Active du/dt filter current analysis

The filter current flowing in the LC circuit during the charge and discharge periods can be solved using the same principle as in solving (3.8), that is, by determining thes-plane LC circuit voltage equation and solving for the current in the LC filter circuit caused by the charging pulse. In the time domain, this analysis yields for the filter current

i_f(t) = A

The maximum filter current during the charging period is at the momentt_1/2, as the supplying voltage is switched off; after that instant the charging current of the inductorL begins to decrease. Based on (3.5) and (3.10), the charging current maximum value can be solved

i_f(t)_max=i(t_1/2) = A pL/Csinπ

3 ≈0.866 A

pL/C. (3.11)

−2

−1 0 1 2

Output Voltage

a)

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

0 0.5 1 1.5 2

Stimulus

Time b)

Figure 3.12. a) Response of the LC circuit. b) Charge and discharge sequences according to the active du/dtmethod are succesfully applied.

As can be seen, the filter peak current is inversely proportional to the square root of the filter inductanceLand proportional to the square root of the filter capacitanceC. Together with theLCconstant, the peak current is an important design consideration, because the IGBT module must withstand the additional current stress caused by the filter current on top of the load current flowing through the output stage.

The filter output voltage and the filter charging current are presented in Figure 3.13 in a normalized form as functions of filter component values and voltage amplitudeA(1 puU_DC of the applied voltage pulses.

The analysis presented in this section concerns the charge pulse, but a similar analysis can be carried out also for the discharge pulse by adding into (3.8) the delayed step functions describing the discharge pulse. The filter voltage and current waveforms are similar for both the charge and discharge pulses, only the direction is different with respect to the zero level.

3.1.5 Different charging schemes for active du/dt filter circuit

Further, the same principle as in the presented charge consisting of a single pulse can be used to derive the filter output voltage and filter current for charge and discharge sequences consisting of several, narrower pulses, with the same duty cycle of 50 %. The output voltage and the filter current can be presented in a general form for a number ofNcharge pulses

3.1 Active du/dtfiltering method 57

Figure 3.13. Generalized filter output a) voltage and b) current waveforms during a charge pulse.

u_out(t) =A

The pulse length, which is equal tot1, has to be solved using the same principle as above. For example, for a charge ofN=2 pulses, there are 2N+1 switching instantst₀,t₁, . . . ,t₄. Now, the output voltage of the filter, which is obtained from (3.12), has to be half of the DC link voltage amplitudeAin the middle of the charge sequence, and full DC link voltage at the last switching instant.

For example, for a case of two charge pulses, these are now att₂andt₄. The pulse lengtht₁ can be generally solved using this method, because for a charge ofNpulses, there are always an odd number of switching instants (2N+1), and therefore, a switching instant in the middle of the charging sequence. For the case of two pulses,t₁can be solved

t₁=1 5π

√

LC. (3.14)

As the number of pulses is increased, analytical solution of (3.12) becomes more difficult, and a numerical solving method may be more feasible. As the pulse width is solved, it can be applied to the discharge sequence because of the symmetry of the sequences.

It should also be noted that the rise time of the voltage slope is slightly increased, as more pulses are used in controlling the filter circuit. The exact value of the rise time can be solved by combining (3.4) and (3.12). The rise time of the filter output voltage depends on the time constant of the LC circuit

whereKdepends on the number of pulses used in the charge period. For the two-pulse charge, it can be obtained from Eq. (3.14) thatt_r= (4π/5)√

LC≈2,513·√ LC.

However, taking the properties of the present semiconductor power switch components into account, the charging scheme consisting of only one charge pulse is the most relevant se-quence because the switching losses increase and the minimum pulse width requirement de-creases as a function of the numberNof pulses used.

Another method for generating longer rise times than the base voltage slope of tr = (2π/3)√

LCis to use a pulse width different from the 50 % duty cycle in the charge and discharge pulses. In this case, instead of charging the filter to the full amplitudeAat once, each individual charge period increases the output voltage by a fraction ofA/M, whereMis the number of individual charge periods. Therefore, the total output voltage slope transition time is increased toMtimes the base transition timet_rby using the same LC circuit. For more on these charging schemes, see publications (Korhonen et al., 2009; Tyster et al., 2009). Nev-ertheless, these pulse sequences are outside the scope of this work and are not studied further here.

3.1.6 Measured example of active du/dt operation

Figure 3.14 illustrates typical operation in an inverter-fed drive. The cable length is 100 meters, and the propagation speed of the wave in the cable is approximately half the speed of light, Reka MCMK. A steep-edged voltage pulse is reflected at the motor terminal, and oscillation occurs. In Figure 3.15, the same situation is presented when active du/dtfiltering is applied. The cable resonance frequency is succesfully filtered, and the cable resonance is eliminated.

As can be seen in Figure 3.15, the LC circuit can be applied to the 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 rising and falling times of the slope depend on the time constant of the LC circuit. It can also be noted that if the voltage is switched off when the LC filter output voltage has reached half the DC link voltage, the output voltage is doubled to equal to the DC

3.1 Active du/dtfiltering method 59

Voltage at 100 meter open−ended cable end

Time [s]

b)

Voltage [V]

Figure 3.14. Measurement of a cable resonance, a) in basic inverter operation, b) for a 100 meter cable.

−1 −0.5 0 0.5 1 1.5 2 2.5 3

Voltage at 100 meter open−ended cable end

Time [s]

b)

Voltage [V]

Figure 3.15. a) Measurement of active du/dtoperation. b) The cable resonance and overvoltage are effectively eliminated.

link voltage. Since the output voltage of the LC circuit was exactly half the step voltage, the total time taken to the full step voltage is double the time of the voltage pulse applied. In this

case, the resonance of the LC circuit is avoided, because the switching instants are conducted exactly as described.

3.2 Active du/dt filter circuit component selection

As presented above, the active du/dt filter circuit consists of a series inductance L and a parallel capacitanceC, which are provided by the filter inductors and capacitors in a real implementation.

However, there are some considerations regarding the design and realization of the filter circuit. First, in the filter component selection, the use of narrow, steep-edged voltage pulses in the order of microseconds has to be taken into account. This sets special requirements for the inductors and capacitors. The inductors have to be designed for high-frequency use, which means that the core material has to be air or high-frequency ferrite material.

In the case of ferrite core, the saturation of the core material has to be avoided, and the filter and motor maximum currents have to considered in the design. The filter maximum current can be obtained from Eq. (3.11). In the filterLandCcomponent value selection for a specific t_r(LCconstant) value, an increase in the inductor value decreases the filter maximum current, whereas an increase in the capacitor value increases the filter maximum current, as seen from Eq. (3.11). The filter maximum current is also an important design consideration, because the current handling capability and power losses of the inverter power modules set limitations on the peak charge and discharge currents. However, the inductance and capacitance values cannot be selected arbitrarily based on the rise time and the filter peak current, because the capacitance value has an effect on the rigidity of the active du/dt filter circuit as a voltage source. The topic will be discussed in more detail later in the next chapter.

Furthermore, the variation in the componentLandCvalues resulting from component tol-erances causes an error in the filter output voltage, if the manufacturing toltol-erances are not taken into account in the design of the charge and discharge sequences. As can be noticed from Eq. (3.5),t_ris proportional to the square root of the component values as follows

t_r∼√

LC. (3.16)

Variation in the component values causes a change proportional to the square root of the designed component value in the correct rise timetr. The amplitude of the resonance, or the error, in the LC circuit is equal to the difference in the filter output (capacitor) and DC link voltages at the instant when the voltage pulse is switched on at the end of the charge or discharge sequence. Therefore, faulty charge according to a wrong rise time causes the filter capacitor to under- or overload, causing a resonating output voltage. Residual LC circuit output oscillations for variousLCconstant errors when compared with the designed value, between 80 and 120 %, are presented in Table 3.1.

However, the filterLCconstant can be detected by generating a voltage step in the inverter