Beware of loop gain
Article By : John Dunn
A look at a simplified op-amp model with the total feedback of a unity gain voltage follower offers a lesson on loop gain.
I have addressed this topic before, but we’ll take a little deeper look at it this time. First off, we are not going to discuss the loop gain situation of Figure 1, even though trying to cope with a troublesome op amp might feel a little bit like that.
Figure 1 This is not the meaning of loop gain we’ll discuss here.
Instead, we will look at a simplified op-amp model with the total feedback of a unity gain voltage follower, as seen in Figure 2.
Figure 2 Let’s look at this simplified op-amp model with full feedback.
Every op amp has at least two poles in its frequency response. One pole is at a low frequency, call that F1, while a second pole is at high frequency, call that F2. Letting C3 be zero so that there is no pole at F3 and using straight line approximations, the total feedback frequency rolloff curve could look like Figure 3.
Figure 3 Here’s the the total feedback frequency rolloff curve for our conditions.
With the pole at F2 higher in frequency than the 0 dB crossover frequency, the slope of this rolloff through that crossover point will be very close to −6 dB per octave. Under that condition, the gain margin and the phase margin will be good and the amplifier will be non-oscillatory. Call that “stable.”
Figure 4 If F2 falls below the 0 dB crossover point, the rolloff slope through that crossover will approach −12 dB per octave.
However, if the higher frequency pole F2 isn’t high frequency enough, if F2 falls below the 0 dB crossover point, the rolloff slope through that crossover will approach −12 dB per octave. Gain margin and phase margin will be poor and oscillatory instability will very likely be the result.
We greybeards can remember how the Fairchild µA709 op amp was made like this. You could not use that op amp as a unity gain voltage follower. The device was not “unity gain stable.” Fairchild’s successor to the µA709 was the µA741, which was unity gain stable.
If we put F2 back up high again but introduce pole F3 as in Figure 5, we can again get a −12 dB per octave slope through the 0 dB crossover point and the same instability issues will arise.
Figure 5 Putting F2 high and introducing pole F3 produces the same instability issues.
Of course, if we were to introduce F3 and put F2 low again, the slope through the 0 dB crossover point would be −18 dB per octave, in which case op-amp oscillation would be absolutely guaranteed to happen. No need to make a drawing for that!
Now we take a look at the practical results of all this. In the non-inverting circuit of Figure 6, the (a) circuit shows a capacitor C3 connected between the inverting and the non-inverting inputs of an op amp and two resistors provide the feedback.
Figure 6 This non-inverting circuit shows the practical results of all this.
Referring again to Figure 6, we can let the variable R go to open circuit to maximize the feedback around the loop and for loop stability analysis we can move C3 from the zero-source-impedance of E1 to the zero-source-impedance of ground, as in (b). Arbitrarily setting an R3 feedback resistance to 10K and using recursive differential equation analysis (one dimensional finite element analysis), we can examine the transient response of the output signal versus variable capacitance values at C3.
Figure 7 Using recursive differential equation analysis, we can examine the transient response of the output signal versus variable capacitance values at C3.
Choosing for the sake of example an op amp with an open loop DC gain of 100 dB, a low frequency pole F1 of 10 Hz, and a high frequency pole F2 of 4 MHz, we obtain a unity gain (0 dB) crossover frequency of 1 MHz. When we let C3 take on several different values, we obtain the following results.
Figure 8 This is the output in response to square wave signal Ein.
Some years ago, we had a seminar here on Long Island delivered by Bob Pease, then of National Semiconductor, where he suggested using a capacitor such as C3, recommending a value of 1000 pF. I raised the question of such a capacitor possibly causing feedback instability due to loss of phase margin. “I’m glad you asked that question!,” he replied. Then he lowered the 1000 pF suggestion down to 100 pF.
Even then, I would still not like to do this. It is preferable to find ways to keep the EMI away from the op amp altogether rather than trying to render that op amp immune to incoming EMI effects by using a C3. With Bob Pease’s original suggestion of 1000 pF, this op amp would not do well. With 100 pF it would do better, but my personal sense is to avoid using any C3 at all.
Find other ways instead to suppress EMI susceptibility problems.
John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).
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- Design PCBs for EMI
- Loop gain measurement error
- Middlebrook’s and Rosenstark’s loop gain measurements
- Loop gain measurements
- Analog great Bob Pease
