The motor parameters of Induction Motors are crucial when the Field Oriented Control Close loop torque/speed control is needed. More specifically, the rotor flux’s absolute position is determined mathematically using known speed, voltage and current, and a model representation of the motor’s main parameters. This blog shows how to determine these parameters.

### Introduction

The motor parameters of Induction Motors are crucial when the Field Oriented Control Close loop torque/speed control is needed. More specifically, the rotor flux’s absolute position is determined mathematically using known speed, voltage and current, and a model representation of the motor’s main parameters shown in the figure below.

These parameters include per phase rotor resistance ‘R_{r}’, rotor leakage inductance ‘L_{lr}’, mutual inductance ‘L_{m}’ and rotor leakage inductance ‘L_{lr}’ (core losses R_{c} are neglected). Furthermore, in order to set the FOC (Flux and Torque) PID gains, the Stator Resistance (R_{s}) and Stator Inductance (L_{ls}) needed, according to the method as described in chapter “FOC Gains Tuning” - Section 8 of Roboteq controllers User Manual v2.0. The most common ways, to manually estimate induction motor parameters through the controller, are to test induction motor under no-load and locked rotor conditions.

### No load testing

The no-load test, like the open circuit test on a transformer, gives information about exciting flux I_{d} current, the magnetizing inductance L_{m} and rotational losses. The test is performed by applying balanced rated voltage on the stator windings at the rated frequency. The small power provided to the motor is due to core losses, friction and winding loses. The motor will consume the necessary flux I_{d} current in order to establish the appropriate magnetic field. Motor will rotate at almost a synchronous speed, which makes slip nearly zero (s≈0). Therefore, the motor equivalent circuit is expressed as follows:

Assuming that the R_{s} (Ω), L_{s}(Η) are much lower than the magnetizing inductance L_{m }(H), the following equation can be extracted:

where V_{s }(V) is the phase stator voltage applied (RMS value), f_{s }(Hz) is the stator frequency and I (A) is the RMS motor amps.

In order to implement the above mentioned no load test with RoboteQ induction motor controllers, please follow these steps:

- Set the operating mode to “Volts per Hertz” mode. encoder feedback action needed.
- Configure the Volt per Hertz setting according to motor nominal voltage (peak value of stator voltage according to manufacturer) and frequency.
- Run the motor without load up to the maximum available voltage (Command=1000), utilizing small acceleration/deceleration value. Be sure that the configured volt per hertz is correct, by checking the “Stator speed RPM” to be approximately equal to manufacturer’s synchronous rated speed.
- Calculate the applied V
_{s}by the following equation:

where m_{a}is the output PWM level (%1000) applied which is equal to Motor Power output measurement in Roborun+ utility (-1000 to 1000 range) and V_{dc}is the battery volts (V). In the considered no-load test the ma is 1000, therefore the applied voltage can be calculated as - Measure the motor RMS current from the utility (Motor Amps). Furthermore, the peak amps value should be configured as “Rotor Flux current” in the configuration tab.
- Calculate the magnetizing inductance for the no load test according to equation (1).

Example:

The following data have been given from the induction motor manufacturer:

P = 400 W (nominal power)

I_{N }= 23 A (nominal stator current)

V_{dc} = 24 V (Input dc voltage at controller)

f_{s} = 90 Hz (Nominal stator frequency)

n_{s }= 2700 rpm (Synchronous speed)

p= 2 (pole pairs)

Therefore, the Volts per Hertz ratio introduced in configuration tab is

Then the motor run volts per Hertz and the respective results from Roborun+ utility are illustrated below.

From the above results, it can been observed that the stator speed is close to the synchronous speed (2700 rpm), while the slip is very small due to the no-load operation.

Therefore, by applying equation (1) the magnetizing inductance is:

Furthermore, the rotor “Rotor Flux current” should be set in the configuration as

### Locked rotor testing

The locked rotor test, like short circuit test on a transformer, provides the information about leakage impedances and rotor resistance. Rotor is at the stand still, while low voltage is applied to stator windings up to rated current. Due to the fact that the magnetizing inductance L_{m} is much higher that leakage phase inductances L_{ls}, L_{lr}, it can be assumed that there is no current is floating to L_{m} parallel branch. Typically, leakage inductances L_{ls}, L_{lr} should be around 2-10% of the magnetizing inductance L_{m}. Since there is no rotation slip (rotor at standstill) s=1, which gives us the following equivalent circuit.

Therefore, the phase L_{ls }(H) , L_{lr} (H), R_{s }(Ω), R_{r }(Ω) motor parameters are calculated as follows:

where P_{s }(W) is the input phase motor power, V_{s }(V) is the phase stator voltage applied (peak value), f_{s }(Hz) is the stator frequency, cosφ is the power factor, I (A) is the motor current (RMS value), Z (Ω) is the equivalent phase impedance at locked rotor test.

According to equations (7) and (8), it is assumed that rotor resistance is equal to the stator resistance, as well as rotor leakage inductance is equal to stator leakage inductance.

In order to implement the above mentioned locked rotor test with induction motor controllers, please follow these steps:

- Set the operating mode to “Volts per Hertz” mode. Encoder feedback action needed.
- Configure the Volt per Hertz setting according to five times lower the motor nominal voltage (peak value of stator voltage according to manufacturer) and frequency (1/5 of nominal V/f). The reason for configuring lower V/f ratio than in no-load test is to appropriately weaken the induced field and reduce the produced torque at startup, in order to be easier to lock the rotor for the test.
- Lock the rotor by appropriate tool/device and increase the command up reaching the 80% of the rated motor current. If the produced torque is high and the rotor cannot be locked, then reduce appropriately the Volt per Hertz at configuration and repeat the test.
- Calculate the applied V
_{s}by utilizing equation (2). - Calculate the input phase motor power P
_{s }by utilizing the following equation:where V

_{dc }(V) is the battery dc volts, I_{dc }(A) is the battery dc current,*η*is the controller efficiency (assume 0.95 efficiency for RoboteQ controllers). Battery volts and amps can be measured from Roborun+ utility.- Measure the I
_{q}(A) current from the Roborun+ utility (FOC Torque Amps). - Calculate the L
_{ls}, L_{lr}, R_{s}, R_{r }motor parameters by applying the equations (3) - (8).

- Measure the I

Example:

For the same induction motor at no load test example, the Volts per Hertz ratio is set 5 times lower than the nominal, that is 0.053. The respective results taken from Roborun+ utility are shown below:

Therefore, the input phase motor power P_{s }is equal to

The power factor is

, while the equivalent phase impedance is Z = 0.081 Ω according to equation (4).

Therefore, the motor stator and rotor resistances are R_{s} = R_{r} = 24 mΩ according to equations (5) and (7) and the leakage inductances according to equation (6) are:

It is noted that the synchronous frequency in locked rotor test is different to the synchronous frequency in no-load test. Finally, the L_{ls} = L_{lr} = 76 uH according to equation (8).

### Optimal slip calculation

After estimating L_{m}, L_{ls}, L_{lr} , R_{s}, R_{r }motor parameters from no-load and locked-rotor tests, the optimal slip where the motor produces the maximum torque s_{maxT} can be estimated as follows:

Next, in order to apply at configuration tab the optimal slip in Hz, the following transformation needed:

Example:

For the calculated L_{m }= 754 uH, motor leakage inductances L_{ls} = L_{lr} = 76 uH and resistance R_{s} = R_{r} = 24 mΩ, the optimal slip is calculated equal to s = 5.1 Hz at rated motor speed.