer connection are lower leakage reactances, lower losses, lower exciting current, increased kVA rating and variable output voltage when a sliding contact is used for the secondary. The disadvantage is the direct connection between the primary and secondary sides.

Three-Phase Transformers

Three-phase Transformers

A three-phase system is used to generate and transmit bulk electrical energy. Three-phase transformers are required to step up or step down voltages in the various stages of power transmission.

Bank of Three Single-phase Transformers (Three-phase Transformer Bank)

A set of three similar single-phase transformers may be connected to forma three-phase transformer. The primary and secondary windings may be connected in either wye (Y) or delta (Δ) configurations. For the Y connection, three terminals of identical polarity are connected together to form the neutral. While for the Δ connection, the windings are connected in series. There are therefore four possible connections for a three-phase transformer and the voltage and current ratings of each transformer depend on the connections used.

Δ-Y: This connection is commonly used to step down a high voltage to a lower voltage. The neutral point on the high-voltage side can be grounded, which is desirable in most cases.

Y-Δ: This connection is commonly used to step up voltage.

Δ-Δ: This connection has the advantage that one transformer can be removed for repair and the remaining two can continue to deliver three-phase power at a reduced rating of 58% of that of the original bank. This is known as the open-delta or V connection.

Y-Y: This connection is rarely used because of problems with the exciting current and induced voltages.

Phase Shift

Some of the three-phase transformer connections will result in a phase shift between the primary and secondary line-to-line voltages. Consider the phasor voltages, for the Y-Δ connections, the line voltage of the primary leads the line voltage of the secondary by 30°. Also Δ-Y connection also provides a 30° phase shift between line-to-line voltages, whereas Δ-Δ and Y-Y connections have no phase shift in their line-to-line voltages.

Single-Phase Equivalent Circuit

If the three transformers are practically identical and the source and load are balanced, the voltages and currents in one phase are the same as those in other phases, except that there is a phase displacement of 120°. Therefore, analysis of one phase is sufficient to determine the variables on the two sides of the transformer. The Y load can be obtained for the Δ load by the well-known Y-Δ transformation. The turns ratio a' of this equivalent Y-Y transformer is

Also, for the actual transformer bank

Therefore, the turns ratio for the equivalent single-phase transformer is the ratio of the line-to-line voltages on the primary and secondary sides of the actual transformer bank. The single-phase equivalent circuit is shown below. This equivalent circuit will be useful if transformers are connected to load or power supply through feeders.

V Connection

The Δ-Δ connection of three single-phase transformers may be employed in an emergency situation when one transformer must be removed for repair and continuity of service is required.

[Right Fig of Fig 2.20 a ]

Transformer windings ab and bc deliver power


and Φ = 0 for a resistive load. Power delivered to the load by the V connection is

With all three transformers connected in delta, the power delivered is


The V connection is capable of delivering 58% power without overloading the transformer.

Three-phase Transformer on a Common Magnetic Core (Three-phase unit transformer)

A three-phase transformer can be constructed by having three primary and three secondary windings on a common magnetic core. Consider three single-phase core-type units. For simplicity, only the primary windings have been shown. If balanced three-phase sinusoidal volt- ages are applied to the windings, the fluxes will also be sinusoidal and balanced. If the three legs carrying these fluxes are merged,the net flux in the merged leg is zero. This leg can therefore be removed.

This structure is not convenient to build. However, if section b is pushed in between sections a and c by removing its yokes, a common magnetic structure is obtained.

This core structure can be built using stacked laminations as shown in the figure below.

Both primary and secondary windings of a phase are placed on the same leg. Note that the magnetic paths of legs a and c are somewhat longer than that of leg b. This will result in some imbalance in the magnetizing currents. However, this imbalance is not significant.

A three-phase transformer of this type weighs less, costs less, and requires less space than a three-phase transformer bank of the same rating. The disadvantage is that if one phase breaks down, the whole transformer must be removed for repair.

Harmonics in Three-phase Transformer Banks

Harmonics in Three-phase Transformer Banks

If a transformer is operated at a higher flux density, it will require less magnetic material. Therefore, from an economic point of view, a transformer is designed to operate in the saturating region of the magnetic core. This makes the exciting current nonsinusoidal. The exciting current will contain the fundamental and all odd harmonics. However, the third harmonic is the predominant one, and for all practical purposes harmonics higher than third can be neglected. At rated voltage the third harmonic in the exciting current can be 5 to 10% of the fundamental. At 150% rated voltage, the third harmonic current can be as high as 30 to 40% of the fundamental.

Consider the system shown in the figure. The primary windings are connected in Y and the neutral point N of the supply is available. The secondary windings can be connected in ∆.

Switch SW₁ Closed and Switch SW₂ Open

Because SW₂ is open, no current flows in the secondary windings. The currents flowing in the primary are the exciting currents. We assume that the exciting currents contain only fundamental and third-harmonic currents.

The current in the neutral line is

Note that fundamental currents in the windings are phase-shifted by 120° from each other, whereas third-harmonic currents are all in phase. The neutral line carries only the third-harmonic current.

Because the exciting current is nonsinusoidal, the flux in the core and hence the induced voltages in the windings will be sinusoidal. The secondary windings are open, and therefore the voltage across a secondary winding will represent the induced voltage.

Both SW₁ and SW₂ Open

In this case the third-harmonic currents cannot flow in the primary windings. Therefore the primary currents are essentially sinusoidal. If the exciting current is sinusoidal, the flux is nonsinusoidal because of nonlinear B-H characteristics of the magnetic core, and it contains third-harmonic components. This will induce third-harmonic voltage in the windings. The phase voltages are therefore nonsinusoidal, containing fundamental and third harmonic voltages.

The line-to-line voltage is

Because v₃s are in phase and have the same magnitude, therefore

The open-delta voltage of the secondary is

The voltage across the open delta is the sum of the three third-harmonic voltages induced in the secondary windings.

Switch SW₁ Open and Switch SW₂ Closed

If switch SW₂ is closed, the voltage will drive a third-harmonic current around the secondary delta. This will provide the missing third-harmonic component of the primary exciting current and consequently the flux and induced voltage will be essentially sinusoidal.

Y-Y System with Tertiary (∆) Winding

For high voltages on both sides, it may be desirable to connect both primary and secondary windings in Y. In this case third harmonic currents cannot flow either in primary or in secondary. A third set of windings, called a tertiary winding, connected in ∆ is normally fitted on the core so that the required third-harmonic component of the exciting current can be supplied. This tertiary winding can also supply an auxiliary load if necessary.

Per-Unit (PU) System

Per-Unit (PU) System

Computations are much simplified when all quantities are expressed in a per-unit (pu) system. The pu quantity is defined as follows

Two major advantages in using a per-unit system are:

1) The parameters and variables fall in a narrow numerical range, this simplifies computations and makes it possible to quickly check the correctness of the computed values.

2) Circuit quantities have no need to be referred, therefore a common source of mistakes is removed.

To establish a per-unit system it is necessary to select base values for any two of power, voltage, current, and impedance. Once base values for any two of the four quantities have been selected, the base values for the other two can be determined from the relationship among these four quantities. Usually base values of power and voltage are selected first and base values of current and impedance are obtained as follows:

Although base values can be chosen arbitrarily, normally the rated volt- amperes and rated voltage are taken as the base values for power and voltage, respectively.

In the case of a transformer, the power base is same for both primary and secondary. However, the voltage base are different on each side, because rated voltages are different for the two sides.

Primary side:

LetSecondary side:


Therefore, the per-unit transformer impedance is the same referred to either side of the transformer. This is another advantage of expressing quantities in a per-unit system.

In a transformer, when voltages or currents of either side are expressed in a per-unit system, they have the same per-unit values.

Transformer Equivalent Circuit in Per-Unit Form

The equation in per-unit form can be obtained by dividing the equation in terms of actual values throughout by the base value of the primary voltage.

It has been shown that the voltages, currents, and impedances in per-unit representation have the same values whether they are referred to primary or secondary. Hence the transformer equivalent circuit in per-unit form for either side is

Note that the values of the voltages are generally close to 1 pu, and this makes the analysis somewhat easier.

Full-load Copper Loss

LetThe full-load copper loss in per-unit form based on the volt-ampere rating of the transformer isHence the transformer resistance expressed in per-unit form also represents the full-load copper loss in per-unit form. The per-unit value of the resistance is therefore more useful than its ohmic value in determining the performance of a transformer.

Question [3.1]

  • 5.0 V
  • 2.4 V
  • 4.3 V

A transformer has 500 turns of the primary winding and 10 turns of the secondary winding.

Determine the secondary voltage if the secondary circuit is open and the primary voltage is 120 V.

Question [3.2]

More problems for practice:

1. An ideal transformer supplying a noninductive resistance R = 10 ohms. The primary winding has N₁ = 100 turns and the secondary winding has N₂ = 50 turns. The instantaneous voltage applied to the primary winding is V₁ = 150 V.

(a) What is the value of the current and its direction through the resistance R?

(b) What is the value of the primary current?

(c) What is the direction of the flux in the core?

(d) What is the rate of change of

(i) The flux in the primary winding ? 
(ii) The flux linkage with the primary winding? 
(iii) The flux in the secondary winding? 
(iv) The flux linkage with the secondary winding?

(e) The transformer and its connected load are replaced by a noninductive resistance which takes the same value of current at 150 v as that in the primary of the transformer. What is the value of the resistance in ohms?

(f) What is the ratio of the resistance in part (e) to that of the resistance R in the secondary? What is this ratio called?

(g) A polarity mark is shown near the upper terminal of the primary winding in Fig. 6-37. Where should the secondary polarity mark be placed?


2. The purpose of this problem is to show, firstly, that a sizeable reduction in frequency without a corresponding reduction in voltage leads to values of exciting current that can exceed the rated current of the transformer, and secondly, that a reduction of the voltage to a value that leads to a safe value of exciting current lowers the rating of the transformer.

A 1,500-w, 240/25-v, 400-cycle transformer operates at a maximum flux density of 36,000 lines per sq in. and the peak value of the exciting current is 0.20 amp.

(a) Determine the peak value of the exciting current when the primary of this transformer is connected to 120-v 60-cycle source. Neglect resistance and leakage flux.

(b) Suppose this transformer were to operate at a frequency of 60 cps and at magnetic flux density B = 90,000 lines per sq in., and the current density in the windings is the same as for rated 400-cps operation. What would the rating of the transformer be in volt-amperes?

  • 3.36 V
  • 4.25 V
  • 8.61 V

The AC source in the circuit shown below has a voltage magnitude of 5 volts, which is divided partly over the 1KΩ resistor and the remainder over the rest of the circuit. Compute the magnitude of the voltage over the 1KΩ.

Electromechanical Energy Conversion [4]

Energy Conversion Process

There are various methods for calculating the force or torque developed in an energy conversion device. The method that will be mentioning is based on the principle of conservation of energy, which states that energy can neither be created nor destroyed; it can only be changed from one form to another. An electromechanical converter system has three essential parts

  • an electric system
  • a mechanical system
  • a coupling field

The energy transfer equation is as follows

The electrical energy loss is the heating loss due to current flowing in the winding of the energy converter. This loss is known as the i²R loss in the resistance (R) of the winding. The field loss is the core loss due to changing magnetic field in the magnetic core. The mechanical loss is the friction and windage loss due to the motion of the moving components. All these losses are converted to heat. The energy balance equation 3.1 can therefore be written as

Now consider a differential time interval dt during which an increment of electrical energy (excluding the i²R loss) flows to the system. During this time, the energy supplied to the field (either stored or lost, or part stored and part lost) and the energy converted to mechanical form (in useful form or as loss, or part useful and part as loss) can be expressed in differential forms as

Core losses are usually small, and if they are neglected, energy supplied to the field will represent the change in the stored field energy. Similarly, if friction and windage losses can be neglected, then all of energy converted to mechanical form will be available as useful mechanical energy output. Even if these losses cannot be neglected they can be dealt with separately. The losses do not contribute to the energy conversion process.

Field Energy

Consider the electromechanical system, the movable part can be held in static equilibrium by the spring. Assume that the movable part is held stationary at some air gap and the current is increased from zero to a value i. Flux will be established in the magnetic system. Obviously,

If core loss is neglected, all the incremental electrical energy input is stored as incremental field energy. Now,

The incremental field energy is shown as the crosshatched area in this figure. When the flux linkage is increased from zero to A, the energy stored in the field is increased from zero to λ, the energy stored in the field is

Other useful expressions can also be derived for the field energy of the magnetic system.



where   A = is the cross-sectional area of the flux path

               B = is the flux density, assumed same throughout

From the equations mentioned above, the energy stored in the field is now;

For the air gap,

The equation of energy stored in the field becomes

Normally, energy stored in the air gap is much larger than the energy stored in the magnetic material. For a linear magnetic system,


Energy, Coenergy

The λ-i characteristic of an electromagnetic system depends on the air gap length and the B-H characteristics of the magnetic material. For larger air gap length, the characteristic is essentially linear. The characteristic becomes nonlinear as the air gap length decreases.

For a particular value of the air gap length, the energy stored in the field is represented by the area A between the λ axis and the λ-i characteristics. The area B between the I axis and the λ-i characteristic is known as the coenergy and is defined as

This quantity can be used to derive expressions for force (or torque) developed in an electromagnetic system.

Note that B has more area than A if the λ-i characteristic is nonlinear and the area of B is equal to A if it is nonlinear.

Mechanical Force in the Electromagnetic System

Consider the system shown, let the movable part move from one position to another position so that at the end of the movement the air gap decreases. The λ-i characteristics of the system for these two positions are shown. The current (i = v/R) will remain the same at both positions in the steady state. If the movable part has moved slowly, the current has remained essentially constant during the motion.

If the motion has occurred under constant-current conditions, the mechanical work done is represented by the shaded area, which is the increase in the coenergy.

Linear System

Consider the electromagnetic system, if the reluctance of the magnetic core path is negligible compared to that of the air gap path, the λ-i relation becomes linear.

For this idealized system

Where L(x) is the inductance of the coil, whose value depends on the air gap length. The field energy is

For a linear system

The total cross-sectional area of the air gap is two times of the air gap. Hence, the force per unit area of air gap, called magnetic pressure Fₘ, is

Rotating Machines

Most of the energy converters produce rotational motion. The essential part of a rotating electromagnetic system is shown below.

The fixed part of the magnetic system is called the stator, and the moving part is called the rotor. The latter is mounted on a shaft and is free to rotate between the poles of the stator. Consider a case when both stator and rotor have windings carrying currents, the current can be fed into the rotor circuit through fixed brushes and the rotor-mounted slip rings.

The stored field energy of the system can be evaluated by establishing the current from the stator and the current from the rotor in the windings keeping the system static, that is, with no mechanical output.

For a linear magnetic system, the flux linkage of the stator winding and the flux linakge of the rotor winding can be expressed in terms of inductances whose values depend on the position θ of the rotor.

Cylindrical Machines

A cross-sectional view of an elementary two-pole cylindrical rotating machine with a uniform air gap is shown below.

The stator and rotor windings are shown as placed on two slots on the stator and the rotor, respectively. In an actual machine the windings are distributed over several slots. If the effects of the slot are neglected, the reluctance of the magnetic path is independent of the position of the rotor. It can be assumed that the self-inductances are constant and therefore no reluctance torques are produced. The mutual inductance varies with rotor position, and the torque produced in the cylindrical machine is

In rotating machines, torque can be produced by variation in the reluctance of the magnetic path or mutual inductance between the windings.

Reluctance machines are simple in construction, but toque developed in these machines is small. Cylindrical machines, although more complex in construction, produce larger torques. Most electrical machines are of the cylindrical type.

Question [4.1]

Linear Machine [5]

Starting the Linear DC Machine

A linear dc machine consists of a battery and a resistance connected through a switch to a pair of smooth, frictionless rails. Along the bed of railroad track is a constant, uniform-density magnetic field directed into the page. A bar of conducting metal is lying across the tracks.

Its behavior can be determined from an application of four basic equations to the machine.

  • The equation for the force on a wire in the presence of a magnetic field:

Where F = force on wire

i = magnitude of current in wire

l = length of wire, with direction of I defined to be the direction of current flow

B = magnetic flux density vector

  • The equation for the voltage induced on a wire moving in a magnetic field:

Where eind = voltage induced in wire

v = velocity of the wire

B = magnetic flux density vector

L = length of conductor in the magnetic field

  • Kirchhoff’s voltage law:

  • Newton’s law for the bar across the tracks:

Starting the Linear DC machine

Starting this machine is simply closing the switch. A current then starts flowing in the bar, which is given by Kirchhoff’s voltage law:

Since the bar is initially at rest, the induced voltage is equal to zero, so the equation above becomes

The current flows down through the bar across the tracks. Current flowing through a wire in the presence of a magnetic field induces a force on the wire. Because of the geometry of the machine, this force is

Therefore, the bar will accelerate to the right by Newton’s Law. However, when the velocity of the bar begins to increase, a voltage appears across the bar. The voltage is given by

The voltage now reduces the current flowing in the bar, since by Kirchhoff’s voltage law, as the induced voltage increases, the current decreases.

The result of this action is that eventually the bar will reach a constant steady-state speed where the net force on the bar is zero. This will occur when the induced voltage has risen and exceed the voltage VB. At the time, the bar will be moving at a speed given by

The bar will continue to coast along at this no-load speed forever unless some external force disturbs it.

The Linear DC Machine as a Motor

Suppose that the linear machine is again operating under no-load steady-state conditions. Now apply a load force to the bar in opposite direction of motion. Since the bar was initially at steady-state, application of the load force will result in a net force on the bar in the direction opposite the direction of motion

The effect of this force will slow the bar. But just as soon as the bar begins to slow down, the induced voltage on the bar drops. As the induced voltage decreases, the current flow in the bar rises. Therefore, the induced force rises too. The overall result of this chain of events is that the induced force rises until it is equal and opposite to the load force, and the bar again travels in steady state, but at a lower speed. There is now an induced force in the direction of motion of the bar, and power is being converted from electrical form to mechanical form to keep the bar moving. The power being converted is

An amount of electrical power is consumed in the bar and is replaced by mechanical power. Since power is converted from electrical to mechanical form, this bar is operating as a motor.

The Linear DC Machine as a Generator

Suppose the linear machine is operating under no-load steady-state conditions. A load force is applied in the direction of motion of the bar. The applied force will cause the bar to accelerate in the direction of motion, and the velocity if the bar will increase. As the velocity increases, the induced voltage will increase and will be larger than the battery voltage. With this conditions, the current reverses direction and is now given by the equation

Since this current now flows up through the bar, it induces a force in the bar

This induced force opposes the applied force on the bar. Finally, the induced force will be equal and opposite to the applied force, and the bar will be moving at a higher speed than before. The battery is charging, the linear machine is now serving as a generator, converting mechanical power into electrical power.

Question [5.1]

Questions: Briefly explain the motor and the generator effects and the laws applied to each effect.

For the answers you can go back to previous lesson and check the correct idea.

Direct-Current Generator [6]

Basic features and properties of DC Generators

Direct-Current Generators

Direct current is mainly produced by electronic rectifiers which can convert the current without using any moving parts. And any dc generator can operate as a motor and vice versa owing to their similar construction.

Generating an ac voltage

The voltage generated in any dc generator is inherently alternating and only becomes dc after it has been rectified by the commutator. An ac generator composed of a coil that revolves at 60 r/min between the N, S poles of a permanent magnet. The coil is connected to two slip rings mounted on the shaft. The slip rings are connected to stationary brushes x and y.

As the coil rotates, a voltage is induced between its terminals A and D. The induced voltage is therefore maximum when the coil is momentarily in the horizontal position cutting across the flux produced by the N, S poles. And no flux is cut when the coil is momentarily in the vertical position; consequently the voltage at these instants is zero. Another feature of the voltage is that its polarity changes every time the coil makes half a turn. The voltage can therefore be represented as a function of the angle of rotation.

Direct-current generator

If the brushes of an ac-generator could be switched from one slip ring to a commutator, we would obtain a voltage of constant polarity across the load. Brush x would always be positive and brush y negative. Due to the constant polarity between the brushes, the current in the external load always flows in the same direction. Therefore, the machine represented in the figure is called a direct-current generator, or dynamo.

Difference between ac and dc generators

These machines only differ in the way the coils are connected to the external circuit: ac generators carry slip rings while dc generators require a commutator. We sometimes build small machines which carry both slip rings and a commutator which can function simultaneously as ac and dc generators.

Improving the waveshape

By increasing the number of coils and segments, we can obtain a dc voltage that is very smooth. The coils are lodged in the slots of a laminated iron cylinder. The coils and the cylinder constitute the armature of the machine. And the voltage differences between the opposite slots are equal to zero. Consequently, no current will flow in the closed loop formed by the coils. This is most fortunate, because any such circulating current would produce a power losses.

Induced voltage

When the armature rotates, the voltage E induced in each conductor depends upon the flux density which it cuts. This fact is upon the equationBecause the density in the air gap varies from point to point, the value of the induced voltage per coil depends upon its instantaneous position. Also, by shifting the brushes, the output voltage decreases. Furthermore, in this position, the brushes continually short-circuited coils. Large currents will flow in the shorted-circulated coils and brushes, and the sparking, that is said to be poor commutation, will result.

Neutral zones

Neutral zones are those places on the surface of the armature where the flux density is zero. When the generator operates at no-load, the neutral zones are located exactly between the poles.

Value of the induced voltage

The voltage induced in a dc generator having a lap winding is given by the equationwhere

Eₒ = voltage between the brushes [V]

Z = total number of conductors on the armature

n = speed of rotation [r/min]

Φ = flux per pole [Wb]

Generator under load: the energy conversion process

The current delivered by the generator also flows through all the armature conductors. The armature conductors under the S pole carry currents that flow into the screen, away from the reader. Conversely, the armature currents under the N pole flow out of the screen, toward the reader. From the direction of current flow and the direction of flux, we find that the individual forces F on the conductors all act clockwise. In effect, they produce a torque that acts opposite to the direction in which the generator is being driven. To keep the generator going, we must exert a torque on the shaft to overcome this opposing electromagnetic torque. The resulting mechanical power is converted into electrical power, which is delivered to the generator load. That is how the energy conversion takes place.

Armature reaction

The current flowing in the armature coils creates a powerful magnetomotive force that distorts and weakens the flux coming from the poles in both motors and generators, called armature reaction.

The figure above shows that the flux in the neutral zone is  no longer zero. A voltage will be induced in the coils that are short-circuited by the brushes. As a result, as the first problem, severe sparking may occur. The second problem created by the armature mmf is that it distorts the flux produced by the poles. In effect, the armature mmf and field mmf produces a different magnetic field as shown below. The neutral zones have shifted in the direction of rotation of the armature. This cause an increase and an decrease of flux under pole tips. As a result, the total flux produced by the N, S poles is less than it was when the generator was running at no-load. Consequently, a corresponding reduction in the induced voltage which could reach to 10% for large machines.

Shifting the brushes to improve commutation

Due to the shift when the generator is under load, the brushes could be moved to reduce the sparking. For generators, the brushes are moved in the direction of rotation. For motors, the brushes are shifted against the direction of rotation. However, if the load fluctuates, the armature mmf rises and falls and so the neutral zone shifts back and forth between the no-load positions. We would therefore have to move the brushes back and forth to obtain sparkless commutation. Note that this procedure is not practical and can only resolve problem for small dc machines.

Communicating poles

To counter the effect of armature reaction in medium- and large-power dc machines, a set of commutating poles are placed between the main poles.

This nullifies the arnature mmf, the flux in the space between the main poles is always zero and so we no longer have to shift the brushes. In practice, the mmf of the commutating poles is made slightly greater than the armature mmf. This creates a small flux in the neutral zone, which aids the commutation process.

Separately excited generator

Instead of using permanent magnets to create the magnetic field, we can use a pair of electromagnets, called field poles. When the dc field current in such a generator is supplied by an independent source, the generator is said to be separately excited. Thus, the dc source connected to terminals a and b causes an exciting current Iₓ to flow. If the armature is driven, a voltage Eₒ appears between brush terminals x and y.

No-load operation and saturation curve

Examine the relationships when a separately excited dc generator runs at no-load.

Field flux vs exciting current

Let us gradually raise the exciting current Iₓ so that the mmf of the field increases, which increases the flux per pole. The saturation curve is therefore obtained whether or not the generator is turning.

Saturation of the iron begins to be important when we reach the "knee" ab of the saturation curve.

If we drive the generator at constant speed. Eₒ is directly proportional to the flux Φ. And a curve whose shape is identical to the saturation curve, called no-load saturation curve of the generator, could be obtained.

Induced voltage vs speed

For a given exciting current, the induced voltage increases in direct proportion to the speed. If we reverse the direction of rotation, the polarity of the induced voltage also reverses. However, if the exciting current is also reverse, the polarity of the induced voltage remains the same.

Shunt generator

A shunt-excited generator is a machine whose shunt-field winding is connected in parallel with the armature terminals, so that the generator can be self-excited. The principal advantages of this connection is that it eliminates the need for an external source of excitation.

When a shunt generator is started up, a small voltage is induced in the armature producing a small exciting current Iₓ in the shunt field. The resulting small mmf acts in the same direction as the remanent flux, causing the flux per pole to increase. And both Eₒ and Iₓ causes each other to increase. This progressive buildup continues until Eₒ reaches a maximum value determined by the field resistance and the degree of saturation.

Controlling the voltage of a shunt generator

We can simply vary the exciting current by means of a rheostat connected in series with the shunt field. Suppose the movable contact p is in the center of the rheostat. If we move the contact toward m, the resistance R₁ between points p and b diminishes, which causes the exciting current to increase. This increases the flux and, consequently, the induced voltage Eₒ. On the other hand, if we move contact p toward n, everything would be increase in the opposite way and Eₒ will fall.

We can determine the no-load value of Eₒ if we know the saturation curve of the generator and the total resistance R₁ of the shunt field circuit between points p and b.We draw a straight line through the origin corresponding to the slope of R₁ and superimpose it on the saturation curve. The point where is intersects the curve yields the induced voltage. Note that the induced voltage suddenly drops to zero if the value of R₁ is greater than the critical value. (Slope is too high that the line does not cut the curve.)

Equivalent circuit

The equivalent circuit of a generator is composed of a resistance Rₒ in series with a voltage Eₒ. The latter is the voltage induced in the revolving conductors, Terminals 1, 2 are the external armature terminals of the machine, and F₁, F₂ are the field winding terminals.

Separately excited generator under load

If we connect a load across the armature, the resulting load current I produces a voltage drop across resistance Rₒ. As we increase the load, the terminal voltage diminishes progressively. In practice, the induced voltage Eₒ also decreases slightly with increasing load, because pole-tip saturation tends to decrease the field flux. Consequently, the terminal voltage E₁₂ falls off more rapidly than can be attributed to armature resistance alone.

Shunt generator under load

The terminal voltage of a self-excited shunt generator falls off sharply with increasing load. The exciting current falls as the terminal voltage drops which is about 15 percent of the full-load voltage, whereas for a separately excited generator it is usually less than 10 percent. The voltage regulation is said to be 15% and 10%, respectively.

Compound generator

The compound generator was developed to prevent the terminal voltage of a dc generator from decreasing with increasing load. It is similar to a shunt generator, except that it has additional field coils connected in series with the armature. These series field coils are composed of a few turns of heavy wire to carry the armature current. The terminal voltage remains practically constant from no-load to full-load. The rise in the induced voltage compensates for the armature IR drop.

Differential compound generator

In a differential compound generator the mmf of the series field acts opposite to the shunt field. As a result, the terminal voltage falls drastically with increasing load. We can make such a generator by simply reversing the series field of a standard compound generator. Differential compound generators were formerly used in dc arc welders, because they tended to limit the short-circuit current and to stabilize the arc during the welding process.

Load Characteristics

The load characteristics of some shunt and compound generators are given in the figure above. The voltage of an over-compound generator increases by 10 percent when full-load is applied, whereas that of a flat-compound generator remains constant. On the other hand, the full-load voltage of a shunt generator is 15 percent below its no-load value, while that of a differential-compound generator is 30 percent lower.

Generator specifications

The nameplate of a generator indicates the power, voltage, speed, and other details about the machine. These ratings, or nominal characteristics, are the values guaranteed by the manufacturer. For example, the following information is punched on the nameplate of a 100 kW generator:

Power                      100  kW     Speed   1200 r/min

Voltage                    250  V        Type      Compound

Exciting current     20  A        Class      B

Temperature rise   50°C

Construction of DC Generators


The field produces the magnetic flux in the machine. It is basically a stationary electromagnet composed of a set of salient poles bolted to the inside of a circular frame. Field coils, mounted on the poles, carry the dc exciting current. The frame is usually made of solid cast steel, whereas the pole pieces are composed of stacked iron laminations. In some generators the flux is created by permanent magnets.

In practice, the number of poles depends upon the physical size of the machine: the bigger it is, the more poles it will have. The shunt field coils are composed of several hundred turns of wire carrying a relatively small current. The coils are insulated from the pole pieces to prevent short-circuits.

The air gap is the short space between the armature and the pole pieces. It ranges from about 1.5 to 5 mm as the generator rating increases from 1 kW to 100 kW. Most of the mmf produced by the field is used to drive the flux across the air gap. Hence, the air gap must be reduced but not too short otherwise the armature reaction effect becomes too great.

If the generator has a series field, the coils are wound on top of the shunt-field coils. The conductor size must be large enough so that the winding does not overheat when it carries the full-load current of the generator.


The armature is the rotating part of a dc generator. It consist of a commutator, an iron core, and a set of coils. The armature is keyed to a shaft and revolves between the field poles. The iron core is composed of slotted, iron laminations that are stacked to form a solid cylindrical core and are coated with insulating film to reduce eddy-current losses. The slots are lined up to provide the space needed to insert the armature conductors. The figures of lamination of a small armature and a cross section view of the slot of a large armature is shown below.

Commutator and brushes

The commutator is composed of an assembly of tapered copper segments insulated from each other by mica sheets, and mounted on the shaft of the machine. Great care is taken in building the commutator because any eccentricity will cause the brushes to bounce, producing unacceptable sparking. The sparks burn the brushes and overheat and carbonize the commutator.

The brushes are made of carbon because it has good electrical conductivity and its softness does not score the commutator. To improve the conductivity, a small amount of copper is sometimes mixed with the carbon. The brush pressure is set by means of adjustable springs. If the pressure is too great, the friction produces excessive heating of the commutator and brushes; on the other hand, if it is too weak, the imperfect contact may produce sparking.

Details of a multiple generator

In order to get a better understanding of multipole generators, let us examine the construction of a 12-pole machine with 72 slots on the armature, 72 segments on the commutator, and 72 coils. Coils A and C are momentarily in the neutral zone, while coil B is cutting the flux coming from the center of the poles.

The voltage generated between brushes x and y is equal to the sum of the voltages generated by the five coils connected to commutator segments 1-2, 2-3, 3-4, 4-5, and 5-6. The voltages between the other brush sets are similarly generated by five coils.

The ideal commutation process

When a generator is under load, the individual coils on the armature carry one-half the load current carried by one brush. The currents flowing in the armature winding next to a positive brush.

If the commutator segments are moving from right to left, the coils on the right-hand side of the brush will soon be on the left-hand side. This means the current in these coils must reverse. The process whereby  the current changes direction in this brief interval called commutation.

Owing to the contact resistance, the conductivity between the brush and commutator is proportional to the contact area. For example, if the area in contact with segment 2 is only one-forth of the total contact area, and so the current from segment 2 is only one-forth of the total current. (25%)

The practical commutation process

In practical, the armature coils have inductance and it strongly opposes a rapid change in current causing the current to reverse slower that it should.

The voltage induced by self-induction is given by

in which

e = induced voltage [V]

L = inductance of the coil [H]

l/∆t = rate of change of current [A/s]

In designing dc motors and generators, every effort is made to reduce the self-inductance of the coils. One of the most effective ways is to reduce the number of turns per coil. But for a given output voltage . The number of coils must be increased and more coils implies more commutator bars.

Direct current generators have a large number of coils and commutator bars not so much to reduce the ripple in the output voltage but to overcome the problem of commutation.

Another important factor in aiding commutation is that the mmf of the commutating poles is always made slightly greater than the armature mmf.

Question [6.1]

Questions with solutions are provided here 



Direct-Current Motor [7]

Basic features and properties of DC Motors


There are three basic types of motors:

  • Shunt motors
  • Series motor
  • Compound motors

Direct-current motors are seldom used in ordinary industrial applications because all electric utility systems furnish alternating current. It is advantageous to transform the alternating current into direct current in order to use dc motors. The reason is that the torque speed characteristics of dc motors can be varied over a wide range while retaining high efficiency.

Counter-electromotive force (cemf)

A voltage is induced in the armature conductors as soon as they cut a magnetic field. The values and polarity of the induced voltage are the same as those obtained when the machine operates as a generator. The induced voltage is therefore proportional to the speed of rotation of the motor and to the flux density as given by

In the case of  a motor, the induced voltage is called counter-electromotive force because its polarity always acts against the source voltage. It acts against the voltage in the sense that the net voltage acting in the series circuit is equal to (Eₛ - Eₒ) volts.

Acceleration of the motor

The net voltage acting in the armature circuit is (Eₛ - Eₒ) volts. The resulting armature current is limited only by the armature resistance so

When the motor is at rest, the induced voltage Eₒ = 0, and so the starting current is

If the starting current are absent, the large forces acting on the armature conductors produce a powerful starting torque and a consequent rapid acceleration of the armature.

As the speed increases, with the result that the value of (Eₛ - Eₒ) diminishes, It follows that the armature current drops progressively as the speed increases.

Although the armature current decreases, the motor continues to accelerate until it reaches a definite, maximum speed. If Eₒ were equal to Eₛ, the net voltage (Eₛ - Eₒ) would become zero and so the current. The driving forces would cease to act on the armature conductors, and the mechanical drag imposed by the fan and the bearings would immediately cause the motor to slow down. As the speed decreases the net voltage (Eₛ - Eₒ) increases and so does the current.

Mechanical power and torque

The power of torque of a dc motor are two of its most important properties.

1. The cemf induced in a lap wound armature is given by

The electrical power supplied to the armature is equal to the supply voltage by the armature current

However, Eₛ is equal to the sum of Eₒ, plus the voltage drop in the armature

The mechanical power of the motor is therefore exactly equal to the product of the cemf multiplied by the armature current

P = Eₒ × I


P = mechanical power developed by the motor [W]

Eₒ = induced voltage in the armature (cemf) [V]

I = total current supplied to the armature [A]

Speed of rotation

When a dc motor drives a load between no-load and full-load, the IR drop due to armature resistance is always small compared to the supply voltage Eₛ. This means that the counter-emf Eₒ is very nearly equal to Eₛ.

When Eₒ = Eₛ,

That is,


n = speed of rotation [r/min]

Eₛ = armature voltage [V]

Z = total number of armature conductors

The equation shows that the speed of the motor is directly proportional to the armature supply voltage and inversely proportional to the flux per pole.

Armature speed control

If the flux per pole is kept constant, the speed depends only upon the armature voltage Eₛ. By raising or lowering Eₛ, the motor speed will rise and fall in proportion.

We can vary Eₛ by connecting the motor armature M to a separately excited variable voltage dc generator G.

The field excitation of the motor is kept constant, but the generator excitation can be varied from zero to maximum, with either positive or negative polarity. Consequently, the motor speed can be varied from zero to maximum in either direction. The generator is driven by an ac motor connected to a 3-phase line. This method of speed control is known as the Ward-Leonard system. It can actually force the motor to develop the torque and speed required by the load.

Another way to control the speed of a dc motor is to place a rheostat in series wit the armature, called Rheostat Speed Control. The current in the rheostat produces a voltage drop which subtracts from the fixed source voltage, yielding a smaller supply voltage across the armature. This method enables us to reduce the speed below its normal speed. The IR drop across the rheostat increases as the armature current increases. This produces a substantial drop in speed with increasing mechanical load.

Field speed control

According to the speed of rotation equation, we can vary the speed of dc motor by varying the field flux. Let us now keep the armature voltage constant so that the numerator is constant. Consequently, the motor speed now changes in inverse proportion to the flux: if we increase the flux, the speed will drop.

This method of speed control is used when the motor has to run above its rate speed called base speed. To control the flux, we connect a rheostat in series with the field.

To understand this method of speed control, suppose the motor is initially running at a constant speed. The counter-emf Eₒ is slightly less than the armature supply voltage, due to the IR drop in the armature. If we suddenly increase the resistance of the rheostat, both exciting current and the flux will diminish. This immediately reduces the cemf Eₒ, causing the armature current I to jump to a much higher value. The current change because the value depends on the very small difference between source voltage and Eₒ. It will accelerate until Eₒ is again equal to source voltage.

To develop the same counter-emf with a weaker flux, the motor must turn faster. The way to increase the speed of the motor is to insert a resistance in series with the field. Broader speed ranges tend to produce instability and poor communication.

Under certain abnormal conditions, the flux may drop to dangerously low values such as if the exciting current of a shunt motor is interrupted accidentally, the only flux remaining is that due to the remanent magnetism in the poles. This flux is so small that the motor has to rotate at a dangerously high speed to induce the required cemf. Safety devices are introduced to prevent such runaway conditions.

Shunt motor under load

Consider a dc motor running at no-load. If a mechanical load is suddenly applied to the shaft, the small no load current does not produce enough torque to carry the load and the motor begins to slow down. This causes the cemf to diminish, it makes a higher current and higher torque. When the torque developed by the motor is exactly equal to the torque from mechanical load, the speed will remain constant. To sum up, as the mechanical load increases, the armature current rise and the speed drops.

The speed of the shunt motor stays a relatively constant from no-load to full-load. In a small motors,it drops a little bit when full load is applied. In big machines, the drop is very less, due in part, to the very low armature resistance. By adjusting the field rheostat, the speed can absolutely constant as the load change.

Typical torque-speed and torque current characteristics of the shunt motor, the speed, torque,and current are given in per unit values. The torque is directly proportional to the armature current.

Series motor

A series motor is identical in construction to a shunt motor except for the field. The field is connected in series with the armature and carry the full armature current. This series field is composed of a few turns of wire having a cross section large enough to carry the current.

In a shunt motor, the flux F per pole is constant at all loads because the shunt field is connected to the line. However, for a series motor, the flux per pole depends on the armature current and upon the load. When the flux is large, the current is also large. Despite these differences, they both use the same basic principle to apply in both machines.

When the series motor operates at full-load, the flux per pole is the same as a shunt motor of identical power and speed. However, when the series motor starts up, the armature current is higher than normal. It follows that the starting torque of a series motor is considerably greater than that of the shunt motor.

On the other hand, if the motor operates at less than full-load, the armature current and the flux per pole are smaller than normal. The weaker field causes the speed to rise in the same way as it would for a shunt motor with a weak shunt field. If the load is small, the speed may rise dangerously high values. For this reason, we never permit a series motor to operate at no-load. The resulting centrifugal forces could tear the winding out of the armature and destroy the machine.

Series motor speed control

When a series motor carries a load. Its speed may have to be adjusted slightly. The speed can be increased by placing a low resistance in parallel with the series field. The field current is then smaller than before which produces a drop in flux and increasing speed.

Conversely, the speed may be lowered by connecting an external resistor in series with the armature and the field. The total IR drop across the resistor and field reduces the armature supply voltage and so the speed must fall.

Typical torque-speed and torque-current characteristic is represented in the form of graph which is quite different from the shunt motor characteristic.

Applications of the series motor

Series motors are used on equipment requiring a high starting torque. They are also used to drive devices which must run at high speed at light loads. The series motor is particularly well adapted for traction purposes such as electric trains. Acceleration is rapid because the torque is high at low speeds. The series motor automatically slow down when the train goes up and turn at the top speed on flat ground. Series motors are also used in electric cranes and hoists.

Compound motor

Compound dc motor carries both series field and a shunt field. In the cumulative compound motor, the mmf of the two fields add. The shunt field is always stronger than the series field.

When the motor runs at no-load, the armature current I in the winding is low and the mmf of the series field is negligible. On the other hand, the shunt field is fully excited by current Ix and the motor behaves like a shunt machine. As the load increases, the mmf of the series field increases but for the shunt field remains constant which make the total mmf and the flux per pole is greater under load than no load.

If the series field is connected opposite the shunt field, we obtain a differential compound motor. For the differential compound motor, the total mmf decrease when the load increases with the same trend as the rising of speed. The differential compound motor has a very few application such as the dc motor in steel mill.

Reversing the direction of rotation

To reverse the direction of rotation of a dc motor, we must reverse the armature connections or reverse both shunt and series field connections. The interpoles are considered to form part of the armature.

The figure above consists of:

Left : Original connections of a compound motor.

Middle : Reversing the armature connections to reverse the direction of rotation.

Right : Reversing the field connections to reverse the direction of rotation.

Starting a shunt motor

If we apply full voltage to a stationary shunt motor, the starting current in the armature will be very high and it will cause some hazards such as

  • Burning out the armature.
  • Damaging the commutator and brushes
  • Overloading the feeder
  • Snapping off the shaft due to mechanical shock
  • Damaging the driven equipment due to the sudden mechanical hammer blow.

All dc motor must limit the starting current to be reasonable values to prevent the risks. To solve the problem, we can connect a rheostat in series with the armature because the resistance is reduced as the motor accelerates and eliminate entirely when the machine has attained full speed. Today, this method uses to limit the starting current and to provide speed control.

Free-plate starter

The diagram of a manual face-plate starter for a shunt motor start from the bare copper contacts are connected to current-limiting resistors. Conducting arm sweeps across the contact when it is pulled to the right by means of insulated handle 2. The arm touches dead copper contact M and the motor circuit will open.

When we draw the handle to the right, the conductor will touch the fixed contact N and the supply voltage Eₛ will immediately cause the current Iₓ to flow but the armature current I is limited by the resistors in the starter box. The motor begin to turn as the induced voltage build up, the armature current falls. When the motor speed increases, the arm is pulled to the next contact by removing resistor 1 from the circuit. To move to the next contact the speed has to drop down. We can see that the contact arm move depends on the speed of the motor. The arm is magnetically held by the small electromagnet which connect in the series to the shunt field.

If the supply voltage is interrupted, the electromagnet will release the arm, allowing it to turn to its dead position under the pull spring. This will prevent the motor from restarting unexpectedly.

Stopping a motor

To stop a dc motor that has a heavy inertia load, it has to use an hour or more to stop the system of the motor. We must apply a breaking torque to ensure a rapid stop because of the amount of time for deceleration  and the circumference of the motor. One way to brake the motor is by simple mechanical friction. A more elegant method is to calculate a reverse current in the armature to brake the motor electrically.

There are two methods to create such an electromechanical brake which are

  • dynamic braking
  • plugging

Dynamic braking

A shunt motor that the field is directly connected to a source voltage and an armature is connected to the same source by double throw switch. The switch connects the armature to either the line or the external resistor. When the motor is running normally, the direction of the armature current and the polarity of the cemf are shown in the figure. If we neglect the IR drop between the resistor, induced voltage will equal to voltage source.

If we open the switch, the motor continue to turn, but the speed will drop because of the friction and windage losses. On the other hand, because of the shunt field is excited, it still has the induced voltage in the system which keep falling according to the rate of speed. Now, the motor become a generator that the armature is on open circuit.

When you close switch on another way, the armature is connected to the external resistor. Induced voltage will produce the current I₂ in the opposite direction to the current I₁ from the previous figure. Then , a reverse torque is developed which its magnitude depends on the value of the current flow I₂. This torque will make the machine to stop.

In practice, resistor is chosen so that the initial braking current will be twice the rated motor current and it will make the initial braking torque twice the torque of the motor. For the dynamic braking, when the motor slows down, the induced voltage will gradually decrease because of the decreasing of current flow I₂. Then the braking torque will become smaller and smaller until it becomes zero when the armature turn. The speed decrease exponentially which you can see that the trend of the graph is similar to the discharge voltage of the capacitor.


Plugging or another name is reverse current braking. It is a reversing of the armature current by reversing the terminals of the sources.

The armature current is given by

If we suddenly reverse the terminals of the sources, the net voltage will also change the direction and becomes (Eₒ+Eₛ). So, the cemf of the armature is no longer counter but add to the supply voltage. From this situation, it makes the net voltage produce high amount of reverse current flow. This current will initiate an arc around the commutator, destroying segments, brushes, and supports before the line circuit breaker to open.

To prevent a catastrophe, we must limit the reverse by adding the resistor R in series to the reversing circuit. As the dynamic braking, the resistor is designed to limit the initial braking current I₂. With the plugging circuit, there will be a reverse torque even  when the armature stop, we must immediately open the armature circuit, otherwise the reverse running circuit will happen.

For the plugging method, the plugging stop the motor completely after 2Tₒ compare to the dynamic braking that still have 25 percents speed for braking. Therefore, most of the application use dynamic braking than plugging.

Dynamic braking and mechanical time constant

For dynamic braking, the speed decreases exponentially with time when the dc motor is stopped. We can speak of a mechanical time constant in the same way as the electrical time constant of the capacitor that discharges into a resistor.

There is a direct mathematical relationship between the conventional time constant T and the half constant Tₒ, it is given by

The mechanical time constant is given by


This equation comes from the assumption that the braking effect due to the energy dissipated in the braking resistor. In general, the motor is subjected to an extra braking torque due to windage and friction.

Armature reaction

Until now we have assumed that the only mmf acting in a dc motor is that due to the field. However, the current flowing in the armature conductors also creates a magnetomotive force that distorts and weakens the flux coming from the poles. This distortion and field weakening takes place in motors as well as in generators. We recall that the magnetic action of the armature mmf is called armature reaction.

Flux distortion due to armature reaction

When a motor runs at no-load, the small current flowing in the armature does not affect the flux that coming from the poles at the first figure. However, when the armature has a normal current, it produces a strong magnetomotive force which would create Φ₂ in the second figure. The net flux between Φ₁ and Φ₂ will give a result as Φ₃.

As you can see, the flux density increases under the left half and decrease under the right half. Because of the unequal flux density, it will cause some effects.

  • the neutral zone will shift which will make poor commutation with sparking at the brushes.
  •  the saturation sets in due to the higher flux density in pole tip.

The increase of flux under the left hand pole is less than the decrease in the right half. Φ₃ at full load is slightly less than Φ₁ at no-load. For large machine, the decreasing of the flux can cause the speed to increase with load and make it unstable. To eliminate this problem is to add series field to increase the flux under load which is called a stabilized-shunt winding.

Commutating poles

To counter the effect of armature reaction and thereby improve commutation, we always place a set of commutating poles between the main poles of medium and large power dc motors. Just like the dc generator, these narrow poles develop a magnetomotive force equal and opposite to the mmf of the armature so that the respective magnetomotive forces rise and fall together as a load current.

In practice, the mmf of the commutating poles is greater than the mmf of the armature. Consequently, a small flux subsists in the region of the commutating poles. The flux is used to induce in the coil undergoing commutation a voltage that is equal and opposite to the self induction voltage.

The neutralization of the armature mmf is restricted to the narrow zone covered by the commutating poles. The flux distribution under the main poles unfortunately remains distorted. This creates no problem for motors driving ordinary loads but for the special case, it necessary to add a compensating winding.

Compensating winding

Some dc motor in the high power range employed in steel mills perform a series of rapid, heavy-duty operations. The amount of armature current produces very sudden changes in armature reaction.

For such motors, the commutating poles and series stabilizing winding do not neutralize the armature mmf. Torque and speed control is difficult under such transient conditions and flash-overs may occur across the commutator. To eliminate this problem, we use special compensating windings to connect in series with the armature. However, because the windings are distributed across the pole faces, the armature mmf is bucked from point to point, which eliminates the field distortion in the first figure below. Also, the field distribution remains essentially undisturbed from no-load to full-load and become the general shape in the second figure.

The addition of compensating windings effect on the design and performance of a dc motor.

  •  The short air gap can be used because we don't have to worry about the demagnetizing effect of the armature. A shorter gap means that the shunt field strength can be reduced and the coil are smaller
  • The inductance of the armature circuit is reduced which the armature current change more quickly and motor give a better response.
  • A motor equipped with compensating windings can develop 3 to 4 times its rated torque. The peak torque will be lower if the armature current is large because the effective flux in the air gap falls off rapidly with increasing current caused by armature reaction.

We conclude that compensating windings are essential in large motors subjected to severe duty cycles.

Basics of variable speed control

The most important outputs of a dc motor are its speed and torque. It is very useful to determine the limits of each as the speed is increased from zero to above base speed. The rated values of the armature current, armature voltage,and field flux must not be exceeded, although lesser values may be used.

The figure on the left side is the ideal separately excited shunt motor that the armature resistance is negligible. All the components in the figure are expressed in per-unit values. If the rated armature voltage happens to be 240 V and the rated armature current is 600 A, they are both given a per-unit value of 1. Similarly,the rated shunt field flux also has a per-unit value of 1. The benefit of the per-unit value is that it renders the torque-speed curve universal.

The per-unit torque is given by the per-unit flux times the per-unit armature current.

Also, the per-unit armature voltage is equal to the per-unit speed times the per-unit flux.

The motor develops rated torque (T=1) at the rated speed (n=1). We called the rated speed as based speed.

In order to reduce the armature voltage to zero, while keeping the rated value of armature current and flux to be constant at their per-unit value of 1. By apply the equation, we get T= 1x1=1 and Ea= n x 1 = n. For the ideal dc shunt motor, it can operate anywhere within the limits of torque-speed curve.

There are the graphs that represent the state of armature voltage, armature current,and flux during motor operation, know as the constant torque mode.

In practice, the actual torque-speed curve may differ from the ideal curve. The curve indicates an upper speed limit 2 but some machine may have the limit at 3 or 4 by reducing the flux. However, the increase of the speed will make the communication problem occurs and it may be dangerous. When the motor run below base speed, the ventilation becomes poorer and the temperature tends to rise. Consequently, the armature current must be reduced that it will lead to the reducing of torque. When the speed is zero, all force ceases and even the field current must be reduced to prevent the heat. As the result, the torque may have the per-unit value just only 0.25.

The drastic fall-off in torque as the speed diminishes can be largely overcome by using an external blower to cool the motor. The torque-speed curve will be ideal if you deliver a constant stream of air.

Permanent magnet motors

We know that the shunt field motors has the energy consumed, the hear produced and the relatively large space taken up by the field poles which are disadvantages of a dc motor. By using the permanent magnet instead of field coils can reduce the disadvantage of the motor. The result is that a small motor has a higher efficiency with the added benefit due to the field failure.

A further advantage of using permanent magnets is that the effective air gap keep increasing because the magnets have a permeability that nearly equal to the air. As the result, the armature mmf cannot create the intense field. The field created by the magnets does not become distorted.

The armature reaction is reduced and commutation is improved as well as the overload capacity of the motor. A further advantage is that the air gap reduces the inductance of the armature to change the armature current flow.

Permanent magnet motors are advantageous in capacities below 5 hp. The magnet are ceramic or rare earth-cobalt alloys to construct the PM motor. Its elongated armature ensures low inertia and fast response when use in the application.

The only drawback of PM motors is the relatively high cost of the magnets and the inability to obtain higher speeds by field weakening.

Question [7.1]

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