# Calculate Hold-Up Capacitor Size For Switching, Linear Regulators

Many power supplies use hold-up capacitors to keep enough energy at the full-wave or bridge rectifier output to supply a load when the rectified voltage drops and rises again at the next half-cycle of the ac supply voltage. When the rectified line voltage goes below the voltage across the hold-up capacitor, the capacitor feeds the power converter circuitry that follows it. As a result, this capacitor ensures smooth, no-dropout operation for the entire power supply.

The hold-up capacitor is charged from the rectifier output, through the very low resistance of the rectifier and ac line, so the voltage across the capacitor follows the rectified line voltage as it charges *(Fig. 1)*. The capacitor then discharges through the power converter supplying the load. There is a big difference in the hold-up capacitor’s required value when it supplies a switching regulator compared to a linear one, so we’ll look at them separately.

**Switching Regulator Supply**

We can neglect power loss in a switching regulator due to its high efficiency.

describes the hold-up capacitor voltage while charging. (The absolute-value sign is used to describe the rectifier action.) V_{in_max} is the peak value of the rectified voltage. The hold-up capacitor voltage, while discharging, is defined by:

where P_{in} is the converter’s input average power, and V_{C} is the present hold-up capacitor voltage. With these parameters, the equivalent discharge resistance r_{disch} can be used to determine C_{h}, the hold-up capacitance. Note that V_{C} is essentially equal to V_{HoldUp}(t) here.

These charge and discharge curves meet up in another cycle, at time t_{0}:

By cancelling out V_{in_max}, we obtain:

To obtain the value of the average voltage V_{C_av} of the hold-up capacitor, define r_{disch} using:

Note that the V_{C} changes while C_{h} discharges, with the same changes as r_{disch}. To simplify the analysis, we have to find out the average value for the discharge resistance. V_{C} changes exponentially from V_{in_max} through V_{disch}. We know the values of V_{in_max} and V_{disch}, and assuming the discharge process is linear, we can find the voltage across average value of C_{h }using:

This voltage value can be used to calculate the discharge-process time constant:

The expression of:

represents the hold-up capacitor discharge time constant.

Solving Equation 4 with respect to C_{h}, we obtain:

This is a transcendental equation with respect to t_{0} and is very hard to solve symbolically, although it would be very interesting to use it to obtain the value for t_{0} directly and then figure out how the “meeting point” t_{0} depends on the capacitance of the hold-up capacitor.

Instead, solve the equation with respect to C_{h}. When C_{h} discharges, it does so until it reaches a pre-determined threshold voltage V_{disch} for the time interval t_{0} using:

which results in:

Substituting Equation 7 into Equation 5, we obtain:

from which we get:

This equation shows that to provide the discharge voltage of V_{disch} at some load value and line frequency ω the hold-up capacitor should have the value of C_{h}.

**Example 1**

With a line voltage V_{line} = 415 V, load power P_{load }= 70 W, and efficiency η =** **97.2%,** **then:

V_{in_max} = √2 × V_{line} = 587 V

Assign:

V_{disch} = √2/2 × V_{in_max} = 415 V at f = 50 Hz, ω = 2πf = 314 sec^{-1}, and P_{in} = P_{load}/η** **= 72 W

From this we obtain a value for V_{C_av}:

which leads to a value for the hold-up capacitor:

with C_{h} = 6.21 × 10^{-6}F, which is a fairly high value for a hold-up capacitor.

We can plot a graph for this example *(Fig. 2)*.

As discharge-margin V_{disch} is lowered, more energy should be supplied to the hold-up capacitor to fully charge it, as seen from:

At this point, the hold-up capacitor starts to charge, as indicated by:

We can calculate the hold-up capacitance value from the point of view of the energy balance:

which shows the amount of energy stored in the hold-up capacitor that should feed the load while the capacitor is discharging, as well as the discharge voltage and the hold capacitor values.

**Linear Regulator Supply**

The situation for a linear regulator is very different than for a switching regulator. With V_{load} as the voltage across the load, and P_{load} as the power to the load, we clearly have:

The linear regulator resistance is given by:

[EQUATION 21]

The time constant is given by:

Substituting this into Equation 4 yields:

and solving with respect to C_{h} we get:

When C_{h} discharges, we find V_{disch} from:

which leads to t_{0}:

Substitute Equation 26 into Equation 24 to obtain:

and then solve with respect to C_{h} via:

**Example 2**

Use P_{load} = 70 W and V_{load} = V_{disch} – 2.5 V. Then determine C_{h }via:

If a schematic has a linear regulator, the hold-up capacitor will have higher value because it should store enough energy to supply power loss across the linear regulator.