The mathematical model of the induction machine: Voltages:Currents: Inductances: Torque:

The synchronous speed is defined by: The steady state analysis of the induction machine: The difference between the synchronous speed and the rotor speed is defined as slip: If the machine rotates at a smaller speed than the synchronous speed, the machine behaves as a motor (the slip is positive). If the speed is higher than the synchronous speed, the machine behaves as a generator (the slip is negative).

The equivalent circuit of one phase of the induction machine: The steady state analysis of the induction machine: The impedance of the machine would be: The phase current would be:

The heating of the machine occurs as a result of power losses inside the machine. Plosses=Pcu+Pfer=Pin-Pout The temperature of the machine can be calculated by the following differential equation: The thermal model of the induction machine: - the heat radiation of the machine. C- thermal capacitance of the machine. These two parameters must be provided by the manufacturer.

The simulation circuit:

Performance of the voltage controller: In order to understand the performance of the voltage controller the following circuit is presented:

The output of the voltage controller for different firing angles: a. b.

1. Definitions: Phase angle is the angle between phase voltage and phase current. When the machine operates as motor, this angle would be:. When the machine operates as generator, this angle would be: The firing angle is the angle between point at which the phase voltage is zero to point of conduction of the appropriate thyristor. 2.

3. Definitions: The delay from the point at which the phase current reaches zero to the point when next thyristor is fired, called the delay angle. The firing angle must be greater than the phase angle. If the firing angle is smaller than phase angle, the delay angle will be negative. In this case, in the steady state, the machine would operate as usual, as if there were no SCRs, and it would not be influenced by the changes in firing angle. However, the transient behavior will be influenced.

Examples: Example 1. Motor: In this case, the currents and the voltages in the steady state will not be influenced by the thyristors. Example 2. Motor:

Matlab simulation In order to perform the simulation in Matlab, two files must be built: 1. The fire.m file. This file is used for definition of parameters for the firing control of thyristors. These parameters are used in the pulse generators in the Simulink file. 2. The Simulink machine.mdl file. In this file the circuit itself is built.

The fire.m file: a=input('enter a:') T=0.02; d=ax*(T/2)/180; pw=(((T/2)-d)/T)*100; when: a- firing angle. T- period (sec). d- phase delay for the pulse generator (sec). Pw- pulse width (% of the period). With the help of this file, the only parameter that the user must insert to the program is the firing angle. Other parameters for pulse generators are calculated automatically by the fire.m file.

The machine.mdl file:

Parameters of the simulated circuit: Three phase voltage source:Induction machine: Rotor type: squirrel cage. Rs=1.435 ohm Rr=1 ohm Ls=2 mHy Lr=2 mHy Lm=49.31 mHy Inertia=0.009 kg*m*m Number of poles=2

The measurements: The measurement of harmonics The measurement of fluxes The measurement of RMS and THD The measurement of Pin The measurement of Pout

The results of the measurements: 1. Firing angle=80 degrees, machine operates as motor. The machine is unloaded, therefore it operates as a motor. In the steady state, the machine would operate as an inductive load. The phase angle in the steady state can be calculated by calculation of the machine's impedance. It is clear that in this case the firing angle is smaller than the phase angle, therefore the delay angle is negative: -5 degrees. The stator currents in the steady state will be continuous.

The stator currents, rotor currents, mechanical speed and torque: I_stator I_rotor wm Torque At t=0 sec, when the source voltages are applied to the machine, the machine's speed is zero and the slip is 1.The steady state begins at t=0.5 sec, when the machine has reached the synchronous speed. The synchronous speed of the motor is rad/sec. The machine's torque is maximal when the speed is low and the torque becomes zero when the machine rotates at the synchronous speed. When the machine has reached the synchronous speed, the slip becomes zero and the resistance Rr/s becomes infinite and the rotor currents become zero.

The harmonics of stator current of phase ‘a’ : First harmonic Second harmonic Third harmonic Fifth harmonic

The RMS, THD of the stator current in phase 'a': i_a rms i_a THD

The fluxes are obtained from the measurement demux block, are in the d-q frame and they must be converted to the regular abc frame. The following block was built in order to perform the conversion: Stator and rotor fluxes:

Converted stator and rotor fluxes: Rotor fluxes Stator fluxes

The results of the measurements: 2. Firing angle=100 degrees, machine operates as motor. Now the machine is loaded by the external load of 6 N*m at the time of t=1.5 sec, when the machine has reached the steady state. At t=1.5 sec, the phase angle is changed from 85 degrees to 72 degrees. The delay angle is now 28 degrees. As it was mentioned before, if the delay angle is higher, the THD will be also higher.

The stator currents, rotor currents, mechanical speed and torque: I_stator I_rotor wm Torque When the machine is loaded at t=1.5 sec, the amplitude of the currents in the steady state jumps from 16.5A to 21.5A From the comparison of stator currents, it is clear that when the delay angle increased, the distortion of the currents also increases. When the machine is loaded, the mechanical speed falls from 314 rad/sec to rad/sec. When the machine is loaded, the induced torque of the machine rises from average zero to average 6 N*m.

The harmonics of stator current of phase ‘a’ : First harmonic Second harmonic Third harmonic Fifth harmonic The fifth harmonic becomes much more dominative after the machine is loaded. This is the reason that the currents become more distorted after the machine is loaded.

The RMS, THD of the stator current in phase 'a': i_a rms i_a THD 54% 14%

Converted stator and rotor fluxes: Rotor fluxes Stator fluxes Rotor fluxes Stator fluxes

The results of the measurements: 3. Firing angle=100 degrees, machine operates as generator. Now the machine is loaded by the negative external load of -6 N*m at the time of t=1.5 sec, when the machine has reached the steady state. In this case the machine is driven at higher speed than the synchronous speed. The machine will deliver the power to the grid. At t=1.5 sec, the phase angle is changed from 85 degrees to 97 degrees. The delay angle is now smaller than in the previous cases: 3 degrees. The distortion of currents must be very low.

The stator currents, rotor currents, mechanical speed and torque: I_stator I_rotor wm Torque When the negative torque is applied, the speed rises from 314 rad/sec (synchronous speed) to rad/sec. The induced torque of generator in the steady state is –6 N*m. After the machine is loaded with negative torque, the stator currents amplitude rises to 20A.

The harmonics of stator current of phase ‘a’ : First harmonic Second harmonic Third harmonic Fifth harmonic The amplitude of fifth harmonic in the generator’s steady state is 0.3A and it almost doesn't influence the sine form of the currents.

The RMS, THD of the stator current in phase 'a': i_a rms i_a THD 14% 2%

Converted stator and rotor fluxes: Rotor fluxes Stator fluxes Rotor fluxes

The input active power Pin: When the machine starts to operate as generator, the input active power becomes negative because now the power is supplied from the machine to the grid.

The output active power Pout and the mean Pout: Pout Mean Pout

The difference between the original thesis simulations to the presented simulations: The original Simulink simulation circuit for my thesis was different from the simulation circuit that was presented. The difference is that in the original simulation was not used the fire.m file. The firing angle control of the thyristors was done by the synchronized 6-pulse generator.

The differences between the simulations: The difference of stator currents in the second case (unloaded machine and firing angle of 100 degrees): The stator current in the original thesis simulation circuit: The peak of stator currents in the first cycle of simulation. The stator current in the original thesis simulation circuit:

The simulation circuit: The Psim simulations: Firing control The stator current in phase ‘ a ’ is measured by the current sensor and from the current sensor is passed to the control part of Psim. The signal is transmitted to the Simulink for RMS, THD and harmonics calculations. Unlike Simulink, in Psim there is no option for measurement of rotor currents. There are two ways to measure the speed of the machine: 1. Mechanical speed can be measured by speed sensor (in rpm) 2. Mechanical speed can be measured by accessing the internal equivalent circuit of the machine ’ s mechanical system. This is done by the mechanical-electrical interface block. The output of this block is the mechanical speed of the machine (in rad/sec). Torque measurement-he internal mechanical system of the machine can be described by the following equation: Three phase wttmeter

The Psim file: Co-simulation between Psim and Simulink: The purpose of this circuit is to simulate resistor of 1 ohm connected through the thyristors to the sine voltage source of 10 v. The firing angle of the thyristors is 100 degrees. The voltage control is performed in Psim. The output voltage of the thyristors is sent to Simulink file, which represents the behavior of the resistor of 1 ohm.

The Simulink file file: The tested circuit:

In order to measure the output voltage of the thyristors, the resistor Rm must be inserted in parallel to the voltage sensor. The resistance Rm must be set to very high value in order to diminish it’s influence on the circuit’s current. When the resistor is set to 1 Mohm, the following current results are obtained: The influence of Rm resistance:

When Rm is set to 1 ohm, the following results are obtained: The influence of Rm resistance: Now the results are logical but Rm has changed the true value of the current, which is supposed to flow for resistor of 1ohm.

If the simulation for Rm=1 Mohm is done only in Psim, without the co-simulation with Simulink, the results are correct. The Psim simulation circuit: The influence of Rm resistance: The conclusion is that there must be a problem with co- simulation of programs for higher values of Rm.

The following parameters will be measured in Psim: 1. RMS, THD and harmonics of the phase ‘a’ stator current. 2. The average of the output active power. In order to perform these measurements, the I_a and Pout signals are sent to the Simulink by Simcoupler. The use in Simcoupler for simulation of case 3:

The measurements results for case 3: Stator currents wm Torque

The harmonics of stator current of phase ‘a’ : First harmonic Second harmonic Third harmonic Fifth harmonic

The RMS, THD of the stator current in phase 'a': i_a rms i_a THD 14% 2%

The input active power and output active power: Pin Pout Mean Pout

The Plecs&Matlab co-simulation:

The contents of Plecs block:

The complete simulation circuit, including measurements:

The measurements results for case 4: wm Induced torque Stator currents Rotor currents

The harmonics of stator current of phase ‘a’ : First harmonic Second harmonic Third harmonic Fifth harmonic

The RMS, THD of the stator current in phase 'a': i_a rms i_a THD 14% 2%

The rotor and stator fluxes: Rotor fluxes Stator fluxes

Powers: Pin Pout Mean Pout

Summary: User interface of the programs: The Psim program has most simple user interface. 1. The control of switches is simpler than in Simulik and Plecs. 2. The elements can be chosen very quickly and easily from the elements library. The Psim program is much more simple in use than Matlab and Plecs. The signals processing and measurement options: Simulink has more options for signal processing and measurements than Psim or Plecs. Therefore, Simulink is often used in co- simulation with other programs.

Summary: Run time of the simulation: Psim has the fastest run time. It took about 10 seconds to simulate the circuit of case 3. Simulink is slower than Psim. It took about 1 minute to simulate the circuit of case 3. In Plecs and Simulink co-simulation, it took 1 hour to simulate the circuit of case 3. Co-simulation with Simulink: Plecs was designed especially for co-simulation with Simulink. ny Plecs circuit can be co-simulated with Simulink, includinig the option of “breaking” the power circuit by controlled current and voltage sources.

Summary: Psim should be co-simulated with Simulink only in the case of signal processing. It is not reccomended to “break” the Psim’s power circuit by controlled current and voltage sources, because there are cases when it won’t work.