Torque in 3 phase induction motor

-

 Torque in a 3 phase induction motor is determined by three key factors:

Firstly the magnitude of rotor current, secondly the flux which interact with the rotor of three phase induction motor and is responsible for producing emf in the rotor part of inuction motor, lastly the power factor of rotor of the three phase induction motor.
By integrating these factors, we derive the torque equation as follows:

Where, T is the torque produced by the induction motor,
φ is flux responsible for producing induced emf,
I2 is rotor current,
cosθ2 is the power factor of rotor circuit.Ezoic

The flux φ produced by the stator is proportional to stator emf E1.
i.e φ ∝ E1
We know that transformation ratio K is defined as the ratio of secondary voltage (rotor voltage) to that of primary voltage (stator voltage).

Rotor current I2 is defined as the ratio of rotor induced emf under running condition , sE2 to total impedance, Z2 of rotor side,

and total impedance Z2 on rotor side is given by ,

Putting this value in above equation we get,

s = slip of inducttion motor

We know that power factor is defined as ratio of reistance to that of impedance. The power factor of the rotor circuit is

Putting the value of flux φ, rotor current I2, power factor cosθ2 in the equation of torque we get,

Combining similar term we get,

Removing proportionality constant we get,

Where, ns is synchronous speed in r. p. s, ns = Ns / 60. So, finally the equation of torque becomes,

How the constant K is derived in the torque equation.
In a 3 phase induction motor copper losses typically occur in the rotor. These rotor copper losses are expressed as
Pc = 3I22R2
We know that rotor current,

Substitute this value of I2 in the equation of rotor copper losses, Pc. So, we get

The ratio of P2 : Pc : Pm = 1 : s : (1 – s)
Where, P2 is the rotor input,
Pc is the rotor copper losses,
Pm is the mechanical power developed.

Substitute the value of Pc in above equation we get,

On simplifying we get,

The mechanical power developed Pm = Tω,

Substituting the value of Pm

We know that the rotor speed N = Ns(1 – s)
Substituting this value of rotor speed in above equation we get,

Ns is speed in revolution per minute (rpm) and ns is speed in revolution per sec (rps) and the relation between the two is

Substitute this value of Ns in above equation and simplifying it we get

Comparing both the equations, we get, constant K = 3 / 2πns

Working Principle of Three Phase Induction Motor – Video

 

Equation of Starting Torque of Three Phase Induction Motor

Ezoic

Starting torque is the torque produced by induction motor when it starts. We know that at the start the rotor speed, N is zero.

So, the equation of starting torque is easily obtained by simply putting the value of s = 1 in the equation of torque of the three phase induction motor,

The starting torque is also known as standstill torque.

Maximum Torque Condition for Three-Phase Induction Motor

In the equation of torque,

The rotor resistance, rotor inductive reactance and synchronous speed of induction motor remain constant. The supply voltage to the 3 phase induction is usually rated and remains constant, so the stator emf also remains the constant. We define the transformation ratio as the ratio of rotor emf to that of stator emf. So if stator emf remains constant, then rotor emf also remains constant.
If we want to find the maximum value of some quantity, then we have to differentiate that quantity concerning some variable parameter and then put it equal to zero. In this case, we have to find the condition for maximum torque, so we have to differentiate torque concerning some variable quantity which is the slip, s in this case as all other parameters in the equation of torque remains constant.
So, for torque to be maximum

Now differentiate the above equation by using division rule of differentiation. On differentiating and after putting the terms equal to zero we get,

Neglecting the negative value of slip we get

So, when slip s = R2 / X2, the torque will be maximum and this slip is called maximum slip Sm and it is defined as the ratio of rotor resistance to that of rotor reactance.
NOTE: At starting S = 1, so the maximum starting torque occur when rotor resistance is equal to rotor reactance.

Equation of Maximum Torque

The equation of torque is

The torque will be maximum when slip s = R2 / X2
Substituting the value of this slip in above equation we get the maximum value of torque as,

In order to increase the starting torque, extra resistance should be added to the rotor circuit at start and cut out gradually as motor speeds up.
Conclusion
From the above equation it is concluded that

  1. The maximum torque is directly proportional to square of rotor induced emf at the standstill.
  2. The maximum torque is inversely proportional to rotor reactance.
  3. Notably, the maximum torque does not depend on the rotor resistance.
  4. The slip at which maximum torque occur depends upon rotor resistance, R2. So, by varying the rotor resistance, maximum torque can be obtained at any required slip.

Comments

Post a Comment

Thank you for visiting😊

Popular posts from this blog

Using MOSFET Body Diodes to Charge Battery in Inverters

-
Image
  Using MOSFET Body Diodes to Charge Battery in Inverters In this post we try to understand how the internal body diodes of MOSFETs could be exploited for enabling the charging of battery through the same transformer which is being used as the inverter transformer. In this article we will investigate a full bridge inverter concept and learn how the in-built diodes of its 4 MOSFETs could be applied for charging an attached battery. What is a Full Bridge or H-Bridge Inverter In few of my earlier posts we have discussed  full bridge inverter circuits  and regarding their working principle. As shown in the above image, basically, in a full-bridge inverter we have a set of 4 MOSFETs connected to the output load. The diagonally connected MOSFET pairs are alternately switched through an external  oscillator , causing the input DC from the battery to transform into an alternating current or AC for the load. The load is normally in the form of a  transformer , whose low ...
Read more

Resistive Transducer

-
Image
  Resistive Transducer Definition:   The  transducer  whose  resistance varies  because of the  environmental effects  such type of transducer is known as the resistive transducer. The  change in resistance  is  measured  by the  ac or dc measuring devices . The  resistive transducer  is used for measuring the  physical quantities like temperature, displacement, vibration etc  . The measurement of the physical quantity is quite difficult. The resistive  transducer  converts the physical quantities into variable  resistance  which is easily measured by the meters. The process of variation in resistance is widely used in the industrial applications. The resistive transducer can work both as the primary as well as the secondary transducer. The primary transducer changes the physical quantities into a mechanical signal, and secondary transducer directly transforms it into an electrical s...
Read more

Solar Cell: Working Principle & Construction (Diagrams Included)

-
Image
  Solar Cell: Working Principle & Construction (Diagrams Included) What is a Solar Cell? A solar cell (also known as a photovoltaic cell or PV cell) is defined as an electrical device that converts light energy into electrical energy through the photovoltaic effect . A solar cell is basically a p-n junction diode . Solar cells are a form of photoelectric cell, defined as a device whose electrical characteristics – such as current , voltage , or resistance – vary when exposed to light. Individual solar cells can be combined to form modules commonly known as solar panels. The common single-junction silicon solar cell can produce a maximum open-circuit voltage of approximately 0.5 to 0.6 volts. By itself this isn’t much – but remember these solar cells are tiny. When combined into a large solar panel, considerable amounts of renewable energy can be generated. Construction of Solar Cell A solar cell is basically a junction diode , although its construction is little bit differen...
Read more