How to easily stabilize an op amp with inductive open loop output impedance?

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How to easily stabilize an op amp with inductive open loop output impedance? [Copy link]

 

Introduction

Some operational amplifiers (op amps) have inductive open-loop output impedances, and stabilizing these types of op amps can be more complicated 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 a closed-loop output impedance that does not require breaking the loop. In this article, I will discuss how to use closed-loop output impedances to stabilize op amps with resistive or inductive open-loop output impedances.

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 factor, 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 bi-inductive, depending on the design of the open-loop output impedance in the op amp. For an op amp with resistive open-loop output impedance, the closed-loop output impedance is resistive and increases with frequency due to the decrease in Aol. As Aol decreases, the closed-loop output impedance becomes inductive. For an op amp with inductive open-loop output impedance, the closed-loop output impedance will have bi-inductive properties.

Figure 1 shows two examples of op amp closed-loop output impedance. On the left is a resistive open-loop output impedance; on the right is the inductive region of the open-loop output impedance. For the resistive open-loop output impedance on the left, notice that above about 10 Hz, Zout increases with frequency and behaves like a 16.4H inductor. The inductive open-loop output impedance example on the right has three regions: capacitive, resistive, and inductive. This makes the closed-loop output impedance resistive, bi-inductive, and inductive, respectively.

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Figure 1: Resistive Zo, resistive and inductive Zout (left); Zo with inductive region, Zout with dual inductive region (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.

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Figure 2: Resistive open-loop output impedance driving a capacitive load

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

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Figure 3: 1-μF capacitor impedance, closed-loop output impedance, and equivalent closed-loop output impedance

Figure 4 shows the large overshoot on the op amp output caused by the resonant frequency. The output of the op amp oscillates around 40 kHz.

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Figure 4: Large overshoot on the output

To correct this instability, an isolation resistor must be added to the circuit as this changes the equivalent closed-loop impedance and eliminates the resonant frequency. Equation 2 gives the minimum resistor value required to calculate a stable circuit:

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

As mentioned previously, Zout appears as an inductor of 16.4μH. For a 1μF capacitive load, an isolation resistor of 8Ω or greater must be used to stabilize the circuit. Figure 5 shows the schematic with the isolation resistor.

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Figure 5: Schematic diagram with isolation resistors

Figure 6 shows the equivalent closed-loop output impedance (Zeq) with the isolation resistor. Note that the resonant peak is eliminated.

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Figure 6: Equivalent closed-loop output impedance with isolation resistor

Figure 7 shows that a significant amount of the overshoot has been eliminated by the added 8Ω isolation resistor.

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Figure 7: Overshoot after using an 8Ω isolation resistor

Operational amplifier with inductive open-loop output impedance

Some op amps have an inductive region in the open-loop output impedance. This makes the closed-loop output impedance doubly inductive, making it difficult to stabilize with capacitive loads. Figure 8 shows the 1-μF capacitor impedance (Zc), closed-loop output impedance (Zout), and equivalent closed-loop output impedance (Zeq) using an op amp with inductive open-loop output impedance. Again, note the peak at about 120 kHz, where the doubly inductive closed-loop output impedance interacts with the capacitive load impedance, causing instability.

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Figure 8: 1-μF capacitor impedance, closed-loop output impedance, and equivalent closed-loop output impedance

Figure 9 shows the large overshoot on the op amp output caused by the Zeq peaking. The output of the op amp oscillates around 120 kHz.

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Figure 9: Large overshoot on the output

To correct this instability, a resistor can be added in the feedback loop to change the open-loop output impedance, thereby eliminating the double inductive region in the closed-loop output impedance. This simplifies the calculation of the isolation resistor to stabilize the op amp. Figure 10 shows a resistor added in the feedback loop to change the open-loop output impedance.

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Figure 10: Schematic diagram of resistors 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 removed. The modified closed-loop output impedance now appears as a 2.32 H inductor above 10Hz.

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Figure 11: Modified open-loop and closed-loop output impedance

Since the open-loop output impedance is now mostly resistive, the same approach as used to stabilize the op amp with a resistive open-loop output impedance can be used. Adding a 3Ω isolation resistor can stabilize the circuit. Figure 12 shows the stabilized circuit using a 100Ω resistor to modify the open-loop output impedance and a 3Ω isolation resistor.

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Figure 12: Schematic diagram of the stabilization circuit with feedback resistor and isolation resistor

Figure 13 shows how much of the overshoot and ringing can be eliminated by adding two resistors to the circuit.

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

in conclusion

Stabilizing an op amp with an inductive open-loop output impedance is much more complex than stabilizing an op amp with a resistive open-loop output impedance. Using the closed-loop output impedance to stabilize the op amp has the added benefit of enabling you to determine if the open-loop output impedance needs to be modified, compared to the "break the loop" approach. Adding a resistor to the feedback loop simplifies the design process for stabilizing an op amp with an inductive open-loop output impedance.

This method significantly reduces the isolation resistor value required to stabilize an op amp compared to the method discussed in the Op Amp video series on TI Precision Labs. So the next time you find yourself having a hard time stabilizing an op amp, consider using the method discussed in this article to see if you need to modify the open-loop output impedance before adding isolation resistors.