Calculate Hold-Up Capacitor Size For Switching, Linear Regulators

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 t₀:

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 t₀ and is very hard to solve symbolically, although it would be very interesting to use it to obtain the value for t₀ directly and then figure out how the “meeting point” t₀ 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 t₀ 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^-6F, 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 t₀:

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.