Damping of PowerConverter FrontEnd Averaging Filters
The input stage of an offline 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 revisit an LC filter for this application. With simple analysis and design examples, we can demonstrate that filter capacitor size and cost in shuntcapacitor 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 outputvoltage 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 powerline 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 outputvoltage 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 lowpass LC filter for a threephase diodebridge rectifier stage of an offline power converter. We compare the performance of an undamped LC filter and the shuntcapacitor damped LC filter under common transient disturbances occurring in an offline 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 filtercapacitor ESR on the circuit.
 Filter cutoff 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 closedloop bandwidth are much higher than the resonant frequency of the frontend filter. Therefore, the input impedance of a switching power supply at the resonant frequency of the frontend 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 threephase 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 linevoltage 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 readymade 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 threephase rectifier operation (using a 400 V RMS line voltage) and 450 V for singlephase 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 parallelbanks for singlephase and in seriesparallel banks for threephase 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 threephase rectifier operating from 400 V RMS, 50 Hz main AC and drawing 10 kW fullload output power (R_{L}/M² = 29.16 W), the inductor current should remain continuous till 20% of full load. The cutoff 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 cutoff 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 seriesparallel 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.
L'(mH)  
C'(µF)  
C_{d}(µF)  
R_{d}(W)  
No. of arms in parallel 
Table 1: Design values for N=1 and N=5
Filter Transfer Function
Figure 4 shows the forward voltage transfer function of the three filters—all three have a cutoff 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 cutoff 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 frontend rectifier and LC filter can vary in sudden steps. The worst case is a sudden switchon of a DCDC power converter (0100% load step) and sudden "throwoff" of the load due to a fault in the converter (1000% load step). Figure 6 shows the outputvoltage 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 noload and fullload 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.
DirectonLine StartUp at No Load
This is an event that does not occur frequently in practice. Some form of inrushcurrentlimiting circuit will be provided at the startup of the rectifier. Nevertheless, such an event may occur following the failure of the inrushlimiting 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 directonline rectifier startup 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 throwoff 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.
Additional Damping 

Table 2: LC filter component values
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 throwoff. 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 offline power supply can be ensured even without providing additional damping to the frontend averaging LC filter. Lowcost, lowvolume inrushcurrentlimiting circuits and/or bleeders can further enhance the safety of these circuits.