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Are there tricks that can improve stability?
In short: no.
National Semiconductor paper
AN-A: The Monolithic Operational Amplifier: A Tutorial Study sections 3 and 4 show the basic problem. A classic transistor op-amp has two gain stages and an ouput buffer. There are three high-frequency roll-offs. A simple universal (not custom-compensated) op-amp must have just one roll-off within the range that feedback is effective (gain bandwidth divided by closed-loop gain). In monolythic op-amps, and most discrete power transistors, the output device(s) gets weak at 1MHz-10mHz and you can't do anything to help it. The low-current input stage is also liable to have low bandwidth.
The solution used on the 101/741 is to put a 3pF-30pF cap across the -second- stage, in such a way that it also loads the first stage. The combined response of the first and second stage is a one-pole (6dB/octave) roll-off crossing unity gain at about 1Mhz, just before the output stage starts to get weak. Now you can apply almost any kind of feedback without much trouble.
That 1Mhz is where the reactance of the 30pF cap equals the dynamic impedance of the input stage emitters. A 101/741 input stage works about 10uA, emitter impedance is then about 3K, or 6K for two transistors. 30pF equals 6K at about 1MHz.
In this scheme, adding an emitter resistor in the second stage is just wasting gain. The 30pF cap from colector to base will set the overall AC gain of first and second stages, so the emitter resistor just reduces DC/LF gain. It is possible a cap across the emitter resistor adds a bump in the gain and phase response; I expect this would be very critical and dependent on the external feedback network.
Given this form of compensation, the slew rate is set. That 30pF cap must charge from the input stage current. 20uA (peak) times 30pF gives 666,667 volts per second, or 0.6 volts per microsecond. If a bipolar op-amp like this is compensated for unity gain at 1MHz, it will slew 0.6V/uS. If you increase input stage current or decrease compensation cap size, you could have 6V/uS with 10MHz GBW, but your output stage must now be solid out past 10mHz.
You can change this relationship with input stage emitter resistors. But that decreases DC gain and increases noise.
In bipolars, gain and current are in fixed relationship, and bipolars have very high gain at a given current. Tubes and FETs have lower gain for a given current, and can be designed for higher slew rate at the same GBW. It isn't clear to me how this differs from adding resistance to a bipolar.
Deane Jensen had an interesting idea: chokes in the input stage emitters. Inside the audio range they have no effect on gain or distortion, but cause gain to fall at high frequency. Problem is that the second stage will also fall at some high frequency. If compensated traditionally, you would have 12dB/Oct at some point and certain stability trouble. Adding a resistor in series with the cap or in shunt with the chokes could fix this.
If you don't need super-low input currents, then it is usually not too hard to get huge bandwidth in first and second stage. Oddly the real problem is the output stage. It looks like "unity gain" but consider: if the large output emitter-followers have an Ft of 10MHz, and a Beta of 50, then they have a constant input impedance only up to 10MHz/50= 200KHz! Above that point their input impedance drops. The second stage gain is roughly proportional to output stage impedance so it wants to drop. The compensation cap internal feedback fights that, but can only do so much.
Let's do numbers. A mike preamp's op-amp might want to be vari-gain from 1 to 100, without changing the compensation. Say we want at least 40dB (100:1) of feedback "over the audio range", meaning to 20KHz. The gain bandwidth product at 20KHz must be 100*100*20,000= 200MHz. If the transistors work at Beta of about 50, and we want Beta to stay constant out to 200MHz, the transistors' Ft must be 200MHz*50= 10 GHZ!!!
That was absurd for most of the discrete-part era, and still unlikely. There are Ft=8GHz parts but generally small, low-power, low-voltage, or very expensive. And at just 200MHz the simple equivalent circuits are complicated by stray capacitance and even lead inductance.
The usual "answer" is to accept less feedback than we would like, maybe only 6dB at 20KHz at gain=100. Now we only "need" 4MHz GBW, and indeed a LOT of audio is built with op-amps about this speed. But we also now know that feedback interacts with distortion, and 6dB feedback is probably more grating to the ear than no feedback at all. We "get away with it" because we don't always crank the preamp to the max, levels at 20KHz are generally low, and in many cases most of the distortion products fall outside the audio range. But it is one reason why "similar" designs can sound quite different on strong complex sounds.
As for your amp: try increasing input emitter resistors to 500 ohms. Or decrease input stage current to about 0.1mA per side (which will also reduce input current and DC error). That may make it "unconditionally stable" down to unity gain (no capacitive loads). The increase in noise voltage will be small for most audio purposes. If you need lowest noise and do not need unity-gain operation, reduce these emitter resistors to 100 ohms or less. Use the highest-Ft output devices that suit your power and cost goals. (Beware: Ft is usually quoted at some medium current where it is "best"; it can be much less at lower current.)
Paralleling small/fast/cheap transistors for outputs is a standard trick. Use separate emitter resistors or one of them will hog all the current and fry.