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Induction machine flux observer for field-oriented control

**Library:**Simscape / Electrical / Control / Observers

The Induction Machine Flux Observer block obtains the synchronous
speed, *ω _{e}*, and electrical angle,

To determine the synchronous speed and electrical angle, the Induction Machine Flux Observer block uses these relationships:

${\lambda}_{dr}^{\text{'}e}={L}_{r}^{\text{'}}{i}_{dr}^{\text{'}e}+{L}_{M}{i}_{ds}^{e}\triangleq {L}_{M}{i}_{mr},$

$0={R}_{r}^{\text{'}}{i}_{dr}^{\text{'}e}-\left({\omega}_{e}-{\omega}_{r}\right){\lambda}_{qr}^{\text{'}e}+\frac{d{\lambda}_{dr}^{\text{'}e}}{dt},$

and

${i}_{qr}^{\text{'}e}=-\frac{{L}_{M}}{{L}_{r}^{\text{'}}}{i}_{qs}^{e},$

in these combined forms:

${i}_{dr}^{\text{'}e}=\frac{{L}_{M}}{{L}_{r}^{\text{'}}}\left({i}_{mr}-{i}_{ds}^{e}\right)$

$\frac{d{i}_{mr}}{dt}=\frac{{R}_{r}^{\text{'}}}{{L}_{r}^{\text{'}}}\left({i}_{ds}^{e}-{i}_{mr}\right)$

and

${\omega}_{e}={\omega}_{r}+\frac{{R}_{r}^{\text{'}}}{{L}_{r}^{\text{'}}}\frac{{i}_{qs}^{e}}{{i}_{mr}}$

where:

${\lambda}_{dr}^{\text{'}e}$ is the

*d*-axis rotor flux.${i}_{mr}$ is the magnetizing current.

${i}_{ds}^{e}$ and ${i}_{qs}^{e}$ are the

*d*-axis and*q*-axis stator currents.${i}_{dr}^{\text{'}e}$ and ${i}_{qr}^{\text{'}e}$ are the

*d*-axis and*q*-axis rotor currents.${\omega}_{e}$ is the synchronous speed.

${\omega}_{r}$ is the mechanical rotational speed.

${R}_{r}^{\text{'}}$ is the rotor resistance, referred to the stator side.

${L}_{r}^{\text{'}}$ is the rotor leakage inductance, referred to the stator side.

${L}_{M}$ the magnetizing inductance.

[1] Vas, P. *Electrical Machines and Drives: A Space-vector Theory
Approach*. New York: Oxford University Press, 1992.

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