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2018.05.24 Transfer Function

Example of Derivation for a Step-Up/Step-Down Converter - 2

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

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This section is the continuation of the previous “Example of Derivation for a Step-Up/Step-Down Converter – 1”. Various control methods are used in step-up/step-down converters. Here, we are focusing on two control methods among these, and in this section we consider the second of these.

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.

Example 2 of Derivation of Transfer Functions for a Step-Up/Step-Down Converter: Step-Up/Step-Down Converter for which tonon'

In the control method considered this time, the ON-times for the step-down switch and for the step-up switch are not the same, and the switches each operate independently. Hence depending on the relation between the input voltage and the output voltage, the behaviors of the step-down switch and the step-up switch vary, and there are transfer functions with a number of patterns. This time we shall derive the transfer functions for a region in which the step-down switch and the step-up switch are both switching.

To begin with, the specific control system is shown in Figure 13.

20170912_graf_12
20170912_graf_13

Depending on the relation between input and output, there are three modes of operation--step-down, step-up/step-down, and step-up. The switching waveforms of SW1 and SW2 are shown. Their switching operation in each of these modes should be clear.

Step-Up/Step-Down Converters for which tonon'

The circuit on the right is a simplified version of the above circuit. The equations are derived in two steps, as we have done before.

20170912_graf_14

●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

20170912_graf_15

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

A calculation example from equations 5-21 and 5-22 above is shown below.

20170912_graf_16

20170912_graf_17

Upon substituting 20170912_graf_20 in equations 5-21 and 5-22, the following is obtained.

20170912_graf_18

Here, we set 20170912_graf_21.

Taking equations 5-25 and 5-26 as a system of simultaneous equations and determining 20170711_graf_12 and 20170711_graf_13, we obtain the following.

20170912_graf_19

As a result, the part indicating the duty is different, but similar results for characteristics are obtained.

Up to this point, we have derived transfer functions for step-down, step-up, and step-up/step-down converters, and we have used a unified approach in all cases. The reader should by all means try deriving the equations and doing the calculations. From the next time, we will consider these transfer functions with other factors included.

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