# How to easily stabilize an operational amplifier with inductive open-loop output impedance?

Introduction

Some operational amplifiers (op amps) have inductive open-loop output impedance, and stabilizing this type of operational amplifier may be more complicated than op amps with resistive output impedance. One of the most commonly used techniques is to use the “open loop” method, which involves breaking the feedback loop of the closed loop circuit and looking at the loop gain to determine the phase margin. A lesser-known method is to use a closed-loop output impedance that does not need to break the loop. In this article, I will discuss how to use closed-loop output impedance to stabilize op amps with resistive or inductive open-loop output impedance.

Equation 1 calculates the closed-loop output impedance Zout, which depends on the open-loop output impedance Zo, the open-loop gain Aol, and the feedback coefficient B. Equation 1 shows that as Aol decreases, Zout increases:

Zout = Zo/(1 + Aol*B)(1)

The closed-loop output impedance can be resistive, inductive and dual-inductive, depending on the design of the open-loop output impedance in the operational amplifier. For operational amplifiers with resistive open-loop output impedance, the closed-loop output impedance is resistive and increases with frequency due to the decrease of Aol. When Aol decreases, the closed-loop output impedance becomes inductive. For an operational amplifier with an inductive open-loop output impedance, the closed-loop output impedance will have double inductance.

Figure 1 shows two examples of the closed-loop output impedance of an operational amplifier. On the left is the resistive open-loop output impedance; on the right is the inductive area in the open-loop output impedance. For the resistive open-loop output impedance on the left, notice that at about 10 Hz, Zout increases with frequency and appears as an inductance of 16.4?H. The inductive open-loop output impedance example on the right has three areas: capacitive, resistive, and inductive. This makes the closed-loop output impedances resistive, dual inductive and inductive respectively. Figure 1: Resistive Zo, resistive and inductive Zout (left); Zo with inductive area, Zout with dual inductive (right)

Operational amplifier with resistive open loop output impedance

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

Figure 3 shows the impedance (Zc), closed-loop output impedance (Zout) and equivalent closed-loop output impedance (Zeq) of a 1μF capacitor. It can be seen that the equivalent impedance has a resonant frequency at about 40 kHz (when the inductive region of Zout intersects the capacitive load). This resonance frequency will cause the op amp output to oscillate, leading to instability. Figure 3: 1-μF capacitor impedance, closed-loop output impedance and equivalent closed-loop output impedance

Figure 4 shows the large amount of overshoot at the output of the op amp caused by the resonant frequency. The output of the operational amplifier oscillates around 40 kHz. Figure 4: Large overshoot on the output

To correct this instability, an isolation resistor must be added to the circuit, because this will change the equivalent closed-loop impedance and eliminate the resonance frequency. Equation 2 gives the minimum resistance value required to calculate a stable circuit:

R>2*sqrt(L/C)(2)

As mentioned earlier, Zout is shown as an inductance of 16.4μH. For a capacitive load of 1μF, an isolation resistance of 8Ω or greater must be used to stabilize the circuit. Figure 5 shows the schematic diagram with isolation resistors. Figure 5: Schematic diagram with isolation resistance

Figure 6 shows the equivalent closed-loop output impedance (Zeq) with isolation resistance. Note that the resonance peak is eliminated. Figure 6: Equivalent closed-loop output impedance with isolation resistance

Figure 7 shows that a large amount of overshoot has been eliminated by the added 8Ω isolation resistor. Figure 7: Overshoot after using an 8Ω isolation resistor

Operational amplifier with inductive open loop output impedance

Some operational amplifiers have an inductive region in the open loop output impedance. This makes the closed-loop output impedance double-inductive, making it difficult for capacitive loads to stabilize. Figure 8 shows the 1-μF capacitive impedance (Zc), closed-loop output impedance (Zout) and equivalent closed-loop output impedance (Zeq) of an op amp with an inductive open-loop output impedance. Note again that there is a peak at about 120 kHz, where the dual inductive closed-loop output impedance and capacitive load impedance will interact, causing instability. Figure 8: 1-μF capacitor impedance, closed-loop output impedance and equivalent closed-loop output impedance

Figure 9 shows the large amount of overshoot at the output of the op amp caused by the Zeq peak. The output of the operational amplifier oscillates at around 120 kHz. Figure 9: Large overshoot on the output

In order to correct this instability, a resistor can be added to the feedback loop to change the open-loop output impedance, thereby eliminating the dual inductive region in the closed-loop output impedance. This simplifies the calculation of the isolation resistance to stabilize the operational amplifier. Figure 10 shows the resistance added in the feedback loop to change the open-loop output impedance. Figure 10: Schematic diagram of the resistance in the feedback loop

Figure 11 shows that by adding a 100Ω resistor in the feedback loop, most of the inductive region in the open-loop output impedance can be eliminated. Now, the modified closed-loop output impedance is shown as a 2.32 H Inductor above 10 Hz. Figure 11: Modified open-loop and closed-loop output impedance

Since the open-loop output impedance is mostly resistive nowadays, the same method as using a resistive open-loop output impedance stabilization op amp can be used. Adding a 3Ω isolation resistor can stabilize the circuit. Figure 12 shows a stable circuit that uses a 100Ω resistor to modify the open-loop output impedance and a 3Ω isolation resistance. Figure 12: Schematic diagram of a stable circuit with feedback resistance and isolation resistance

Figure 13 shows the elimination of a large amount of overshoot and reverberation by adding two resistors to the circuit.

Figure 13: Overshoot after adding a resistor in series with an isolation resistor in the feedback loop

in conclusion

Stabilizing an op amp with an inductive open-loop output impedance is much more complicated than stabilizing an op amp with a resistive open-loop output impedance. Compared with the “open loop” method, using a closed-loop output impedance to stabilize the op amp adds additional benefits, allowing you to determine whether you need to modify the open-loop output impedance. Adding a resistor in the feedback loop simplifies the design process of stabilizing an op amp with an inductive open-loop output impedance.

Compared with the method discussed in the Operational Amplifier Video Series on TI’s Precision Labs, this method significantly reduces the isolation resistance required to stabilize the operational amplifier. So next time when you find it is difficult to stabilize the op amp, you can consider using the method discussed in this article to see if you need to modify the open-loop output impedance before adding an isolation resistor.

references

1. “Zout-based closed-loop analysis of the stability of load inductance amplifiers.” Texas Instruments Application Report SLYA029, October 2017. 