Meaning behind DC/DC converter frequency response

I am currently reading an article about control methods of DC-DC converters which contains the following diagram:

This diagram seems to show the magnitude (gain) and phase of the output signal of the following transfer function:

$$Gp(s) = y(s)/u(s) = \frac{V/CL}{s^2 + s/RC + 1/CL}$$

Where $$\y(s)\$$ is the voltage output and $$\u(s)\$$ is the duty cycle.

I cannot seem to understand how to generate the graphs from the transfer function or what they really mean. Does the frequency here represent the the switching frequency? Does the gain represent the ratio between voltage and duty cycle when reaching steady state?

$$\text{Gain} = \frac{V_\text{out}}{D}$$

• Looks like maybe it's the (idealized) control to output response of a voltage-mode converter. (No compensation.) Do you know how to generate a Bode plot from a transfer function in general? Commented Feb 2, 2023 at 17:13
• I don't see what "V" in the numerator represents for your 1st formula? Commented Feb 2, 2023 at 17:32
• "An article". Please edit your question with a link to or proper citation of the article you're working off of. While you're at it, what are the inputs and outputs of the transfer function? Commented Feb 2, 2023 at 19:57
• Many EEs struggle with these, not the least because while there are many examples online of feedback loops, they tend to centre around naive "s^3 + s^2 + s +1" exponential series, which is completely useless for dealing with power supply controls. The magic you're looking for is (in Octave) is this simple line: s = tf('s'); and with that special "s" you can now just type in the equations you find in datasheets etc. For plotting bode plots from octave you have to jump through some hoops, though, octave doesn't know about Hz bode plots. Commented Feb 2, 2023 at 21:58

1 Answer

The plot you show seems to be what is called the control-to-output transfer function of a dc-dc converter. It characterizes in magnitude and phase the way a stimulus - $$\D(s)\$$ - applied to the control input propagates through the converter to form a response - $$\V_{out}(s)\$$. It is a transfer function, a mathematical relationship linking a response to a stimulus.

Look at the below picture which represents a simplified buck converter operated in voltage-mode control. The error voltage coming from the control section goes through the pulse-width modulator (PWM) which sets a duty ratio - $$\D\$$ - and controls the power stage to deliver a dc output voltage:

By adjusting the control voltage $$\V_{err}\$$, you directly set the duty ratio (which is a discrete value) for regulation purposes: $$\V_{out}\$$ is kept constant regardless of perturbations like the input voltage $$\V_{in}\$$ and the output current $$\I_{out}\$$.

Because you deal with a control system, you need to stabilize the feedback loop by selecting a crossover frequency with adequate operating margins (phase and gain). These parameters will define the speed and precision of the system when reacting to a perturbation, i.e. its response time to a step load on its output. However, before thinking about a possible compensation strategy, you need this control-to-output transfer function to see the characteristics of the system (often called the plant - Yes, like Robert) you want to stabilize. This is the formula given in the bottom of the picture from which you extract the Bode plot of the right side. From this picture, you see if your compensator will generate gain, attenuation, if it needs one or two zeroes to meet your time-domain response etc.

The Bode plot graphs in the vertical axes the magnitude and phase variations of the transfer function versus frequency which appears in the log-compressed x-axis. In your example, you have rad/s but you will see hertz (Hz) in a practical implementation, for instance when using a frequency response analyzer or FRA.

The converter I used for illustration purposes, is coming from one of the 80+ ready-made templates you can freely download from my web page. They work with the demo version of SIMPLIS and you can simulate immediately many dc-dc or ac-dc converters. Finally, if you are interested in small-signal modeling in particular, you can check my free seminars on the subject that you can download here or have a look at my last book describing transfer functions of switching converters.