# Induction motor torque and voltage relationship

### Variable Speed Drives: Variable Voltage Fixed Frequency Induction Motor Operation

The relationship between a motor's electrical characteristics and mechanical performance can be calculated as such (note: this is the analysis. T-Nr characteristics with variable stator voltage. If voltage applied to the stator of the induction motor is varied, developed torque will vary with a relation T o Vs. 2. In an induction motor frequency is speed, current is torque and voltage is the capacity to make torque. What is the relation between speed and current in single phase induction motor? What is the relationship between stator current and rotor resistance in a three-phase induction.

### Induction motor:Effect of Change in Supply Voltage on Torque and Speed. | electric equipment

The back-emf models the voltage generated by the moving electric current in the magnetic field basically a DC electric motor can function as a generator.

It's also possible to model the inherent inductance of the motor by adding an inductor in series, however for the most part I've ignored this and assumed the motor is at quasi steady state electrically, or the motor's time response is dominated by the time response of the mechanical systems instead of the time response of the electrical systems. This is usually true, but not necessarily always true.

The generator produces a back EMF proportional to speed of the motor: The current flowing through the motor can then be calculated: The torque generated by the motor is proportional to the amount of current flowing through the motor: Also, at the no load speed the motor has no torque and no current flowing through it. When does the motor produce the most power?

Well, power can be calculated one of two ways: So all things considered, how does the motor voltage stack up? For the same motor, ideally if you apply double the voltage you'll double the no-load speed, double the torque, and quadruple the power.

**Torque Equation of 3 Phase Induction Motor**

In this work, a new relation that improves the speed response of the IM is presented. The proposed relationship is defined by a frequency factor that expresses the relation of the operating frequency of the IM to its nominal value.

The variations in the reactances of the IM are considered and defined by the frequency factor. Although the IM is operated in open loop, the goal, by using the proposed relationship, is to maintain a constant torque at each operating frequency.

The experiments are carried out for different motor velocities with and without mechanical load applied to the motor shaft. The results using the proposed relation are compared to those obtained with the commonly used method, In a further work, this new relationship will be used by a control system to fulfill the EV demands.

## Torque Equation of an Induction Motor

The profile required is shown in figure 1 [2, 4, 12]. It can be seen that the electric motor develops a constant torque from the motor start up until it reaches its nominal speed.

If the motor speed continues increasing, after the nominal value, the torque decreases. The main desired characteristics of electric motors driving EV are [2, 4, 13]: The permanent magnet motor PMM has high torque and power density, develops constant torque at low speeds and is small in volume and weight.

However, at high speeds, this type of motor presents appreciable losses with a consequent low efficiency. Additionally, the DCM has low power density and torque [2, 14]. In applications for electric propulsion of EV, IM is the preferred type due to its reliability, ruggedness, low maintenance and ability to operate in wide ranges of speed [15].

The SRM can operate at extremely high speeds and, at low speeds, their torque is large. Their main disadvantages are acoustic noise generation, torque ripple and excessive electromagnetic interference EMI [1, 2, 12]. The squirrel cage Induction Motor The IM is composed of a group of thin laminated steel sheets arranged into a cylinder with slots. Coils are inserted into the slots.

Each coil group constitutes an electromagnet.

## Torque equation of three phase induction motor

The number of poles depends on the internal connection of the stator coils. The squirrel cage rotor is a cylinder made of aluminum bars.

The bars are connected mechanically and electrically by ending rings. Considering the stator connected to the source, the generated magnetic field of the stator rotates at Ssync expressed in rpm ; therefore, the rotor is inside of an electromagnetic field.

An electromotive force EMF is produced in the rotor. The rotor current, produced by the rotor induced EMF, generates an electromagnetic field that has opposed polarity with respect to the stator electromagnetic field. Consequently, the interaction between both fields produces an electromagnetic torque in the rotor which makes it rotate in the direction of the stator electromagnetic field.

There is a difference between Ssync and the rotor speed Sr. The speed difference is named slip speed and the sliding, is expressed, according to equation 1as relative error with respect to Ssync[5]: Ssync is given by equation 2 [5]: From equation 1we can derivate equation 4 in order to express the rotor speed as: This equation indicates that rotor speed can be adjusted with the frequency of the IM source.

The frequency of the induced EMF in the rotor changes in inverse proportion to Sr.

The frequency of the rotor induced EMF is expressed by equation 5: The rotor bars have low electrical resistance, as well as an inductance Lr and inductive reactance Xr properties.

The inductive reactance of the rotor is computed with the rotor blocked [3, 16]. The EMF induced in the rotor, when its blocked, is defined by equation 7 [5]: The torque with the rotor blocked can be determined by the current in the rotor according to equation 9 [5]: The equivalent circuit of the single cage IM is shown in figure 2.

The circuit is a model in steady state and allows obtaining the equations that define the IM behavior [17, 18]. The mutual reactance jXm represents the difference between stator and rotor inductances.

The impedance of the stator is represented by Rs and Xs.

### New Relation to Improve the Speed and Torque Characteristics of Induction Motors

The s term takes into account the apparent increment of Rr when the rotor is moving. From figure 2 ait can be seen that the rotor current magnitude Ir, with the rotor blocked, is expressed according to equation 10 [5]: By substituting equation 10 into equation 9 and considering that equation 11 expresses the initial value of the rotor torque [5]: For any value of s, considering equations 6 and 8equation 12 indicates that rotor current is: The term, at any value of s, is Hence, equation 13 expresses that, for a given sliding the torque of the IM is [5]: The equation 14 can be derived considering equations 7 and 8: Therefore, the stator magnetic field, expressed by equation 15is: By substituting equation 15 into equation 13the torque magnitude is expressed by quation 16 as: As stated before, a region of constant torque is demanded for speeds below the nominal.

Commonly, the voltage magnitude of the motor power source is modified in direct proportion to the frequency. When the frequency is larger than the nominal, the voltage magnitude is the nominal value. To determine the real value of E1rand the real value of the torque, Rs and Ls should be considered. The equivalent circuit of the IM can be reduced to the circuit shown in figure 2 b.

The magnitude of the equivalent impedance is expressed by equation According to the circuit of figure 2 bthe magnitude of V can be expressed by equation Hence, by substituting equation 20 into equation 19Er is given by equation 21 as: The equation 22 expresses Er when the rotor is blocked: