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 industrysimple 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

120 esync

fn rpm

P

• A rotating magnetic field with constant magnitude is produced, rotating with a speed

Where fe is the supply frequency ;

P is the no. of poles and nsync 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:

ቑ

𝐹𝑎 𝜃, 𝑡 = 𝑁𝑖𝑎 𝑡 cos 𝜃

𝐹𝑏 𝜃, 𝑡 = 𝑁𝑖𝑏 𝑡 cos(𝜃 − 120°)

𝐹𝑐 𝜃, 𝑡 = 𝑁𝑖𝑐 𝑡 cos(𝜃 − 240°)

Where 𝑖𝑎 𝑡 , 𝑖𝑏 𝑡 , 𝑖𝑐 𝑡 are balanced three phase stator currents:

ቑ

𝑖𝑎 𝑡 = 𝐼𝑚 cos 𝜔𝑒𝑡

𝑖𝑏 𝑡 = 𝐼𝑚cos(𝜔𝑒𝑡 − 120°)

𝑖𝑐 𝑡 = 𝐼𝑚cos(𝜔𝑒𝑡 − 240°)

Where 𝜔𝑒 is the angular frequency of the input power

Rotating magnetic field - Analytical method

⇒ 𝐹 𝜃, 𝑡 = 𝐹𝑎 𝜃, 𝑡 + 𝐹𝑏 𝜃, 𝑡 +𝐹𝑐 𝜃, 𝑡

= 𝑁𝑖𝑎 𝑡 cos 𝜃 + 𝑁𝑖𝑏 𝑡 cos 𝜃 − 120° +𝑁𝑖𝑐 𝑡 cos(𝜃 − 240°)

= 𝑁𝐼𝑚 cos𝜔𝑒𝑡 𝑐𝑜𝑠𝜃 + 𝑁𝐼𝑚 cos(𝜔𝑒𝑡 − 120°)cos(𝜃 − 120°)+𝑁𝐼𝑚 cos(𝜔𝑒𝑡 − 240°)cos(𝜃 − 240°)

=3

2𝑁𝐼𝑚cos(𝜔𝑒𝑡 − 𝜃)

Motion of resultant MMF

Induced voltage on Rotor- rotor standstill

• Flux density distribution in air gap:𝐵 𝜃, 𝑡 = 𝐵𝑚 cos 𝜔𝑒𝑡 − 𝜃

• The air gap flux per pole is:

∅𝑝(𝑡) = න−𝜋/2

𝜋/2

𝐵 𝜃, 𝑡 𝑙𝑟𝑑𝜃 = 2𝐵𝑚𝑙𝑟 cos𝜔𝑒𝑡

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:

𝑒𝑎 = −𝑁𝑑∅𝑝(𝑡)

𝑑𝑡= 2𝜔𝑒𝑁𝐵𝑚𝑙𝑟 sin𝜔𝑒𝑡 = 𝐸𝑚sin𝜔𝑒𝑡

𝑒𝑏 = 𝐸𝑚sin 𝜔𝑒𝑡 − 120°𝑒𝑐 = 𝐸𝑚sin 𝜔𝑒𝑡 − 240°

Air gap flux density distribution

Starting torque - rotor stand still

• Since rotor is short-circuited, current establishes

𝑖𝑎𝑟 𝑡 =𝑒𝑎 𝑡

𝑍𝑟= 𝐼𝑚𝑟 sin 𝜔𝑒𝑡 − 𝛾

𝑖𝑏𝑟 𝑡 = 𝐼𝑚𝑟sin(𝜔𝑒𝑡 − 𝛾 − 120°)

𝑖𝑐𝑟 𝑡 = 𝐼𝑚𝑟sin(𝜔𝑒𝑡 − 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:

𝑓𝑎𝑟(𝑡) = 𝐼𝑚𝑟 sin 𝜔𝑒𝑡 − 𝛾 ∙ 𝐵𝑚 cos 𝜔𝑒𝑡 − 𝜃 ∙ 𝑙

=1

2𝐼𝑚𝑟𝐵m𝑙 sin 2𝜔𝑒𝑡 − 𝛾 − 𝜃 + sin 𝜃 − 𝛾

𝑓𝑏𝑟 𝑡 =1

2𝐼𝑚𝑟𝐵m𝑙 sin 2𝜔𝑒𝑡 − 𝛾 − 𝜃 − 120° + sin 𝜃 − 𝛾

𝑓𝑏𝑟(𝑡) =1

2𝐼𝑚𝑟𝐵m𝑙(sin 2𝜔𝑒𝑡 − 𝛾 − 𝜃 − 240° + sin(𝜃 − 𝛾))

• The direction of all the forces are clock wise.

• The torques on each conductions add together𝑇 = 3𝐼𝑚𝑟𝐵𝑚𝑙𝑠𝑖𝑛(𝜃 − 𝛾)

• No time component, constant torque, rotor start to rotate from standstill

θ

Induced voltage and toque – rotor rotating

• If rotor rotates at speed ωr, the relative speed between the rotating field and the rotor is: 𝜔𝑒 − 𝜔𝑟

• Re-derive the induced voltage:

• 𝐸𝑚 𝑖𝑠 𝑝𝑟𝑜𝑝𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 (𝜔𝑒 − 𝜔𝑚)

• Torque has the same form of expression, but the angle will be different

𝑇 = 3𝐼𝑚𝑟𝐵𝑚𝑙𝑠𝑖𝑛 𝜃 − 𝛾

• 𝐼𝑚𝑟 𝑖𝑠 𝑝𝑟𝑜𝑝𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 (𝜔𝑒 − 𝜔𝑚)

𝑒𝑎= 𝐸𝑚sin(𝜔𝑒−𝜔𝑟)𝑡

𝑒𝑏 = 𝐸𝑚sin (𝜔𝑒−𝜔𝑟)𝑡 − 120°

𝑒𝑐 = 𝐸𝑚sin (𝜔𝑒−𝜔𝑟)𝑡 − 240°

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∅

𝑑𝑡is zero. No voltage induced in the rotor, no current induced in the

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

Where nslip= slip speed

nsync= speed of the magnetic field

nm = mechanical shaft speed of the motor

slip sync mn n n

The Slip

sync m

sync

n ns

n

Where s is the slip

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