Chapter 5 Three phase induction machine (1) Shengnan Li

Chapter 5 Three phase induction machine (1) Shengnan Li

Main content Structure of three phase induction motor Operating principle of three phase induction motor Rotating magnetic field Graphical representation Analytical representation Induced rotor voltage and current

Introduction Three-phase induction machine are the most common and frequently encountered machines in industry simple design, rugged, low-price, easy maintenance wide range of power ratings: fractional horsepower to 10 MW Mostly used as motor instead of generator run essentially as constant speed from no-load to full load Its speed depends on the frequency of the power source not easy to have variable speed control requires a variable-frequency power-electronic drive for optimal speed control 3

Construction- stator Consisting of a steel frame that supports a hollow, cylindrical core Core, constructed from stacked laminations (why?), having a number of evenly spaced slots, providing the space for the stator winding

Construction - rotor composed of punched laminations, stacked to create a series of rotor slots, providing space for the rotor winding Copper bars shorted together at the ends by two copper rings, forming a squirrel-cage shaped circuit (squirrel-cage) (not cover in this course) conventional 3-phase windings made of insulated wire (wound-rotor), similar to the winding on the stator

Rotating magnetic field - Graphical method Provided by stator winding current Balanced three phase windings, mechanically displaced 120 degrees form each other, fed by balanced three phase source t0

Rotating magnetic field -Graphical method t1

Rotating magnetic field- field-graphical method A rotating magnetic field with constant magnitude is produced, rotating with a speed n sync 120 f e P rpm Where f e is the supply frequency ; P is the no. of poles and n sync is called the synchronous speed in rpm (revolutions per minute)

Rotating magnetic field - Analytical method Each phase winding provide MMF not only along winding axis but also along angle θ: F a θ, t = Ni a t cos θ F b θ, t = Ni b t cos(θ 120 ) ቑ F c θ, t = Ni c t cos(θ 240 ) Where i a t, i b t, i c t are balanced three phase stator currents: i a t = I m cos ω e t i b t = I m cos(ω e t 120 ) ቑ i c t = I m cos(ω e t 240 ) Where ω e is the angular frequency of the input power

Rotating magnetic field - Analytical method F θ, t = F a θ, t + F b θ, t +F c θ, t = Ni a t cos θ + Ni b t cos θ 120 + Ni c t cos(θ 240 ) = NI m cos ω e t cosθ + NI m cos(ω e t 120 )cos(θ 120 ) +NI m cos(ω e t 240 )cos(θ 240 ) = 3 2 NI mcos(ω e t θ) Motion of resultant MMF

Induced voltage on Rotor- rotor standstill Flux density distribution in air gap: B θ, t = B m cos ω e t θ The air gap flux per pole is: p (t) = න π/2 π/2 B θ, t lrdθ = 2B m lr cos ω e t Where l is the axial length; r is the radius of the stator at the air gap Since rotor stand still, the voltage induced in rotor winding aa is: e a = N d p(t) = 2ω dt e NB m lr sin ω e t = E m sinω e t e b = E m sin ω e t 120 e c = E m sin ω e t 240 Air gap flux density distribution

Starting torque - rotor stand still Since rotor is short-circuited, current establishes i ar t = e a t Z r = I mr sin ω e t γ i br t = I mr sin(ω e t γ 120 ) i cr t = I mr sin(ω e t γ 240 ) Electromagnetic force (Lorentz force) on rotor conductors: Ԧf = Ԧi B l The magnitude of the forces on rotor winding a,b,c are: f ar (t) = I mr sin ω e t γ B m cos ω e t θ l = 1 2 I mrb m l sin 2ω e t γ θ + sin θ γ θ f br t = 1 2 I mrb m l sin 2ω e t γ θ 120 + sin θ γ f br (t) = 1 2 I mrb m l(sin 2ω e t γ θ 240 + sin(θ γ)) The direction of all the forces are clock wise. The torques on each conductions add together T = 3I mr B m lsin(θ γ) No time component, constant torque, rotor start to rotate from standstill

Induced voltage and toque rotor rotating If rotor rotates at speed ωr, the relative speed between the rotating field and the rotor is: ω e ω r Re-derive the induced voltage: E m is propotional to (ω e ω m ) e a = E m sin(ω e ω r )t e b = E m sin (ω e ω r )t 120 e c = E m sin (ω e ω r )t 240 Torque has the same form of expression, but the angle will be different T = 3I mr B m lsin θ γ I mr is propotional to (ω e ω m )

Induced voltage and toque synchronous speed If rotor rotates at speed ωe, the relative speed between the rotating field and the rotor is zero d is zero. No voltage induced in the rotor, no current induced in the dt rotor No force and torque generated No load condition

Induction motor speed IM runs at a speed lower than the synchronous speed The difference between the motor speed and the synchronous speed is called the Slip n n n slip sync m Where n slip = slip speed n sync = speed of the magnetic field n m = mechanical shaft speed of the motor

The Slip Where s is the slip s n sync n sync Notice that : if the rotor runs at synchronous speed s = 0 if the rotor is stationary s = 1 Slip may be expressed as a percentage by multiplying the above eq. by 100, notice that the slip is a ratio and doesn t have units n m