# Induction Machine Direct Torque Control (Single-Phase)

Single-phase induction machine direct torque control

**Libraries:**

Simscape /
Electrical /
Control /
Induction Machine Control

## Description

The Induction Machine Direct Torque Control (Single-Phase) block implements a single-phase induction machine direct torque control structure.

### Equations

This diagram shows the direct torque control architecture for single-phase machines

The torque and flux estimation is based on machine voltage equations. The discrete-time voltage equations using the backward Euler discretization method, are:

$${\psi}_{a}=({v}_{a}-{i}_{a}{R}_{as})\frac{{T}_{s}z}{z-1}$$

$${\psi}_{b}=({v}_{b}-{i}_{b}{R}_{bs})\frac{{T}_{s}z}{z-1}$$

where:

*R*and_{as}*R*are the main winding resistance and the auxiliary winding resistance, respectively._{bs}*i*and_{a}*i*are the main winding current and the auxiliary winding current, respectively._{b}*v*and_{a}*v*are the main and auxiliary winding voltage, respectively._{b}*ψ*and_{a}*ψ*are the main and auxiliary winding flux, respectively._{b}

The torque and flux are obtained from:

$$T=-p(a{\psi}_{a}{i}_{b}-\frac{1}{a}{\psi}_{b}{i}_{a})$$

$${\psi}_{s}=\sqrt{{\psi}_{a}{}^{2}+{\psi}_{b}{}^{2}}$$

where:

`p`

is the number of pole pairs.`a`

is the auxiliary to main windings turn ratio.

Employing simple hysteresis comparators detects the status of flux and torque errors. The following figures illustrate the hysteresis comparators and the switching sectors.

The table shows the optimum switching table (inverter high side).

c,
_{ψ}c,
_{T}S(θ) | S_{0} | S_{1} | S_{2} | S_{3} | |
---|---|---|---|---|---|

`c` =
1 | `c` = 1 | 1, 0 | 1, 1 | 0, 1 | 0, 0 |

`c` = 0 | 1, 1 | 0, 1 | 0, 0 | 1, 0 | |

`c` =
0 | `c` = 1 | 0, 0 | 1, 0 | 1, 1 | 0, 1 |

`c` = 0 | 0, 1 | 0, 0 | 1, 0 | 1, 1 |

The torque reference can be provided as an input, or, in the case of speed control, be generated internally using a PI speed controller.

The flux reference is generated internally using:

$${\psi}_{s}{}^{*}=\frac{2\pi {f}_{n}{\psi}_{n}}{p\mathrm{min}(|{\omega}_{r}|,\frac{2\pi {f}_{n}}{p})}$$

where,

*ω*is the rotor angular mechanical speed in rad/s._{r}*f*is the rated frequency._{n}*ψ*is the rated flux._{n}

### Limitations

The power inverter dead-times are not considered in this block. For hardware implementation, add the dead-time externally.

## Ports

### Input

### Output

## Parameters

## References

[1] Takahashi, I., and T. Noguchi. "A
New Quick-Response and High-Efficiency Control Strategy of an Induction Motor."
*IEEE Transactions on Industry Applications*. Vol. IA-22,
Number 5, 1986, pp. 820 - 827.

## Extended Capabilities

## Version History

**Introduced in R2018b**

## See Also

### Blocks

- Induction Machine Current Controller | Induction Machine Direct Torque Control | Induction Machine Direct Torque Control with Space Vector Modulator | Induction Machine Field-Oriented Control | Induction Machine Field-Oriented Control (Single-Phase) | Induction Machine Flux Observer | Induction Machine Scalar Control