Phase Margin vs. Rate of Closure: Part 4 of 7

Rate of Closure: Same Circuit, Different Curves

In this blog post, I’d like to introduce the “Rate of Closure” method for determining stability.  Our “Open the Loop” (aka BigL/BigC from Part 3) schematic works well for this analysis method.  But we need to plot a new curve to use this method: 1/β.

1/β Explained

Let's look at that new curve: (1/β) represents the noise gain, where β is the feedback factor.  Using the BigL/BigC example from the previous blog, we can plot A and 1/β together.  The rate of closure is measured where the two curves intersect.  See Figure 1.

Figure 1: Rate of Closure Measured at the Intersection of A and 1/β

Download the zip file to follow along. Be sure to unzip before opening in LTspice:

Open the Loop - BigL-BigC.zip

Open the schematic Open the Loop – BigL-BigC.asc and run the simulation.  To plot A as shown in Figure 1 in green, plot V(abeta), then right-click on the plot title and change the expression to V(abeta)/V(beta).  To plot 1/β (in blue), start by plotting V(beta).  Then right-click on the plot title and change the expression to 1/V(beta).  To simplify the plots and remove the phase portion of the plot, right-click on the phase axis and click Don’t plot phase on the Right Vertical Axis settings window (Figure 2).

Figure 2: Disabling Phase Curve

Rate of Closure Explained

Simply stated, we measure the rate of closure at the intersection of A and 1/β as the difference of the slopes of the two curves in dB/decade.  In our example in Figure 1, the slope of A at the intersection point is -20dB/decade, and the slope of 1/β at that same point is 20dB/decade.  The rate of closure is the absolute value of the difference of these two slopes:

\left| \text{slope of } A - \text{slope of } \frac{1}{\beta} \right| = \left| (-20) - 20 \right| = 40\,\text{dB/decade}

When the rate of closure is 40dB/decade, this indicates an unstable circuit, which we can also observe with both the transient simulation in Part 1 and the phase margin analysis in Part 3.

Be sure to watch Part 1 of this video for more discussion on the rate of closure method.

Figure 3: 40dB/decade Rate of Closure for Unstable Example

Using 1/β to Identify Root Cause of Stability Issues

In Part 1, we showed that this example circuit behaves better when we change Rf and Rg from 5kΩ to 250Ω.  Making that same change here (right-click on the resistor symbol to change its value), we can observe a change in the rate of closure (Figure 4).

Figure 4: Rate of Closure for Stable Example

In this case, the slope of 1/β at the intersection is very close to 0 (we’ll use 0 to keep the math simple):

\left| \text{slope of } A - \text{slope of } \frac{1}{\beta} \right| = \left| (-20) - 0 \right| =20\,\text{dB/decade}

When the rate of closure is 20dB/decade, this indicates a stable circuit.  We already knew this example circuit (Rf = Rg = 250Ω) is stable from previous blog posts, but rate of closure is another method for visualizing this.

This example illustrates the benefit of using rate of closure to analyze stability. By plotting A and 1/β separately, it is easier to identify compensation techniques that can be made in the feedback network (β).  In this example, we lowered Rg and Rf to push the pole in the β curve out to a higher frequency. This ensured that the slope of 1/β was close to zero when it intersected with A.

Using Rate of Closure to Determine Compensation for a TIA

The Amplifier Stability Essentials blog series (a separate blog series from this one) provides some excellent stability training using rate of closure.  Part 2 explores a photodiode transimpedance amplifier (TIA) example, which is also plagued by input capacitance at the op amp input.

Also be sure to click here for more material on this topic.

Capacitive Loading Example

The example we’ve been exploring in this series so far demonstrates stability issues related to parasitic op amp input capacitance.  But what do these rate of closure curves look like when your stability problems are related to capacitive loading at the output?  Part 3 of the Amplifier Stability Essentials blog series has you covered!  You can also explore compensation solutions detailed here.

What’s Next?

At this point in the series, you should have the tools needed to analyze op amp stability in LTspice, including some guidance on how to apply those techniques to fully differential amplifiers.  In future blog posts, I will introduce some other competing methods and explore the pros and cons of each.

Read all the blogs in the LTspice Amplifier Stability series.