Example of Derivation for a Step-Up/Step-Down Converter – 1

We have derived transfer functions for step-down converters and step-up converters. From here, we will embark on derivation of the transfer functions for a step-up/step-down converter. There are a number of control methods used in step-up/step-down converters; here we consider two among these in deriving transfer functions. This time, we focus on the first of these methods.

As we have noted a number of times, the transfer functions for the step-up/step-down converters considered this time are similar to those for step-down and for step-up converters; that is, the transfer functions to be derived are and , and the transfer functions are similarly derived in two steps.

Example 1 of Derivation of Transfer Functions for a Step-Up/Step-Down Converter: Step-Up/Step-Down Converter for which t_on＝t_on’

As stated above, there are various control methods used in step-up/step-down converters. A reason for this is that it is not a single parameter that determines the output voltage for an input voltage. Here we take a moment to explain this briefly.

In the case of a step-down converter, the relation between the input voltage and the output voltage takes the form of the following equation.

As the equation above indicates, if V_IN, V_out, and T (the period) are determined for the step-down converter, t_on is uniquely determined. However, in the case of a step-up/step-down converter, the following equation obtains, and even if V_IN and V_out are determined, there is no unique result.

Hence there are various control methods. As one of these, in this section we derive the transfer functions for a step-up/step-down converter for which t_on＝t_on’.

Step-Up/Down Converters for which t_on＝t_on’

The circuit on the right uses a control method in which the step-up switch is turned on in sync with the step-down switch. In this case, similarly to the cases of a step-down and a step-up converter, we set up the equations in two steps.

●Step 1: Consider the stable states of the system

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

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

A calculation example from equations 5-15 and 5-16 above is shown below.

Upon substituting in equations 5-15 and 5-16, the following is obtained.

Taking equations 5-19 and 5-20 as a system of simultaneous equations and determining and , we obtain the following.

Upon comparing with the transfer functions for the step-up converter in the previous section, we see that ultimately, characteristic results similar to those for a step-up converter are obtained.

Next time, we will derive transfer functions for a step-up/step-down converter for which t_on≠t_on’.