INDUCTION MOTOR steady-state model (squirrel cage) MEP 1523 ELECTRIC DRIVES INDUCTION MOTOR steady-state model (squirrel cage)

CONTENTS Construction and Principle of Operation Per-phase Equivalent Circuit Power And Torque Steady-State Performance

Stator – 3-phase winding Rotor – squirrel cage / wound Construction and principle of operation 120o a c’ b’ c b a’ Stator – 3-phase winding Rotor – squirrel cage / wound

Construction and principle of operation Single N turn coil carrying current i Spans 180o elec Permeability of iron >> o → all MMF drop appear in airgap a  a’   /2 -/2 - Ni / 2 -Ni / 2

Construction and principle of operation Distributed winding – coils are distributed in several slots Nc for each slot  (3Nci)/2 (Nci)/2 - -/2 /2  

Distributed winding (full-pitch) Construction and principle of operation Distributed winding (full-pitch) The resultant MMF is the total contribution of MMF from each coil Considering only the space-fundamental component, Concentrated Distributed Concentrated space fundamental Distributed space fundamental

IDEAL case: sinusoidal flux Construction and principle of operation Phase a – sinusoidal distributed winding conductor density  Turns/rad OR conductors/rad If Ns is the number of turns per pole, Hence for sinusoidal distributed winding, IDEAL case: sinusoidal flux In PRACTICE: negelect harmonics and only consider the fundamental component

Phase a – sinusoidal distributed winding Construction and principle of operation Phase a – sinusoidal distributed winding IDEAL case: sinusoidal flux In PRACTICE: negelect harmonics and only consider the fundamental component

Phase a – sinusoidal distributed winding Construction and principle of operation Phase a – sinusoidal distributed winding

Phase a – sinusoidal distributed winding Construction and principle of operation Phase a – sinusoidal distributed winding

Combination of 3 standing waves resulted in MMF wave rotating at: Construction and principle of operation Sinusoidal winding for each phase produces space sinusoidal MMF and flux Sinusoidal current excitation (with frequency s) in a phase produces space sinusoidal standing wave MMF Combination of 3 standing waves resulted in MMF wave rotating at: p – number of poles f – supply frequency

Construction and principle of operation

Rotating flux induced: Construction and principle of operation Rotating flux induced: emf in stator winding (known as back emf) Emf in rotor winding Rotor flux rotating at synchronous frequency Rotor current interact with flux producing torque Rotor ALWAYS rotate at frequency less than synchronous, i.e. at slip speed: sl = s – r Ratio between slip speed and synchronous speed known as slip

✔ CONTENTS Construction And Principle Of Operation Per-phase Equivalent Circuit Power And Torque Steady-State Performance ✔

Per-phase equivalent circuit Stator voltage equation: Vs = Rs Is + j(2f)LlsIs + Eag Eag – airgap voltage or back emf Eag = k f ag Rotor voltage equation: Er = Rr Ir + js(2f)Llrlr Er – induced emf in rotor circuit Er /s = (Rr / s) Ir + j(2f)Llrlr

Rs – stator winding resistance Rr – rotor winding resistance Per-phase equivalent circuit Llr Lls Ir Rs + Vs – + Eag – + Er/s – Is Lm Rr/s Im Rs – stator winding resistance Rr – rotor winding resistance Lls – stator leakage inductance Llr – rotor leakage inductance Lm – mutual inductance s – slip

We know Eg and Er related by Per-phase equivalent circuit We know Eg and Er related by Where a is the winding turn ratio The rotor parameters and variable referred to stator are: 𝑅 𝑎 ′ = 𝑎 2 𝑅 𝑟 𝐿 𝑙𝑟 ′ = 𝑎 2 𝐿 𝑙𝑟 𝐼 𝑟 ′ = 𝐼 𝑟 𝑎 rotor voltage equation Er /s = (Rr / s) Ir + j(2f)Llrlr becomes  Eg = (Rr’ / s) Ir’ + j(2f)Llr’ Ir’

Rs – stator winding resistance Per-phase equivalent circuit Rr’/s + Vs – Rs Lls Llr’ Eag Is Ir’ Im Lm Rs – stator winding resistance Rr’ – rotor winding resistance referred to stator Lls – stator leakage inductance Llr’ – rotor leakage inductance referred to stator Lm – mutual inductance Ir’ – rotor current referred to stator

✔ ✔ CONTENTS Construction And Principle Of Operation Per-phase Equivalent Circuit Power And Torque Steady-State Performance ✔ ✔

Converted to mechanical power = (1–s)Pag Power and Torque Power is transferred from stator to rotor via air–gap, known as airgap power Lost in rotor winding Converted to mechanical power = (1–s)Pag  Pm = (1-s)Pag

Mechanical power, Pm = Tem r Power and Torque Mechanical power, Pm = Tem r But, ss = s - r  r = (1-s)s  Pag = Tem s Therefore torque is given by (based on approximate equivalent circuit):

sm Tem Pull out Torque (Tmax) Trated r 0 rated s s 1 0 Power and Torque Tem Pull out Torque (Tmax) Trated r 0 rated s sm s 1 0

✔ ✔ ✔ CONTENTS Construction And Principle Of Operation Per-phase Equivalent Circuit Power And Torque Steady-State Performance ✔ ✔ ✔

Steady state performance The steady state performance can be calculated from equivalent circuit, e.g. using Matlab Rr’/s + Vs – Rs Lls Llr’ Eag Is Ir’ Im Lm

e.g. 3–phase squirrel cage IM V = 415 V Rs= 0.25  Rr=0.2  Steady state performance Rr’/s + Vs – Rs Lls Llr’ Eag Is Ir’ Im Lm e.g. 3–phase squirrel cage IM V = 415 V Rs= 0.25  Rr=0.2  Llr = Lls = 1.6mH Lm= 95.5mH f = 50Hz p = 4

Steady state performance : m code Rs=0.25; Rr=0.2; Lr=(0.0971)-(0.0955); Ls=Lr; Lm=0.0955; f=50; p=4; V=(415/sqrt(3)); Zs=Rs+(2*pi*f*Ls)*i; Xm=2*pi*f*Lm; Zth = Zs*Xm*i/(Zs + (Xm*i)); Vth = V*(Xm*i)/(Zs + (Xm*i)); s=linspace(0,1,1000); Ir=Vth./(Zth + 2*pi*f*Lr*i + Rr./s); Is=Ir.*(((Rr./s) + (2*pi*f*(Lr+Lm)*i))/(i*Xm)); Te= (p/2)*3*(abs(Ir).^2)*Rr./(s*2*pi*f); wr=(1-s)*(2*pi*f)/(p/2); figure(1); subplot(3,1,1); plot(wr,Te,'k','LineWidth',2);grid; xlabel('rotor speed (rad/s)'); ylabel('Torque (Nm)'); subplot(3,1,2); plot(wr,abs(Is),'k','LineWidth',2);grid; ylabel('Stator current (A)'); subplot(3,1,3); plot(wr,abs(Ir),'k','LineWidth',2);grid; ylabel('Rotor current (A)');

Steady state performance

Steady state performance