Effect of the ON-Resistance of a Switch on the Transfer Functions

The last two sections were devoted to Part 1 and Part 2 of “Examples of Transfer Function Derivation for Step-Up/Step-Down Converters”. This time, we will study the “Effect of the ON-Resistance of a Switch on the Transfer Functions”.

In this study as well, we will take the same approach as used before. As before, the transfer functions to be derived are and , and once again, the transfer functions are derived in two steps.

Effect of the ON-Resistance of a Switch on the Transfer Functions

In transfer function derivations up till now, the effect of the ON-resistance of a switch (switching transistor) has not been considered. However, it is well known that in actuality a switch has an ON-resistance, and that the ON-resistance affects the actual operation. Here, we consider the effect of this parameter of the switch ON-resistance.

We start with the step-up/step-down converter for which ton≠ton’ in ” Example of Transfer Function Derivation for a Step-Up/Step-Down Converter – 2 “, and use a similar procedure.

The circuit on the right is the simplified circuit for the step-up/step-down converter described the last time, with the ON-resistance for the MOSFET that is the switch indicated by R_ONp and R_ONn.

●Step 1: Consider the stable states of the system

① The coil current does not change over
    one period
② The capacitor charge amount does
    not change over one period

Terms (in red) relating to the ON-resistance are added to the equation.

●Step 2: Determine change amounts for an external disturbance, and describe the transfer functions

A calculation example from equations 5-27 and 5-28 above is shown below.
Upon substituting , , in equations 5-27 and 5-28, the following is obtained.

Taking equations 5-31 and 5-32 as a system of simultaneous equations and determining and , we obtain the following.

As can be seen from the results, the first-order imaginary terms differ considerably. This is as was explained in the chapter “What are Transfer Functions“; here, we present only the characteristic results in the graphs that follow.

Finally, the total transfer functions with and without the ON-resistance added are summarized below.