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The basics of op-amp loop-stability analysis: Breaking the loop

After my last Signal Chain Basics article, The basics of op amp loop-stability analysis: A tale of twin loop-gains, I received questions about how to generate the open-loop SPICE simulation curves that I reviewed. Although there are many ways to do this, the method I’ve always used is to open or “break” the loop while injecting a small signal into a high-Z node and look at the response at different points in the loop. But you may have additional questions about where to break the loop, the method used to break the loop and how this method compares with other more formal loop-stability methods.

Let’s use Figure 1 as a starting point to dig into this method; I’ll also explain why I’m comfortable using it and where you may run into challenges. One of the most important parts of this process is to understand the component interactions that must take place for an accurate loop-gain simulation. To make these visualizations easier, Figure 1 shows the operational amplifier’s (op-amp) open-loop output impedance, ZO , and input capacitance, CIN , represented outside the amplifier with discrete components.

Note that CIN is simplified from two common-mode capacitances and a differential capacitance into a single, lumped capacitance. Modifications to the circuit’s open-loop gain curve will take place due to interactions between ZO and the output load, CL . Therefore, you should not break the loop in a way that isolates ZO from CL or from other loads in the system.

The second interaction that needs to occur is between the feedback components, RF and RI , and CIN . The feedback component interactions cause modifications in the inverse feedback factor (1/ β) curve. So, you should not break the loop in a way that isolates CIN from the other components.

Figure 1

Typical op amp circuit with ZO and CIN represented outside the op amp

Typical op amp circuit with ZO and CIN represented outside the op amp

Figure 2 shows the most common places where you could break the loop. The options in the first row are not effective and prevent proper interactions between the output load and ZO or between the amplifier feedback network and CIN , respectively. The options in the second and third rows are effective at capturing the primary interactions that occur with the op amp’s ZO and CIN . The option in the second row misses subtle interactions between ZO and the feedback network that can occur in higher-bandwidth amplifiers (>10-50 MHz) with reactive output impedances. It is possible to implement this break without modifying the primary circuit topology and since it captures the primary interactions, it is the method recommended most often.

The options in the third row capture all possible circuit interactions, but require the creation of models of the op amp’s ZO or CIN outside the op amp’s macromodel, which in turn requires that you have knowledge about these components and how to model them.

The bottom-right option in the third row is common with more advanced circuits containing multiple feedback loops, and only requires external modeling of the op-amp input capacitances. These input capacitances are usually available in the product data sheet and can be modeled with a single capacitor, shown by CIN in Figure 2.

Figure 2

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Different circuit locations where you could break the loop

Different circuit locations where you could break the loop

The next step is to maintain a proper DC operating point while performing the open-loop simulations. To obtain accurate small-signal open-loop results, the op-amp circuit must be biased in a linear DC operation region. An op amp with an open feedback loop at DC will produce an output voltage that saturates into one of the output rails based on which input voltage is larger, operating as a comparator. The small-signal open-loop analysis will not be correct when biased in this saturated condition because the internal circuit components will be saturated and won’t behave as they would in their linear operating regions. The method that breaks the loop must still provide a valid DC operating point while acting like an open circuit for AC frequencies.

The method I was taught uses a large inductor and capacitor. The large inductor provides a very low impedance (short circuit) at DC; its large inductance value provides a very large impedance (open circuit) for AC frequencies of interest (>0.01 Hz). The large capacitor provides the opposite effect, and presents a very large impedance (open circuit) to the circuit at DC and a very small frequency (short circuit) for AC frequencies of interest. These effects are represented in Figure 3, using a simple op-amp buffer circuit as an example. Switches SW1 and SW2 represent the inductor and capacitor at DC and AC frequencies, respectively.

Figure 3

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Breaking the loop on a buffer circuit and showing the effects of L1/C1 at DC and AC frequencies

Breaking the loop on a buffer circuit and showing the effects of L1/C1 at DC and AC frequencies

Using these methods, Figure 4 breaks the feedback loop in the original circuit from Figure 1 in two ways. The left circuit uses the more common method and will properly capture the interactions between the op-amp model’s ZO and CIN parameters with the circuit load and feedback network without adding them externally. The right circuit breaks the loop at the input, which is a slightly more robust method. It captures the slight interactions between the output impedance and feedback network, but requires that you add the CIN component externally in order to account for its interactions with the feedback network impedance. You should use this method for circuits with multiple feedback loops, such as active filters, most servo loops and some capacitive load-drive circuits.

Figure 4

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Examples breaking the loop in the feedback (left) and input (right)

Examples breaking the loop in the feedback (left) and input (right)

The equations in Figure 5 calculate the AOL , 1/β and AOL β using the VOUT and VFB probes in the simulation circuits.

Figure 5

Equations to calculate the open-loop circuit parameters from the simulation probes

Equations to calculate the open-loop circuit parameters from the simulation probes

Figure 6 shows the results for the respective circuit breaks. The results show that both methods produce nearly identical phase loop-gain magnitude and phase responses, confirming that both options work in most cases. I’ve compared the results obtained from this method to others several times in my career and have found breaking the loop to be robust and accurate, providing similar results. Other methods certainly work as well, but require multiple simulations and often more advanced calculations where you must paste the results into a spreadsheet for processing.

Figure 6

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Open-loop curve results from the circuits in Figure 4

Open-loop curve results from the circuits in Figure 4

For accurate simulation results, take care when breaking the loop on circuits such that you can maintain a proper DC operating point and preserve the important component interactions. More advanced circuits with feedback to both inputs require differential analysis, which uses a similar but slightly modified method that breaks the loop at both inputs while injecting the signal differentially. The simulated results have also been confirmed many times to match well with bench results provided that you correctly model the op amp’s AOL , ZO and CIN parameters, and will resolve most stability problems in simulation before the hardware is built.

References

1. TI training, “TI Precision Labs – Op Amps: Stability”, Section 10.

2. Analog Engineer’s Circuit: Amplifiers: “Noninverting amplifier circuit

3. “Analog Engineer’s Circuit Cookbooks

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