1. What causes amplifier stability issues???
So what is the issue at the root of all stability issues?

2. Poles and Bode Plots Pole Location = fP Magnitude = -20dB/Decade Slope
Slope begins at fP and continues down as frequency increases
Actual Function = -3dB fP
Phase = -45°/Decade Slope through fP
Decade Above fP Phase = -84.3°
Decade Below fP Phase = -5.7°

3. Zeros and Bode Plots Zero Location = fZ Magnitude = +20dB/Decade Slope
Slope begins at fZ and continues up as frequency increases
Actual Function = +3dB fZ
Phase = +45°/Decade Slope through fZ
Decade Above fZ Phase = +84.3°
Decade Below fZ Phase = 5.7°

4. Capacitor Intuitive Model
Our Intuitive Capacitive Model for AC Stability Analysis is defined above and contains three distinct operating areas. At “DC” the capacitor will be viewed as an open circuit. At “High Frequency” the capacitor will be viewed as a short circuit. In between the capacitor will be viewed as a frequency controlled resistor with a 1/Xc decrease in impedance as frequency increases.

5. Inductor Intuitive Model

6. Op-Amp Intuitive Model
Our Intuitive Op Amp Model for AC Stability Analysis is defined above. The differential voltage between the IN+ and IN- terminals will be amplified by x1 and converted to a single-ended AC Voltage Source, VDIFF. VDIFF is then amplified by K(f). K(f) represents the data sheet Aol Curve (Open Loop Gain vs Frequency Plot). This resultant voltage, VO, is then followed by the Op Amp Open Loop, AC Small Signal, Output Resistance, RO. After passing through RO the voltage appears as VOUT.

7. Op-Amp Loop Gain Model VOUT/VIN = Acl = Aol/(1+Aolβ)
If Aol >> 1 then Acl ≈ 1/β
Aol: Open Loop Gain
β: Feedback Factor
Acl: Closed Loop Gain
The lower diagram is the traditional control loop model which represents an op amp circuit with feedback. The top diagram depicts the sections of a typical op amp circuit with feedback which correspond to the control loop model. This model of an op amp circuit with feedback we will call the Op Amp Loop Gain Model. Notice that the Aol is the Op Amp data sheet parameter Aol, and is the open loop gain of the op amp. β (Beta) is the amount of output voltage from VOUT which gets fed back as feedback. The β network in this example is a resistor feedback network.
In the derivation of VOUT/VIN we see that the closed loop gain function is directly defined by Aol and β.

8. Amplifier Stability Criteria
VOUT/VIN = Aol / (1+ Aolβ)
If: Aolβ = -1
Then: VOUT/VIN = Aol / 0  ∞
If VOUT/VIN = ∞  Unbounded Gain
Any small changes in VIN will result in large changes in VOUT which will feed back to VIN and result in even larger changes in VOUT  OSCILLATIONS  INSTABILITY !!
Aolβ: Loop Gain
Aolβ = -1  Phase shift of +180°, Magnitude of 1 (0dB)
fcl: frequency where Aolβ = 1 (0dB)
Stability Criteria:
At fcl, where Aolβ = 1 (0dB), Phase Shift < +180°
Desired Phase Margin (distance from +180° Phase Shift) > 45°
From our Op Amp Loop Gain Model we can derive the criteria for a stable closed loop op amp circuit.
At the frequency fcl, where Loop Gain (Aolβ) goes to 1 or 0dB, if the Loop Gain Phase Shift is +/-180° then we have instability!!
At fcl the distance the Loop Gain Phase Shift is from 180° is called the Loop Gain Phase Margin. Our desired Loop Gain Phase Margin is >45° for a critically damped well-behaved closed loop response.
In the derivation of VOUT/VIN we see that the closed loop gain function is directly defined by Aol and β.

9. Fundamental Cause of Amplifier Stability Issues
Too much delay in the feedback network
There is too much delay between the output and the fed-back input of the circuit. This prevents the amplifier from being able to sense changes in the output quick enough to react to them and as a result the circuit begins to uncontrollably oscillate back and forth.

10. Cause of Amplifier Stability Issues
Example circuit with too much delay in the feedback network
So here is a practical circuit representation of delay in the feedback of an amplifier. A few RC elements are sufficient to delay the feedback enough to cause an uncontrollable oscillation. This seems a little unreasonable though right? Who would ever design a circuit that looked like that?

11. Cause of Amplifier Stability Issues
Real circuit translation of too much delay in the feedback network
Well what if we re-draw the previous circuit to look like a non-inverting amplifier driving a heavy capacitive load with a stray capacitance combining with the amplifier’s input capacitance? There are still two RC delays in this circuit, one from Ro and Cload and one from R1 and Cin+Cstray

12. Cause of Amplifier Stability Issues
Same results as the example circuit
Retesting shows that the results are the same as the “unrealistic” example circuit.

13. How do we determine if our system has too much delay??
So how do we measure and determine if the feedback network of our amplifier has too much delay?

14. Phase Margin Phase Margin is a measure of the “delay” in the loop
Open-Loop
The Phase Margin of an amplifier circuit dictates how much phase shift is left in the circuit before the phase shift equals 180 degrees.
The LoopGain (AOL*B) Phase shift is a measure of how many degrees of phase shift there are from the output to the input of an amplifier. Phase Margin is a measure of the difference in degrees between the Loop-gain phase shift and 180 degrees when the LoopGain Magnitude equals 0dB. A phase shift of 180 degrees or more is completely unstable and will likely oscillate uncontrollably. So typically it is desired that the LoopGain have a phase shift less than 135 degrees which yields a phase margin of 45 degrees or more. Having at least 45 degrees of phase margin allows for a good safety margin due to standard shifts in AOL over process variations, temperature, and loading, or component value shifts, or any other occurrences that could cause the phase margin to decrease from the standard values.

15. Small-Signal Overshoot vs. Phase Margin
90°
80° 2%
70° 5%
60°
10%
50°
16%
40°
25%
30°
37%
20°
53%
10°
73%
Phase margin is not very easy to measure in the lab because it requires the user to break the loop with some type of transformer that will allow the tester to inject a small signal into the loop and observe the gain at different parts of the loop.
Luckily as mentioned earlier all we need is an oscilloscope and a step generator!! If we can measure the overshoot of the circuit with a small signal input that produces an output change of between 50mV – 500mV then we can measure the overshoot and relate it back to phase margin using the chart listed.
From: Dorf, Richard C. Modern Control Systems. Addison-Wesley Publishing Company. Reading, Massachusetts. Third Edition, 1981.

16. Damping Ratio vs. Phase Margin
Phase margin is not very easy to measure in the lab because it requires the user to break the loop with some type of transformer that will allow the tester to inject a small signal into the loop and observe the gain at different parts of the loop.
Luckily as mentioned earlier all we need is an oscilloscope and a step generator!! If we can measure the overshoot of the circuit with a small signal input that produces an output change of between 50mV – 500mV then we can measure the overshoot and relate it back to phase margin using the chart listed.
From: Dorf, Richard C. Modern Control Systems. Addison-Wesley Publishing Company. Reading, Massachusetts. Third Edition, 1981.

17. AC Peaking vs. Damping Ratio
Phase Margin
AC Peaking @Wn
90°
-7dB
80°
-5dB
70°
-4dB
60°
-1dB
50°
+1dB
40°
+3dB
30°
+6dB
20°
+9dB
10°
+14dB
Similar to the overshoot measurement, an AC peaking measurement can be converted back to phase margin by comparing the AC peaking to the damping ratio which can be converted back to phase margin as well.
From: Dorf, Richard C. Modern Control Systems. Addison-Wesley Publishing Company. Reading, Massachusetts. Third Edition, 1981.

18. Rate of Closure
Rate of Closure: Rate at which 1/Beta and AOL intersect
ROC = Slope(1/Beta) – Slope(AOL)
ROC = 0dB/decade – (-20dB/decade) = 20dB/decade
Phase margin only shows the final result while rate of closure will help the designer determine if the issues lies within the feedback and 1/B or in the loaded AOL curve.

19. Rate of Closure and Phase Margin
So a pole in AOL or a zero in 1/Beta inside the loop will decrease AOL*B Phase!!
So if we have a pole in the AOL plot or a zero in the 1/Beta plot or some other combination of the two can cause the ROC to be >20dB/decade and the system will be unstable.

20. Rate of Closure and Phase Margin
Relationship between the AOL and 1/Beta rate of closure and Loop-Gain (AOL*B) phase margin
In a system where the rate of closure is 20dB/decade we can infer that there is at least 45degrees of phase margin remaining. This is because we can assume every amplifier circuit begins with 180 degrees of phase margin and 90 degrees is removed with the dominant low-frequency pole. Therefore if we have 90 degrees remaining for until the loop is closed and we do not want to give up more than another 45 degrees then we need to make sure that there is not a pole in AOL or zero in 1/B closer than 1 decade to the ROC. At one decade away the pole or zero that causes the phase margin to decrease will only cause an additional 45 degrees of phase shift.
Using the same logic if there is a known reaction in the system that causes the phase margin to decrease by an additional 90 degrees, such as a capacitive load interacting with the Zo of the amplifier, then we must place a compensation at least 1 decade before the ROC so we can ensure the phase is boosted at least back to 45 degrees by the time the loop closes.

21. Rate of Closure and Phase Margin
AOL Pole
1/Beta Zero
So if we have a pole in the AOL plot or a zero in the 1/Beta plot or some other combination of the two can cause the ROC to be >20dB/decade and the system will be unstable.

22. Testing for Rate of Closure in SPICE
Break the feedback loop and inject a small AC signal
Short out the input source
Break the loop with L1 at the inverting input
To determine the open-loop parameters of their closed-loop system the loop must be broken!
Break the loop by inserting an extremely large inductance which will be a short at DC and will open the circuit up for all AC frequencies. Then inject an AC signal into the loop through an extremely large capacitor that will be an open at DC and a short at all AC frequencies.
Inject an AC stimulus through C1

23. Breaking the Loop DC AC
To determine the open-loop parameters of their closed-loop system the loop must be broken!
Break the loop by inserting an extremely large inductance which will be a short at DC and will open the circuit up for all AC frequencies. Then inject an AC signal into the loop through an extremely large capacitor that will be an open at DC and a short at all AC frequencies.

24. Plotting AOL, 1/Beta, and Loop Gain
AOL = Vo/Vin
1/Beta = Vo/Vfb
AOL*B = Vfb/Vin
With AOL, 1/Beta, and AOL*B we can determine the rate of closure and phase margin of this system. In this case the ROC is 20dB/decade and the PM is 88 degrees. This is a very stable system.
Note 1/Beta = 6dB or 2V/V. The circuit we are analyzing was an inverting amplifier with a gain of -1V/V. However, the noise gain is taken from the positive input so the 1/Beta is 1+ |G| = 2V/V = 6dB.

25. Both circuits have a NOISE GAIN (NG) of 2.
Understanding Noise Gain vs. Signal Gain
Inverting Gain, G = -1
Non-Inverting Gain, G = 2
All stability analysis is performed based on the “Noise Gain” of the circuit which is the gain seen by the non-inverting terminal of the amplifier. This is because noise sources are always modeled at the non-inverting input and if the non-inverting input sees an unstable system then the noise will cause the amplifier to oscillate uncontrollably.
So the circuit on the left has a signal gain of -1V/V. The same circuit when the input is applied to the non-inverting terminal has a signal gain of 2V/V.
Since noise gain is always taken from the non-inverting terminal both of these circuits have a noise gain of 2V/V.
NG = 1 + ΙGΙ = 2
NG = G = 2
Both circuits have a NOISE GAIN (NG) of 2.

26. Noise Gain Noise Gain vs. Signal Gain Gain of -0.1V/V, Is it Stable?
Inverting Gain, G = -0.1
Noise Gain, NG = 1.1
This is just a quick slide to debunk a common myth that a circuit with a gain less than unity is unstable. As long as the amplifier is unity-gain stable then since the noise gain is always 1+X then no matter how small the attenuation factor is, the noise gain will always be greater than 1. Ex: /100 = 1.01.
If it’s unity-gain stable then it’s stable as an inverting attenuator!!!

27. Capacitive Loads

28. Capacitive Loads Unity Gain Buffer Circuits Circuits with Gain
As mentioned previously, the most common stability issues that we encounter involve capacitive loads. There are several ways to solve issues with capacitive loads, but the if the circuit must remain a unity-gain buffer then the options are limited. Therefore this section will be divided into two parts, unity-gain buffer circuits and circuits with gain.

29. Capacitive Loads – Unity Gain Buffers - Results
Determine the issue:
Pole in AOL!!
ROC = 40dB/decade!!
Phase Margin 0!!
NG = 1V/V = 0dB
Unity-gain buffers are actually the most difficult circuits to compensate because the loopgain is not reduced and is open to the full frequency range of the AOL. Therefore a pole in AOL or zero in 1/B at higher frequencies may affect a unity-gain circuit where it would not have affected a circuit with a higher gain.
Breaking the loop shows us that an interaction with the capacitive load is causing a pole in the AOL block that is causing our 1/Beta with a slope of 0dB/decade to intersect the AOL with a slope of -40dB/decade at a rate of closure of 40dB/decade. As explained previously this tells us that the phase margin will be reduced by the pole in the loop. Checking the phase margin shows that the pole was more than a decade from the frequency of intersection resulting in a phase margin of 0degrees. This explains why we observed the heavy overshoots and ringing shown on the previous slide.

30. Capacitive Loads – Unity Gain Buffers - Theory
Looking at a simplified model of the OPA627 we can see that there is a non-zero open-loop output impedance of roughly 50 ohms. Quick reminder that the amplifier is open-loop for all AC frequencies as shown with the open inductor and closed capacitor. Since the loop is open the output impedance and capacitive load appear as a load on the AOL block which can be modeled as shown in the image on the right. Looks like a very familiar RC low-pass filter now, huh?

31. Capacitive Loads – Unity Gain Buffers - Theory
A quick analysis of the circuit shows that the transfer function from AOL to the actual amplifier output now is now comprised of a single-pole response with the pole location at 1/(2*pi*Ro*Cload) Plotting this shows a reduction in the magnitude response with a -3dB point at the pole location. The phase response shows that the phase begins to decrease a decade before the pole frequency and continues to decrease until a decade after the pole totaling a 90 degree phase shift.

32. Capacitive Loads – Unity Gain Buffers - Theory
AOL
AOL Load X
So to look at the effect on the final AOL graphically, we have the initial AOL of the OPA627 on the left multiplied by the response of the AOL output load. A multiplication is an addition in the log-domain which is how we obtain the final response shown in the bottom circuit. The pole begins reducing the AOL at the pole frequency and the phase is shifted to basically 0 degrees for a unity-gain system.
Loaded AOL =

33. Stabilize Capacitive Loads – Unity Gain Buffers

34. Stability Options
Unity-Gain circuits can only be stabilized by modifying the AOL load

35. Method 1: Riso
The second method that we will look at uses an isolation resistor again to break capacitive load from the amplifier output. A high-frequency feedback is taken directly around the amplifier making a buffer driving Riso+Cload at high frequencies. The DC loop is closed on the other side of the isolation resistor so there isn’t a DC drop even with a load.

36. Method 1: Riso - Results
Theory: Adds a zero to the Loaded AOL response to cancel the pole
Unity-gain buffers are actually the most difficult circuits to compensate because the loopgain is not reduced and is open to the full frequency range of the AOL. Therefore a pole in AOL or zero in 1/B at higher frequencies may affect a unity-gain circuit where it would not have affected a circuit with a higher gain.
Breaking the loop shows us that an interaction with the capacitive load is causing a pole in the AOL block that is causing our 1/Beta with a slope of 0dB/decade to intersect the AOL with a slope of -40dB/decade at a rate of closure of 40dB/decade. As explained previously this tells us that the phase margin will be reduced by the pole in the loop. Checking the phase margin shows that the pole was more than a decade from the frequency of intersection resulting in a phase margin of 0degrees. This explains why we observed the heavy overshoots and ringing shown on the previous slide.

37. Method 1: Riso - Results
When to use: Works well when DC accuracy is not important, or when loads are very light
The first method we will look at to stabilize the capacitive load is to add a series resistor or “isolation resistor” , Riso, between the amplifier output and the capacitive load. This will add a zero to cancel the pole in the Loaded AOL.

38. Method 1: Riso - Theory
Looking at a simplified model of the OPA627 we can see that there is a non-zero open-loop output impedance of roughly 50 ohms. Quick reminder that the amplifier is open-loop for all AC frequencies as shown with the open inductor and closed capacitor. Since the loop is open the output impedance and capacitive load appear as a load on the AOL block which can be modeled as shown in the image on the right. Looks like a very familiar RC low-pass filter now, huh?

39. Method 1: Riso - Theory
A quick analysis of the circuit shows that the transfer function from AOL to the actual amplifier output now is now comprised of a pole-zero response with the zero location at 1/(2*pi*Riso*Cload) and the pole location at 1/(2*pi*(Ro+Riso)*Cload)) Plotting this shows a reduction in the magnitude response with a -3dB point at the pole location and it flattening back off to 3dB above it’s final value of (Riso / (Riso + Ro) at the zero location. The phase response shows that the phase begins to decrease a decade before the pole frequency and continues to decrease until the zero begins to increase counteract the pole a decade before the zero location. The final result is a net 0degree phase shift through this network.

40. Method 1: Riso - Theory X =
So to look at the effect on the final AOL graphically, we have the initial AOL of the OPA627 on the left multiplied by the response of the AOL output load. A multiplication is an addition in the log-domain which is how we obtain the final response shown in the bottom circuit. The pole begins reducing the AOL at the pole frequency and the phase is shifted to basically 0 degrees for a unity-gain system. =

41. Method 1: Riso - Design Ensure Good Phase Margin:
1.) Find: fcl and f(AOL = 20dB)
2.) Set Riso to create AOL zero: Good: f(zero) = Fcl for PM ≈ 45 degrees.
Better: f(zero) = F(AOL = 20dB) will yield slightly less than 90 degrees phase margin
fcl = kHz
f(AOL = 20dB) = 70.41kHz
1.5 decades ~ Fp * 35

42. Method 1: Riso - Design Ensure Good Phase Margin: Test fcl = 222.74kHz
f(AOL = 20dB) = 70.41kHz
→ Riso = 2.26Ohms
fcl = kHz
→ Riso = 0.715Ohms
1.5 decades ~ Fp * 35

43. Method 1: Riso - Design Prevent Phase Dip:
Place the zero less than 1 decade from the pole, no more than 1.5 decades away
Marginal: Decades: F(zero) ≤ 35*F(pole) → Riso ≥ Ro/34 → 70° Phase Shift Desirable: 1 Decade: F(zero) ≤ 10*F(pole) → Riso ≥ Ro/ → 55° Phase Shift
Reminder that the pole does move with the zero.
1.5 decades ~ Fp * 35

44. Method 1: Riso – Design Summary
Ensure stability by placing Fzero ≤ 10* Fpole
Reminder that the pole does move with the zero.
1.5 decades ~ Fp * 35

45. Method 1: Riso - Disadvantage
Voltage drop across Riso may not be acceptable
Showing the major negative factor about the isolation resistor which is the voltage drop when there is a load on the output.

46. Method 2: Riso + Dual Feedback
The second method that we will look at uses an isolation resistor again to break capacitive load from the amplifier output. A high-frequency feedback is taken directly around the amplifier making a buffer driving Riso+Cload at high frequencies. The DC loop is closed on the other side of the isolation resistor so there isn’t a DC drop even with a load.

47. Method 2: Riso + Dual Feedback
Theory: Features a low-frequency feedback, Rf, to cancel the Riso drop and a high-frequency feedback, Cf, to create the AOL pole and zero.
The second method that we will look at uses an isolation resistor again to break capacitive load from the amplifier output. A high-frequency feedback is taken directly around the amplifier making a buffer driving Riso+Cload at high frequencies. The DC loop is closed on the other side of the isolation resistor so there isn’t a DC drop even with a load.

48. Method 2: Riso + Dual Feedback
When to Use: Only practical solution for very large capacitive loads ≥ 10uF
When DC accuracy must be preserved across different current loads
The second method that we will look at uses an isolation resistor again to break capacitive load from the amplifier output. A high-frequency feedback is taken directly around the amplifier making a buffer driving Riso+Cload at high frequencies. The DC loop is closed on the other side of the isolation resistor so there isn’t a DC drop even with a load.