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2018.06.21 Transfer Function

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

DC/DC Converters: Sharing of Transfer Functions among Control Systems

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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 20170711_graf_12 and 20170711_graf_13 , 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 RONp and RONn.

●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

UL23_graf01

UL23_graf02

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.

UL23_graf06

Taking equations 5-31 and 5-32 as a system of simultaneous equations and determining 20170711_graf_12 and 20170711_graf_13, we obtain the following.

UL23_graf08

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.

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