the mathematical model of the induction machine: voltages:currents: inductances: torque:
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The mathematical model of the induction machine:
Voltages:
gFrmwkwGdtgFrd
girRrgVr
gFskwGdtgFsd
gisRsgVs
*)(**
***
Currents:
FsKmFrKrir
FrKmFsKsis
**
**
Inductances:
LrlLmLr
LslLmLs
Torque:
dsiqsFqsidsFsisFGpsisFpTe **)*(**4
3)(**
4
3
lTmwmBdtmdw
JTe *
lslLrlLmLslLmLrl
LmKm
LrlLslLrlLmLslLm
LsKr
lslLrlLmLslLmLrl
LsKs
***
***
***
The synchronous speed is defined by:
The steady state analysis of the induction machine:
[rpm] *60
p
Fsyncn [rad/sec]
60
*2***2 syncn
p
Fsync
The difference between the synchronous speed and the rotor
speed is defined as slip:
syncnmnsyncn
s
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:
rjXsrR
mjBsjXsReqZ
)(
11
The phase current would be:eqZphV
phI
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:
radiationPlossesPdt
dTC
- the heat radiation of the machine.C- thermal capacitance of the machine.These two parameters must be provided by the manufacturer.
radiationP
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. 60 100b
.
1.
Definitions:
angle phase 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: 18090
The firing angle is the angle between point at which the phase
voltage is zero to point of conduction of the appropriate
thyristor.
2. angle firing
900
3.
Definitions:
angledelay 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:
40
80
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:
100
80
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:
vcV
vbV
vaV
,120300
,120300
,0300
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 fluxesThe 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.
,8518.16
628.0)01
(
11
49.151
1628.0435.1
2)2(
11
11
j
jj
jXsRmjB
jXReqZ
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 314.2 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:
dF
qF
cFbFaF
*
2
3
2
1-
2
3-
2
1-
0 1
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 265.5 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
TorqueWhen the negative torque is applied, the speed rises from 314
rad/sec (synchronous speed) to 318.6 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:
loadTemTdtmdw
loadJmachineJ *)(
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
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 :
PinPout
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.
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