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Bode Plots and Compensation Networks

An editor of a technical magazine once told me that subject matter recirculates every eight months as a method of informing an ever-changing audience. It’s been a while since I addressed Bode plots as a main subject so I thought this would be a good opportunity for a refresher on a very basic stability tool.

Bode plots are used to measure gain and frequency versus a logarithmic frequency scale. In doing so, they provide a wealth of information about system performance including stability and response time. Ringing and other possibilities are displayed in the performance curve.

Bode Plots Relate Directly to Transient Response of a System (Image courtesy of Reference 2)

Bode Plots Relate Directly to Transient Response of a System (Image courtesy of Reference 2)

The nice thing about Bode plots is they can mostly be created using basic algebra. In order to do so, we have a built-in transition via the Laplace transform that eases the math. As shown in this buck regulator transfer function from Microsemi1

.

Click here for larger image

The omega, ωo , terms are the corner frequencies of the circuit components and are addressed later in this blog. Bode plots can be estimated by looking at the terms in the equation and understanding a few simple rules:

  • Singular power “s” terms [those raised to the power of one] in the denominator
    • Gain rolls off at 20 dB per decade starting at the location of the corner frequency
    • Phase rolls off at 45 degrees dB per decade starting a decade above the corner frequency and continuing a decade after the corner frequency
  • Squared “s” terms in the denominator
    • Gain rolls off at 40 dB per decade starting at the location of the corner frequency
    • Phase rolls off at 90 degrees dB per decade starting a decade above the corner frequency and continuing a decade after the corner frequency

Squared terms are “double poles” with 180 degrees of total phase lag.

Singular terms are “single poles” with 0 degrees of total phase lag.

For terms in the numerator, the graph increases instead of decreases. The exception in terms of gain and phase is the Right Half Plane Zero which has increasing gain and decreasing phase due to a negative “s” value.

RHP Zero Equation and Graph (Courtesy of Reference 4)

RHP Zero Equation and Graph (Courtesy of Reference 4)

The basic L [inductor] and C [capacitor] components are shown in the following figure.

Basic Buck Regulator Power and Control Components (Image courtesy of Reference 3)

Basic Buck Regulator Power and Control Components (Image courtesy of Reference 3)

From these plots, a compensation network is developed. The goal of the compensation network is to have adequate phase margin at the point where the gain crosses zero dB [a gain of 1]. For most pulse width modulated converters, this is well below the Nyquist rate of half the switching frequency. The phase margin is simply the difference in degrees between 180 degrees of phase shift. A phase margin of 45 degrees is considered to be stable in most cases. 30 degrees of phase margin is possible however it can be conditionally stable.

Typical compensation networks are labelled Type 1, Type 2, and Type 3 as shown in the following Figure.

Three Common Types of Compensation Networks (Image courtesy Reference 1)

Three Common Types of Compensation Networks (Image courtesy Reference 1)

Basic Op Amp Compensation (Image courtesy Reference 5)

Basic Op Amp Compensation (Image courtesy Reference 5)

Sometimes op amp calculations are a bit confusing especially for first timers. The trick is to look at the negative op amp input as a solid DC voltage and therefore consider it an AC ground. Next, replace resistances with impedances so that Rf becomes Zf and Rin becomes Zin. Knowing that the AC current flows from Vin to Vout the equation becomes:

AC current = Vin/Zin = -Vout/Zf.

Solving for the gain results in Vout/Vin =-Zf/Zin. Simply replace the impedances in the Z values based on the compensation Type [1, 2, or 3 above]. It helps to know that the impedances are:

  • Inductor = sL =jωL
  • Capacitor = 1/sC = 1/jωC
  • Resistors are simply R

By laying the compensation Bode Plot over the power stage bode plot and adding the two plots, one can program stability rather easily. An example of an Excel file I modified follows:

Click here for larger image

This gain combination is the open loop gain of the system and it is used to determine stability and transient response. The references go into more detail.

I like to plot the individual branch impedances to see where the AC “corners” or “shorts” occur between the resistors and capacitors. You can see the frequencies move for these by changing the component values. More importantly, changing the values moves the Bode plots around. This helps you get an idea as to how the system will behave. The goal here is to have a -1 slope [-20 dB/decade] around the cross over frequency where the plot crosses zero dB. Once the gain plot looks good, add the phase plots for the compensation network and main power stage then combine the plots by adding them. Check the phase margin at the cross over frequency to ensure stable operation and you’re done!

References

  1. Voltage-Mode, Current-Mode (and Hysteretic Control), Microsemi Technical Note TN-203 Sanjaya Maniktala, 2012
  2. Control Loop Design, Lloyd Dixon, [Unitrode sometime in the 1980s] now Texas instruments copyright 2001
  3. Stabilize the Buck Converter with Transconductance Amplifier, eeweb posting Thursday, February 04, 2016
  4. Transfer Functions of PWM DC-to-DC Converters, slideplayer website, Published by Damon Walker, no date given
  5. Electronics/Electronics Formulas/Op Amp Configurations, wikibooks website

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