## MMF – Magnetomotive force

Table of Contents

MMF – Magnetomotive force

## What is *Magnetomotive force* (MMF)

The effort required to move a unit Magnetic pole once around the Magnetic circuit is known as the Magnetomotive force (MMF). Magnetic pressure has the Tendency to create Magnetic flux in a Magnetic circuit. The mmf is Calculated Mathematically as the sum of a coil’s current and turns.

So Consider the coil Consisting of N turns and an electric current of I amperes flowing in it. Therefore, the mmf is given by,

Magnetomotive force, MMF = 𝑁𝐼 … (1)

## Unit of MMF

Since MMF is a product of current in amperes (A) and number of turns (T) in a coil, thus **the unit of Magnetomotive force is**

Unit of MMF = Ampere *Turns (or AT)

## Magnetomotive Force Formula (MMF)

- In any magnetic circuit, if H being the magnetising force and
*l*is the length of the magnetic circuit, then the mmf for the magnetic circuit is given by,

**MMF = 𝐻 × 𝑙 … (2)**

- In any magnetic circuit, if Φ being the magnetic flux and S is the magnetic reluctance of the magnetic circuit. Then, the mmf required is given by,

**MMF=Magnetic flux(ϕ)×Reluctance(S)**

⇒MMF=ϕ×(l/aμ0μr) = (ϕ/a)×(1/μ0μr)

((∵S=1/μ0μr ) and ( ϕ / a) = B(Magnetic flux density))

∴MMF=B×(1/μ0μr) =H×l…(3)

(∵H=B/μ0μr)

## MMF – Magnetomotive force Numerical Example (1)

An Electromagnet Consists of a coil of 1500 turns, if 2 A current flows in the coil. Determine the mmf of the Electromagnet.

**Solution** −

MMF = NI = 1500 × 2 = 3000 AT

### MMF Of Distributed Windings

### 1. Alternating Field Distribution

While the field strength amount Fluctuates Periodically with the current Frequency, the spatial field distribution and zero crossings remain constant. Alternating field is the name given to this type of field.

So Fourier analysis can be used to identify the Square-wave Function’s Fundamental wave. Inversely Proportional Decreasing Amplitude with ordinal numbers and an Infinite count of single waves of odd ordinal numbers are the outcomes. Fundamental wave and Harmonic Amplitudes exhibit Proportional Dependence on current, but zero crossings are Unaffected. Standing waves are what they are. The spatial distributions of the Windings are what cause Harmonics to exist. The current that is Generated is entirely Sinusoidal and devoid of Harmonics. It is necessary to Differentiate between

**wave**: Spatiotemporal Behaviour,

**oscillation**: pure time Dependent behavior

**2. Rotating field**

Rotating fields appear as spatial Distributed fields of constant form and amount, Revolving with angular speed *w*1:

It is possible to divide a Sinusoidal Alternating field into two Sinusoidal Rotating fields. Their angular speeds are Sign-opposed, their peak value is half that of the Corresponding Alternating field.

**3. Three phase winding**

Most simple Arrangement of a Three-phase stator consist of:

Score stack composed of laminations with approximately 0,5 mm thickness, mutual insulation for a reduction of eddy currents

2. The number of pole pairs is *p=1* in Fig.138. In case of *p*>1, the configuration repeats *p*-times along the circumference.

**4. Determination of slot mmf for different moments**

- quantity of slot mmf is applied over the circumference angle.
- line integrals provide enveloped mmf, dependent on the circumference angle.
- total mmf is shaped like a staircase step function, being constant between the slots. At slot edges, with slots assumed as being narrow, the total mmf changes about twice the amount of the slot mmf, the air gap field results from the total mmf

**Magnetic Fields In Rotating Machines**

### 1. Winding factor

The effective number of windings appears to be less than it actually is if w windings per phase are not placed in two opposing slots but rather are distributed over multiple slots (zone winding) and return conductors are returned under an electric angle less than 180°.

So This means is utilized for a supression of harmonics, which cause parasitic torques and losses, influencing proper function of a machine..Actually there is no machine with *q =1 *Only zoning and chording enable disregarding harmonics.

**Rotating Magnetic Field**

It only takes three coils to create a symmetric rotating magnetic field. A symmetric three-phase AC sine current system will be required to drive the three coils, so each phase will be 120 degrees out of phase with the others. The magnetic field is assumed in this example to be a linear function of the coil current.

Sine wave current in each of the coils produces sine varying magnetic field on the rotation axis. Magnetic fields add as vectors. Vector sum of the magnetic field vectors of the stator coils produces a single rotating vector of resulting rotating magnetic field.

So A single rotating vector is produced by adding three sine waves with a phase angle of 120 degrees on the motor’s axis. A constant magnetic field surrounds the rotor. The magnetic field of the stator’s S pole will move toward the N pole of the rotor, and vice versa. This magneto-mechanical attraction generates a force that propels the rotor to synchronously follow the rotating magnetic field.

##### In such a field, a permanent magnet will rotate to keep itself aligned with the external field. Early alternating current electric motors made use of this effect. Two orthogonal coils with an AC current phase difference of 90 degrees can be used to create a rotating magnetic field. However, in reality, a three-wire setup with different currents would be used to supply such a system.

The standardisation of conductor size would face serious difficulties as a result of this inequality. Three-phase systems are used to get around this, where the three currents have an equal magnitude and a 120 degree phase difference. In this instance, the rotating magnetic field will be produced by three similar coils with 120 degree geometrical angles between them. One of the main reasons why three phase systems predominate in the world’s electric power supply systems is their capacity to produce the rotating field required by electric motors.

Induction motors also employ rotating magnetic fields. Induction motors use short-circuited rotors (instead of a magnet) that follow the rotating magnetic field of a multicoiled stator because magnets deteriorate over time. The rotor is moved by Lorentz force in these motors because the short circuited turns of the rotor create eddy currents in the rotating field of the stator. These kinds of motors typically lack synchronous operation and require some degree of “slip” in order to generate current from the relative movement of the field and the rotor.

The single coil of a single phase induction motor does not produce a rotating magnetic field, but a pulsating 3-φmotor runs from 1-φ power, but does not start.

##### Another view is that the single coil excited by a single phase current produces two counter rotating magnetic field phasor, coinciding twice per revolution at 0o (Figure above-a) and 180o (figure e). When the phasor rotate to 90o and -90o they cancel in figure b. At 45o and -45o (figure c) they are partially additive along the +x axis and cancel along the y axis. An analogous situation exists in figure d. The sum of these two phasor is a phasor stationary in space, but alternating polarity in time. Thus, no starting torque is developed.

However, if the rotor is rotated forward at a bit less than the synchronous speed, It will develop maximum torque at 10% slip with respect to the forward rotating phasor. Less torque will be developed above or below 10% slip. The rotor will see 200% – 10% slip with respect to the counter rotating magnetic field phasor. Little torque (see torque vs. slip curve) other than a double frequency ripple is developed from the counter rotating phasor.

Thus, the single phase coil will develop torque, once the rotor is started. If the rotor is started in the reverse direction, it will develop a similar large torque as it nears the speed of the backward rotating phasor. Single phase induction motors have a copper or aluminum squirrel cage embedded in a cylinder of steel laminations, typical of poly-phase induction motors.

**2. Distribution factor(k**_{d}**)**

All w/p windings per pole and phase are distributed over q slots. Any of the

w/pq conductors per slot show a spatial displacement of.

The resulting number of windings wresper phase is computed by geometric addition of all q partial windings w/pq. The vertices of all q phasors per phase, being displaced by ;N (electrically), form a circle. The total angle per phase per phase adds up to q;N.

**Purpose: **The purpose of utilizing zone winding is to aim

i. slot mmf fundamental waves adding up

ii. harmonics compensating each other, as they suppose to do.

**Kd = ER when coils are distributed / ER when coils are concentrated**

**= 2Rsin(mβ/2) / 2mRsin(β/2)**

**= sin(mβ/2) / msin(β/2)**

Where

m= Slots per pole per phase

β = Slot angle =180 o/ n

n = Slots per pole

The distribution factor is also called **winding factor** or** breadth factor.**

**3. Pitch factor**

**Kc = ER when coil is short pitched / ER when coil is full pitched**

**= 2Ecos(****a****/2) / 2E**

**= cos(****a****/2)**

a = Angle of short Pitch

**Rotating Mmf Waves**

The Principle of operation of the Induction machine is based on the generation of a Rotating Magnetic field. Let us understand this idea better.

Consider a cosine wave from 0 to 360◦. This sine wave is plotted with unit Amplitude.

• So Now allow the Amplitude of the sine wave to vary with respect to time in a Sinusoidal fashion with a Frequency of 50Hz. Let the maximum value of the Amplitude is, say, 10 units. This waveform is a pulsating sine wave.

Now consider a second sine wave, which is Displaced by 120◦ from the first (lagging).

• and allow its Amplitude to vary in a similar manner, but with a 120◦time lag. Similarly consider a third sine wave, which is at 240◦ lag.

• And allow its Amplitude to change as well with a 240◦ time lag. Now we have three Pulsating sine waves. Let us see what happens if we sum up the values of these three sine waves at every angle.

##### The result really speaks about Tesla’s genius. What we get is a constant amplitude travelling sine wave!

In a three phase Induction machine, there are three sets of Windings,phase A winding, phase B and phase C Windings. These are excited by a Balanced Three-phase voltage supply.

Three phase current that is balanced would result from this. Keep in mind that there is a 120-second gap between them. In addition, not all of the windings in an induction machine are situated in the same location. They are scattered throughout the machine 120 apart from one another (more about this in the section on alternators). The correct terminology would be to state that the axes of the windings are 120 apart in space. Because waves also travel at a 120° angle in space, the phases A, B, and C are used. mmfs are produced by the coils when currents flow through them.

Since mmf is Proportional to current, these Waveforms also show the total mmf as well as the mmf produced by the coils. Additionally, because the machine contains magnetic material (iron), these mmfs produce magnetic flux that is proportional to the mmf (we may assume that iron is Infinitely permeable and non-linear effects such as hysterisis are neglected). The flux produced inside the machine would then be represented by the waveforms seen above. As we can see, the overall effect is a moving flux wave. The machine’s space angle as it rotates around the air gap would be represented by the x-axis.

##### The first pulsating waveform seen earlier would then represent the a-phase flux, the second represents the b-phase flux and the third represents the c-phase. This may be better visualized in a polar plot. The angles of the polar plot represent the space angle in the machine, i.e., angle as one travels around the stator bore of the machine.

The pulsating wave is depicted in this plot at the zero-degree axes. At 0° axes, the amplitude is at its maximum, and at 90°, it is zero. The waveform’s positive portions are represented by red and its negative ones by blue. It should be noted that the waveform pulses at the 0180 axis and that red and blue alternate on each side. The sinewave current’s polarity is changing as a result. Keep in mind that the sinewave’s maximum amplitude is only attained along the 0180 axis.

The amplitude does not increase to this value at any other angles. It does, however, reach a maximum value that is lower than the peak located at the 0–180 axis. More specifically, the peak at the 0–180 axis would costimes the maximum reached at any space angle. Additionally, the time variation is sinusoidal at any space angle, with the excitation’s frequency, phase lag, and amplitude corresponding to the angle.

##### This plot shows the pulsating waveforms of all three cosines. Note that the first is pulsating about the 0 − 180◦ axis, the second about the120◦− 300◦axis and the third at 240◦− 360◦axis.

The trajectory of the travelling wave is depicted here as circular. A Pulsating wave’s maximum Amplitude is 10, but the Resultant wave has a maximum Amplitude of 15. If the flux Waveform’s Amplitude in each phase is f1, then. The following ideas are worth Reflecting on.

1. what is the interpretation of the Pulsating plots of the animation? If one wants to know the ‘a’ phase flux at a particular angle for all instants of time, how can it be obtained?

2. What will this time variation look like? It is obviously periodic. What will be the amplitude and frequency?

**1. Voltage Induction caused by influence of Rotating field**

Voltage in Three-phase Windings Revolving at Variable speed, induced by a Rotating field is subject to Computation in the following:

The flux linkage of a coil is the result of the spatial integration of the air gap field. The flux linkage with respect to time is then used to derive induced voltage. Induced voltages in the stator and rotor can be discussed using a transfer onto three-phase windings and the definition of slip. The fundamental wave is the only factor taken into account in the following statements.

**2. Flux linkage**

The air gap field is created in the three-phase winding of the stator, characterized by the number of windings *w1* and current *I1*:

So First of all, only one single rotor coil with number of windings *w2* and arbitrary position angle of twist is taken into account. Flux linkage of the rotor coil results from spatial integration of the air gap flux density over one pole pitch.

**3. Induced voltage, slip**

Induced voltage in a rotor coil of Arbitrary angle of twist *a(t)*, which is flowed through by the air gap flux density, computes from variation of the flux linkage with Time. Described variation of flux linkage can be caused by both variation of currents *iu(t), iv(t), iw(t)*with time, inside the exciting Three-phase winding and also by rotary motion *a(t)* of the coil along the air gap Circumference.

- · Some aspects regarding induced voltage dependencies are listed below: the amplitude of the induced voltage is proportional to the line frequency of the statorand to the according slip.
- · frequency of induced voltage is equal to slip frequency at rotor standstill (s=1), frequency of the induced voltage is equal to line frequency.
- · when rotating ( s < 1 ), voltage of different frequency is induced by the fundamental wave of the stator windings.
- · no voltage is induced into the rotor at synchronous speed (s=0).
- · phase displacement of voltages to be induced into the rotor is only dependent from the spatial position of the coil, represented by the (elec.) angle p R a .Is a rotor also equipped with a three-phase winding, instead of a single coil similar to the stator arrangement with phases being displaced by a mechanical angle, a number of slots per pole and phase greater than 1 (q>1) and the resulting number of windings w2;2 then follows for the induced voltage of single rotor phases.