# The basics of op amp loop-stability analysis: A tale of twin loop-gains

This article showcases the advantages of an op amp loop-stability analysis method that I use myself and recommend to others. This method looks at the behavior and rate of closure of the open-loop gain (Aol) and inverse feedback factor (1/β) curves in addition to the loop-gain (Aol β) phase margin. This method applies to general control systems but was championed for op amp circuit analysis by Jerald Graeme and later taught to me by my mentors when I joined Texas Instruments.

The strength of this method lies in the ability to visually identify the cause of the stability issue in the circuit feedback network or output network. Once you’ve identified the root cause of the stability issue, you can implement an appropriate compensation scheme.

Let’s use the two circuits in Figure 1 to demonstrate this stability analysis method. The circuits share the same component arrangement, but the component values selected in the two circuits have led to the two most common stability issues I see designers accidentally create in op amp circuits.

Figure 1

Twin unstable circuits with different component values and different stability issues

The transient responses for these two circuits in Figure 2 show that they are both unstable, with significant overshoot and ringing. Using these circuits as MUX buffers, reference buffers, analog-to-digital converter (ADC) input drivers or other applications where transient settling time is important would result in poor circuit performance because of the unpredictable transient responses. The differences in transient-response behavior between the two circuits are based on the locations and pairing of poles and zeros in the transfer function, which goes beyond the scope of this article.

Figure 2

Transient step response for the two circuits

Standard loop-stability analysis focuses on the loop-gain magnitude and phase. The difference between the circuit phase shift and 180 degrees when Aol crosses 0dB is where the recognizable “phase-margin” stability measurement comes from. However, analyzing the loop gains of the two circuits in Figure 3 shows that they have nearly identical loop-gain magnitude and phase responses, with crossover frequencies of roughly 875kHz and phase margins of less than 9 degrees. I’ve already hinted that the circuits have two different stability issues, but by only looking at the loop-gain responses, there isn’t any indication of how to tailor the compensation scheme based on the cause of the issues for each circuit.

Figure 3

The twin circuits have twin loop-gain (Aol β) magnitude and phase responses

Plotting the Aol and 1/β curves along with loop gain enables you to identify if the stability issues are coming from the feedback network or output network. In Figure 4, Circuit 1 results show that the Aol response is standard but that the 1/β response has an unwanted zero, which decreases the loop-gain phase. Circuit 2 results show a flat 1/β but the Aol has an additional pole, which decreases the loop-gain phase. Both circuits have a 40dB/decade rate of closure between the Aol and 1/β responses, which is a first-order indication that the circuit will have stability issues. Issues in 1/β originate from interactions between the components in the feedback network; compensation schemes should address these interactions. Issues in Aol arise from interactions with the amplifier output impedance and the circuit loads, most commonly capacitive loads.

Figure 4

Plotting Aol and 1/β allow you to determine the cause of the stability issue

The results in Figure 4 enable you to determine that adding a capacitor in parallel with the feedback resistor will compensate Circuit 1. You could also have stabilized Circuit 1 by decreasing the value of the feedback resistors or by selecting a lower bandwidth amplifier. Increasing the resistance between the output and load capacitance will compensate Circuit 2. Decreasing the load capacitance or selecting an amplifier with lower open-loop output impedance, and therefore better capacitive load drive, would have also solved the stability challenges in Circuit 2. The compensated circuits are shown in Figure 5. The compensation used in this example was not necessarily optimized for a particular goal (bandwidth, noise, etc.), other than to be stable. To learn more about the selection and design of compensation methods, see the TI Precision Labs Op Amps videos on stability.

Figure 5

Updated circuits with compensation

The loop-stability analysis for the two updated circuits in Figure 6 shows that the zero in 1/β in Circuit 1 has been canceled with a pole and the pole in Aol in Circuit 2 has been compensated with an added zero. Both circuits now have high phase margins, confirming that compensation was successful.

Figure 6

Transient step response for the two circuits

Figure 7 shows the transient responses for the two compensated circuits. The compensated circuits both have transient responses free from large overshoots and ringing.

Figure 7

Transient step response for the two circuits

I hope that showing how two circuits can have two different stability issues, but identical loop-gain responses emphasize the advantages of performing Aol and 1/β rate-of-closure stability analysis to determine the cause of the stability issue. My next Signal Chain Basics article will cover more details on the simulation method used to generate these curves and how it produces results that match the results from other stability simulation methods I often see recommended.

References

Jerald Graeme books:

TI.com content:

## 5 comments on “The basics of op amp loop-stability analysis: A tale of twin loop-gains”

1. D Feucht
September 6, 2018

Colin,

1. There is nothing controversial about feeedback analysis as applied here. Thus, there are no issues . They are problems instead. The word issues is being widely misapplied nowadays; let's try to correct that.

2. I find it easier to do feedback analysis using the loop gain, G H, (or A*beta in the old nomenclature). Then the slope of GH as it crosses a gain of one tells much about stability. Separate G and H are not needed to determine stability.

3. The output-node impedance transformation that occurs above the open-loop bandwidth of GH can introduce additional poles or zeros that affect stability. This  is why capacitive loading of amplifiers can cause them to oscillate. The amplifier  output resistance increases with frequency from the closed-loop value as loop gain decreases with frequency, causing the amplifier output to appear inductive. This inductance resonates with the external load capacitance.

2. CollinWells
September 6, 2018

Hi Dennis!  Thanks for reading and posting.

#1:  Glad you agree with the analysis.  I used “problem” in my responses to 2/3 🙂

#2:  No controversy here. We can definitively conclude that both loop-gain responses (GH) in Figure 3 are unstable based on both the slope at the cross-over frequency (40dB/decade) as well as the phase-margin (8 degrees).  The main point of the article and analysis method is that by loop-gain alone you can't determine where to apply compensation unless you have inside knowledge about capacitive loading, input impedance interactions, and why they're causing stability problems in the circuit.  Experts such as you will be able to quickly identify the root cause of the problem but beginners don't always know where to apply compensation once they've discovered they have a loop-gain problem.  Analyzing if problem is from Aol (G) or 1/Beta (H) helps target the compensation methods to apply to fix the problem.

#3:  Agreed.  Another way I commonly describe this is that the open-loop output resistance of the op amp and capacitive load form an RC network which results in an additional pole in the loop-gain response, degrading the circuit phase margin.

3. D Feucht
September 8, 2018

Collin (with both “l”s this time!),

re: your #3: That's true too, but beginners might become confused into thinking that it is the same effect as the closed-loop resonance caused by impedance gyration of the open-loop output resistance into an inductance in the frequency region between loop-bandwidth and its unity-gain bandwidth. (Maybe I am anticipating your next article with this?)

And by the way, for closed loops with two-pole rolloff (or -40 db/dec unity-gain crossover), I am still interested in doing a joint article with you on this, but I must confess that I haven't worked out the considerably harder impedance-gyrated equivalent circuits at the ouput yet. Maybe you will! It would be a significant feat.

4. ritesh1347
September 15, 2018

Hi Feucht,

I really like your point 3. That's really new way to think of it. Similarly we can say that increasing R is providing enough damping.(which is making it stable).

But I agree with Collin on point 2. Mostly while working with opamps, in case of unstability, we feel that our root cause of unstability is Beta path and we don't even think about Aol.

Sometimes, we put our opamp in undesired conditions (as in circuit 2) and we still  consider that Aol is going to be same as given in datasheet. But, may be in that condition, our Aol is changing its behaviour as shown in figure and then , your loop gain is going to  get change because of 'Aol' which you may not find or guess if you are only checking loop gain. In that case, it's good practice to check Aol, beta and Aol*beta as well. This will also give you an idea about your loading condition.

And also truly speaking, you can analyse the circuit in so many ways. If I consider output impedance as a part of Beta path , then in that case Aol is going to be same as in normal load conditon and then in that case its your Beta path (which is now going to include your load , feedback path and output impedance)  which has an issue.

But, I really feel that it's a good practice to break your loop gain to get better understanding of real issue.(even you can break it into so many smaller blocks).

5. analog.freak
February 14, 2019

I just found an interesting article, whose Fig. 3 provides two alternative viewpoints of the same problem.  It always pays to be able to affort more than one viewpoint of the same problem.