Analysis of the active du/dt filtering method

The rise timet_rfor the single pulse charge described previously can be derived from the single phase equivalent circuit of the three-phase filter, Figure 3.7.

L C

u

u

Figure 3.7. Single-phase ideal equivalent circuit of the proposed LC filter for active du/dtcontrol.

As presented above, the filter circuit is an LC filter, in which the inductorLand the capacitor Care in series. The output voltage of the filter is the voltage of the capacitor, and both the load and filter current flow through the inductor. Before the theory for the control of active du/dtcan be developed, the operation of the LC circuit during transients must be analyzed.

First, the response of the LC circuit shown in Figure 3.7 is analyzed. Deriving from the s-plane transfer functionH(s)of a second-order system

H(s) = ω_n²

s²+2ζ ω_ns+ω_n², (3.1) yields that thes-plane transfer function for the active du/dt filter circuit shown in Figure 3.7 is

LCandζ =0, because in this simplified analysis, resistance is assumedR=0 and the circuit is at resonance at the frequency when the reactances of both the inductor and the capacitor are the same, that is, when the conditionXL=XCis satisfied.

As presented, the feeding voltage must be switched off, when the output voltage of the filter reaches half the DC link voltage. Based on (3.2), the step response of the presented LC circuit for the step of an amplitudeAcan be transformed into the time domain. The output voltage of an ideal LC circuit for a step of an amplitudeAis

u_out(t) =A·

√LC/3. Because the instant at which the charge, that is, the rising voltage slope, is complete and the feeding voltage is switched on again ist2=2t_1/2, the rise timetrof the charge is

3.1 Active du/dtfiltering method 51

At the momentt₂,u_outequals the amplitudeAof the output voltage pulse, which in this case is equal to the DC link voltage. As we can see from (3.5), the voltage slope rise time depends on theLCconstant of the circuit. The pulse widths of the charge sequence also depend on theLCconstant of the circuit and thereby on the target voltage transition time. It can also be noted that for fast voltage transition times, the inverter output stage must be able to produce pulses in the order of the desired transition time. For example, if the target is a 2 µs voltage slope, the output pulse width in the charge sequence equals 1 µs. However, as the motor cable length increases, the longer are the required voltage slopes, and therefore the situation is easier for the inverter output stage. This is also the situation at which the motor overvoltage problems are most evident.

Because of the symmetricity of the charging and discharging sequences, the pulse widths are the same for both the sequences in an ideal case. In a real implementation, various delays between the control logic, gate drivers, output stage power modules, and also the dead times, turn-on and turn-off delays of the actual power switches have to be taken into account in a successful implementation. However, a sufficient requirement is that the pulses produced by the inverter output stage are of correct length and pulse width, despite the internal implemen-tation of the charge and discharge pulse generation.

Because a common two-level inverter has only two voltage levels, it is the positive and neg-ative DC bus rails, to which the output phase can be connected through the inverter bridge.

Thus, half of the DC voltage cannot be directly generated. However, half the DC link voltage can be generated in the same way as different voltage levels are normally generated using pulse width modulation in the inverter, as stated earlier. This introduces a new edge modula-tion in a faster time domain compared with the normal inverter PWM modulamodula-tion. In addimodula-tion to the normal phase voltage modulation at the switching frequency, at every turn-on switching action of the inverter output stage, the edge modulation has to be carried out for the voltage step for successful active du/dtfiltering.

Based on Figure 3.5, if the voltage applied to the LC circuit is cut at the moment when the voltage is at the half of the DC link voltage, the LC circuit will double the output voltage to the full DC link voltage. By solving from Eq. (3.2) and by using the stimulus described, the output of the LC filter circuit in the time domain can be obtained from

u_out(t) =A

whereAequals the DC link amplitude,εis the Heaviside step function,ε(t−t₁)is the step function delayed byt₁, andt₁is the moment, at which the output voltage of the LC circuit is half the DC link step applied to the circuit. The stimulus and the output voltage of the LC circuit are presented in Figure 3.8 for an amplitude ofA=1, which can be considered to be 1 puU_DC.

The behavior presented in Figure 3.8 can be explained by the fact that the voltage is cut at the moment when the output voltage is at the half of the voltage step, and the LC circuit doubles

−2

The response of the LC circuit for a single charge pulse

0

Figure 3.8. a) Response of the LC circuit for a single pulse, the pulse width of which is adjusted so that the pulse is turned off at the instant when the output voltage of the LC circuit is half the DC link amplitude. Therefore, the output voltage of the LC circuit increases to the potential of the DC link, at a rise time set by the time constant of the LC circuit, as presented. b) The LC circuit impulse response is also shown as a comparison.

the voltage applied. Hence, the voltage maximum is the amplitude of the DC link voltage, not twice the DC link voltage as for a plain step as in Figure 3.5. However, as previously noted, the LC circuit will resonate at twice the amplitude of the voltage step, if the damping factorζ is zero. Therefore, the oscillation amplitude in this case is twice the amplitude of the applied voltage step, as in Figure 3.5, but now the output voltage resonates around zero instead instead between zero and twice the DC link voltage. Further, the stimulus approximates roughly the Dirac delta (impulse) function,δ(t). The impulse response of the LC circuit can be solved from Eq. (3.2):

which is also presented in Figure 3.8.

It can be noted that the curves in Figure 3.8 have a similar form, but neither of the voltage waveforms are useful in the generation of the filter output voltage. However, we can see from Figure 3.8 that if the stimulus voltage to the LC circuit is switched back on exactly at the instant when the output voltage of the circuit is at the same voltage as the DC link voltage, no transient will occur, and the output voltage will remain at the DC link voltage applied.

3.1 Active du/dtfiltering method 53

Based on (3.2) and (3.5), the output voltage of the filter can be solved for the pulse sequence described. The stimulus consists of a sum of step functions of amplitudeA, of which two are delayed byt₁andt₂

By transforming (3.8) into the time domain, the output voltage of the filter can be written as

u_out(t) =A·

whereAagain equals the DC link voltage andεis the Heaviside step function.t₂is the instant at which the output voltage of the LC circuit has doubled to the full step voltage. Ideally,t₂is two timest₁, because att₁the output voltage of the LC circuit is at half the DC link voltage.

The waveform is presented in Figure 3.9.

0

The response of the LC circuit for active du/dt charge

0

Figure 3.9. a) Response of the LC circuit. b) Charge sequence, according to the active du/dtmethod is applied.

As can be seen from Figure 3.9, the LC circuit can be employed in generation of an output voltage, which consists of several delayed step responses of the LC circuit in order to produce

a rising voltage slope. The rise time of the slope depends on the time constant of the LC circuit.

By definition, the LC circuit doubles the modulated voltage applied to full voltage, but in this case the resonance of the LC circuit is avoided, if the switching instants are conducted exactly as described. If there is variation from the ideal timing of the switching instants, resonance will be induced in the LC circuit, resulting in residual oscillation. The amplitude of the residual oscillation depends on the amount of inaccuracy, as will be presented later in this chapter. Therefore, accurate control of the voltage pulses fed to the LC circuit is essential, since inaccurate control does not bring any benefits.

The method described above is called charging the filter, and the pulse sequence in Figure 3.9 is known as the charge pulse. In addition to this, generation of the charging pulse can be thought to consist of several delayed steps, in this analysis unit steps (1 puU_DC). If the steps are correctly timed, the step responses, as in Figure 3.5, are superimposed in the LC circuit in a way that produces a voltage slope of desired length. This idea is illustrated in Figure 3.10.

−2 0 2

Step responses

a)

The response of the LC circuit for active du/dt charge

−2

Figure 3.10. a) Individual step responses of the LC circuit for the steps applied as presented in the active du/dttheory. b) The delayed steps are shown to produce c) a voltage slope as a combined response.

The rise time of the slope depends on the resonance frequency of the LC circuit, as seen from a), and thereby on the actualLandCcomponent values.

As stated before,t₁andt₂correspond toπ/3 and 2π/3, respectively. Therefore, the phase shift between the individual responses has to beπ/3 for zero residual oscillation. Inaccurate timing causes error in the phase shift and, therefore, oscillating filter voltage.

In addition, the pulse sequence can also be applied to produce a falling voltage slope in addition to the presented rising voltage slope. The falling slope is achieved by using a similar

3.1 Active du/dtfiltering method 55

but reversed pulse pattern as in the charge pulse, as was presented above in Figure 3.6. If the filter circuit is not succesfully discharged, LC circuit resonance will occur, as presented in Figure 3.11. An example of a successful charge and discharge sequence is presented in Figure 3.12.

The response of the LC circuit for a falling voltage step

0

Figure 3.11. a) The response of the LC circuit. The effect of an unmodulated falling step is also presented. b) Charge sequence according to the active du/dtmethod is applied.