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Suppose one is designing a full-wave rectifier bridge, with a known maximum current load, and a known input inductance. Assume three-phase for this example, though the same concept applies to single-phase:

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The ripple current seen by the output capacitor is critical. If that current is too high, the capacitor will heat up, and its lifespan will be reduced. But how does one compute the ripple seen by this capacitor?

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Assume the system is already precharged and operating in a steady state. The bridge has two discrete states: either the capacitor is charging (a diode pair is forward biased), or the capacitor is discharging. Call the period P, the charge time DP, and the discharge time (1-D)P.

During the charge cycle, we can approximate the current entering the capacitor as a triangle, starting at 0, and rising to a peak. $$ 1: I_{charge}(t) = \frac{t I_{peak}}{DP}\\ $$ Assume that the output capacitance is large enough that its voltage ripple is small, meaning the current out of the cap during the discharge time is fixed. $$ 2: I_{discharge}(t) = I_{load}\\ $$

Computing the RMS: $$ 3: I_{RMS}=\sqrt{\frac{\int_0^{DP}I_{charge}^2(t) dt + \int_{DP}^{P}I_{discharge}^2(t) dt}{P}} $$

Evaluating the integral: $$ 4: I_{RMS}=\sqrt{\frac{I_{peak}^2D}{3} + I_{load}^2(1-D)} $$

Since we're in a steady state, the total charge into the capacitor during the charge cycle must be equal to the total charge leaving the capacitor during its discharge time: $$ 5: Q_{charge}=Q_{discharge} $$

The total charge entering the capacitor is the area of the current triangle: $$ 6: Q_{charge}=\frac{I_{peak}DP}{2}. $$

The charge leaving the capacitor during the discharge cycle is the product of the fixed current and time: $$ 7: Q_{discharge} = I_{load}(1-D)P. $$

Which gives us: $$ 8: \frac{I_{peak}DP}{2} = I_{load}(1-D)P $$

Solve for peak current: $$ 9: I_{peak}=\frac{2I_{load}(1-D)}{D} $$

Substitute into equation 4: $$ 10: I_{RMS}=I_{load}\frac{\sqrt{D^3-5D^2+4D}}{D\sqrt{3}} $$

From this we see that the ripple current seen by the output capacitor is a function of the load current and the fraction of the AC period spent charging the capacitor. As D approaches 0, the ripple current approaches infinity. As D approaches 1, the ripple current approaches 0. Longer charge times reduce the ripple.

Consider the choke currents and capacitor voltages during a charge cycle: $$ 11: V_{choke} = L\frac{di}{dt}\\ 12: I_{cap} = C\frac{dv}{dt} $$

During the charge cycle, we have approximated the current through the choke into the capacitor as a triangle with a height of I_peak. The average current into the capacitor during the charge cycle is half this peak. The length of the charge cycle is DP. The voltage across the choke starts at 0, rises to a peak approximately equal to the ripple voltage dv, then falls back to zero. We can approximate the average voltage across the choke as half the ripple voltage. $$ di = I_{peak}\\ dt = DP\\ I_{cap} = \frac{I_{peak}}{2}\\ V_{choke} = \frac{dv}{2} $$ Substituting into 11 and 12: $$ 13: \frac{dv}{2} = L\frac{I_{peak}}{DP}\\ 14: \frac{I_{peak}}{2} = C\frac{dv}{DP} $$ Solve both equations for dv, then solve for D: $$ 15: \frac{2LI_{peak}}{DP} = \frac{DPI_{peak}}{2C}\\ 16: D = \frac{2\sqrt{CL}}{P} $$ Substitute into equation 10 to find the RMS current seen by the capacitor.

So the length of the charge cycle is twice the time constant of the LC resonant circuit. Increasing the size of the choke spreads the charge cycle over a longer time, reducing the RMS current (and improving line harmonics). Increasing the size of the capacitor lengthens the time the choke is forward-biased. And increasing the frequency (decreasing the period) means each charge pulse can be smaller and deliver the same current. Thus, three-phase rectifiers have lower ripple current on their output capacitors than single-phase. This math indicates that for a fixed capacitor ripple current, a three-phase rectifier run with a single-phase input can only run ~30% of the three-phase load current.

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    \$\begingroup\$ i regret that i have but one vote to give for so much math. \$\endgroup\$ Mar 13 '13 at 17:43
  • \$\begingroup\$ Not sure I understand where L in (11) came from. Could you please elaborate a bit more on the choke voltage? \$\endgroup\$
    – jippie
    Mar 13 '13 at 17:52
  • \$\begingroup\$ @jippie L is the AC line inductance seen in the schematic. That could be the inductance of the line itself, or a choke added purposefully to reduce the capacitor ripple and improve line harmonics. Alternately, a DC link choke could be present between the diodes and the capacitors. \$\endgroup\$ Mar 13 '13 at 18:04
  • \$\begingroup\$ Great post Stephen, I have some few remarks. (1) I'm not sure to understand the assumption "During the charge cycle, we can approximate the current entering the capacitor as a triangle, starting at 0, and rising to a peak". Is this not the contrary ? assume source inductance (Ls) is 0, the current in the capacitor get "instantaneously" to a peak and fall "linearly" to 0 ... (2) I'm not convinced that capacitor current can be approximated by a triangle for high values of Ls. \$\endgroup\$
    – user95873
    Jan 1 '16 at 21:14
  • \$\begingroup\$ (3) I don't find probing results for the computing of D=f(C,L). The obtained value is very different from the simulation, for non negligible values of Ls. \$\endgroup\$
    – user95873
    Jan 1 '16 at 21:14

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