Editor's note: this series consists of three parts:

• Part 1 looks at Bode Plots, power op amp behavior versus frequency, and a first-order check for stability, click here
• Part 2 looks at four compensating techniques, including phase, feedback zero, noise gain, and isolation resistor compensation, click here
• Part 3 provides examples of compensation techniques, including feedback zero, feedback zero and noise gain, compensation, click here

It is unlikely that a power operational amplifier will always be employed to drive a purely resistive load. For if that were the case, stability would rarely become a problem. On the other hand, a load which is largely capacitive does present stability issues, as will become clear in this article. In a purely capacitive load, the voltage lags the charging current by 90°, so there is a large phase delay in the loop response of the voltage feedback circuit. And such a phase delay can be a significant contributor to instability.

Consider a multistage amplifier as depicted by the plot in Figure 1. The open loop gain (A_OL) will begin to exhibit a roll off, and an increase in phase delay, at higher frequencies because of the low-pass filters formed by nodes with finite source impedances driving capacitive loads within the amplifier stage, Reference 1. So a load that is largely capacitive, together with the capacitance within a multistage amplifier, can bring about instability since the phase lag may be approaching, and perhaps exceed 180°. It is this phase delay that is a key contributor to instability.

Figure 1: Stable or Unstable? Modifying the closed loop gain (1/β) is one of several effective ways of achieving stable operation.
(Click on image to enlarge)

If instability occurs, it can inadvertently transform a power amplifier into an oscillator, and the device is likely to become quite hot and fail in as little as one second. In the discussion to follow, there are some simple techniques to manage phase and gain relationships to maintain a stable circuit.

Bode Plots in a nutshell
The Bode Plot of a power op amp's open loop gain (A_OL) is usually a graph of the log of the magnitude plotted as a function of the log of the frequency, as depicted in Figure 2. The magnitude axis is usually expressed in decibels. This transforms multiplication into a matter of simply adding vertical distances on the plot. Since this is an examination of a multistage amplifier's behavior, the open loop gain (A_OL) response is comparable to a single-pole, low-pass filter, as depicted in Figure 1.

As shown in Figure 2, the Bode Plot of a low-pass filter can be approximated by straight lines. Note that as the straight-line approximation approaches the corner frequency, the true value, as shown by the dotted line, departs from the straight-line approximation and is actually 3.01 dB below the junction of the two intersecting straight line segments.

Figure 2: A Bode Plot representation: A logarithmic amplitude plot versus the log of frequency is a basic tool for evaluating stability
(Click on image to enlarge)

Note that for frequencies much lower than the corner frequency, the approximation is a horizontal straight line which covers the lower frequencies where the capacitive reactance is large. For frequencies much larger than the corner frequency the curve is approximated by the straight line descending at a rate of 20 dB per decade. This is due to the diminishing capacitance reactance at higher frequencies.

A Bode Plot for the log of the phase relationship between the input and output could also be prepared, but the analysis employed in this article will skip this step. Note that power op amps often exhibit more than one pole in their open loop gain (A_OL) Bode Plots, as shall be demonstrated in this examination of various power op amp configurations.

Power op amp behavior as signal frequency rises
In Figure 3, there is a critical frequency (F_CL) at which the loop gain (A_OL–1/β) becomes zero and the power op amp simply degenerates into an open loop amplifier with an open loop gain of A_OL. If that occurs, the amplifier no longer behaves as a power amplifier and its benefits simply vanish.

Also note in Figure 3 that the closed loop (1/β) line–the dashed blue line–is depicted as a constant. In this case, the components of β are independent of the frequency–or purely resistive.

Figure 3: When the loop gain Runs out of gas
(Click on image to enlarge)

Is the power amplifier unstable? That depends on the phase relationship between the input and output for the amplifier for frequencies of F_CL and above, as discussed below.

Some definitions
Oscillation can occur if at a phase shift of 180° the loop gain (1/β) is one (0 dB), or more. Consequently, the 'Phase Margin' is defined as how close the phase shift is to a full 180° when the loop gain is 0 dB. This is also called "The Point Of Intersection". A good rule of thumb is to design for a maximum phase shift of 135°, which constitutes a minimum phase margin of 45°. This will allow for events that occur during power up and power down–as well as other transient conditions which may cause changes in the open loop gain (A_OL) curve that could initiate transient oscillations.

Note that it is the included angle formed at the Point Of Intersection (Closed loop gain = 0 dB) which denotes stability (20 dB/decade) or instability (40 dB/decade, or more). The 'Usable Bandwidth' is usually defined as the frequency at which the loop gain (A_OL - 1/β) falls to 20 dB. The reason is that above this frequency the error rises above 10% so that the performance of the op amp is degraded significantly.

A first-order check for stability
The loop gain (1/β) is the amount of signal available for feedback to reduce errors and non-linearities. A first-order check for stability is to make sure that when the closed loop gain (1/β) reaches the open loop gain (A_OL) line the phase shift is less than 180°.

Shown in Figure 4 is a plot of an open loop gain (A_OL) which, at frequencies of approximately 100 Hz to 80 kHz, exhibits a descent rate of 20 dB per decade. This denotes a phase shift of 90°. At approximately 80 kHz the descent rate changes to 40 dB per decade. So above this frequency, the phase shift is 180°.

Note that for a loop gain (1/β₁), the plot intersects the open loop gain (A_OL) plot where its descent rate is 20 dB per decade at X₁. So this curve passes the first check for stability. For a complete check, what is the phase shift of the open loop gain (A_OL) response over frequency? However, with a loop gain of 1/β₂, the plot intersects the open loop gain (A_OL) plot where its rate of intersection is 40 dB per decade at X₂. So this curve fails the first check for stability.

Figure 4: A first-order check of stability
(Click on image to enlarge)

References
1. The Art of Electronics, Second Edition, Paul Horowitz and Winfield Hill, Cambridge University Press, 1989, p. 242
2. Application Note APEX - AN47, Techniques for Stabilizing Power Op Amplifiers, Section 4, Cirrus Logic, www.cirruslogic.com

Bibliography
1. Network Analysis And Feedback Amplifier Design, H.W. Bode, D. Van Nostrand., 1945
2. Intuitive Operational Amplifiers, Thomas M. Fredericksen, McGraw-Hill Book Co., 1988

About the author
Sam Robinson is Marketing and Applications Manager for the Apex Precision Power™ product family at Cirrus Logic, Inc. His role involves management of product development and marketing, as well as overseeing the applications technical support for this high performance, high precision analog product family. He holds a BSEE from the University of Alabama, Huntsville. Sam has enjoyed a 15+ year tenure working in the power-analog market space.