2017.04.27

Points of this article

・Subharmonic oscillations occur when the duty cycle is above 50%.

・Although not directly related to transfer functions, it is important to understand this analytic solution for subharmonic oscillation.

In a step-down converter with a fixed frequency and with peak current control, the inductor current flows continuously, and it is well known that if the duty cycle is 50% or higher, subharmonic oscillations occur. In nearly all recent ICs for use in power supplies, a slope compensation circuit is provided to address this issue.

This time, we stray somewhat from derivation of transfer functions for DC-DC converters to explain an analytic solution for subharmonic oscillation, an understanding of which is important.

The coil (inductor) current waveform in a current-mode step-down converter is similar to that in Fig. 10. Here the current value at a given time is I_{n}, and the current value one period later is I_{n+1}.

Further, the turn-on time is t_{ON}(n), the turn-off time is t_{OFF}(n), the slope of the coil current while turned on is m_{1}, and the slope of the coil current while turned off is m_{2}.

These can be used to express I_{n+1} in terms of I_{n} as in equation 3-15 below.

Here, m_{1} and m_{2} can be expressed by equations 3-16 and 3-17 respectively.

Fig. 11 below shows a PWM input waveform corresponding to Fig. 10. If the current sense gain is R_{S}, then immediately after turning on the PWM input is R_{S}I_{n}.

Further, the current that has increased during t_{ON} becomes m_{1}t_{ON}(n), and so the PWM input peak voltage V_{C} is as expressed by equation 3-18.

Also, ｔ_{OFF}(n) in equation 3-15 can be rewritten as in equation 3-19 below. Here T is the period.

From equation 3-18, t_{ON}(n) is found, and upon substitution into equation 3-15 and calculating, equation 3-20 is obtained.

｛I_{n+1}-I_{n}} is a geometric progression, and so the condition for not causing subharmonic oscillation is that this geometric progression converges to 0 as n → ∞. In other words, the condition is expressed by equation 3-21 below.

Substituting equations 3-16 and 3-17 into 3-21, we obtain the following conditional formula 3-22.

Thus the condition for not causing subharmonic oscillation is that the duty cycle D be 1/2, that is, 50% or lower. Referring to our statement at the outset, then, we have derived the analytic solution indicating that at a duty cycle above 50%, subharmonic oscillations occur.

Downloadable materials, including lecture materials from ROHM-sponsored seminars and a selection guide for DC-DC converters, are now available.

- What are transfer functions?
- What is the Transfer Function of an Amplifier?
- What is the Transfer Function of a Slope?
- What is the switching transfer function?
- Transfer function of each converter
- Example of Derivation for a Step-Down Converter
- Example of Derivation for a Step-Up/Step-Down Converter – 1
- Example of Derivation for a Step-Up/Step-Down Converter – 2
- Switching Transfer Functions: Derivation of Step-down Mode Transfer Functions Serving as a Foundation