Impedance Question?

Maybe this was mentioned and I missed it but is there a simple mathematical relationship between the value of the coupling capacitors and the output impedance or other variables are in play?

There is-  Zout+C, where C is the impedance of the cap at the specified frequency, and Zout is the output impedance of the circuit driving the cap.

At 1000Hz, typically the cap has no influence on the output impedance. But as the frequency is decreased, the output coupling cap begins to have a greater and greater influence as the cutoff (-3db point) frequency is approached, such that it is the major component of the output impedance rather than the circuit driving it.

Keep in mind that this is a bit oversimplified as there can be complicating aspects that do affect output impedance, such as the application of loop feedback which will reduce some of the effects of the output coupling cap.

I’m curious to know what these issues are with direct-coupling are since I may go this path with the SLP-98.

The Cary employs direct-coupling but not at its output- there it uses a conventional coupling capacitor.

But to answer the question, direct-coupling issues are
1) the unit must not make DC that could damage an amp or speaker
2) the DC must be able to stabilize during warmup without damage to amp or speaker
3) This requires a servo control; the servo circuit must be stable at all input signal conditions and not impart any sound of its own

The tricky bit is actually 2) above, in many solutions a protection relay is used and the relay itself can contribute (or rob from; depends on how you look at it) to the sound. We patented a circuit that is reliable enough that a protection relay is not needed.

I may have overshot here- in case you were only asking about the direct-coupling used prior to the output of the Cary. If that is the case, the big issue is power supply stability, as the circuit, except for the output, has bandwidth to DC. Power supplies, even though they are a source of DC, have a timing constant that is something just shy of actual DC like a battery has. If the circuit can go lower than the power supply, low frequency instability can result. 'Motorboating' which is a problem in amps and preamps with failing power supplies, is an example of this phenomena.

Why would they not use a higher value capacitor if that improves the sound and provides more pairing options? Cost can't be the reason since the price differences is not much. There must be other reasons or design tradeoffs.

You are correct. The bigger the cap, the more coloration it will impart, regardless of the dielectric. As a result, you want the smallest cap to do the job right, if low coloration is your goal. A 0.47uf cap sounds noticeably better than a 5.0uf cap; capacitors are wound on a machine so they have not only capacitive qualities but also inductance and resistive qualities as well. The dielectric (Teflon, paper, polypropylene, etc) affects the sound as does the length and diameter of the part.

This is the main reason that some tube preamps don't play nice with some solid state amps. The coupling cap is too small, and one big enough imparts more coloration than the manufacturer is willing to accept. So the manufacturer solves the problem by declaration (example: ARC limits their preamps to power amps with 30K or more input impedance). We solved it by direct-coupling; others use output transformers (which have a different set of issues).

jmcgrogan2 has the best answer +1

Our tube preamps don't have an output coupling cap- so they can drive 600 ohms let along 10,000 ohms with ease. The output impedance is the same at 5Hz as it is at 1KHz or 30KHz.

Generally speaking though, most tube preamps are likely not to play bass right driving 10K, simply because of the output coupling cap being too small.