Mangesh Borage, Sunil Tiwari, and S. Kotaiah
Centre for Advanced Technology, India
TechOnline

The input stage of an off-line power converter usually contains an LC averaging filter. A common practice is to damp the LC filter using additional reactive/dissipative components. In this article, we re-visit an LC filter for this application. With simple analysis and design examples, we can demonstrate that filter capacitor size and cost in shunt-capacitor damping is larger than with an undamped filter. The performance of an undamped filter is comparable with that of a damped filter with respect to output-voltage excursions upon line and load steps. Our results suggest that bulky and costly damping may not be justified for the improvement in performance.

• If the filter's Q is too high, an increase in emission can occur at the resonant frequency of the filter.
• A high Q in a power-line filter may cause stability problems with switching power supplies.
• At a resonant frequency, the filter will have a voltage gain such that the output voltage is Q times the input voltage.
• Input and output impedances are affected by filter Q. If the Q is greater than 1, the input impedance will be lower than and output impedance will be greater than the characteristic impedance by a factor of Q.
• A step change in line or load will give output-voltage overshoots and undershoots that are proportional to Q.

All these reasons require a designer to minimize or control the filter's Q to be within acceptable limits by incorporating suitable damping circuits. However, the damping circuits, while minimizing the aforementioned adverse effects, are bulky, lossy, and costly.

We will address the classical low-pass LC filter for a three-phase diode-bridge rectifier stage of an off-line power converter. We compare the performance of an undamped LC filter and the shunt-capacitor damped LC filter under common transient disturbances occurring in an off-line power converter. However, a practical filter is never completely undamped. The ESR of a filter capacitor partially damps the filter to some extent. In this study, we signify an undamped filter as a filter without additional damping; however, we still consider the effect of filter-capacitor ESR on the circuit.

• Filter cut-off or resonant frequency,     (1a)
  
  
• Characteristic impedance,     (1b)
  
  
• Filter Q,     (1c)

Figure 1:  Classical LC filter including filter capacitor ESR

Figure 2 shows the forward voltage transfer function and output impedance of the LC filter. Note that both gain and output impedance increase at the filter's resonant frequency. The gain peak amplifies the input voltage at the resonant frequency. The peak in output impedance may cause stability problems with negative incremental input impedance of a constant power load (switching power supply).

Figure 2:  Forward voltage transfer function and output impedance of the classical LC filter with the filter capacitor's ESR

Peak output impedance of LC filter of Figure 1 is:

    (2)

Middlebrook shows that the input impedance of a switching power supply is minimum at the resonant frequency of the output averaging filter and the low frequency value is equal to:

    (3)

where R_L is load resistance of the power converter and M is:

    (4)

Both the resonant frequency of output averaging filter and filter's closed-loop bandwidth are much higher than the resonant frequency of the front-end filter. Therefore, the input impedance of a switching power supply at the resonant frequency of the front-end filter is equal to the value given by Equation 3.

For stable operation, the following condition must be satisfied:

    (5)

If we apply another design parameter—resonant frequency of the filter—in conjunction with the condition of Equation 5, we get following theoretical limiting values of L and C for stable operation.

    (6)

The Practical Case
In practice, another constraint is imposed on the design of an LC filter for a three-phase rectifier. A continuous current in the filter's inductor is desirable for many reasons, and a value for critical inductance for continuous current in the inductor is given by Mohan et al,

    (7)

where V_LL is the RMS line-voltage of the AC main line and w is the line frequency in rad/sec. Having derived the required value of L > L_crit, we can now find a value of C from Equation 1 as:

    (8)

For stable operation, the required R is then given from Equation 6 as,

    (9)

The values of L and C obtained from these equations need to be modified to suit component availability. In most cases, there is no ready-made inductor available, requiring one to be designed and fabricated for a particular application. The designer, therefore, has control over the inductor's value. However, a limit exists on the range of commercially available capacitor values. Therefore, it is quite practical to slightly increase or decrease the value of the capacitor from the one given by Equation 8 to be able to use available capacitors, and then modify the inductor value accordingly.

For example, the voltage rating of filter capacitors should be 900 V for three-phase rectifier operation (using a 400 V RMS line voltage) and 450 V for single-phase rectifiers (using a 230 V RMS line voltage), including approximately a 50% safety margin. Therefore, a commercially available capacitor with voltage rating of 450 V can be standardized and applied in parallel-banks for single-phase and in series-parallel banks for three-phase rectifier applications.

Let C' be the nearest value of capacitance realizable using commercially available components. To keep the resonant frequency of the filter unchanged, the inductor value must be changed to:

    (10)

Let R' be the effective ESR bank of capacitor C'. Then, from Equation 9:

    (11)

If this condition is not satisfied, the designer needs to externally connect a suitable value of resistor in series with the capacitor.

Design Example
For a three-phase rectifier operating from 400 V RMS, 50 Hz main AC and drawing 10 kW full-load output power (R_L/M² = 29.16 W), the inductor current should remain continuous till 20% of full load. The cut-off frequency of the LC filter is 30 Hz. We calculate the following component values:

L = 4.47 mH
C = 6300.87 µF
R > 24.33 mW (for stability)

We use a 470 µF/450 V electrolytic capacitor with an ESR of 0.7 W. With series connection of two such capacitors, we get a voltage rating of one "arm" of 900 V and capacitance of 235 µF. The nearest realizable capacitor using this arm is 6345 µF by connecting 27 arms in parallel. The effective ESR of this bank is 0.052 W. To keep our cut-off frequency unaffected, we change the inductor to 4.44 mH. Thus:

L' = 4.44 mH
C' = 6345 µF
R' > 24 mW (for stability)

With the proposed series-parallel capacitor bank, we get the effective ESR as (0.7*2/27) = 52 mW, which is greater than limiting value of 24 mW shown.

Figure 3:  Shunt capacitor damping of an LC filter

The most popular method of damping an LC filter introduces additional shunt damping resistance, R_d, in parallel with the load. A blocking capacitor, C_d, is placed in series with R_d to avoid power loss in the resistor. If C_d > C, the reactance of C_d becomes less at w_o and R_d effectively damps the filter's Q. This shunt capacitor damping of LC filter is illustrated in Figure 3. For given ratio, N = C_d/C, there exists an optimum value of R_d which minimizes peaks in the forward voltage transfer function, output impedance, or input impedance. For instance, a relationship that gives an optimum value of R_d to minimize the peak in the forward voltage transfer function is:

    (12)

We design the filter of prior design example with shunt capacitor damping and add a constraint of an available capacitor arm of 235 µF/ 900 V. The design values for two cases, N=1 and N=5, are given in Table 1.

Filter Transfer Function
Figure 4 shows the forward voltage transfer function of the three filters—all three have a cut-off frequency of 30 Hz.

Figure 4:  Transfer functions of the three filters of the design exam

Size of Filter Capacitor
For the undamped filter, the number of paralleled capacitor arms is 27, whereas for damped filter the number is 36 for N=1 and 48 for N=5. Clearly, the capacitor volume, weight, and cost are higher for damped filters. All three designs have almost identical inductor values.

Figure 5:  Comparison of capacitor arms for damped and undamped filters

Figure 5 shows the required number of parallel capacitor arms for a filter with a 30 Hz cut-off frequency and maintaining continuous current in the filter inductor until 20% of full load. The increase in capacitor volume at high output power is apparent. In fact, we additionally need approximately 1 arm per kW for N=1 and 2 arms per kW for N=5 compared to an undamped filter.

Sudden Load Steps

Figure 6:  Output voltage excursion due to a sudden change in load

The load on the front-end rectifier and LC filter can vary in sudden steps. The worst case is a sudden switch-on of a DC-DC power converter (0-100% load step) and sudden "throw-off" of the load due to a fault in the converter (100-0% load step). Figure 6 shows the output-voltage changes following these load steps. The output voltage of undamped filter exhibits larger overshoots and undershoots that eventually dampen after few cycles. However, these excursions are well within safe limits with respect to capacitor voltage ratings and are not significantly greater than those observed with damped filters.

Sudden Line Steps

Figure 7:  Output voltage excursions due to sudden changes in line voltage

The supply line voltage can also vary in the form of sudden steps. Commonly specified line steps are ±2%. We simulate the effect of a 50 V line step under no-load and full-load conditions. Figure 7 shows the results of these steps. Again, the output voltage excursions of damped and undamped filters are not appreciably different and are within safe limits.

Direct-on-Line Start-Up at No Load
This is an event that does not occur frequently in practice. Some form of inrush-current-limiting circuit will be provided at the start-up of the rectifier. Nevertheless, such an event may occur following the failure of the inrush-limiting circuit. Figure 8 shows simulated output voltage excursions under these conditions. Note that with undamped filter the capacitor voltage exceeds the safe voltage limit of 900 V.

Figure 8:  Output voltage overshoots on direct-on-line rectifier start-up rectifier damped and undamped filters

Experiment with Laboratory Prototype
To validate the simulation results, we performed an experiment with a prototype rectifier. The output voltage excursion on sudden load throw-off was recorded with a damped filter (C_d=4C) and with a filter without additional damping. Figure 9 shows no significant difference was observed between the two cases. The component values of the LC filter are shown in Table 2.

We implemented each 500 µF / 900 V capacitor by connecting two 1000 µF / 450 V capacitors in series. The ESR of each 1000 µF / 450 V capacitor is 0.35 W.

Figure 9:  Output voltage excursion on sudden load throw-off. Ref A used a damped filter. Ref B used a filter without additional damping. The rectifier was operated at 20% of nominal line voltage for the experiment.

We proved that the stable and safe operation of an off-line power supply can be ensured even without providing additional damping to the front-end averaging LC filter. Low-cost, low-volume inrush-current-limiting circuits and/or bleeders can further enhance the safety of these circuits.