characteristic impedance/impedance matching

Eh wasn't entirely sure if this warranted its own thread, but I stumbled across this youtube video from AlphaPhoenix which I think does a very good job explaining characteristic impedance and impedance matching and should prompt some discussion for audio use:



I was able to follow like 82% of this, and there this audio gut understanding of "reflected thingies bad, make bouncy in places where no want bouncy".

 

A lot of the information in that video went over my head, but that was incredibly fascinating.

 

I was thinking about this over the past week, but enlightenment didn't hit until I was doing some amp simulation today...

The video did great explaining the transmission line theorem on the physics side. On the other hand, the rule of thumb on the engineering side is not to worry about it if your cables are shorter than 1% of the wave length. 1% of wavelength of 20 kHz wave at the speed of light is 150 m or 492.125984 ft, so we don't have to worry about it unless we are running cables to the neighbors down the blocks, right?

A huge butt is that your amps typically has bandwidth into the MHz region, 1% wavelenght at 10 MHz is 0.3m or 0.984251969 ft, or roughly 2.3 penises long. I believe we all have some analog cables longer than that in our system.

Say we have a perfectly 150 ohm headphone + a typical 1.8 m or 6 ft long cable with 50 ohm characteristic impedance. The ringings in the video can be modeled as inductance and capacitance kinda like the reverse EMF stuff. At 10 MHz, the cable is around 0.12x the wavelength. With some RF Black Magic (pic from here):
The impedance should be around 50 -50j ohms. The -50j ohms is like 320 pF capacitance at 10 MHz.
Yeah I eyeballed it in MS paint outta my ass, the numbers aren't exact but should be close in magnitude.

We got capacitance, so what now? Let's see how would an opamp handles it. Here's the overshoot characteriscs of OPA1656:

At 50 ohms and 320pf, an OPA1656 buffer would have around 38% overshoot. Take that as what you will. I wish I could derive out the phase margins and stability stuff, but datasheets don't always have everything.

On the DAC side, the input impedances of your amps are typically around 10k-100k ohms, so they are overmatched to your cable. They'll be inductive if you keep your cables short, but they could turn into very capacitve if your cables are long or frequencies are high, which could be a problem depending on your DAC's output stage.
Edit: crap I got rusty with the RF stuff, 10k-100k ohms load is going to be quite capacitive even with reasonably short cables. The situation is worse than the headphones due to more severe mismatch. You better pray your snake oil RCA/XLR cables are either really short or have very low capacitance, or your DAC has really powerful output stage. For example, LME49724 is rated at 100 pF with 5% overshoot, while many cables have around 100+ pF/m capacitance ratings (measured only at 1 kHz though).


Here are the typical remedies to solve this stability problem:
1. Match the cable to the load or just use shorter cables.
2. Ditch the opamps in the output stage, or only use it in very high gain for more phase margin.
3. Or ditch the global feedbacks that's feeding the capacitive phase delay back into the input stage.
4. Use biggest assed discrete buffers for the output stage to fight the capacitance.
5. Way over compensate the miller cap to have more phase margin.

I think these solutions align quite well with audiophile preferences. Maybe this theory could explain the amp preference better than the reverse EMF effect, which aren't really that bad on the output stages. Anyway, I sorta pulled all these out of my ass without real data. I think the next thing to do is to look at what the headphone impedances actually are at MHz frequencies. I hope there are chirstmas deals for vector analyzers

 

Phew, ok we're about one and a half levels above my paygrade, but a question about the overshoot... is it "ideal" to have overshoot closer to 0% or is a little bit desirable in order to reach the target sooner?

I'm reminded of this kind of graph discussing damping factor (I know, damping ratio is not the same as the capacitive thingy discussed above, but the "idea" of it is what I'm going for... though maybe I'm correlating/conflating things that shouldn't be)

 

Epic.

Although I do suspect the high frequency issue is likely not a problem for good quality gear. If you have a whole heap of signal in the MHz range from noise, oscillations or poor filtering, interactions with cable capacitance are the least of your worries. That said, it's also why I like Blue Jeans LC-1 at 12.2 pf/ft (40 pf/m) and have never had a lick of cable nervosa.

 

If a number has to be picked, the "ideal" damping ratio is usually at 0.707 aka the butterworth response for a 2nd order system. It's the quickest to reach the target without causing overshoot. Since the problem is the resonance at MHz level, we probably won't benefit anything at all even if our amp settles an impulse nano seconds quicker. There's just no audio at that frequency with that speed. The main issue here is stability with NFB amps.

I think your chart's overshoot vs phase margin is for a different system. The math might be a bit different for feedback system. Here's the chart for feedback stability:

Usually 45-60 degrees of phase margin is good enough. I was aiming at 60-90 degrees in my amp simulation, which I found was a bit trouble with capacitive loads. They could eat away the phase margin pretty quickly, especially when the amp's phase compensation is pushed to the edge to allow for the most open loop gain, coupled with weak sauce output stage it could be a nightmare. Heck, my simulated NFB amp had many watts, but it could only drive up to 10 nF without causing overshoot. If all the phase margin is used up, it'll turn negative feedback into positive feedback and kill itself with maximum oscillation. I don't think our chip amps are that bad yet, but they could be having some anomalies at MHz level and farting craps into the signal chain. The oscillation caused by improperly bypassing power supply to the ground would be similar effect. Since that oscillation is already widely addressed, maybe it's time to look into the phase margin oscillations. Square wave testing amps and DACs loaded with 1nF caps might tell some story.

 

Put up some code to show the capacitance curves of cables and loads. Hope it'll help visualize what's going on. The estimation I did last night was in serial combination, but I'm doing parallel combination now, so the number will be different. Paralleled capacitance to the load shows closer to what the output stages are dealing with.

First up is the 150 ohm headphone case:

It adds 100ish pF undamped capacitance to your output stage. Still would cause 40+% overshoot in the OPA1656 datasheet with Riso = 0 ohm. Damp it out with higher Riso could reduce the overshoot, but won't eliminate it as shown by the blue arrow:




Next up is pretty much the worst case for headphones. Say you are using 3 meter long of RF grade 50 ohm cables for your HD650, and thinking it would be an upgrade:

Yeah you just added 420 pF load to your amplifier at 22 MHz. Better hope it can handle it. It still has 200 pF below 10 MHz, which could be a problem for some weak amps.

Say your amp has 20 ohm damper added in series with the output:

It's just enough to damp out the peak. Your amp still has 170 pF to drive below 20 MHz, and serial resistors degrades sound quality for unknown reasons.



Here's a typical situation for your DACs' output stages, 6 ft RCA cables into amps with 10 kohms potentiometers:

It's sunshine and roses below 20 MHz with less than 100 pF that the LME49724 can handle. However it resonates like hell at 42 MHz with 3409 pF pure capacitance. Damper resistors are pretty much mandatory now. Say your DAC adds a 75 ohm damper at the output stage to match with the cable:

Fortunately that peak is gone. This is why your DACs list their output impedance in their specifications. The output stage still needs to drive 80 pF though, which is still stressful for some opamps. Keep in mind there will be more capacitance as the cable gets longer, so 6ft is probably the longest cable you should get for DACs with LME49724 output stages or alike.


What about under matching the load to the cable? Here's what happens when you hook up 8 ohm drivers with 50 ohm cable:

It's very inductive below 20 MHz, but I didn't plot it out cuz parallel inductance shouldn't be a problem at high frequncies. However, the capacitance still shows up at higher frequencies, peaking around 200 pF near 50 MHz. The problem exacerbates as your cable gets longer:




So yeah, there's always gonna be capacitance unless your cables are really short or matched to the loads. All the simulations aren't taking into account of the parasitic inductance and capacitance in the driver, so the actual scenario could be better or worse, but probably not very far off. If the output stage can drive 1 nF without overshoot and oscillation, it will do just fine in all the scenario above.

 

Maybe a little off topic but I always wanted to know how you pick the right potentiometer value for an amp and why tube amps used high impedance potentiometers. Like the Bottlehead Crack uses a 100kΩ pot. I was always under the impression that high resistance added noise but that doesn’t seem like a problem for tubes

 

Traditionally, a tube amp would be fed by a tube output stage with high output impedance and low current drive capability. Larger pot, less load. And the larger the input impedance, the smaller any coupling caps have to be.

There are much more intricate design considerations, in terms of the source impedance seen by the input stage that affect input bias currents and the like. But that's well above my pay grade.

 

Now what if the load is very high impedance and sorta like a capacitor ala stax?

 

Great call, here's what happens with a typical Stax load:

I plotted inductance here too. There appears to be a resonance at 40 MHz. I don't think that's going to be a problem for electrostatic amps, because they usually have very high gain and lots of phase margin. This plot does prove why electronics are getting smaller, because we can't have nice capacitors with long leads. Inductance and capacitance are all over the place at high frequencies. The resistance in the load cannot damp out this effect.

Just in case of resonance phobia, putting a 33 ohm damper at the amp's output pretty much damps out the resonance:


Edit: Crap, what I did was 100pf in series with 150 kohm. Wasn't sure if they are series or parallel. Just did it again with parallel, there's not really much difference:

 

@cameng318 Tossing this bit of info in for reference as well by the guy who literally wrote the book on electrostatic loudspeakers:
http://sanderssoundsystems.com/technical-white-papers/54-cables-white-paper

So he says for electrostats an ideal cable is low inductance, low capacitance, and mild resistance which seems to track with what you have above. He suggests a coax cable interestingly enough.

 

Interesting read. ESL isn't my field of strength, but reading this article helped me undestand the ESL systems more. I think our focuses were different though. My focus was the cable capacitance at MHz level knocking 120+dB NFB opamps into oscillation, while his focus is to flatten out the LC resonance with the output transformer around kHz level, where matching characteristic impedance isn't the main problem. It's interesting that we arrived to similar conclusions from different paths.

I was just thinking about matching cables to 8 ohm drivers (which isn't necessary at all with properly designed amps), and found twited pair cables won't ever have characteristic impedance that low. Coaxial cables offer much more design freedom, they can probably be optimized for pretty much any use cases.