Some operational amplifiers (op amps) have an inductive open-loop output impedance, which can be more complex to stabilize than op amps with resistive output impedances. One of the most common techniques is to use the “break the loop” method, which involves breaking the feedback loop of a closed-loop circuit and looking at the loop gain to determine the phase margin. A lesser-known method is to use the closed-loop output impedance, which does not require breaking the loop. In this article, I’ll discuss how to use the closed-loop output impedance to stabilize op amps with resistive or inductive open-loop output impedances.

The equation below calculates the closed-loop output impedance, Z_{OUT}, which depends on the open-loop output impedance (Z_{O}), open-loop gain, A_{OL}, and the feedback factor, B. **Equation 1** shows that as A_{OL} decreases, Z_{OUT} increases:

Closed-loop output impedances can be resistive, inductive, or double inductive, depending on how the design of its open-loop output impedance. For an op amp with a resistive open-loop output impedance, the closed-loop output impedance is resistive and increases with frequency as A_{OL} begins to decrease. As A_{OL} decreases, the closed-loop output impedance becomes inductive. For an op amp with an inductive open-loop output impedance, the closed-loop output impedance will appear double inductive.

**Figure 1** displays two examples of a closed-loop output impedance of an op amp. On the left is a resistive open-loop output impedance; on the right is an inductive region in the open-loop output impedance. For the resistive open-loop output impedance on the left, notice that above approximately 10 Hz, Z_{OUT} increases with frequency and appears as a 16.4 µH inductor. The inductive open-loop output impedance example on the right has three regions: capacitive, resistive and inductive. This causes the closed-loop output impedance to be resistive, double inductive and inductive, respectively.

**Op amps with resistive open-loop output impedance**

**Figure 2** shows an op amp with a resistive open-loop output impedance driving a capacitive load.

**Figure 3** displays the impedance of a 1-µF capacitor (Z_{C}), closed-loop output impedance (Z_{OUT}) and equivalent closed-loop output impedance (Z_{EQ}). You can see that the equivalent impedance has a resonant frequency at approximately 40 kHz where the inductive region of Zout and the capacitive load meet. This resonant frequency causes oscillations on the output of the op amp, resulting in instability.

**Figure 4** shows the large overshoots seen on the output of the op amp, which is caused by the resonant frequency. The output of the op amp is oscillating at approximately 40 kHz.

To correct for this instability, you must add an isolation resistor to the circuit, which will alter the equivalent closed-loop impedance and eliminate the resonant frequency. **Equation 2** calculates the minimum resistor value required to stabilize the circuit:

As stated earlier, Z_{OUT} appears as a 16.4-µH inductor. For a 1-µF capacitive load, you must use an isolation resistor of 8 Ω or greater to stabilize the circuit. **Figure 5** displays the schematic with the isolation resistor.

**Figure 6** displays the equivalent closed-loop output impedance (Z_{EQ}) with the isolation resistor. Note the elimination of the resonant peak.

**Figure 7** shows that large overshoots have been eliminated by adding the 8-Ω isolation resistor.

**Op amps with inductive open-loop output impedance**

Some op amps have a region in the open-loop output impedance that is inductive. This causes the closed-loop output impedance to be double inductive and makes capacitive loads complicated to stabilize. **Figure 8** displays the impedance of a 1-µF capacitor (Z_{C}), closed-loop output impedance (Z__ _{OUT}__) and equivalent closed-loop output impedance (Z

_{EQ}) using a device with an inductive open-loop output impedance. Notice again that there is peaking located at approximately 120 kHz. This is where the double inductive closed-loop output impedance interacts with the impedance of the capacitive load, which causes instability.

**Figure 9** shows the large overshoots seen on the output of the op amp due to Z_{EQ} peaking. The output of the op amp is oscillating at approximately 120 kHz.

To correct for this instability, you can add a resistor inside the feedback loop to alter the open-loop output impedance, thus eliminating the double inductive region in the closed-loop output impedance. This simplifies the calculation of an isolation resistor to stabilize the op amp. **Figure 10** shows a resistor added inside the feedback loop to alter the open-loop output impedance.

**Figure 11** shows that by adding a 100-Ω resistor inside the feedback loop, you can remove a majority of the inductive region in the open-loop output impedance. Now the modified closed-loop output impedance appears as a 2.32-µH inductor above 10 Hz.

With the open-loop output impedance now mostly resistive, you can apply the same approach used to stabilize an op amp with a resistive open-loop output impedance. Adding a 3-Ω isolation resistor stabilizes the circuit. **Figure 12** shows the stabilized circuit using the 100-Ω resistor to modify the open-loop output impedance and 3-Ω isolation resistor.

**Figure 13** shows the elimination of large overshoot and ringing by adding two resistors to the circuit.

**Conclusion**

Stabilizing op amps with an inductive open-loop output impedance can be more complex than op amps with a resistive open-loop output impedance. Using the closed-loop output impedance to stabilize an op amp adds an extra benefit when compared to the “break the loop” method, enabling you to see if the open-loop output impedance needs modifying. Adding a resistor inside the feedback loop simplifies the design process for stabilizing op amps with inductive open-loop output impedance.

This method significantly reduces the isolation resistor value required to stabilize the op amp compared to the method discussed in the op amp video series on TI Precision Labs. So the next time you find it difficult to stabilize an op amp, consider using the approach discussed in this article to see if the open-loop output impedance needs modifying before adding your isolation resistor.

**Reference**

- “Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using Zout.” Texas Instruments application report SLYA029, October 2017.

I think R1 in Figure 5 should be 8 Ohms based off the preceding paragraph and Figure 7.

I was also wondering if you could expand on how you used equation 2 to find the value for R1? Or possibly point me to another source?

Many thanks!

Actually, I see in the application note the formula for R is actually; R = 2 * sqrt(L/C).

Thats great