Stability, compensation, oscillators and slew rate
To stabilize a follower, adding a new pole leaves less than 1 kHz of bandwidth. Lowering the pole that was already there leaves 845 kHz with the same phase margin.
01When a feedback amplifier oscillates
The denominator \( 1 + T(s) \) vanishes if at some frequency the loop gain is exactly −1: magnitude 1 and phase −180°. A signal of that frequency goes around the loop and returns unchanged, with no excitation at all. The Nyquist criterion gives the complete condition; for everyday amplifiers, with a loop gain that decreases with frequency, the margins seen in the previous unit are enough:
02Compensation
To compensate is to modify the gain without feedback so that \( |T| \) crosses 1 before the phase approaches −180°. There are two ways with a single capacitor:
| Method | What it does | Cost |
|---|---|---|
| Added dominant pole | A new RC network adds a very low pole | The bandwidth falls to almost nothing |
| Shifting the first pole | A large capacitor at the node of the first pole lowers it | Much smaller: the original pole stops adding phase |
| Miller (pole splitting) | A capacitor between the input and output of the gain stage | It lowers the first pole and also raises the second |
Miller compensation is that of almost all integrated op amps: the Miller effect that in frequency response was a problem is used to advantage. A capacitor of a few picofarads, multiplied by the gain of the second stage, creates the dominant pole, and the local feedback it introduces pushes the second pole out to higher frequencies.
The amplifier of the previous unit: gain \( 10^5 \) and poles at 1 kHz, 1 MHz and 10 MHz. It is compensated by adding a pole at \( f_d \) or by lowering the 1 kHz one down to \( f_d \). The gray curve is the original gain; the blue one, the compensated one; the green one, the feedback one, and the dotted yellow one, \( 1/\beta \).
03Slew rate
Compensation has another consequence. In an op amp with Miller compensation, the maximum current that the input stage can deliver to the capacitor \( C_c \) limits the speed at which the output can change:
Above \( f_{\max} \) the sine wave becomes a triangle. It is a large-signal limitation, which does not appear in the small-signal analysis: an amplifier can have bandwidth to spare and distort all the same.
04Low-frequency oscillators
An oscillator is an amplifier that purposely meets the oscillation condition at a single frequency: the Barkhausen criterion, \( T(j\omega_0) = 1 \). In the Wien bridge, a series–parallel RC network has zero phase and a transfer of 1/3 at one frequency, and the amplifier provides a gain of 3:
The phase-shift oscillator uses three RC sections that add up to 180° and an inverter with a gain of 29. The audio-frequency oscillators in laboratories are Wien-bridge types: the first company of Hewlett and Packard was born with one, stabilized with a lamp. The introduction with the equipment is in oscillators, in the technician program.
05In the lab
Apply a large square wave to a follower and measure the slope of the output. With a 10 V peak sine wave, raise the frequency until the triangle appears and compare with \( f_{\max} \).
Build a Wien-bridge oscillator with diode stabilization and measure the frequency and the distortion. Change the gain and observe the start-up.
With an uncompensated op amp, such as those that have pins for the compensation capacitor, measure the overshoot as a follower with different capacitors.
06Common mistakes
- Compensating for high gain and using the amplifier as a follower. Compensation is designed for the lowest gain of use.
- Confusing small-signal bandwidth with full-power bandwidth. The latter is set by the slew rate and depends on amplitude.
- Setting the Wien gain at exactly 3. The oscillator does not start or dies out; a margin and an amplitude control are needed.
- Forgetting parasitic capacitances. A long cable at the output adds a pole and eats into the phase margin.
07Self-assessment
What full-power bandwidth does a 13 V/μs op amp have at 10 V peak?
\( 13/(2\pi \cdot 10) \) MHz ≈ 207 kHz.
A Wien bridge uses R = 10 kΩ and C = 10 nF. At what frequency does it oscillate?
\( 1/(2\pi \cdot 10^{-4}) \approx 1.59 \) kHz.
Why does shifting the first pole give more bandwidth than adding a new pole?
Because the original 1 kHz pole stops adding phase: the crossover can be near the second pole instead of below the first.
If Imax = 20 μA and Cc = 30 pF, what is the slew rate?
0.67 V/μs.
08Further reading
- Paul R. Gray, Paul J. Hurst, Stephen H. Lewis and Robert G. Meyer. Analysis and Design of Analog Integrated Circuits. 5th ed., Wiley, 2009. Miller compensation, pole splitting and the slew rate of the two-stage op amp.
- Sergio Franco. Design with Operational Amplifiers and Analog Integrated Circuits. 4th ed., McGraw-Hill, 2015. Stability with capacitive loads and oscillators with op amps.
- Jerald G. Graeme. Optimizing Op Amp Performance. McGraw-Hill, 1997. Stability and compensation of op amp circuits, analyzed with the loop gain.