Error caused by the induction motor current

The error in the operation of the active du/dtfilter, caused by the load current of the induction motor, can be analyzed using a simplified equivalent circuit as presented in Figure 4.1.

L C

u

_{i n v}

u

_{o u t}

Z

L

I

L

I

f

+ I

L

Figure 4.1. LC filter for active du/dt control presented with the loading impedance of the induction motor on a per-transition basis.

The impedanceZL is used to model the loading impedance of the induction motor for the analysis from a viewpoint of a single transient. From this viewpoint, for a pulse-width-modulated voltage waveform, the rate of change in the motor current is slow. For example, in a typical case, the period of the motor current is in the order of tens of milliseconds (tens of hertz) and the pulse width modulation at the switching frequency in the order of a hundred microseconds (corresponds to 10 kHz), as the edge modulation of each switching transient of the PWM-switched voltage is carried out in a time plane that is in the order of a microsecond.

Therefore, in the analysis of the effect of load current, the instantaneous value of the slow motor current can be approximated as a constant current, when the edge modulation of a single voltage transient is considered.

Second, the inductance visible from the asynchronous machine for a single voltage transient is the transient inductanceL⁰_s, which is defined as (Pyrhönen et al., 2008)

L⁰_s=Lsσ+ LrσLm

L_rσ+L_m≈Lsσ+L_rσ, (4.1)

whereL_sσis the stator leakage inductance,L_rσ is the rotor leakage inductance, andL_mis the magnetizing inductance. The transient inductance is in a major role to filter the motor current in an inverter drive, and it mainly consists of the stator and rotor flux leakages (Pyrhönen et al., 2008). For typical one-phase asynchronous machine equivalent circuit parameters and transient inductances for various motor sizes, see Appendix C.

If the load impedanceZ_Lis considered as the transient inductance,L⁰_s, it can be stated for a single transient that

4.1 Effects of an electric motor on the active du/dtfiltering method 67

L_f<<LL. (4.2)

Since the equivalent loading resistance of the induction machine, regardless of the loading condition of the machine, resides behind the transient inductance, the electric motor can thereby be regarded as a constant current source for the single transient, and thus, for a single active du/dtcharge or discharge pulse. This is also equal to the analysis above, because the time constant and hence the transient behavior is considerably slower in the system presented by the higher inductance, that is, the induction motor.

The voltage and current waveforms of the active du/dtfiltering, together with the correspond-ing gate control signals in basic operation as presented above for one inverter phase, are shown in Figure 4.2. As seen, dead times are not taken into account in the simplified analy-sis.

When the power switch corresponding to the voltage step transition direction is turned on, the absolute value of the current begins to increase at a rate defined by theLCconstant, and the capacitor begins to either charge or discharge depending on the slope direction. When the half of the total voltage transition is reached, the power switch is turned off. At that moment, both the power switches are in a nonconducting state, and the current flows through the freewheeling diodes in the output stage. The absolute current begins to decrease at the same rate at which it increased, and the output voltage of the filter continues to rise or fall depending on the direction of the voltage, as the transient behavior of the LC circuit presented in Chapter 3 defines. After this, the same power switch is turned on, when the current has returned to the same value as it was before the transient, and the voltage has reached either the negative or positive DC link rail potential. In the case of no external loading of the filter, as presented in Figure 4.2, the initial current before the transition is zero.

In the figure, there is no load at the output of the filter, but the only load to the inverter output stage is the LC filter circuit. In Figures 4.3 and 4.4, the rising and falling voltage slopes and the corresponding filter currents are shown in two cases: for positive and negative load currentsI_L.

As can be seen from Figures 4.3 and 4.4, the load, or the base current, at which the filter charge or discharge is carried out, will cause an error depending directly on the idle current instantaneous value related to the magnitude of the filter current during the operation of the filter. Therefore, the residual oscillation in the filter output, and also the correction method, is a function of load current.

If the load currentI_L is greater than zero, generation of the rising voltage slope is not af-fected, and the filter will operate normally in the freewheeling mode in the same way as in the zero idle current situation. However, the falling voltage slope is affected, because the edge modulation pattern of the falling slope will cause crossing of the zero current. In the freewheeling mode, the current will not return to the same value as in the beginning of the edge modulation sequence, and the current will thus remain at the zero current level. As the filter inductor current drops to zero, the inductive load impedance will start to drain charge

g a t e c o n t r o l s i g n a l s o f o n e i n v e r t e r l e g

gate control

u p p e r s w i t c h

l o w e r s w i t c h t i m e

voltage

o u t p u t v o l t a g e o f t h e L C f i l t e r

current

o u t p u t c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r

a )

b )

c )

i n v e r t e r p h a s e o u t p u t v o l t a g e

output

d )

Figure 4.2. Basic active du/dtoperation: a) filter output voltage and b) filter current. c) The gate control signals of the inverter leg are also shown along with d) the inverter output voltage.

4.1 Effects of an electric motor on the active du/dtfiltering method 69

t i m e

voltage

t i m e

current

t i m e

voltage

o u t p u t v o l t a g e o f t h e L C f i l t e r

t i m e

current

o u t p u t c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r IL > 0

a ) b )

c ) d )

Figure 4.3. a) and c) Operation of the active du/dtmethod, when the load current instantaneous value is greater than zero (towards the motor), and b) and d) less than the filter peak current. As shown, zero end current d) will result in oscillation c).

t i m e

voltage t i m e

current

t i m e

voltage

o u t p u t v o l t a g e o f t h e L C f i l t e r

t i m e

current

o u t p u t c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r IL < 0

a ) b )

c ) d )

Figure 4.4. a) and c) Operation of the active du/dtmethod, when the load current instantaneous value is less than zero (towards the inverter), and b) and d) less than the filter peak current. As shown, zero end current d) will result in oscillation c).

4.1 Effects of an electric motor on the active du/dtfiltering method 71

from the filter capacitor, causing the filter phase output voltage to turn into negative.

The error in the current waveform is related to the instantaneous value of the load current.

In order to correct this error, the filter inductor current must be returned to the initial value, which is carried out using the opposing inverter switch. Similarly as in the basic operation of active du/dt with no load, the capacitor voltage must be at the target value and the filter inductance current must be at the initial value at the end of the sequence to avoid residual filter oscillation.

In the contrary case, if the currentILis less than zero, the falling voltage slope is not affected, but the rising voltage slope will be erroneous for the same reason: the filter inductor current will stay at zero current instead of returning to the initial negative current. The correction is carried out in the same way as in the case of positive idle current, using the opposite inverter switch in comparison with the basic active du/dtoperation presented in Chapter 3. The idea of the correction sequence is presented in Figure 4.5 for both the cases requiring the current correction pulse described above.

t i m e

current

t i m e

gate control

u p p e r s w i t c h

l o w e r s w i t c h IL < 0

c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r

t i m e

current

t i m e

gate control

u p p e r s w i t c h

l o w e r s w i t c h IL > 0

c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r

g a t e c o n t r o l s i g n a l s o f o n e i n v e r t e r l e g g a t e c o n t r o l s i g n a l s o f o n e i n v e r t e r l e g

a ) b )

c ) d )

Figure 4.5. Principle of the current correction pulse to correct the operation of the active du/dtmethod.

a) and b) show the effect of the current correction pulse, c) and d), on the filter current.

The idea of the current correction pulse is presented in Figure 4.6. As the load current|I_L|

increases as a result of the fundamental modulation, depending on the potential the phase voltage is switched to, either the falling or rising voltage edge modulation must include a current correction pulse. As the absolute value of the idle current increases, the compensating current correction pulse extends from the end of the charge or discharge period toward the start of the period. Therefore, the minimum value of the pulse length is zero, at zero load current, which also means that the correction pulse is absent. As a result of the 50 % duty cycle of the basic active du/dtvoltage level transition edge modulation, the ideal maximum length of the current correction pulse is half of the charging or dicharging period, because otherwise inverter leg short circuit would occur. This situation is also equal to the instant at which the filter current is at its maximum value and the current correction pulse will last for the entire period.

The pulse length in the ideal case can be derived from the filter current equation (3.10) based on the principle presented in Figure 4.6.

The length of the current correction pulset_corris indicated in Figures (4.6) and (4.7). Equa-tion (3.10) can be divided into parts in the same way as the voltage equaEqua-tion presented in Figure 3.10: The parts of the current that have an effect on the different phases of the filter current are also indicated in Figure 4.7. The length of the current correction pulse can be determined by solving the equation

i_f(t) =|I_L|, (4.4)

because of the symmetricity of the filter current waveform, only the part (1) of Eq. (4.3) has to be taken into account in the solution. Therefore, the solution for the length of the current correction pulse in the ideal case is

t_corr=√

whereI_Lis the load current instantaneous value andAis the amplitude of the inverter DC link voltage.

The cases in which the absolute value of the load current is between zero and the filter max-imum current have been discussed above. The case in which the load current is greater than the filter current is shown in Figure 4.8.

4.1 Effects of an electric motor on the active du/dtfiltering method 73

t i m e

current

t i m e

gate control

u p p e r s w i t c h

l o w e r s w i t c h c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r

IL < 0

IL

Îf

t i m e

current

t i m e

gate control

u p p e r s w i t c h

l o w e r s w i t c h IL > 0

c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r

IL

g a t e c o n t r o l s i g n a l s o f o n e i n v e r t e r l e g g a t e c o n t r o l s i g n a l s o f o n e i n v e r t e r l e g

a ) b )

c ) d )

i n v e r t e r p h a s e v o l t a g e

output

i n v e r t e r p h a s e v o l t a g e

output

e ) f )

Figure 4.6. Principle of the current correction pulse to correct the operation of the active du/dtmethod.

As the load current absolute value increases, a) and b), a current correction pulse of increasing length must be applied, c) and d). Inverter leg output voltages are shown, e) and f) for the the gate control signals, c) and d), of the individual power switches.

t i m e

current

c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r IL < 0

| IL| if(t) = |IL| tc o r r

( 1 ) , ( 1 ) + ( 2 ) , ( 1 ) + ( 2 ) + ( 3 )

Figure 4.7. Principle of the derivation of the current correction pulse length. (1), (2), and (3) refer to the parts of Eq. (4.3).

In this case, the current correction pulse must be half of the total charging or discharging period, which is also the maximum length of the correction pulse. Now the absolute value of the load current is greater than the filter maximum current, and hence, no zero crossing takes place in the current waveform. The actual edge modulation of the active du/dt modulation is not necessary in this case either, because the absolute value of the current will start to decrease in the freewheeling mode, when both the switches of the inverter leg are turned off. The filter inductance current is restored to the initial idle current value using only the current correction pulse, which is half of the period. In this case, the edge modulation pattern is similar to the basic active du/dtmodulation pattern; the pattern itself is the same, but the inverter switch used is the opposite. In addition, the current correction, for any load current, can be carried out using the full-length current correction pulse, if ideal switches are used.

However, the current correction idea based on the actual commutation instant was presented as a basis for implementation on a real inverter.

However, implementing the current correction pulse in an actual inverter is not as straight-forward as presented here, because the properties of the inverter output stage, for example the losses, minimum pulse lengths, and required dead times, will all have a significant effect on the final result of the active du/dt modulation. Implementation of the current correction modulation in a real inverter should be based on the idea presented above, taking into account the limitations defined by the actual IGBT modules in the output stage, and it is outside the scope of the work presented in this dissertation.

4.1 Effects of an electric motor on the active du/dtfiltering method 75

t i m e

current

t i m e

gate control

u p p e r s w i t c h

l o w e r s w i t c h IL > |Îf|

c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r t i m e

current

t i m e

gate control

u p p e r s w i t c h

l o w e r s w i t c h I_L < - |Îf|

c u r r e n t o f o n e p h a s e o f t h e i n v e r t e r

g a t e c o n t r o l s i g n a l s o f o n e i n v e r t e r l e g g a t e c o n t r o l s i g n a l s o f o n e i n v e r t e r l e g

i n v e r t e r p h a s e v o l t a g e

output

i n v e r t e r p h a s e v o l t a g e

output

a ) b )

c ) d )

e ) f )

Figure 4.8. a) and b) Principle of the current correction pulse to correct the operation of the active du/dt method, when the instantaneous load current is greater in amplitude than the filter peak current. c) and d) The charge and discharge pulses are eliminated by the freewheeling operation of the circuit, and only the current correction pulse is required to restore the current of the filter reactor to the starting value.

Inverter leg output voltages, e) and f), are shown for the gate control signals of the individual power switches, c) and d).

In document Active du/dt Filtering for Variable-Speed AC Drives (sivua 66-76)