Induction motors modelling in ABC frame of reference

The modelling of induction motors in the ABC frame of reference can provide useful insight into the time-domain current-voltage-power relationships in the motor. Many induction motors operate from three-phase power; analyzing the time-dependent current/voltage relationships in each phase (labeled A, B, and C) of the three-phase circuit can provide insights into the transient and steady-state characteristics of the motor.

Introduction

In an induction motor the electromagnetic energy is transferred by inductive coupling of the stator winding to rotor winding, which are separated by a small air gap.

In the mathematical model of an induction motor, the main variables of concern are Phase currents for the stator and rotor, output torque and speed with respect to time. Three phase supply voltages are the applied variables, which are generally sinusoids with a phase displacement of  ^2π⁄₃ radians for each phase, if the motor is directly supplied from electricity grid. In case of an inverter-supplied motor, the voltages may be other functions of time. Additionally, motor parameters such as stator and rotor winding resistances, and self- and mutual-inductance values are required to complete the model.

Model

Let e_A, e_B, e_C represent the phase voltages in the stator windings; and e_a, e_b, e_c the phase voltages in the rotor windings. After the application of these voltages the currents in the stator and rotor windings are i_A, i_B, i_C, i_a, i_b, i_c respectively. The voltage equations for the stator and the rotor can be written as

$e_A = i_A r_A + { {d \varphi_A} \over {dt}}$

$e_B = i_B r_B + { {d \varphi_B} \over {dt}}$

$e_C = i_C r_C + { {d \varphi_C} \over {dt}}$

$e_a = i_a r_a + { {d \varphi_a} \over {dt}}$

$e_b = i_b r_b + { {d \varphi_b} \over {dt}}$

$e_c = i_c r_c + { {d \varphi_c} \over {dt}}$

Where φ_x represents the flux linkage for the respective phase winding.

In matrix notation these equations can be written as

$e=i[R]+p[\varphi]$

Where e is the vector of the phase voltages and i is the vector of phase currents for stator and rotor respectively, and p is the derivative operator d/dt . [R] is the matrix for winding resistances. The total flux which linking for a particular winding is due the currents in all the windings, so flux for a particular phase in terms of currents in the same and all other phases is

$\varphi=[L]i$

Where [L] is the inductance matrix for machine. φ can be rewritten as

$\varphi = \begin{bmatrix} \varphi_{stator} \\ \varphi_{rotor} \end{bmatrix}$

$\varphi_{stator}=[L_{ss} ] i_{stator}+[L_{sr}]i_{rotor}$

$\varphi_{rotor}=[L_{rs} ] i_{stator}+[L_{rr}]i_{rotor}$

Where

$[L_{ss}] = \begin{bmatrix} L_{AA} & L_{AB} & L_{AC} \\ L_{BA} & L_{BB} & L_{BC} \\ L_{CA} &L_{CB} &L_{CC} \end{bmatrix}$

$[L_{sr}] = \begin{bmatrix} L_{Aa} & L_{Ab} & L_{Ac} \\ L_{Ba} & L_{Bb} & L_{Bc} \\ L_{Ca} &L_{Cb} &L_{Cc} \end{bmatrix}$

$[L_{rs}] = \begin{bmatrix} L_{aA} & L_{aB} & L_{aC} \\ L_{bA} & L_{bB} & L_{bC} \\ L_{cA} &L_{cB} &L_{cC} \end{bmatrix}$

$[L_{rr}] = \begin{bmatrix} L_{aa} & L_{ab} & L_{ac} \\ L_{ba} & L_{bb} & L_{bc} \\ L_{ca} &L_{cb} &L_{cc} \end{bmatrix}$

i_stator and i_rotor are the vectors for the phase currents in the stator and rotor respectively. L_ss is the stator self-inductance matrix. Diagonal elements of stator self-inductance matrix are the self inductances of the respective phases, and non-diagonal elements are the mutual inductances between any two stator phases of the machine.

$L_{AA} = L_{BB} = L_{CC} = L_{ls} + L_{ms}$

$L_{AB} = L_{BA} = L_{AC} = L_{CA} = L_{BC} = L_{CB} = - \frac{1}{2} L_{ms}$

L_ms is the per-phase magnetizing inductance for the stator winding, and L_ls is the leakage inductance for the stator winding. L_sr and L_rs are the stator and rotor mutual inductance matrices, which are functions of rotor position θ_r at any time. L_rr is the rotor self-inductance matrix

$L_{aa} = L_{bb} = L_{cc} = L_{lr} + L_{mr}$

$L_{ab} = L_{ba} = L_{ac} = L_{cb} = L_{bc} = L_{ca} = - \frac{1}{2} L_{mr}$

L_mr is the per-phase magnetizing inductance for rotor winding, and L_lr is the leakage inductance for the rotor winding.

Now stator and rotor voltage equations can be solved to find the stator and rotor phase currents. Stator and rotor voltage equations are first order differential equations and to solve these equations, the Runge-Kutta fourth order technique can be used.

The expression for developed electromagnetic torque T_e is as follows

$T_e = [ { i_A ( i_a -\frac{i_b}{2} - \frac{i_c}{2} ) + i_B ( i_b -\frac{i_a}{2} -\frac{i_c}{2} ) + i_C ( i_c -\frac{i_b}{2} -\frac{i_a}{2} ) } \sin \theta_r + \frac{\sqrt{3}}{2} {i_A ( i_b - i_c ) + i_B ( i_c - i_a ) + i_C ( i_a - i_b ) } \cos \theta_r ]$

To find the speed of the rotor, ω_r, the following differential equation is solved. For this equation the inputs are the load torque, T_L, moment of inertia of the rotor J and friction and damping coefficient B_m.

${\displaystyle {\frac {d\omega _{r}}{dt}}={\frac {T_{e}-B_{m}\omega _{r}-T_{L}}{J}}}$

$\omega_r=\frac{d\theta_r}{dt}$

See also

• $\alpha \beta \gamma$ transform
• Adaptive control
• Control engineering
• Control theory
• Dqo transformation
• Eigenvalues and eigenvectors
• Extended Kalman filter
• Filter (signal processing)
• Frequency response
• Hilbert transform
• Impulse response
• Kalman filter
• Robust control
• Root locus
• Perturbation theory
• Signal-flow graph
• Small signal model
• Sliding mode control
• State observer
• State space representation
• Symmetrical components
• Systems analysis
• Transient response
• Transfer function
• Vector control (motor)

References

• P.S. Bimbhra "Generalised Theory of Electrical Machines", Khanna Publishers
• P.C. Krause,O. Wasynczuk, S. D. Sudhoff, "Analysis of Electric Machinery and Drives System", Second edition