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These equations give you the basics of DC/DC

In part 1 of this series, we described the basic principle of operation of a switching voltage regulator SWR to manage the power inside a SoC. The articles covered creating an output voltage by stepping down or up a supply input voltage depending on the voltage levels required inside the integrated circuit to feed all the sub-circuits. We also covered managing the flow of power in an effective mode by avoiding the waste of energy that otherwise would reduce battery life.

We introduced in part 2 some basic equations of the switching voltage regulator (SWR):

Now we will use the basic equations (1),(2),(3) to obtain the the conversion ratio of each type of SWR.

**Step-down (or Buck) DC-DC switching voltage regulator**

This type of SWR is shown in **Figure 1**.

In the buck converter of Fig. 1 during time t_{ON}, S1 is a short circuit and S2 is an open circuit. The voltage across the inductor is:

Instead, during the t_{OFF} S2 is a short circuit and S1 is an open circuit, so the voltage across the inductor is equal to:

From equation (2) we see that the integral over a switching period is null. Therefore, applying equations (4) and (5), we obtain:

Recalling that, for equation (3), we have:

Replacing (7) in equation (6) we obtain the conversion ratio of the SWR regulator of the buck converter or step-down type:

**Step-up (or Boost) DC-DC switching voltage regulator**.

This type of SWR is shown in **Figure 2**.

In the boost converter of **Figure 2** during the t_{ON} S1 is a short circuit and S2 is an open circuit and the voltage across the inductor is:

Instead, during the t_{OFF} S2 is a short circuit and S1 is an open circuit, so the voltage across the inductor is equal to:

From equation (2) it is obtained that the integral over a switching period is null therefore, applying the equations (9), (10) we obtain:

Recalling that, for equation (3), we have:

Replacing (12) in equation (11) we obtain the conversion ratio of the SWR regulator of the boost converter or step-up type:

**Step-up/Step-down (or Buck/Boost) DC-DC switching voltage regulator**.

This type of SWR is shown in **Figure 3**.

In the buck/boost converter of Fig. 3 during the t_{ON} S1 is a short circuit and S2 is an open circuit and the voltage across the inductor is:

Instead, during the t_{OFF} S2 is a short circuit and S1 is an open circuit, so the voltage across the inductor is equal to:

From equation (2) it is obtained that the integral over a switching period is null therefore, applying the equations (14), (15) we obtain:

Recalling that, for equation (3), we have:

Replacing (17) in equation (16) we obtain the conversion ratio of the SWR regulator of the boost converter or step-up type:

From equation (18) it is obtained that the conversion ratio of the boost / buck regulator, being a function of the duty cycle (D between 0 and 1), is such as to make the module of the output voltage greater than that of the voltage of input | V_{OUT} |> | V_{IN} | when the duty cycle is greater than 0.5 instead we have | V_{OUT} | <| V_{IN} | when the duty cycle is less than 0.5, then by appropriately adjusting the duty cycle the regulator can be operated as the boost converter or as the buck, hence the name of step-up and step-down.

The value of duty cycle can be programmed automatically depending on the specification of the SoC, as we will describe in the next part of this series.