What are Induction Machines?
The induction machine was invented by Nikola Tesla in 1888. It is available from fractional horsepower ratings to
megawatt levels. It finds very wide usage in all various application areas. The induction
machine is an AC electromechanical energy conversion device. The machine interfaces with
the external world through two connections (ports) one mechanical and one electrical. The
mechanical port is in the form of a rotating shaft and the electrical port is in the form of
terminals where the AC supply is connected. There are machines available to operate from three-phase or single-phase electrical input.
What is an Induction Motor?
About 90% of the ac motor used for utility and industrial purposes are induction motors due to their robust construction and excellent running characteristics. These are singly excited motors. Three-phase induction motors are of two types Slip Ring induction motors and Squirrel Cage induction motors. The name of these motors is given based on the rotor used in their construction.
Principle of Operation:
The stator is provided with 3-phase winding which is 120 degrees displaced in slots in the inner periphery and the rotor is provided with short-circuited bars at both ends to develop emf. When 3 phase supply is given to the stator RMF (rotating magnetic field ) is generated which passes through short-circuited secondary ie. the rotor and rotor start rotating to oppose the cause in the direction of RMF at slip speed. The speed of stator RMF is synchronous speed Ns and rotor speed is sNs ie. slip times of synchronous speed where s is slip.
where Slip=Ns-N
 |
| Induction Machine |
Rotor Design:
Basically, two types of rotors are used in induction motors: Squirrel cage and Slip Ring type rotor.
Squirrel Cage Rotor:
It consists of a cylinder of steel laminations with aluminum or copper conductors embedded in its surface. In operation, the non-rotating "stator" winding is connected to an alternating current power source; the alternating current in the stator produces a rotating magnetic field. The rotor winding has current induced in it by the stator field and produces its own magnetic field. The interaction of the two sources of magnetic field produces torque on the rotor. The motor rotor shape is a cylinder mounted on a shaft. Internally it contains longitudinal conductive bars (usually made of aluminum or copper) set into grooves and connected at both ends by shorting rings forming a cage-like shape. The name is derived from the similarity between this rings-and-bars winding and a squirrel cage.
 |
| Squirrel Cage Rotor |
Slip Ring Rotor:
The rotor windings are connected through slip rings to external resistances. Adjusting the resistance allows control of the speed/torque characteristic of the motor. Wound-rotor motors can be started with a low inrush current, by inserting high resistance into the rotor circuit; as the motor accelerates, the resistance can be decreased. the rotor of the slip ring motor has more winding turns; the induced voltage is then higher and the current lower, than for a squirrel-cage rotor. During the start-up, a typical rotor has 3 poles connected to the slip ring. Each pole is wired in series with a variable power resistor. When the motor reaches full speed the rotor poles are switched to a short circuit. During start-up, the resistors reduce the field strength at the stator. As a result, the inrush current is reduced. Another important advantage over squirrel-cage motors is higher starting torque.
 |
| Slip Ring Rotor |
Circuit and Phasor Diagram:
 |
| Circuit Diagram of Induction Machine |
 |
| Phasor Diagram of Induction Machine |
Torque Equation:
The torque produced by 3 phase induction motor depends upon the following three factors:
Firstly the magnitude of the rotor current, secondly the flux which interacts with the rotor of three phase induction motor and is responsible for producing emf in the rotor part of the induction motor, and lastly the power factor of the rotor of the three-phase induction motor.
Combining all these factors together we get the equation of torque as-
T ∝ 𝜑I₂ cosθ₂
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 the rotor circuit.
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).
K = E₂ / E₁
or, K = E₂ / φ
or, E₂ = φ
Rotor current I₂ is defined as the ratio of rotor-induced emf under the running condition, sE₂ to total impedance, Z₂ of rotor side,
i.e I₂ = sE₂ / Z₂
and total impedance Z2 on the rotor side is given by,
Z₂ = √{ (R₂)² + ( sX₂ )² }
Putting this value in the above equation we get,
I₂ = sE₂ / √{ (R₂)² + ( sX₂ )² }
s = slip of Induction motor
We know that the power factor is defined as the ratio of resistance to that of impedance. The power factor of the rotor circuit is
cosθ₂ = R₂ / Z₂ = R₂ / √{ (R₂)² + ( sX₂ )² }
Putting the value of flux φ, rotor current I₂, and power factor cosθ₂ in the equation of torque we get,
T ∝ E₂ [ sE₂ / √{ (R₂)² + ( sX₂ )² }][ R₂ / √{ (R₂)² + ( sX₂ )² } ]
Combining similar terms we get,
T ∝ sE₂²[ R₂ / √{ (R₂)² + ( sX₂ )² }]
Removing the proportionality constant we get,
T = KsE₂²[ R₂ / √{ (R₂)² + ( sX₂ )² }]
Where K= 3 / 2πnₛ
Where ns is the synchronous speed in r. p. s, nₛ = Nₛ / 60. So, finally, the equation of torque becomes,
T = sE₂²[ R₂ / √{ (R₂)² + ( sX₂ )² }][ 3 / 2πnₛ ]
Derivation of K in torque equation.
In the case of three phase induction motor, there occur copper losses in the rotor. These rotor copper losses are expressed as
Pc = 3(I₂)²R₂
We know that rotor current,
I₂ = sE₂ / √{ (R₂)² + ( sX₂ )² }
Substitute this value of I₂ in the equation of rotor copper losses, Pc. So, we get
Pc = 3R₂[ sE₂ / √{ (R₂)² + ( sX₂ )² }]²
On simplifying Pc = 3R₂ s²(E₂)² / { (R₂)² + ( sX₂ )² }
The ratio of P₂: Pc: Pₘ = 1: s : (1 - s)
Where P₂ is the rotor input,
Pc is the rotor copper loss,
Pₘ is the mechanical power developed.
Pₘ = (1 - s)Pc / s
Substitute the value of Pc in the above equation we get,
Pₘ = (1/s) [ (1 - s) 3R₂ s²(E₂)² / { (R₂)² + ( sX₂ )² }]
On simplifying we get,
Pₘ = (1 - s) 3R₂ s(E₂)² / { (R₂)² + ( sX₂ )² }
The mechanical power developed Pₘ = Tω,
ω = 2πN/60
Pₘ = 2πNT/60
Substituting the value of Pₘ
(1/s) [ (1 - s) 3R₂ s²(E₂)² / { (R₂)² + ( sX₂ )² }] = 2πNT/60
or T = (1/s) [ (1 - s) 3R₂ s²(E₂)² / { (R₂)² + ( sX₂ )² }] (60/2πN)
We know that the rotor speed N = Nₛ(1 - s)
Substituting this value of rotor speed in the above equation we get,
T = (1/s) [ (1 - s) 3R₂ s²(E₂)² / { (R₂)² + ( sX₂ )² }] {60/2πNₛ(1 - s)}
Ns is the speed in revolution per minute (rpm) and ns is the speed in revolution per sec (rps) and the relation between the two is
Nₛ/60 = nₛ
Substitute this value of Nₛ in the above equation and simplify it we get
Torque, T = [s(E₂)²R₂ / { (R₂)² + ( sX₂ )² }] (3/ 2πNₛ)
T = Ks(E₂)²R₂ / { (R₂)² + ( sX₂ )² }
Comparing both equations, we get, the constant K = 3 / 2πnₛ
Equation of Starting Torque of Three-Phase Induction Motor
slip s = (Nₛ - N)/ Nₛ becomes 1
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,
T = [(E₂)²R₂ / { (R₂)² + (X₂)² }] (3/ 2πNₛ) N-m
The starting torque is also known as the standstill torque.
Maximum Torque Condition for Three-Phase Induction Motor
T = [s(E₂)²R₂ / { (R₂)² + ( sX₂ )² }] (3/ 2πnₛ)
The rotor resistance, rotor inductive reactance, and synchronous speed of the induction motor remain constant. The supply voltage to the three-phase induction motor is usually rated and remains constant so the stator emf also remains constant. The transformation ratio is defined 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 with respect to 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 with respect to some variable quantity which is slip, s in this case as all other parameters in the equation of torque remain constant.
So, for torque to be maximum
dT/ds = 0
T = Ks(E₂)²R₂ / { (R₂)² + ( sX₂ )² }
Now differentiate the above equation by using the division rule of differentiation. On differentiating and after putting the terms equal to zero we get,
s² = (R₂)² / (X₂)²
Neglecting the negative value of slip we get
s² = (R₂)² / (X₂)²
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.
The equation for maximum torque is
T = s(E₂)²R₂ / { (R₂)² + ( sX₂ )² }
The torque will be maximum when slip s = R₂ / X₂
Substituting the value of this slip in the above equation we get the maximum value of torque as,
Tmax = K(E₂)² / 2X₂ N-m
In order to increase the starting torque, extra resistance should be added to the rotor circuit at the start and cut out gradually as the motor speeds up.
Observation:
From the above equation, it is concluded that
- The maximum torque is directly proportional to the square of rotor-induced emf at the standstill.
- The maximum torque is inversely proportional to rotor reactance.
- The maximum torque is independent of rotor resistance.
- The slip at which maximum torque occurs depends upon rotor resistance, R₂. So, by varying the rotor resistance, maximum torque can be obtained at any required slip.